MeGRAW-HICC SERIES IN HIGHER MATHEMATICS
EDWIN H. SPANIER CONSULTING EDITOR
auslander and Mackenzie Introduction to Differentiable Manifolds
curry Foundations of Mathematical Logic
goldberg Unbounded Linear Operators
guggenheimer Differential Geometry
Rogers Theory of Recursive Functions and Effective Computability
rudin Real and Complex Analysis
spanier Algebraic Topology
valentine Convex Sets
ALGEBRAIC TOPOLOGY
ALGEBRAIC TOPOLOGY!
i
i
>
!
I
EDWIN H. SPANIER
Professor of Mathematics
University of California, Berkeley	i
\ ' 11 • 
1 -'.'F’*
McGRAW-HILL BOOK COMPANY
New York San Francisco St. Louis
Toronto London Sydney

PREFACE
THIS BOOK IS AN EXPOSITION OF THE FUNDAMENTAL IDEAS OF ALGEBRAIC
topology. It is intended to be used both as a text and as a reference. Particular
emphasis has been placed on naturality, and the book might well have been
titled Functorial Topology. The reader is not assumed to have prior knowledge
of algebraic topology, but he is assumed to know something of general topology
and algebra and to be mathematically sophisticated. Specific prerequisite
material is briefly summarized in the Introduction.
Since Algebraic Topology is a text, the exposition in the earlier chapters
is a good deal slower than in the later chapters. The reader is expected to
develop facility for the subject as he progresses, and accordingly, the further
he is in the book, the more he is called upon to fill in details of proofs.
Because it is also intended as a reference, some attempt has been made to
include basic concepts whether they are used in the book or not. As a result,
there is more material than is usually given in courses on the subject.
The material is organized into three main parts, each part being made up
of three chapters. Each chapter is broken into several sections which treat
vii
viii	PREFACE ;
individual topics with some degree of thoroughness and are the basic organi-
zational units of the text. In the first three chapters the underlying theme is	!
the fundamental group. This is defined in Chapter One, applied in Chapter	;
Two in the study of covering spaces, and described by means of generators I
and relations in Chapter Three, where polyhedra are introduced. The concept !
of functor and its applicability to topology are stressed here to motivate
interest in the other functors of algebraic topology.	|
Chapters Four, Five, and Six are devoted to homology theory. Chapter	:
Four contains the first definitions of homology, Chapter Five contains further	.
algebraic concepts such as cohomology, cup products, and cohomology oper-
ations, and Chapter Six contains a study of topological manifolds. With each
new concept introduced applications are presented to illustrate its utility.
The last three chapters study homotopy theory. Basic facts about homo- ;
topy groups are considered in Chapter Seven, applications to obstruction ,
theory are presented in Chapter Eight, and some computations of homotopy
groups of spheres are given in Chapter Nine. Main emphasis is on the appli- ;
cation to geometry of the algebraic tools introduced earlier.
There is probably more material than can be covered in a year course.
The core of a first course in algebraic topology is Chapter Four. This contains
elementary facts about homology theory and some of its most important
applications. A satisfactory one-semester first course for graduate students
can be based on the first four chapters, either omitting or treating briefly
Secs. 5 and 6 of Chapter One, Secs. 7 and 8 of Chapter Two, Sec. 8 of
Chapter Three, and Sec. 8 of Chapter Four. A second one-semester course
can be based on Chapters Five, Six, Seven, and Eight or on Chapters Five, [
Seven, Eight, and Nine. For students with knowledge of homology theory and
related algebraic concepts a course in homotopy theory based on the last
three chapters is quite feasible.
Each chapter is followed by a collection of exercises. These are grouped ’
into sets, each set being devoted to a single topic or a few related topics.
With few exceptions, none of the exercises is referred to in the body of the
text or in the sequel. There are various types of exercises. Some are examples
of the general theory developed in the preceding chapter, some treat special
cases of general topics discussed later, and some are devoted to topics ,
not discussed in the text at all. There are routine exercises as well as more '
difficult ones, the latter frequently with hints of how to attack them. Occa- ‘
sionally a topic related to material in the text is developed in a set of exercises :
devoted to it.	'
Examples in the text are usually presented with little or no indication of
why they have the stated properties. This is true both of examples illustrating
new concepts and of counterexamples. The verification that an example has ।
the desired properties is left to the reader as an exercise.	;
The symbol  is used to denote the end of a proof. It is also used at the !
end of a statement whose proof has been given before the statement or which
follows easily from previous results. Bibliographical references are by footnotes
preface	jx
in the text. Items in each section and in each exercise set are numbered con-
secutively in a single list. References to items in a different section are by
triples indicating, respectively, the chapter, the section or exercise set, and the
number of the item in the section. Thus 3.2.2 is item 2 in Sec. 2 of Chapter
Three (and 3.2 of the Introduction is item 2 in Sec. 3 of the Introduction).
The idea of writing this book originated with the existence of lecture
notes based on two courses I gave at the University of Chicago in 1955. It is
a pleasure to acknowledge here my indebtedness to the authors of those notes,
Guido Weiss for notes of the first course, and Edward Halpern for notes of
the second course. In the years since then, the subject has changed substan-
tially and my plans for the book changed along with it, so that the present
volume differs in many ways from the original notes.
The final manuscript and galley proofs were read by Per Holm. He made
a number of useful suggestions which led to improvements in the text. For
his comments and for his friendly encouragement at dark moments, I am
sincerely grateful to him. The final manuscript was typed by Mrs. Ann
Harrington and Mrs. Ollie Cullers, to both of whom I express my thanks for
their patience and cooperation.
I thank the Air Force Office of Scientific Research for a grant enabling
me to devote all my time during the academic year 1962-63 to work on this
book. I also thank the National Science Foundation for supporting, over a
period of years, my research activities some of which are discussed here.
Edwin H. Spanier

CONTENTS
INTRODUCTION
I	Set theory 1
2	General topology 4
3	Group theory 6
4	Modules 7
5	Euclidean spaces 9
I HOMOTOPY AND THE FUNDAMENTAL CROUP	12
I	Categories	14
2	Functors	18
3	Homotopy	22
4	Retraction and deformation 27
5	H spaces	33
6	Suspension 39
7	The fundamental groupoid 45
8	. The fundamental group 50
Exercises 56
xi
XU
CONTENTS
60
2 COVEBIIVG SPACES ANB FIBItATIOIVS
I	Covering projections 62
2	The homotopy lifting property 65
3	Relations with the fundamental group 70
fl The lifting problem 74
S The classification of covering projections ' 79
® Covering transformations 85
7 Fiber bundles 89
8 Fibrations	96
Exercises	103
3 POLYHEDISA
106
I Simplicial complexes 108
2	Linearity in simplicial complexes 114
3	Subdivision 121
4	Simplicial approximation 126
S	Contiguity classes 129
6	The edge-yath groupoid 134
7	Graphs 139
8	Examples and applications 143
Exercises 149
4 HOMOLOGY
1	Chain complexes 156
2	Chain homotopy 162
3	Иге homology of simplicial complexes 167
4	Singular homology 173
5	Exactness 179
6	Mayer-Vietoris sequences 186
7	Some applications of homology 193
8	Axiomatic characterization of homology 199
Exercises 205
154
CONTENTS
xiii
g PBODPCTS
210
1	Homology with coefficients 212
2	The universal-coefficient theorem for homology 219
3	The Kiinneth formula 227
4	Cohomology 236
5	The universal-coefficient theorem for cohomology 241
6	Cup and cap products 248
T Homology of fiber bundles	255
8	The cohomology algebra 263
9	The Steenrod squaring operations	269
Exercises 276
6 GENERAL COHOMOLOGY THEORY AND DUALITY	284
I	The slant product 286
2	Dualitу in topological manifolds 292
3	The fundamental class of a manifold 299
4	The Alexander cohomology theory 306
5	The homotopy axiom for the Alexander theory 311
6	Tautness and continuity 315
7	Presheaves 323
8	Fine presheaves 329
9	Applications of the cohomology of preshea ves	338
IО	Characteristic classes 346
Exercises 356
7 HOMOTOPY THEORY	302
I Exact sequences of sets of homotopy classes 364
t Higher homotopy groups 371
3	Change of base points 379
4	The Hurewicz homomorphism 387
5	The Hurewicz isomorphism theorem 393
6	CW complexes 400
7	Homotopy functors 406
8	Weak homotopy type 412
Exercises 418
xiv
CONTENTS
8
OBSTRUCTION THEORY
422
I	Eilenberg-MacLane spaces 424
2	Principal fibrations	432
3	Moore-Postnikov factorizations 437
4	Obstruction theory	445
5	The suspension map 452
Exercises 460
9
SPECTRAE SEQUENCES ANO HOMOTOPY
GROUPS OF SPHERES
404
I	Spectral sequences 466
2	The spectral sequence of a fibration 473
3	Applications of the homology spectral sequence 481
4	Multiplicative properties of spectral sequences 490
5	Applications of the cohomology spectral sequence 498
6	Serre classes of abelian groups 504
7	Homotopy groups of spheres 512
Exercises 518
INDEX
521
ALGKBItAIC TOPOLOGY
CONTENTS
xiv
U OBSTRUCTION THEORY
42a
I Eilenberg-MacLane spaces 424
2	Principal fibrations	432
3	Moore-Postnikov factorizations 437
4	Obstruction theory	445
5	The suspension map 452
Exercises 460
9 SPECTRAE SEQUENCES AN» HOMOTOPY
GROUPS OF SPHERES
404
1	Spectral sequences 466
2	The spectral sequence of a fibration 473
3	Applications of the homology spectral sequence 481
4	Multiplicative properties of spectral sequences 490
S	Applications of the cohomology spectral sequence 498
6	Serre classes of abelian groups 504
7	Homotopy groups of spheres 512
Exercises 518
INDEX
521
ALGEBRAIC topology
INTRODUCTION
THE READER OF THIS BOOK IS ASSUMED TO HAVE A GRASP OF THE ELEMENTARY
concepts of set theory, general topology, and algebra. Following are brief
summaries of some concepts and results in these areas which are used in this
book. Those listed explicitly are done so either because they may not be
exactly standard or because they are of particular importance in the subse-
quent text.
SET TlEORYi
The terms “set,” “family,” and “collection” are synonyms, and the term
class” is reserved for an aggregate which is not assumed to be a set (for
example, the class of all sets). If X is a set and P(x) is a statement which is
either true or false for each element x £ X, then
1	As a general reference see P. R. Halmos, Naive Set Theory, D. Van Nostrand Company, Inc.,
Princeton, N.J., 1960.
2	INTRODUCTION
{хЕХ|Р(.т)}
denotes the subset of X for which P(x) is true.
If J = {;} is a set and {A,} is a family of sets indexed by J, their union is
denoted by U Aj (or by U;,j Aj), their intersection is denoted by П Aj (or by
Cl^jAj), their cartesian product is denoted by X Aj (or by Xj(j A;), and
their set sum (sometimes called their disjoint union) is denoted by V Aj (or
by Vj(j Aj) and is defined by V Aj = U (/ X Aj). Incase J = {1,2, . . . ,n},
we also use the notation Ai U Aj U  •  U An, Aj П A2 П  •  П An,
Ax X A2 X •   X An, and Aj v A2 v    v A„, respectively, for the union, !
intersection, cartesian product, and set sum.
A function (or map) f from A to В is denoted by /: Л B. The set of all :
functions from A to В is denoted by B1. If A' C A, there is an inclusion map '
i: A' —> A, and we use the notation i: A' C A to indicate that A' is a subset of 
A and i is the inclusion map. The inclusion map from a set A to itself is called
the identity map of A and is denoted by 1л. If J' C J, there is an inclusion
map	.
fl: V Aj (A V Aj
i:J‘ i(J
An equivalence relation in a set A is a relation — between elements of A •
which is reflexive (that is, a ~ a for all a £ A), symmetric (that is, a — a' ;
implies a' ~ a for a, a’ E A), and transitive (that is, a — a' and a' ~ a"
imply a ~ a" for a, a', a" E A). The equivalence class of a £ A with
respect to — is the subset {<7' £ A | a — a'}. The set of all equivalence classes ;
of elements of A with respect to ~ is denoted by А/ ~ and is called a
quotient set of A. There is a projection map A А/ —- which sends a £ A to
its equivalence class. If J' is a nonempty subset of J, there is also a projection map ;
pjc XAj—> X Aj
jtJ j(J‘	/
(which is a projection map in the sense above).	
Given functions f: A В and g: В C, their composite g ° f (also de- ‘
noted by gf) is the function from A to C defined by (g 0 f)(a) = g(f(a)) for ;
a £ A. If A' C A and f: A B, the restriction of f to A' is the function >
f\A’: A' В defined by (/| A')(a') = f(a') for a' £ A' (thus /| A' = /° i, j
where i: A' C A), and the function / is called an extension off\ A' to A. j
An injection (or injective function) is a function /: A -> В such that i
flcfl) = f(a2) implies (i\ = a2 for щ, a2 £ A. A surjection (or surjective •'
function) is a function'/: A В such that b E В implies that there is a E A .
with fla) = b. A bijection (also called a bijective function or a one-to-one
correspondence) is a function which is both injective and surjective.	j
A partial order in a set A is a relation < between elements of A which is 5
reflexive and transitive (note that it is not assumed that a < a' and a' < a 
imply a — a'). A total order (or simple order) in A is a partial order in A such j
that for a, a' £ A either a < o' or a' < a and which is antisymmetric •
(that is, a < a' and a' < a imply a = a'). A partially ordered set is a set with 1
a partial order, and a totally ordered set is a set with a total order.	i
SEC. 1 SET THEORY	3
I zojrn’s lemma A partially ordered set in which every simply ordered
subset has an upper bound contains maximal elements.
A directed set A is a set with a partial-order relation < such that for
e, / E Л there is у £ A with a < у and fl < y. A direct system of sets
{A“,/o/J} consists of a collection of sets {A“} indexed by a directed set
A = {a} allcla collection of functions/,/1: A“ At for every pair a < / such
that
(a)	fla = Li«: A ° C A“ for all a £ A
(b)	fly = /А ° //= A“ -> Ar for a < fl < у in A
The direct limit of the direct system, denoted by lim , {A“}, is the set
of equivalence classes of V A" with respect to the equivalence relation
~ at if there is у with a < у and fi < у such that fayaa = flyat. For each
a there is a map ia: A“ —> lim , {A“}, and if a < // then fl — ip ° flt.
2	Given a direct system of sets {Aflflt} and given a set В and for every
a E A a function g„: An В such that ga = g/: ° flt if a < fl, there is
a unique map g: lim _ {A"} В such that g ° fl = ga for all a £ A.
3	With the same notation as in theorem 2, the map g is a bijection if and
only if both the following hold:
(a)	B=U ga(Aa)
(b)	ga(aa) = gflcd) if and only if there is у with a < у and / < у such
that fly (aa) = fl flat)
Let {A,} be a collection of sets indexed by J — {/}. Let A be the
collection of finite subsets of J and define a < / for a, ft £ A if a C fl
Then A is a directed set and there is a direct system {A“} defined by
A“ = Vj€tt Aj, and if a < />, then flt: A“ At is the injection map.
Let ga: A ° Vj, j A - be the injection map.
1 With the above notation, there is a bijection g: lirn , {A“} —> VjejAj
such that g ° itt = gtt (that is, any set sum is the direct limit of its finite
partial set sums).
An inverse system of sets (Aa,flt) consists of a collection of sets {Att} in-
dexed by a directed set A = {a} and a collection of functions flt-. Ap —> Aa
for a < j8 such that
(°) fla = 1л„: Aa C Att for a E A
(b) fly = fl11 ° fly- Ay Aa for a < fl < у in A
The inverse limit of the inverse system, denoted by lim.. {Att}, is the subset of
X Aa consisting of all points (oa) such that if a < fl, then aa = fltap. For
each a there is a map pa: liny {Aa} Am and if a < fl, then pa = flt ° pp.
5 Given an inverse system of sets {Aa,flt) and given a set В and for every
<у E. A a function ga: В Aa such that ga = flt ° gp if a < fl, there is
a unique function g: В —» lim{Aa} such that ga = pa ° g for all a E A.
4	INTRODUCTION ]
6	With the same notation as in theorem 5, the map g is a bi jection if and,’
only if both the following hold:	j
(a)	ga(b) = gQ(b/) for all a £ A implies b = b'	*
(b)	Given (aa) £ X Aa such that aa = fa^ap if a < ft, there is b £ В
such that ga(b) = aa for all a 6 A	;
Let (A>) be a collection of sets indexed by J = {/}. Let A be the collection ;
of finite nonempty subsets of J, and define a < ft for a, ft £ A if a C ft,,;
Then A is a directed set and there is an inverse system {A,,.} defined byf
Aa — XjeaA>, and if a < ft, fj3: Ap Aa is the projection map. For each/
a £ A let ga: Xy./A' Aa be the projection map.	j
7	With the above notation, there is a bijection g: XjejAt liin. {An} i
such that g, = p, ° g (that is, any cartesian project is the inverse limit of its'-
finite partial cartesian products').	;
i
2 GENERAL TOPOLOGVi	'
A topological space, also called a space, is not assumed to satisfy any separation ;
axioms unless explicitly stated. Paracompact, normal, and regular spaces will •
always be assumed to be Hausdorff spaces. A continuous map from one ;
topological space to another will also be called simply a map.
Given a set X and an indexed collection of topological spaces { -fy-fy j and J
functions fi: X Xj, the topology induced on X by the functions {])} is the :
smallest or coarsest topology such that each / is continuous.	!
1 The topology induced on X by functions (f: X —> Xj} is characterized r
by the property that if Y is a topological space, a function g: Y —> X is j
continuous if and only iff ° g: Y -» Xj is continuous for each j £ J-	i
A subspace of a topological space X is a subset A of X topologized by ;
the topology7 induced by7 the inclusion map А С X. A discrete subset of aj
topological space X is a subset such that every7 subset of it is closed in X. The |
topological product of an indexed collection of topological spaces {Xyfy j is |
the cartesian product X X}, given the topology induced by the projection j
maps рр X Xj Xj for j £ J. If {Ха}оед is an inverse system of topologicalf
spaces (that is, Xa is a topological space for a £ A and//': Xf: Xa is con--
tinuous for a < ft) their inverse limit lirn, {Xn} is given the topology7 induced;
by7 the functions pa: linu {Xft} —> Xa for a £ A.	7
Given a set X and an indexed collection of topological spaces {Xj}je/i
and functions gy. Xj X, the topology coinduced on X by7 the functions {g;] j
is the largest or finest topology7 such that each g; is continuous.	j
1 As general references see J. L. Kelley, General Topology, D. Van Nostrand Company, Inc., i
Princeton, NJ., 1955, and S. T. Hu, Elements of General Topology, Holden-Day, Inc., ;
San Francisco, 1964.	•	j
SEC. 2 GENERAL TOPOLOGY	5
2	The topology coinduced on X by functions {gy Xj -> X} is characterized
by the property that if Y is any topological space, a function f. X —> Y is con-
tinuous if and only if f ° gy Xj Y is continuous for each j / J.
A quotient space of a topological space X is a quotient set X' of X topol-
ogized by the topology7 coinduced by the projection map X -> X'. If А С X,
then X/A will denote the quotient space of X obtained by7 identifying all of A
to a single point. The topological sum of an indexed collection of topological
spaces {X/fy j is the set sum V Xj, given the topology coinduced by7 the
injection maps у: Xj V Xj for j £ J. If {Х“}аед is a direct system of topo-
logical spaces (that is, X“ is a topological space for a £ A and//': X" XP is
continuous for a < ft) their direct limit linu {X") is given the topology7
coinduced by7 the functions i„: Xa linu {Xtt} for a £ A.
Let if = {A} be a collection of subsets of a topological space X. X is said
to have a topology coherent with if if the topology7 on X is coinduced from the
subspaces {A} by7 the inclusion maps А С X. (In the literature this topology
is often called the weak topology with respect to Cf.)
3	A necessary and sufficient condition that X have a topology coherent
with C? is that a subset В of X be closed (or open') in X if and only if В П A
is closed (or open) in the subspace A for every A £ If.
4	If d is an arbitrary open covering or a locally finite closed covering of X,
then X has a topology coherent with (f.
5	Let Xbe a set and let {A;} be an indexed collection of topological spaces
each contained in X and such that for each j and j', Aj П Ay is a closed (or
open) subset of Aj and of and the topology induced on А} П Ay from A}
equals the topology induced on Aj П Ay from Ar. Then the topology coin-
duced on X by the collection of inclusion maps (Aj С X} is characterized by
the properties that Aj is a closed (or open) subspace of X for each j and X has
a topology coherent with the collection {A;}.
The topology7 on X in theorem 5 will be called the topology coherent
fivith {Aj}. A compactly generated space is a Hausdorff space having a topology7
coherent with the collection of its compact subsets (this is the same as what
is sometimes referred to as a Hausdorff k-space).
<» A Hausdorff space which is either locally compact or satisfies the first
axiom of countability is compactly generated.
7 If X is compactly generated and Y is a locally compact Hausdorff space,
X X Y is compactly generated.
If X and Y are topological spaces and А С X and В C Y, then <A;B>
denotes the set of continuous functions /: X —> Y such that f(A) С B.
Iх denotes the space of continuous functions from X to Y, given the coinpact-
open topology (which is the topology7 generated by7 the subbase {<K;(7>},
where К is a compact subset of X and U is an open subset of Y). If А С X
6	INTRODUCTION }
(
and В C У, we use (У,7^)(л'’л| to denote the subspace of Ул' of continuous J
functions f: X У such that /(A) С B. Let E-. Ул X X	У be the evalua-
Hon map defined by E(fix) = fix). Given a function g: Z	Yf the composite |
z yy X у	;
is a function from Z X X to У.
8	theorem of exponential correspondence If X is a locally compact ,
Hausdorff space and Y and Z are topological spaces, a map g: Z —> Ул is con- ;
tinuous if and only if E ° (g X !)•’ Z X X Y is continuous.	!
9	exponential law If X is a locally compact Hausdorff space, Z is ;
a Hausdorff space, and Y is a topological space, the function fi. (YAr)z —> YZXA i
defined by f(g) = E ° (g X 1) is a homeomorphism.	j
10	If Xis a compact Hausdorff space and У is metrized by a metric d, then ’
Yx is metrized by the metric d' defined by	j
d'(fig) = SUP {d(f(xlg(x)) I x £	t
3 GROUP THEOBTI	i
A homomorphism is called a monomorphism, epimorphism, or isomorphism, }
respectively, if it is injective, surjective, bijective. If (G;)7ejis an indexed cob ;
lection of groups, their direct product is the group structure on the cartesian (
product X Gj defined by (g;)(g;) — (gjgff If is an inverse system of'"|
groups (that is, G„ is a group for each a andfi/1-. Gfi —> G„ is a homomorphism !
for a < /j), their inverse limit lim. {Ga} (which is a set) is a subgroup of X G«-;!
Let A be a subset of a group G. G is said to be freely generated by A and A i
is said to be a free generating set or free basis for G if, given any function j
f A H, where H is a group, there exists a unique homomorphism tp: G —> H \
which is an extension of f. A group is said to be free if it is freely generated *
by some subset. For any set A a free group generated by A is a group F(A)J
containing A as a free generating set. Such groups F(A) exist, and any two агет
canonically isomorphic.	“	j
I
1 Any group is isomorphic to a quotient group of a free group.	j
A presentation of a group G consists of a set A of generators, a set j
В C F(A) of relations, and a function f: A —> G such that the extension of f |
to a homomorphism rp: F(A) -ч> G is an epimorphism whose kernel is the nor- *
1 As a general reference for elementary group theory see G. Birkhoff and S. MacLane, A j.
Survey of Modern Algebra, The Macmillan Company, New York, 1953, For a discussion of free j
groups see R. H. Crowell and R. H. Fox, Introduction to Knot Theory, Ginn and Company, j
Boston, 1963.
1
SEC. 4 MODULES	у
mal subgroup of F(A) generated by B. If A and В are both finite sets, the pres-
entation is said to be finite and G is said to be finitely presented.
4 MODUbKSi
We are mainly interested in R modules where R is a principal ideal domain.
However, we shall begin with some properties of R modules where R is
a commutative ring with a unit which acts as the identity on every module.
If у: A В is a homomorphism of R modules, then we have R modules
ker ф = {a € A | <p(a) = 0} C A
im <p = (b 6 В | b — q>(a) for some a £ А} С В
coker ф — B/im rp
I noether isomorphism theorem Let A and В be submodules of a
module C and let A + В be the submodule of C generated by A U B. The
inclusion map A C A -(- В sends A Pl В into В and induces an isomorphism
ofA/(A П B) with (A B)/B.
И {A;}Jej is an indexed collection of R modules, their direct product
X Aj is an R module and their direct sum (J) A; is an R module (@ Aj is the
submodule of X A; consisting of those elements having only a finite number
of nonzero coordinates). The inverse Emit lim< {A,,} of an inverse system of
R modules (and homomorphisms ff: Ap —> Aa for a < ft) is an R module, and
the direct limit of a direct system of R modules (and homomorphisms) is an
В module.
2	Any R module is isomorphic to the direct limit of its finitely generated
submodules directed by inclusion.
If A and В are R modules, their tensor product A ® В (also written
A ® B) is an R module. For a g A and b E B, there is a corresponding element
o®hpA®B. A®Bis generated by the elements {a® b | aE A, b £ B)
with the relations (for a, a' E A, b, b' E B, and r, Z E R)
(ra + /a') ® b = r(a ® b) /(a' ® h)
a ® (rb + /&') = r(a ® b) + /(a ® b')
In case A or В is also an B' module, then so is A ® B.
R
3	For any R module A the homomorphisms a —> a ® 1 and a—> 1 ® a
define isomorphisms of A with A R and R ® A.
JAs general references see H. Cartan and S. Eilenberg, Homological Algebra, Princeton
University Press, Princeton, N.J., 1956 and S. MacLane, Homology, Springer-Verlag OHG,
Berlin, 1963.
8	INTRODUCTION '
4	For R modules A and В there is an isomorphism of A ® В with В ® A }
taking a' ® b to b ® a.	;
5	If A and В are R modules and В and C are R' modules, there is an iso- >
morphism of (A ® В) ® C with A ® (В ® C) (both being regarded as R and ।
R' modules) taking (a ® b) ® c to a ® (b ® c).
If A and В are R modules, their module of homomorphisms Hom (A,B) [
[also written IIomK (A,B)] is an R module whose elements are R homomor- ;
phisms A B. In case A or В is also an R' module, then so is lloms (A,B). ;
6 If A and В are R modules and В and C are R' modules, there is an iso- ’
morphism of Honi(1« (A ® В, C) with Нотв (A, Homjr (B,C)) (both being
regarded as R and R' modules) taking an R' homomorphism q>-. A ® В C .
to the R homomorphism <p': A Нотв- (B,C) such that <p'(a)(b) = q>(a ® b).
A subset S of an R module A is said to be a basis for A (and A is said to i
be freely generated by S) if any function f: S B, where В is an В module, :
admits a unique extension to a homomorphism ср: A -ч> B. If a module has a
basis, it is said to be a free module. For any set S the free module generated 
by S, denoted by-FR(S), is the module of all finitely nonzero functions from ;
S to В (with pointwise addition and scalar multiplication) and with s £ S ’
identified with its characteristic function. FR(S) contains S as a basis, and any i
module containing S as a basis is canonically isomorphic to FR(S).	'
7 Any R module is isomorphic to a quotient of a free R module.	!
« If A' is a submodule of A, with A/A' free, then A is isomorphic to the ]
direct sum A' ® (A/A').	;
We now assume that В is a principal ideal domain (that is, it is an ;
integral domain in which every ideal is principal). If A is an В module, its -
torsion submodule Tor A is defined by	:
Tor A = {« £ A | ra = 0 for some nonzero r £ B}	<
A is said to be torsion free or without torsion if Tor A = 0.	i
9 Over a principal ideal domain, a submodule of a free module is free. |
10 Over a principal ideal domain, a finitely generated module is free if andf,
only if it is torsion free.	j
1
11	Over a principal ideal domain, А/Tor A is torsion free.	j
If A is a finitely generated module over a principal ideal domain B, its i
rank p(A) is defined to be the number of elements in a basis of the quotient I
module А/Tor A.	’
12	If A' is a submodule of a finitely generated module A (over a principal. ;
ideal domain), then	'
p(A) = p(A') + p(A/A')	. ’
SEC. 5 EUCLIDEAN SPACES	9
Let q>: A -a> A be an endomorphism of a finitely generated module (over
a principal ideal domain B). The trace of <p, Tr <p, is the element of В which
is the trace of the endomorphism <p' induced by q> on the free module
А/Tor A [that is, if А/Tor A has a basis a^, . . . , a„, then <p'(af = 5 гцО]
and Tr q> — S qj.
13	Let rp be an endomorphism of a finitely generated module A and let A'
be a submodule of A such that <p(A') C A'. Then <p | A' is an endomorphism
of A' and there is induced an endomorphism q>" of A/A'. Their traces satisfy
the relation
Tr <p = Tr (<p | А') у Tr <p"
A module with a single generator is said to be cyclic. Over a principal
ideal domain В such a module A is characterized, up to isomorphism, by the
element rA £ В which generates the ideal of elements annihilating every
element of А (гл is unique up to multiplication by invertible elements of B).
14	STRUCTURE THEOREM FOR FINITELY GENERATED MODULES Over a principal
ideal domain every finitely generated module is the direct sum of a free
module and cyclic modules Ai, . . . , Aq whose corresponding elements
t'h , rq В have the property that rt divides r;+i for 1 < i < q. The
elements гъ . . . , rq are unique up to multiplication by invertible elements
of В and, together with the rank of the module, characterize the module up
to isomorphism.
5 Kl’CLIDEAN SPACES
We use the following fixed notations:
0	=	empty set
Z	=	ring of integers
Z!n	=	ring of integers modulo m
R	=	field of real numbers
C = field of complex numbers
Q = division ring of quaternions
R!! = euclidean n-space, with ||x|| =	x,-2 and (x,y) = S x^y,
0	=	origin ofR"
I	=	closed unit interval
t	=	{0,1} C I
In	=	n-cube = {x £	Rn | 0 < xj < 1 for 1 < i < n)
ln = {x £ I” | for some i, x, — 0 or x, =1}
En = n-ball — {x £ Rn | ||x|| < 1}
fin-i — (n _ l)-Sphere = (x £ R” | ||x|| = 1}
Pn = projective n-space — quotient space of Sn with x and — x identified
for all x £ S”
10
INTRODUCTION « OTHER BOOKS
11
If x and у are points of a real vector space, the closed line segment join- j
ing them, denoted by [x,y], is the set of points of the form fir + (1 — t)y for]
0 < t < 1 (thus I — [0,1]). If x -f- y, the line determined by them is the set;
(tx 4- (1 — t)y | t £ R). A subset C of a real vector space is said to be!
an affine variety if whenever x, у E C, with x ty y, then the line determined;
by x and у is also in C. A subset C is said to be convex if x, у E C imply ]
[x,y] С C. A convex body1 in R" is a convex subset of R" containing a non-;
empty open subset of R” (thus In and En are convex bodies in R»).
4
1 If C is a con vex body in Rn and C is a convex body in R”’, then С X C'l
is a con vex body in R" x R™ = Rn+m.	i
2 Any two compact convex bodies in R” are homeomorphic.	!
A subset S of a real vector space is said to be affinely independent^
if, given a finite number of distinct elements Xo, Xj, . . . , x,„ E S and;
to, ti, ... , tm E R such that S h = 0 and S tpxi = 0, then fi,- = 0 for ;
0 < i < m (this is equivalent to the condition that	у
I
X’l — Xq, X2 — Xo, . . . ,	Xo	:
be linearly independent).	i
3 There exist affinely independent subsets of R,! containing n + 1 points, j
but no subset ofH" containing more than n -f- 1 poin ts is affinely independent, j
4 Given points x0, xb . . . , xm E R!!, the convex set generated by them isi
the set of all points of the form 2 fijxj, with 0 < h < 1 and 2 fij - 1. The sett
{xo,xi, . . . ,xm} is affinely independent if and only if every point x in the\
convex set generated by this set has a unique representation in the formi
x = 2 tjXi, with 0 < h <1 for 0 < i < m and 2 fi; = 1.	j
j
I
ОТИЕВ BOOKS ON ALGEBBAIC TOPOLOGY	(
f
Alexandroff, P., and H, Hopf: Topologie, Springer-Verlag OHG, Berlin, 1935.	i
Bourgin, D. G.: Modem Algebraic Topology, The Macmillan Company, New York, 19631
Cairns, S. S.: Introductory Topology, The Ronald Press Company, New York, 1962.	1
Eilenberg, S., and N. E. Steenrod: Foundations of Algebraic Topology, Princeton Univer-f
sity Press, Princeton, N.J., 1952.	:
Godement, R.: Topologie algebrique et theorie des faisceaux, Hermann & Cie, Paris, 1958, i
Hilton, P. J., and S. Wylie: Homology Theory, Cambridge University Press, London, I960,;
Hocking, J. G,, and G. S. Young: Topology, Addison-Wesley Publishing Company, Inc,, 
Reading, Mass., 1961.	1
{
J
1 For general properties of convex sets see F. A. Valentine, Convex Sets, McGraw-Hill Booli
Company, New York, 1964.	•
Ди S, T.: Homotopy Theory, Academic Press, Inc., New York, 1959.
Lefschetz, S.: Algebraic Topology, American Mathematical Society Colloquium Series,
vol. 27, New York, 1942.
Lefschetz, S.: Introduction to Topology, Princeton University Press, Princeton, N.J., 1949.
Pontryagin, L. S.: Foundations of Combinatorial Topology, Graylock Press, Rochester,
N.Y., 1952.
Schubert, H.: Topologie, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1964.
Seifert, H., and W. Threlfall: Lehrbuch der Topologie, B. G. Teubner Verlagsgesellschaft,
Leipzig, 1934.
Steenrod, N. E,: The Topology of Fibre Bundles, Princeton University Press, Princeton,
N.J., 1951.
Wallace, A. H.: An Introduction to Algebraic Topology, Pergamon Press, London, 1957.
Wilder, R. L.: Topology of Manifolds, American Mathematical Society Colloquium
Series, vol. 32, New York, 1949.
CHAPTER ONES
HOMOTOPY AM> THE:
FUNDAMENTAL GHOUPi
t
i
i
!
r
i
t
t.
)
I
TOPOLOGY IS THE STUDY OF TOPOLOGICAL SPACES AND CONTINUOUS FUNCTIONS
between them. A standard problem is the classification of such spaces and
functions up to homeomorphism. A weaker equivalence relation, based on
continuous deformation, leads to another classification problem. This latter
classification problem is of fundamental importance in algebraic topology,
since it is the one where the tools available seem to be most successful.
As a working definition for our purposes, algebraic topology may be
regarded as the study of topological spaces and continuous functions by means
of algebraic objects such as groups, rings, homomorphisms. The link from
topology to algebra is by means of mappings, called functors. For this reason,
Secs. 1.1 and 1.2 are devoted to the basic concepts of category and functor.
In Secs. 1.3 and 1.4 the concept of continuous deformation, known tech-
nically as homotopy, is introduced. We then define the homotopy category
and certain functors on this category, all of which are important for the sub-
ject. Sections 1.5 and 1.6 are devoted to a study of conditions under which
these functors on the homotopy category7 take values in the category of groups.
As examples, the homotopy group functors are briefly mentioned.
13
14	HOMOTOPY AND THE FUNDAMENTAL GROOT CHAP. 1 I
The first functor considered in detail is the fundamental group functor, j
introduced and discussed in Secs. 1.7 and 1.8. This is an intuitively appealing !
example of the kind of functor considered in algebraic topology. Some appli- 7
cations of this functor are presented in the exercises at the end of the chapter. \
In Chapter Two this functor is used in a systematic study and classification of 
covering spaces.
1 catecoibies	:
j
An algebraic representation of topology is a mapping from topology to algebra, J
Such a representation converts a topological problem into an algebraic one to ;
the end that, with sufficiently many representations, the topological problem >
will be solvable if (and only if) all the corresponding algebraic problems are
solvable.	<
The definition of a representation, formally called a functor, is given in j
the next section. This section is devoted to the concept of category, because ;
functors are functions, with certain naturality properties, from one or several •
categories to another.	•
A category may be thought of intuitively as consisting of sets, possibly j
with additional structure, and functions, possibly preserving additional struc- ;
ture. More precisely, a category (2 consists of	j
(a)	A class of objects	j
(b)	For every ordered pair of objects X and Y, a set hom (X,Y) of .
morphisms with domain X and range Y; if f £ hom (X, Y), we write ;
f: X —> Y or X 4 Y	'
(c)	For every ordered triple of objects X, Y, and Z, a function associating j
to a pair of morphisms f: X Y and g: Y —> Z their composite	;
gf—g°f:X-^Z	J
4
These satisfy the following	two axioms:	1
Associativity. If f: X Y, g: Y —> Z, and h: Z W, then	'
h(gf) = (hg)f.X^W	j
Identity. For every object Y there is a morphism ly: Y —> Y such that if 1
f: X -ч> Y, then lYf =	f, and if h: Y -ч> Z, then hlY =	h.	‘
If the class of objects	is a set, the category is said to be small. For most J
of our purposes we could restrict our attention to small categories, but it !
would be inconvenient to have to specify a set of objects before obtaining a |
category. For example, we should like to consider categories whose objects!
are sets or groups, and we prefer to consider the class of all sets or groups, !
rather than some suitable set of sets or groups in each instance.	|
From the two axioms it follows that 1Y is unique (see lemma 1 below)» >
SEC. 1 CATEGORIES	Jg
and it is called the identity morphism of Y. Given morphisms f: X Y and
g: Y —» X such that gf = lx, g is called a left inverse of f and f is called a
right inverse of g. A two-sided inverse (or simply an inverse) of f is a
morphism which is both a left inverse of f and a right inverse of f. A morphism
f: X -> his called an equivalence, denoted by f: X Y, if there is a morphism
g: У -> X which is a two-sided inverse of f. If g': Y X is a left inverse of f
and g": Y—» X is a right inverse of f, then
g' = g'ly = g'(fg") = (g'f )g" = Ixg" = g"
showing that g' — gf. Therefore we have the following lemma.
1	lemma If f: X Y has a left inverse and a right inverse, they
are equal, and f is an equivalence. 
In particular, it follows that an equivalence f: X Z Y has a unique in-
verse, denoted by f~b Y —> X, and f 1 is an equivalence. If there is an
equivalence f: X Sr Y, X and Y are said to be equivalent, denoted by X sr Y.
Because the composite of equivalences is easily seen to be an equivalence,
the relation X ~ Y is an equivalence relation in any set of objects of Q.
We list some examples of categories.
2	The category of sets and functions [that is, the class of objects is the class
of all sets, and for sets X and Y, hom (X, Y) equals the set of functions from
Xto Y]
3	The category of topological spaces and continuous maps
4	The category of groups and homomorphisms
5	The category of R modules and homomorphisms
(» The category of normed rings (over R) and continuous homomorphisms
7 The category of sets and injections (or surjections or bijections)
It The category7 of pointed sets (a pointed set is a nonempty set with a dis-
tinguished element) and functions preserving distinguished elements
fl The category of pointed topological spaces (a pointed topological space
is a nonempty topological space with a base point) and continuous maps
preserving base points
10 The category of finite sets and functions
11 Given a partial order < in X, there is a category whose objects are the
elements of X and such that hom (x,x) is either the singleton consisting of the
ordered pair (x,x') or empty, according to whether x < x' or x x'
12 The category of groups and conjugacy classes of homomorphisms (that is,
a morphism G —> G' is an equivalence class of homomorphisms from G to G',
two homomorphisms being equivalent if they differ by an inner automorphism
ofG')
16	HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP. 1 j
A subcategory (2' С C1 is a category such that	)
(a)	The objects of I?' are also objects of (2
(b)	For objects X' and Y' of G', how (-',Y') C hony (X,Y)	'
(c)	If X' Y' and g': Y' ~^> Z' arc morphisms of (2', their composite
in (S' equals their composite in (2	.
(S' is called a full subcategory of 6 if (S' is a subcategory of (S and for ob- ,
jects X' and Y' in G', honV (X',Y') = home (X,Y). The category in example 7 ‘
above is a subcategory of the one in example 2, and the category in example '
10 is a full subcategory of the one in example 2. The categories in examples )
3, 4, 5, 6, and 8 are not subcategories of the category of sets, because each ;
object of one of these categories consists of a set, together with an additional :
structure on it (hence, different objects in these categories may have the same '
underlying sets). In examples 11 and 12, the morphisms in the respective ’
categories are not functions, and so neither of these categories is a subcate- '
gory of the category of sets.	;
A diagram of morphisms	such as the square	
X Д Y	:
f'
X' r	J
is said to	be	commutative	if any	two composites of morphisms in the diagram*
beginning at the same place and ending at the same place are equal. This,
square is commutative if and only if hf — f'g.	;
Following are descriptions of some categories which are associated to fi
given category. Given a category (2, there is an associated category called:
the category of morphisms of G. Its objects are morphisms X 4 Y, and.
its morphisms with domain X -U Y and range X' -4 Y' are pairs of morphisms.
g: X X' and h: Y Y' such that the square	'
is commutative. In a similar way, diagrams of morphisms in G more general ;
than X 4 Y are the objects of a suitable category associated to (2.	j
Let C be a category' whose objects are sets with additional structures;
(such as distinguished elements or topologies) and whose morphisms are|
functions preserving the additional structures. For example, (2 might be any)
of the categories in examples 2 through 10. There is a category associated to(
(?, called the category of pairs of G, whose objects are injective morphisms;
A X (because each morphism in such a category' is a function, it is mean-j
ingful to consider those which are injective) and whose morphisms are com-j
mutative squares	(
А 4 X	i
H Й	j
Y
SEC. 1 CATEGORIES	J у
Thus the category7 of pairs of Gis a full subcategory of the category of morphisms
of G. The notation (X,A) will denote the pair consisting of X and i: А С X,
and the notation fi (X,A)	(Y,B) will mean that fi. X -> Y is a morphism of
G such that/(i(A)) C j(B). The category of pairs of (2, therefore, has as objects
the pairs (X,A) and has as morphisms the morphisms fi (X,A)
If C i and t2 are categories, their product c'i X (?2 is the category whose
objects are ordered pairs (YbY2) of objects Yi in (2г and Y2 in G2 and whose
morphisms (XbX2) (Yi,Y2) are ordered pairs of morphisms (/i,/2), where
fi'. Xi Yi in ti and f2: X2 Y2 in G2. Similarly7, there is a product of an
arbitrary indexed family of categories.
Given a category (2, there is an opposite category (2* whose objects Y*
are in one-to-one correspondence with the objects Y of (2 and whose morphisms
f* = Y* -> X * are in one-to-one correspondence with the morphisms fi X Y
[with/*g* defined to equal (gf)* for X4 Y4 Z in (?]. We identify ((2*)*
with G, so that (X *) * = X and (f*) * = f.
We next show how to interpret sums and products, as well as direct and
inverse limits in arbitrary categories. An object X in a category (2 is said to be
an initial object if for each object Y in (2 the set hom (X, Y) contains exactly
one element. Dually, an object Z of (2 is said to be a terminal object if for each
Y of (2 the set hom (Y,Z) contains exactly7 one element. Note that any two
initial objects of (2 are equivalent and any7 two terminal objects of (2 are
equivalent. In examples 2 and 3 the empty set is an initial object and any
one-point set is a terminal object. In example 4 the trivial group is both an
initial and a terminal object. In example 7 the category of sets and bijections
has neither an initial object nor a terminal object.
Let {Yj}jeJ be an indexed collection of objects of a category (2. Let S{ Y;}
be the category7 whose objects are indexed collections of morphisms of
(2 having the same range and whose morphisms with domain { f}: Yj Z}
and range {//: Y} Z'} are morphisms g: Z Z' of (2 such that gfi = fffor
every7 j £ J. An initial object of 5{ YJ is called a sum of the collection {Y;}. A
given collection may7 or may7 not have a sum in (2. The set sum is a sum in
the category of sets, the topological sum is a sum in the category of topologi-
cal spaces, the free product is a sum in the category of groups, and the direct
sum is a sum in the category of R modules. In the category7 of finite sets, in
general only7 finite collections have a sum. Similarly7, in the category of finitely
generated R modules, in general only7 finite collections have a sum.
Dually, given an indexed collection of objects (Yf}jej in (2, let ^{YJ be
the category7 whose objects are indexed collections of morphisms {g/};ej of (2
having the same domain and whose morphisms with domain {gg X Yfi
and range {g': X' Y;} are morphisms fi. X X' of G such that gf = g, for
every / £ J. A terminal object of l?(Y;} is called a product of the collection
{Yj}. The cartesian product of sets is a product in the category of sets, the
topological product is a product in the category of topological spaces, and the
direct product is a product in the category of groups, or R modules. In the
category7 of finite sets (or finitely7 generated R modules), in general only finite
collections have a product.
18
HOMOTOPY AND THE FUNDAMENTAL GROOT CHAP. 1
A direct system {Y",f,./:} in a category (2 consists of a collection of
objects { Y“] indexed by a directed set A = {a} and a collection of morphisms
{ f,./1'. Ya —> Y®} in (2 for a < /3 in A such that
(a)	faa = ly" for a £ A
(b)	fy = f/YfJ3'- Y" -> Yy for a < {3 < у in A
There is then a category' dir {¥“,//} whose objects are indexed collections of
morphisms {g(1: Y“ Z},I( A such that g„ = gpff if a < (3 in A and whose
morphisms with domain {ga: Ya —> Z) and range {g'a: Ya —> Z'} are
morphisms h: Z —> Z' such that hga = g'a for a 6 A. An initial object
of dir	is called a direct limit of the direct system {Y^ff}. The direct
limits of sets, topological spaces, groups, and R modules are examples of
direct Emits in their respective categories.
Dually, an inverse system {Y„,ff:} in £ consists of a collection of objects
{Ytt} indexed by' a directed set A = {«} and a collection of morphisms
{fa13-. Y/3 Ya) in (2 for a < (3 in A such that
(fl) faa = ly„ for a E A
(b)	fay = fapf(3y. Y7 Ya for a < (3 < у in A
There is then a category' inv ( Ya,fap} whose objects are indexed collections of
morphisms {ga: X —> Уй)й(л such that ga = ffgn if a < /? in A and whose
morphisms with domain {ga: X —> Yf,} and range { g(,: X' -y Ya) are morphisms
h: X —s> X' of ё such that g,',/i = ga for o f A. A terminal object of inv {Ya,/a^}
is called an inverse limit of the inverse system { Y„,/'/: }. The inverse limits of
sets, topological spaces, groups, and R modules are examples of inverse limits
in then respective categories.
By similar- considerations it is possible to define a direct or inverse Emit
for an arbitrary indexed collection of objects in a category' (2 and an indexed
collection of morphisms in (2 between these objects. We omit the details.
2 HfflCTOKS
Our main interest in categories is in the maps from one category to another.
Those maps which have the natural properties of preserving identities and
composites are called functors. This section is devoted to the definition
of functors of one or more variables, some examples and appfications, and
the definition of natural transformations between functors.
Let (2 and 4) be categories. A covariant functor (or contravariant functor)
T from ё to bD consists of an object function which assigns to every' object X
of ё an object T(X) of 'f) and a morphism function which assigns to every' mor-
phism f. X Y of (2 a morphism T(/): T(X) T(Y) [or T(/): T(Y) T(X)]
of such that
(fl) T(lf) = 1ад
(b) W) = T(g)T(/) [or T(gf) = T(/)T(g)]
SEC. 2 FUNCTORS	19
We list some examples of functors.
I There is a covariant functor from the category of topological spaces and
continuous maps to the category of sets, and functions which assigns to every
topological space its underlying set: This functor is called a forgetful functor
because it “forgets” some of the structure of a topological space.
>	There is a covariant functor from the category of sets and functions to
the category of R modules and homomorphisms which assigns to every set
the free R module generated by it.
3	Given a fixed R module Mo, there is a covariant functor (or contravariant
functor) from the category of R modules and homomorphisms to itself which
assigns to an R module M the R module Нотд(Мо,Л1) [or Нотд(М,М0)].
4	For any category (2 and object Y of (2 there is a covariant functor тгу (or
contravariant functor тгy) from (2 to the category' of sets and functions which
assigns to an object Z (or X) of G the set TryfZ) = horn (Y,Z) [or тгг(Х) =
horn (X,Y)] and to a morphism h: Z Z' [or/: X X'] the function
hom (Y,Z) horn (Y,Z') [or /#: hom (X',Y) —> hom (X,Y)]
defined by h#(g) = h ° g for g: Y Z [or /# (g') = g' ° / for g': X' -> Y]
5 There is a contravariant functor C from the category of compact Hausdorff
spaces and continuous maps to the category7 of normed rings over R and con-
tinuous homomorphisms which assigns to X its normed ring of continuous
real-valued functions.
6	There is a covariant functor Ho from the category of topological spaces
and continuous maps to the category of abelian groups and homomorphisms
such that H0(X) is the free abelian group generated by the set of components
of X, and if f: X Y, then II ff): Ho(X) -ч> Ho(Y) is the homomorphism such
that if C is a component of X and C is the component of Y containing f(C),
then H0(/)C = C.
7	A direct system (or inverse system) in a category G is a covariant functor
(or contravariant functor) from the category of a directed set (defined as in
example 1.1.11) to G.
8	For any category (2 there is a contravariant functor to its opposite cate-
gory 6* which assigns to an object X of 6 the object X* of (2* and to
a morphism /; X Y of e' the morphism /*: Y* X*.
Note that any contravariant functor on (2 corresponds to a covariant
functor on 6*, and vice versa. Therefore any functor can be regarded as co-
variant on a suitable category. Despite this, we shall find it convenient to con-
sider contravariant as well as covariant functors on (2, rather than consider
only covariant functors on two categories.
Any functor from the category of topological spaces and continuous
maps to an algebraic category (such as the category of abelian groups and
.20
HOMOTOPY AND THE FUNDAMENTAL CROUP CHAP, 1
homomorphisms) is a representation of the topological category by an alge-
braic one. Algebraic topology is the study of such functors; we show that
simple remarks about functors can be used to obtain necessary conditions for
the solvability of topological problems.
9	theorem Let T be a functor from a category 8 to a category 4). Then
T maps equivalences in Q to equivalences in °P.
proof Assume that T is a covariant functor (the ar gument is similar if T is
contravariant). Let f: X —у У be an equivalence in £ Then fff = 1%.
Therefore
W) = T(ty) = T(tyi)T(/)
Similarly, T(f)T(f r) — ty(V). Therefore T(f ]) is a two-sided inverse of T(f'),
and T(/l is an equivalence in cf>. 
In particular, if T is an algebraic functor on the category of topological
spaces and continuous maps, a necessary condition that X be homeomorphic
to У is that T(X) be equivalent to T(Y). Thus the functor Ho of example 6
shows that the real line R and the real plane R2 are not homeomorphic
[if they were homeomorphic, then R — 0 would be homeomorphic to R2 — p
for some p 6 R2, but Ho(R — 0) is a free abelian group on two generators,
while //()(R2 — p) is a free abelian group on one generator]. This is a trivial
example. However, the homology functors HQ defined in Chapter 4 generalize
Ho and can be used in much the same way to prove that R” and Rm are not
homeomorphic if n ty m.
In applications of algebraic functors to topological problems the algebra
will frequently play an essential role. For example, let To(X) be the functor
obtained by composing the functor Ho with the forgetful functor, which
assigns to every abelian group its underlying set. The functor To contains less
information than the functor Ho and does not give as strong a necessary con-
dition for homeomorphism [for example, T0(R — 0) and To(R2 — p) are both
countably infinite sets and are equivalent in the category of sets and func-
tions], For this reason it is important to provide functors with as much alge-
braic structure as possible. Later we shall consider functors which depend on
a chosen topological space. These functors take values in the category of sets
and functions, but some of them, depending on properties of the particular-
spaces which define them, are functors to the category of groups and homo-
morphisms. The added algebraic structure in such cases will prove useful.
To show how functors can be applied to another problem, let A be
a subspace of a topological space X and let /: A У be continuous. The ex-
tension problem is to determine whether f has a continuous extension to X—
that is, whether the dotted arrow in the triangle
А С X
corresponds to a continuous map making the diagram commutative.
[ SEC. 2 FUNCTORS	21
I	I<> theorem Let T be a covariant functor (or contravari’ant functor') from
i	the category of topological spaces and continuous maps to a category £ A
necessary condition that a map f: A Y be extendable to X {where i: А С X)
is that there exist a morphism tp: T(X) T(Y) [or q>: T(Y) T(X)] such that
\	<p°T(i) = T(f) [or T(f) = T(f) о <р].
‘ proof Assume that/': X -ч> У is an extension of f. Then f'i = f. Therefore
T(f') ° T(i) — T(f) [or T(f) = T(i) ° T(f')], and T(f') can be taken as the
; morphism <p.	
The above result can be applied to prove that the identity map of I can-
! not be extended to a continuous map I I. We use the functor Ho and ob-
tain the necessary condition that there must exist a homomorphism
<p: Ho(I) H0(t) such that <p ° Ho(f) — Ho(lj) (where i: icfj. Because H0(I) is a
i	free abelian group on two generators and Ho(I) is a free abelian group on one
J	generator, there is no such homomorphism <p. Again, this is a trivial example,
i	but it illustrates the method, and the general homology functors Hq defined
i	later can be used in the same way to show that there is no continuous map
;	En+1 —> S” that is the identity map on S” .
Thus we see that a functor yields necessary conditions for the solvability
of topological problems. There are situations in which these necessary con-
' ditions are also sufficient. For example, the functor C of example 5 gives a
; necessary- and sufficient condition for homeomorphism—that is, two compact
;	Hausdorff spaces X and У are homeomorphic if and only if C(X) and C(Y)
j	are isomorphic.1 This is not a particularly useful result, however, because it
!	seems to be no easier to determine whether or not two normed rings are iso-
	morphic than it is to determine whether or not two compact Hausdorff
i	spaces are homeomorphic. We seek functors to categories that are somewhat
-	simpler than the category of topological spaces, so that the algebraic problems
;	that arise in these categories can be effectively7 solved. One big problem of
;	algebraic topology7 is to find, and compute, sufficiently7 many7 such functors
7	that the solvability of a particular topological problem is equivalent to the
; solvability of the corresponding (and simpler) algebraic problems.
i We shall also have occasion to compare functors with each other. This is
I done by7 means of a suitable definition of a map between functors. Let 7) and
1 T2 be functors of the same variance (either both covariant or both contravariant)
 from a category C to a category7 °D. A natural transformation ф from 7] to
; Tg is a function from the objects of (3 to morphisms of °D such that for every7
J morphism f: X -ч> Y of & the appropriate one of the following diagrams
( is commutative:
>	Ti(X)	Tr(Y)	ВД Ta(Y)
[	9>(X)|	|<р(У)	9>(X)|	]у(У)
j	T2(X)	t2(Y)	t2(x) <^fL T2(Y)
?	?i, T2 covariant	Ti, T2 contravariant
|	1 See Theorem D on page 330 of G. F. Simmons, Introduction to Topology and Modem Analy-
J sis, McGraw-Hill Book Company, New York, 1963.
г
1
22	HOMOTOPY AND THE FUNDAMENTAL CROUP CHAP. 1	[
If Ф is a natural transformation from Tf to T2 such that flX) is an cquiv- (
alence in '’D for each object X in (?, then <p is called a natural equivalence. !
As an example of a natural transformation, let Yr and Y2 be objects of a i
category (? and let g: Yr Y2 be a morphism in t1. There is a natural trans- 1
formation g# from the covariant functor rty, t° the covariant functor чту: and ;
a natural transformation g# from the contravariant functor tX* to the contra- '
variant functor rX, If g is an equivalence in F, both these natural transforma- (
tions are natural equivalences.	f
It is also of interest to consider functors of several variables. Thus, if ;
l?2, and are categories, a covariant functor from Fi X F2 to °? is called a <
functor of two arguments covariant in each. A covariant functor from
Fi X I?* to regarded as a function from ordered pairs (Xi,X2), where Xi is '
an object of Qr and X2 is an object of F2, is called a functor of two arguments \
covariant in the first and contravariant in the second. In a similar fashion, i
functors of more arguments with mixed variance are defined.	i
If F is any category, there is a functor of two arguments in F to the cate- ;
gory of sets and functions which is contravariant in the first argument and t
covariant in the second. This functor assigns to an ordered pair of objects X ;
and Y of F the set hom (X, Y) and to an ordered pair of morphisms f: X’ X [
and g: Y Y' in F the function f#g# = g#f#'. hom (X,Y) —> hom (X', Y').	।
3 homotopy	!
The problem of classifying topological spaces and continuous maps up to '•
topological equivalence does not seem to be amenable to attack directly by :
computable algebraic functors, as described in Sec. 1.2. Many of the comput- j
able functors, because they are computable, are invariant under continuous J
deformation. Therefore they cannot distinguish between spaces (or maps) that (
can be continuously deformed from one to the other; the most that can be (
hoped for from such functors is that they characterize the space (or map) up /
to continuous deformation.	'
The intuitive concept of a continuous deformation will be made precise ‘
in this section in the concept of homotopy. This leads to the homotopy cate- *
gory which is fundamental for algebraic topology. Its objects are topological !
spaces and its morphisms are equivalence classes of continuous maps (two <
maps being equivalent if one can be continuously deformed into the other). ?
For technical reasons we consider not just the homotopy category of topologi- <
cal spaces, but rather the larger homotopy category of pairs.	?
A topological pair (X,A) consists of a topological space X and a subspace I
А С X. If A is empty, denoted by 0, we shall not distinguish between the [
pair (X, 0) and the space X. A subpair (X',AZ) G (X,A) consists of a pair with I
X' G X and A' C A. A map f: (X,A)	(Y,B) between pairs is a continuous j
function f from X to Y such that f(A) С B, and as in Sec. 1.1, there is 5
sec. 3 homotopy	23
a category of topological pairs and maps between them which contains as full
subeategories the category of topological spaces and continuous maps, as well
as the category of pointed topological spaces and continuous maps.
Given a pair (ХД), we let (X,A) X I denote the pair (X X I, A X !) Let
X' G X and suppose that/о, fat (X,A) -ч> (Y,B) agree on X' (that is, fo|X' =
Д | X'). Then fa is homotopic to fa relative to X', denoted by fa ~ /1 rel X', if
there exists a map
F: (X,A) X I (Y,B)
such that F(x,0) = fa(x) and F(x,l) = fa(x) for x 6 X and F(x,t) = fa(x) for
x E X' and t G I. Such a map F is called a homotopy relative to X' from fa to
fa and is denoted by F: fa ~ fa rel X'. If X' = 0, we omit the phrase “rela-
tive to 0.” Clearly, fa ~ fa rel X' implies fa ~ fa rel X" for any X" G X'. A
map from X to Y is said to be null homotopic, or inessential, if it is homotopic
to some constant map.
For f 6 I define ht: (X,A)	(X,A) X I by ht(x) = (x,t). If F: fa fa rel X',
then Fho = fa, Fhr = fa, and Fht | X' = fa | X' for all t El. Therefore the collec-
tion	is a continuous one-parameter family of maps from (X,A) to
(Y,B), agreeing on X', which connects/о = Fho to fa = Fhfa. Hence fa ~ fa
rel X' corresponds to the intuitive idea of continuously deforming fa into fa
by maps all of which agree on X'. Note that if fa ~ fa rel X' there will usually
be many maps F which are homotopies relative to X' from fa to fa (see
example 3 below).
1	example Let X = Y = R” and define fa(x) = x and /i(x) = 0 for
x £ R’! (that is, fa = 1ц» and fa is the constant map of Rn to its origin).
If F: В'1 X I —> R” is defined by
F(x,t) = (1 - t)x
then F: fa ~ fa rel 0.
2	example Let X = Y = I and define fa(f) = t and faff) = 0 for t £ I. If
F: I X I I is defined by
F(t,t') = (1 - t’)t
then F: fa ~ fa rel 0.
3	• example Let X = Y — E2 = (z £ C |z = reie, 0 < r < 1} and let
A = В = S1 = (z £ C|z = eie}. Define fa-. IFfaS1) -> (EfaS1) to be the
identity map and fa: (E2^1)	(E^S1) to be the reflection in the origin [that
is, У1(геге) = reW+’d], Define a homotopy F: fa ~ fa rel 0 by F(rele,t) =
Another homotopy F': fa c^fa rel 0 is defined by F' (rele,t) = re1^*^.
1 Л one-parameter family ft: (X,A) —> (Y,B) for t ( I is continuous is jointly continuous
in t and x, in which case the function (x,t) fi(x) is a homotopy from jo to ji. The correspond-
ing function t fi from I to	is always continuous [where (Y,B)<X’A1 = {g: (X,A)
(Y,B)} topologized by the compact-open topology]. Conversely, in case X is a locally compact
Hausdorff space, it follows from theorem 2.8 in the Introduction that for any continuous map
</>: I (Y,B)<r^> the one-parameter family <p(t) is continuous and defines a homotopy from <f(0)
to ¥>(!)•
24
HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP. 1	! SEC. 3 HOMOTOPY
25
4 example Let X be an arbitrary space and let У be a convex subset of
R". Let /0, fi: X —> Y be maps which agree on some subspace X' С X. Then ;
fo — ft rel X', because the map F: X X I Y defined by	।
F(x,t) = tfy(x) + (1 - #)/0(x)	;
is a homotopy relative to X' from fo to /1.
Example 4 is a generalization of examples 1 and 2. In example 3 the space E2 ;
is convex, but the homotopy between fo and/i cannot be taken to be a partic- •
ular case of the homotopy in example 4, because it must keep S1 mapped into ,
itself at all stages, and S1 is not convex.
To define the homotopy category we need the following easy results. j
5 theorem Homotopy relative to X' is an equivalence relation in the set 
of maps from (X,A) to (Y,B).	_
proof Reflexivity. For/: (X,A)	(Y,B) define F: f ~ /rel X by F(x,t) = /(x).
Symmetry. Given F: /0 ~ /1 rel X', define F': fy ~ f0 rel X' by F'(x,t) — г
F(%, i -1).	:
Transitivity. Given F: f0 ~ fy rel X' and G: /1 ~ f2 rel X', define 
H: fo ~ /2 rel X' by	;
и(г	0 < t < У2	1
(	~ I G(x, 2/ — 1) У2 < t < 1	I
Note that H is continuous because its restriction to each of the closed sets i
X X [0,Уг] and X X [У2Д] is continuous. 	1
It follows that the set of maps from (X,A) to (Y,B) is partitioned into dis- / •
joint equivalence classes by the relation of homotopy relative to X'. These . J
equivalence classes are called homotopy classes relative to X'. We use [
[X,A; Y,B]x' to denote this set of homotopy classes. Given/: (X,A)	(Y,B), we •
use [/]x' to denote the element of [X,A; Y,B]x' determined by/. Homotopy j
classes relative to the empty set will be denoted by omitting the subscript X'. ,j
G theorem Composites of homotopic maps are homotopic.	;
proof Let fo, fy. (X,A)	(Y,B) be homotopic relative to X' and let go, gy.
(Y,B)	(Z,C) be homotopic relative to Y', where /i(X') C Y'. To show that
go/o, gi/i; (X,A) —> (Z,C) are homotopic relative to X', let F: /0 fy rel X'
and G: go ~ gi rel Y'. Then the composite
(X,A) X / -> (1Д) -> (Z,C)
is a homotopy relative to X' from go/о to g0/i, and the composite	
(X,A) x I —(Y,B) x I Я (Z,C)	1
is a homotopy relative to fy~1(Y') from g0/i to gyfy. Since X' C /1"1(Y'), we j
have shown that go/o =: go/i rel X' and go/r ~ gyfy rel X'. The result now (
follows from theorem 5. 	t
The last result shows that there is a homotopy category of pairs whose
objects are topological pairs and whose morphisms are homotopy classes
(relative to 0). This category contains as full subcategories the homotopy
category of topological spaces (also shortened to homotopy category) and the
homotopy category of pointed topological spaces. There is a covariant functor
from the category of pairs and maps to the homotopy category of pairs whose
object function is the identity map and whose mapping function sends a map
/to its homotopy class [/]. As pointed out at the beginning of the section,
most of the algebraic functors we consider will be defined from the appro-
priate homotopy category. A diagram of topological pairs and maps is said to
be homotopy commutative Я it can be made a commutative diagram in the
homotopy category (that is, when each map is replaced by its homotopy
dass).
As in example 1.2.4, for any pair (P,Q) there is a covariant functor
(or a contra variant functor	from the homotopy category of pairs to the
category of sets and functions defined by tt(pq) (X,A) = [F,Q; X,A] (or
(X,A) = [X,A; P,Q]), and if /: (X,A)	(Y,B), then ([/]) = f#
(or -n™ ([/]) = /#), where/#[g] = [/g] for g: (P,Q)	(X,A) (or/#[h] = [hf]
for h: (Y,B)	(P,Q))- If (P,Q) ff'&f there is a natural transformation
a# from to 77(p,q) and a natural transformation a# from nU’-Q) to rr<p'>Q"i.
A map/: (X,A)	(Y,B) is called a homotopy equivalence if [/] is an
equivalence in the homotopy category of pahs. A map g: (Y,B) —> (X,A)
is called a homotopy inverse of /if [g] = [/]-1 in the homotopy category.
Pairs (X,A) and (Y,B) are said to have the same homotopy type if they are
equivalent in the homotopy category.
The simplest nonempty space is a one-point space. We characterize the
homotopy type of such a space as follows. A topological space X is said to be
contractible if the identity map of X is homotopic to some constant map of X
to itself. A homotopy from lx to the constant map of X to x’o € X is called a
contraction of X to xo. Examples 1 and 2 show that Rn and I are contractible,
and example 4 shows that any convex subset of R" is contractible. The fol-
lowing lemma may be regarded as a generalization of the result of example 4.
7 lemma Any two maps of an arbitrary space to a contractible space are
homotopic.
proof Let Y be a contractible space and suppose ly ~ c, where c is a con-
stant map of Y to itself. Let /0, fy.X—>Y be arbitrary. By theorem 6,
/0 = ly/o c/o, and similarly, /1 ~ cfy. Since c/o = cfy, it follows from
theorem 5 that f0 ~ fy. 
8 corollary If Y is contractible, any two constant maps of Y to itself
are homotopic, and the identity map is homotopic to any constant map of Y
to itself. 
It is interesting to observe that lemma 7 cannot be strengthened to the
case of relative homotopy. That is, if /0 and fy are maps of X into a contract-
27
26	HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP. I
ible space Y which agree on X' С X, it need not be true that /0 — fi rel X'
(although example 4 shows this to be true for convex subsets of R'1). The fol-
lowing example illustrates this and will be referred to again later.
9 example The comb space Y illustrated in the diagram
(0,1)	(l/n,l)	(%,1)	(1,1)
I
(0,0)	(1/П.0)	(V2.0)
Comb space
(1,0)
SEC. 4 RETRACTION AND DEFORMATION
J 1 corollary Two contractible spaces have the same homotopy type, and
any continuous map between contractible spaces is a homotopy equivalence.
proof The first part follows from theorem 10 and the transitivity of the
relation of having the same homotopy type. The second part follows from the
first part and lemma 7 (and from the obvious fact that any map homotopic to
a homotopy equivalence is itself a homotopy equivalence). »
The next result establishes an important relation between homotopy and
the extendability of maps.
12 theorem Let po be any point of Sn and let fi. Sn Y. The following
are equivalent:
(o) f is null homotopic
(b) f can be continuously extended over En+1
(c) f is null homotopic relative to po
proof (a) => (b). Let F: f ~ c, where c is the constant map of Sn to yo £ Y.
Define an extension f of f over En+1 by
_ f y°	0 < ||x|| < %
1	~ [f(x/||x||, 2 — 2||x||)	%< ||x|| < 1
Since F(x,l) = y0 for all x £ S'1, the map f' is well-defined, f' is continuous
because its restriction to each of the closed sets {x £ E"+1|0 < ]|x|| < У2)
and {x £ E"+1| У2 < ||x|| < 1} is continuous. Since F(x,0) = fix) for x £ S'1,
f' I S« = f and f' is a continuous extension of f to En+1.
(bfr>(c). If f has the continuous extension f'; En+1 Y, define
is defined by
i
a;
Let F: Y X i -> Y be defined by F((x,y), t) = (x, (1 — t)y). Then F is
homotopy from Ip to the projection of Y to the x axis. Since the latter map isL
homotopic to a constant map, Y is contractible. Let c: Y У be the constant^
map of Y to the point (0,1). By corollary 8, ly ~ c, but even though these!
two maps agree on (0,1), there is no homotopy relative to (0,1) between tlieni. f
The following theorem shows that contractible spaces are homotopically 1
as simple as possible.	)
10 theorem A space is contractible if and only if it has the same homotopy |
type as a one-point space.	|
proof Assume that X is contractible and let F: X X I X be a contraction!
of X to a point Xo £ X. Let P be the one-point space consisting of xo and Idj
f : X P and j: P С X. Then fl — Ip and F: Ly jf. Therefore [/] = [ /1
and f is a homotopy equivalence from X to P.
Conversely, if X has the same homotopy type as a one-point space P, let
f: X P be a homotopy equivalence with homotopy inverse g: F —> X. Then 
Ij- ~ gf. Because gf is a constant map, X is contractible. 
F(x,t) = f'((1 - t)x + tp0)
Then F(x,0) = /'(x) = fix) and F(x,l) = f'(po) for x £ Sn. Since F(p0,t) = f'(po)
for t £ I, F is a homotopy relative to po from / to the constant map to f'(po)-
(c) ==> (a). This is obvious. 
Combining theorem 12 with lemma 7, we obtain the following result.
13 corollary Any continuous map from Sn to a contractible space has a
continuous extension over En+1. 
4 RETRACTION АЯВ DEFORMATION
This section is concerned mainly with inclusion maps. We consider whether
such a map has a left inverse, a right inverse, or a two-sided inverse in either
the category of topological spaces and continuous maps or the homotopy
category.1
*-Many of the results in this section can be found in R. H. Fox, On homotopy type and de-
formation retracts, Annals of Mathematics, vol. 44, pp. 40-50, 1943 (see also H. Samelson,
Remark on a paper by R. H. Fox, Annals of Mathematics, vol. 45, pp. 448-449, 1944).
28
HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP. 1
A subspace A of X is called a retract of X if the inclusion map i: А С X
has a left inverse in the category of topological spaces and continuous maps.
Hence A is a retract of X if and only if there is a continuous map г: X —> A >
such that ri = 1A [that is, r(x) = x for x £ А]. Such a map r is called a retrac- I
tion of X to A.	;
A subspace A of X is called a weak retract of X if the inclusion map !
i: А С X has a left homotopy inverse (that is, a left inverse in the homotopy •
category). Thus A is a weak retract of X if and only if there is a continuous 1
map г: X —> A such that ri ~ 1A. Such a map r is called a weak retraction of f
X to A.	j
Any one-point subspace is a retract of any larger space containing it. A •
discrete space with more than one point is never a weak retract of a connected •
space containing it. If A is a retract of X, it is a weak retract of X. The con- j
verse is not true, as is shown by the following example.	I
1 example Let X be the closed unit square I2 in R2 and let А С X be j
the comb space of example 1.3.9. Then A and X are both contractible, and j
by corollary 1.3.11, the inclusion map А С X is a homotopy equivalence. >
Therefore A is a weak retract of X. However, it can be shown that A is not a i
retract of X.	(
Despite the fact that, in general, a weak retract need not be a retract, I
these concepts do coincide when A is a suitable subspace of X. This occurs ’
frequently enough to warrant special consideration and will prove of use later. ?
Let (X,A) be a pair and Y be a space. (X,A) is said to have the homotopy ex- |
tension property with respect to Y if, given maps g: X Y and G: A X I Y f
such that g(x) = G(x,0) for x £ A, there is a map F; X X I s Y such that '
F(x,0) = g(x) for x £ X and F| A X I = G. If g is regarded as a map of X x 0 ;•
to Y, the existence of F is equivalent to the existence of a map represented by >
the dotted arrow which makes the following diagram commutative:	:
A X	0	C	A	X	I	!
G/	'
(T Y П	j
g/	\	|
X X	о	C	X	X	I	|
If (X,A) has the homotopy extension property with respect to Y and fa 1
fa. A —> Y are homotopic, then if f0 has an extension to X, so does/i; for if J
g: X —> Y is an extension of fo and G: A X I Y is a homotopy from fo to j
fa the homotopy extension property implies the existence of a map I
F: X X I Y which is an extension of G, therefore F(x,l) is an extension!
of fa It follows that whether or not a map A —> Y can be extended over X is t
a property of the homotopy class of that map. Therefore the homotopy j
extension property implies that the extension problem for maps A —> Y is a j
problem in the homotopy category.	1
Sl«.. 4 RETRACTION AND DEFORMATION	29
Of particular importance is the case when (X,A) has the homotopy
extension property with respect to any space. More generally, a map/: X' X is
called a cofibration if, given maps g: X Y and G: X' X i -> Y (where Y is
arbitrary) such that g(/(%')) = G(x',0) for x' £ X', there is a map F: X X I—» Y
such that F(x,0)=(f(x) for x £ X and F(/(x'), t) = G(x',t) for x' £ X' and t £ I.
If g is regarded as a map of X X 0 to Y, the existence of F is equivalent to
the existence of a map represented by the dotted arrow which makes the fol-
lowing diagram commutative:
X' X 0 C X' x 1
XxO C Xxl
Thus an inclusion map i: А С X is a cofibration if and only if (X,A) has the
homotopy extension property with respect to any space.
2	theorem If (X,A) has the homotopy extension property with respect to
A, then A is a weak retract of X if and only if A is a retract of X.
proof We show that any weak retraction г: X A is, in fact, homotopic to
a retraction. Let i: А С X; then ri 1A. Let G: A X I —> A be a homotopy
from ri to 1д; then G(x,0) = r(x) for x £ A, Because (X,A) has the homotopy
extension property with respect to A, there is a map F: X X I A which
extends G such that F(x,0) = r(x) for x £ X. If ?•': X —» A is defined by
r'(x) = F(x,l), then ?' is a retraction of X to A, and F is a homotopy from
r to /. 
We can just as well consider inclusion maps with right homotopy
inverses as those with left homotopy inverses. This leads to the following
definitions. Given X' С X, a deformation D of X' in X is a homotopy
D: X' x I X
such that D(x',O) = x' for x' £ X'. If, moreover, Z?(X' x 1) is contained in a
subspace A of X, D is said to be a deformation of X' into A and X' is said to
be deformable in X into A. A space X is said to be deformable into a subspace
A if it is deformable in itself into A. Thus a space X is contractible if and only
if it is deformable into one of its points.
3	lemma A space X is deformable into a subspace A if and only if the
inclusion map i: А с X has a right homotopy inverse.
proof If i has a right homotopy inverse f: X A, then if ~ lx. Let
F: X x I —» X be a homotopy from lx to if; then F(x,0) = x, so F is a defor-
mation of X, and F(X X 1) = tflfa C A, so X is deformable into A.
Conversely, if X is deformable into A, let D; X X I —> X be a deforma-
tion such that D(X X 1) C A. Let f: X —> A be defined by the equation
if(x) = D(x,l) x £ X
30	HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP. 1
Then D; lj ~ if, showing that f is a right homotopy inverse of i. 
Note that an inclusion map i: А С X never has a right inverse in the
category of topological spaces and continuous maps except in the trivial case |
A = X.	J
We now consider inclusion maps which are homotopy equivalences. A J
subspace А С X is called a weak deformation retract of X if the inclusion I
map i: A C Xis a homotopy equivalence. From lemma 1.1.1 and lemma 3 I
above we obtain the following result.	I
4	lemma A is a weak deformation retract of X if and only if A is a weak ?
retract of X and X is deformable into A. 	;
As was the case with the concept of weak retract, there are more useful 1
concepts than that of weak deformation retract. The subspace A is a strong {
deformation retract of X if there is a retraction r of X to A such that if
i: А С X, then lx ir rel A. If F: lx ~ ir rel A, F is called a strong deforma- |
lion retraction of X to A.	
There is an intermediate concept useful in comparing the weak and ;
strong forms already defined. A subspace A is called a deformation retract of •
X if there is a retraction r of X to A such that if i: А С X, then lx ~ ir. If j
F: lx ~ ir, F is called a deformation retraction of X to A. A homotopy i
F: X X I X is a deformation retraction if and only if F(x,0) = x for i
x £ X, F(X X 1) C A. and F(x,l) = x for x E A. It is a strong deformation
retraction if and only if it also satisfies the condition F(x,t) = x for x £ A and [
t € I.	j
5	example It follows from example 1.3.4 that any one-point subset of a j
convex subset	of R’1 is a strong deformation retract of the	convex	set.	j
6	example	Sn is a strong deformation retract of Rn+1	— 0.	In	j
map F: (R’l+1	— 0) X 1 —> R”+1 — 0 defined by	:
F(x,t) = (1 - #)x + -Д- x £ R»+i - 0, t	E I	!
is a strong deformation retraction of R’l+1 — 0 to S”.	[
It is clear that a strong deformation retract is a deformation retract, and a |
deformation retract is a weak deformation retract. The following examples I
show that neither of these implications is reversible.	J
7	example As in example 1 above, let X be the closed unit square and A '
be the comb space. As pointed out in example 1, the inclusion map А С X is j
a homotopy equivalence, but A is not a retract of X. Therefore A is a weak [
deformation retract of X which is not a deformation retract of X.	|
8	example Let X be the comb space and A be the one-point subspace of I
X consisting of the point (0,1). Because X is contractible, there is a homotopy t
F from lx to the constant map of X to A. Such a map F is a deformation re-1
SEC* 4 RETRACTION AND DEFORMATION	31
fraction of X to A. However, as was remarked in example 1.3.9, there is no
homotopy relative to A from lx to the constant map to A; therefore A is a
deformation retract of X which is not a strong deformation retract of X.
In the presence of suitable homotopy extension properties the three con-
cepts of deformation retract coincide, and we shall now prove this.
9	lemma If X is deformable into a retract A, then A is a deformation re-
tract of X.
proof Let г: X —> • A be a retraction and let i: А С X. Then r is a left
homotopy inverse of i. Because X is deformable into A, it follows from
lemma 3 that i has a right homotopy inverse. By lemma 1.1.1, r is also a right
homotopy inverse of i. Since lx ir, A is a deformation retract of X. 
Combining lemma 9 with theorem 2 yields the following corollary.
10	corollary If (X,A) has the homotopy extension property with respect
to A, then A is a weak deformation retract of X if and only if A is a defor-
mation retract of X. 
11	theorem If (X x I, (X X 0) U (A x I) U (X X 1)) has the homotopy
extension property with respect to X and A is closed in X, then A is a defor-
mation retract of X if and only if A is a strong deformation retract of X.
proof If A is a deformation retract of X, let F: X X I X be a homotopy
from lx to ir, where г: X -a A is a retraction and i: А С X. A homotopy
G: [(X X 0) U (A X I) U (X X 1)] X/^X
is defined by the equations
G((x,0), t') = x
G((x,t), f) = F(x, (l - t')t)
G((x,l), t') = F(r(x), 1 - t')
G is well-defined, because for x E A
G((x,0), f) - x = F(x,0)
by the first two equations and
G((x,l), f) = Fix, 1 - F) = F(?-(x), 1 - t')
by the last two equations. G is continuous because its restriction to each of
the closed sets (X X 0) X I, (A X I) X I, and (X X 1) X I is continuous. For
(x,t) E (X X 0) U (A X I) U (X X 1), G((x,t), 0) = F(x,t) [because F(x,0) ~ x,
and since ?'is a retraction, F(r(x), 1) = ir(r(x)) = ?'(x) = F(x,l)]. Therefore G re-
stricted to [(X X 0) U (A X I) U (X X 1)] X 0 can be extended to (X X 1) X 0.
From the homotopy extension property in the hypothesis, G restricted to
[(X X 0) U (A X I) U (X x 1)] X 1 can be extended to (X x I) X к Let
G': (X X I) X 1 X be such an extension, and define H: X X I X
x£X,f E I
x £A-,t,t’El
x E X, I/ E I
32
HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP. J|
by H(x,t) = G'((x,f), 1). Then we have the equations
H(x,0) =	G'((x,0), 1)	=	G((x,0), 1)	= x	x £ X
H(x,l) =	G((x,l), 1)	=	F(r(x),0) =	r(x)	x £ X
H(x,t) =	G((x,t), 1)	=	F(x,0)	—	x	x £ A, t £	I	;
Therefore H is a homotopy relative to A from lx to ir, so A is a strong defer-i
mation retract of X. 	’
The next result asserts that any map is equivalent in the homotopy!
category to an inclusion map that is a cofibration. Let f: X Y and let Zfi
denote the quotient space obtained from the topological sum of X X I and У"
by identifying (x,l) £ X X I withf(x)’£ Y. Zf is called the mapping cylinder'
of f and is depicted in the diagram	:
Y	;
Mapping cylinder	‘
We use [x,#] to denote the point of Zf corresponding to (x,t) £ X X I under;
the identification map and [y] to denote the point of Zf corresponding;
to у £ Y (thus [x,l] = [f(x)] for x £ X). There is an imbedding i: X Zp
with i(x) = [x,0] and an imbedding /: Y Zf with ;(y) = [у]. X and Y are;
regarded as subspaces of Zf by means of these imbeddings. A retraction?
r: Zt —> Y is defined by r[x,t] = [ /(x)] for x £ X and t £ I and r[y] = [y] for'
y£Y.	‘	‘	;
12 theorem Gwen a map fi. X Y, there is a commutative diagram >
X Zf	i
A Z	!
¥	i
such that	i
(o)	lzf	F rel	Y	i
(b)	i is	a cofibration	j
proof By definition, ri = fi and the triangle is commutative.	|
(o) A homotopy F: Zf x I Zf is defined by	|
F([x,t], t’) = [x, (1 - t’)t + t'] X £ X; f £ /
= [у]	у £ Y, f £ I
Then F: lzf — p’ rel Y.
SEC. 5 H SPACES
33
(b)	Let g:	Zf	W and G: X x I W be such that g([x,O])	= G(x,O)
for x £ X. If H:	Zf x	I —> W is	defined by the equations
= g[y]	У € X t' € I
_	№> (2# -	П/(2 - #')] 0 < t’ < 2t <	2, x	£	X
ц ’J’	) -	_	2f)/^ _ f)) о < 2t < t’ <	i, %	€	x
then H([x,t], 0) = g[x,t] and H([y],0) = g[y], and H(X X I = G. 
It follows that the map i: X C Zf is a cofibration equivalent in the
homotopy category to the map f: X —> Y. The mapping cylinder can be used
to prove the following amusing result.
13 theorem Two spaces X and Y have the same homotopy type if and
only if they can be imbedded as weak deformation retracts of the same
space Z.
proof If X and Y can be imbedded as weak deformation retracts of the
same space Z, then X and Y each have the same homotopy type as Z. There-
fore X and Y have the same homotopy type.
Conversely, if f: X Y is a homotopy equivalence, it follows from
theorem 12 that if Zf is the mapping cylinder of f, then the composite
X Zf Y is a homotopy equivalence. Because r is a homotopy equiva-
lence, this implies that i is a homotopy equivalence. By theorem 12a, /: Y Zf
is a homotopy equivalence. Therefore X and Y are imbedded as weak defor-
mation retracts in Zf. 
All the foregoing concepts can also be considered for pair's. For example,
a pair (X',A') C (X,A) is a strong deformation retract if there is a map
F: (X,A) x I (X,A) such that F(x,0) = x for x € X, F(X X 1) С X',
F(A X 1) C A'; and F(x',t) = x! for x' £ X' and t £ I. The mapping cylinder
of a map f: (X,A)	(Y,B), where A is closed in X, is the pair’ (Z^,Zf2), where
Zf, is the mapping cylinder of the map Д: X —> Y defined by f and Zf2 is the
mapping cylinder of the map f2: A -ч> В defined by f. A map f: (X',A')
(X,A) is a cofibration if, given maps g: (X,A)	(Y,B) and G: (X',A') x I —>
(Y,B) [where (Y,B) is arbitrary] such that G(x',0) = gf(x') for x' £ X', there
exists a map F: (X,A) X I -> (Y,B) such that F(x,0) = fix) for x £ X and
G(x',t) = F(fix'), t) for x' £ X' and t £ I. All the results remain valid when
suitably formulated for pairs.
5 H SPACES
In some cases it is possible to introduce a natural group structure in the set
of homotopy classes of maps from one space (or pair) to another. In this
section we consider spaces P such that [X;P] admits a group structure for all
X. It is not surprising that there is a close relation between natural group
structures on [X;F] for all X and “grouplike” structures on P.
34
HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP. 1
We shall work in the homotopy category of pointed topological spaces,
although much of what we do is also valid in the homotopy category of
topological spaces. If X and Y are pointed topological spaces, [X;Y ] will de-
note the set of base-point-preserving homotopy classes of continuous maps
X —>Y (with all homotopies understood to be relative to the base point).
Thus [X; Y [ is the set of morphisms from X to Yin the homotopy category of
pointed topological spaces.
One method of obtaining a group structure on [X;P] is to start with a
group structure on P. Thus, let P be a topological group with identity element
as base point. There is a law of composition in the set of all base-point-
preserving continuous maps from X to P defined by pointwise multiplication
of functions. That is, if gi, g2: X -> P, then gjg2: X P is defined by
gig2(x) = gi(T)g2(x), where the right-hand side is the group product in P. With
this law of composition, the set of base-point-preserving continuous maps
from X to P is a group (which is abelian if P is abelian). The law of composi-
tion carries over to give an operation on homotopy classes such that [gi |[g2] =
[gig2], and we have the following theorem.
I theorem If P is a topological group, ttp is a contravariant functor from
the homotopy category of pointed topological spaces to the category of groups
and homomorphisms. 
We give two examples.
2 S1 is an abelian topological group (the multiplicative group of complex
numbers of norm 1). Therefore [XgS'1] is an abelian group, and if f: X—» Y,
then/#: fY;S' | [XjS1] is a homomorphism.
3 S3 is a topological group (the multiplicative group of quaternions of
norm 1). Therefore [X;S3] is a group, and if ft X Y, then/#: [Y;S3]	[X;S3]
is a homomorphism.
This group structure on [X;P[ was deduced from a group structure on
the set of base-point-preserving continuous maps from X to P. There are situ-
ations in which [X;P] admits a natural group structure, but the set of base-
point-preserving continuous maps from X to P has no group structure. For
example, if P is a pointed space having the same homotopy type as some
topological group P', then ttf is naturally equivalent to тг7''. Therefore rrp can
be regarded as a functor to the category of groups. The following definitions
will be used to describe the additional structure needed on a pointed space P
in order that rrp take values in the category of groups and homomorphisms.
If /: X —> Y and g: X -ч- Z, we define
(/,g): X —> Y X Z
to be the map (/,g)(x) = ( J(x),g(x)) for x £ X.
An H space consists of a pointed topological space P together with a con-
tinuous multiplication
pt P X P P
SEC. 5 H SPACES
35
• for which the (unique) constant map с: P P is a homotopy identity, that is,
each composite
I	P 44 p x p Д p and p ДД p x P P
i is homotopic to Ip. The multiplication p is said to be homotopy associative if
I the square
j	PX PX PPXP
! 1X,'| |f<
i
I	P X P	—>	P
| is homotopy commutative, that is, p ° (p, X I) p ° (1 X p). A continuous
i function <p: P > P is called a homotopy inverse for P and p if each of
! the composites
j	p IM P x P P and P 4Д P x P Л P
i
!is homotopic to с: P P.
A homotopy-associative H space with a homotopy inverse satisfies the
i group axioms up to homotopy. Such a pointed space is called an H group.
i Clearly, any topological group is an H group.
!	A rnultiplication p in an H space is said to be homotopy abelian if the
l triangle
I	p x p 4 p x p
I	Z Z
j	p
I where T(pi,pg) = (рг,Р1), is homotopy commutative. An H group with
homotopy-abelian multiplication is called an abelian Я group.
If P and P' are H spaces with multiplications p and p', respectively, a
continuous map a: P P' is called a homomorphism if the square
i
I	Px p 4 p
j	1“
p' %P' 4. p
! is homotopy commutative.
i
	4 Theorem A pointed space having the same homotopy type as an
: H space (or an H group) is itself an H space (or H group) in such a way that
! the homotopy equivalence is a homomorphism.
36
HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP. 1
proof Let f:P—*P' and g: P' P be homotopy inverses and let P be an
H space with multiplication /z: P x P P. If p': P' X P' P' is defined to
be the composite
P' x P' PxP P Л P
then p' is a continuous multiplication in P' and the composite P' (1,c) >
P' '/ P -> P equals the composite P' A P —fhd > Ру PP PP P, which
is homotopic to the composite P к Л P'. Because fg ~ Ip*, the map
p' ° (l,c') is homotopic to lj-. Similarly, the map // ° (c',1) is homotopic to
lp'. Therefore P> is an H space. Because the square
P' X P' 4 P'
exel
PXP P
is homotopy commutative, g is a homomorphism (and so is / ). If /z is homotopy
associative or homotopy abelian, so is ju,', and if <p: P P is a homotopy
inverse for P, then frpg: P' —» P' is a homotopy inverse for P'. 
Given an H space P, for any pointed space X there is a law of composi-
tion in [X;P] defined by [gi][g2] = [p ° (gugz)]. If P is an H group, [X;P|
becomes a group with this law of composition, and if /: X —> Y, then
f#: [Y,P] —> [X;P] is a homomorphism. Therefore we have the following
theorem.
5 theorem If P is an H group, rrp is a contravariant functor from the
homotopy category of pointed topological spaces with values in the category
of groups and homomorphisms. If P is an abelian H group, this functor takes
values in the category of abelian groups. 
It is interesting that the following converse of theorem 5 is also valid.
6 theorem If P is a pointed space such that np takes values in the cate-
gory of groups, then P is an H group (abelian if rrp takes values in the
category of abelian groups). Furthermore, for any pointed space X, the group
structure on rrp(X) is the same as that given by theorem 5.
proof Let pi: P X P —* P and ;z2: P X P —> P be the projections, and let
p: P X P —> P be a map such that [/z] = [pj * [p2], where is the law
of composition in the group [P X P', Р]. For any maps f, g: X P,
(f,g)#: [P X P; P] —> [X;P] is a homomorphism and
[p ° (f>g)] = (f>g)#W = (/,g)#([pJ * Ы)
= (f,gMPJ * (/,g)#[p2] = [f] * [g]
This shows that the multiplication in [X;P] is induced by the multiplication
map p.
Let X be a one-point space. The unique map X —> P represents the
identity element of the group [X;P]. Because the unique map P X induces
SEC. 5 H SPACES	37
a homomorphism [X;P] —> [P;P], it follows that the composite P X —> P,
which is the constant map с: P —> P, represents the identity element of [P;P],
It follows that // ° (W) ~ Ip and // ° (c,lp) ~ Ip. Therefore P is an H space.
To prove that p is homotopy associative, let qr, q2, q3: P X P X P —> P
be the projections. Then
[p ° (i x m)] = (i x = (i x m)#Cpi] * (i x м)#Ы
= Ы * Ip(.q2,q3)l = [<71] * ([<72] * [73])
Similarly,
[ft “ (ft X 1)] = ([<71] * [<72]) * [<7з]
Because [P X P X P; P] has an associative multiplication, p 0 (1 X p) —
fi0 (M X 1).
To show that P has a homotopy inverse, let <p: P —> P be such that
[Ip] * [<p] — [c]; then	c. Also, [<p] * [1P] = [c], and so p(<p,lp) ~ c.
Therefore <p is a homotopy inverse for P.
This proves that P is an H group and that the multiplication in тгр is in-
duced from that on P. If [P X P; P] is an abelian group, a similar argument
shows that P is an abelian H group. 
The following complement to theorems 5 and 6 is easily established by
similar methods.
7 theorem Let a: P P' be a map between H groups. Then a# is
a natural transformation from rrp to rrp' in the category of groups if and only
if a is a homom orphism. 
We describe a particularly useful example of an H group. Let Y be
a pointed topological space with base point y0. The loop space of Y (based at
yf), denoted by BY [or by B(Y,y0)], is defined to be the space of continuous
functions w: (1,1) —> (Y,yo) topologized by the compact-open topology. BY is
regarded as a pointed space with base point w0 equal to the constant map of
I to y0. There is a map
p:BY x BY —> BY
defined by
/	fw(2t)	0 < t < ’/2
Tb prove that p is continuous, let E: BY X I Y be the evaluation map. By
theorem 2.8 in the Introduction, it suffices to show that the composite
ЙУ X SYy I BY X I Д Y
is continuous. The formula which defines p shows that this composite is con-
tinuous on each of the closed sets BY x BY X [0,%] and BY X BY x [^,1].
We construct a number of homotopies to show that BY is an H group.
F(a>,t)(t') =
Уо
38	HOMOTOPY AND THE FUNDAMENTAL GBOUP CHAP. 1
Similar formulas will be used again in Sec. 1.7 to define homotopies of (non-
closed) paths in a topological space.
To prove that the map w g(w,wo) is homotopic to the identity map of
£2Y, define F: QY X I -+ £2Y by
< e < i
2 ~ -
This formula shows that E(F X 1): (£2Y X I) X f —> Y is continuous; there-
fore F is continuous and is a homotopy from the map w —> g(w,w0) to Iny,
Similarly, the map w ft(w0,w) is homotopic to l<>y. Therefore £2 Y is an
H space with multiplication ft.
To show that g is homotopy associative, define
G: £2Y x £2Y X £2Y X 7	£2Y
by the formula
w
w'(4t' — t — 1)
E(G X l)(w,w',w",t,f') =
u -	-	4
L±l<tl<±±l
4	-	-	4

4
(lnY X fi), showing that fi is homotopy associative,; 1
inverse <p: £2Y £2Yby <p(oj)(/) = w(l — t). Then’ |
Then G: fi ° (fi X lny) ~ ft °
We define a homotopy
we define H: £2Y X fl Y by
Уо
<x(2t' — t)
w(2 - 2t' - f)
Уо
E(H X l)(w,f,t') =4
o<<-<4	i
2 ~	~ 2	I
i
2 ~ S 2	|
j
H is a homotopy from the map w —> g(w,<p(w)) to the constant map of £2Y to
itself. Similarly, there is a homotopy from the map w /z(<p(cj),cj) to the con-
stant map of O Y. Therefore <p is a homotopy inverse for BY, and £2 Y is an
H group.
If h: Y —> Y' preserves base points, there is a map
£2/i: £2Y^ £2Y'
SEC. 6 SUSPENSION	39
defined by £2h(w)(f) = 7i(w(#)). Clearly, Qh is a homomorphism, and we sum-
marize these remarks about loop spaces as follows.
H theorem The loop functor £2 is a covariant functor from the category
of pointed topological spaces and continuous maps to the category of H
groups and continuous homomorphisms. 
The functor £2 also preserves homotopies. That is, if h0, 1ц: Y -> Y' are
homotopic by a homotopy ht, then £2h0, Uhr: £2Y	£2Y' are homotopic by a
homotopy £2/?d which is a continuous homomorphism for each t £ I.
6 SUSPENSION
This section deals primarily with results dual to those of Sec. 1.5. We consider
pointed spaces Q such that ttq is a covariant functor from the homotopy cate-
gory of pointed spaces to the category of groups and homomorphisms, and
this leads to the concept of H cogroup, dual to that of H group. An important
example of an H cogroup is the suspension of a pointed space, a concept dual
to that of the loop space. The homotopy groups of a space defined in the sec-
tion are examples of groups of homotopy classes of maps from suspensions to
(lie space.
If X and Y are pointed topological spaces, their sum in the category of
pointed topological spaces will be denoted by X v Y. If X has base point Xo
and Yhas base point уо, X v Y may be regarded as the subspace X X у о U Xq X Y
of X X Y If f: X —> Z and g: Y Z, we let (f,g): X v Y Z be the map de-
fined by the characteristic property of the sum [that is, (J,g)| X = f and
(f>g) I = g]-
An H cogroup consists of a pointed topological space Q together with a
continuous comultiplication
r; Q —> Q v Q
such that the following properties hold:
Existence of homotopy identity. If c: Q Q is the (unique) constant
map, each composite
and ()4
is homotopic to 1q.
Ifomotopy associativity. The square
Q	QvQ
'1	|1VI’
Qv Q ----> QvQvQ
is homotopy commutative.
40	HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP. 1 ’
Existence of homotopy inverse. There exists a map if: Q Q such that
each composite
is homotopic to c: Q —> Q.
If X is any pointed space and Q is an H cogroup, there is a law of
composition in [Q;X] defined by [fi][fz] = [(fi>fz) ° ?] which makes [Q;X] a
group.
An H cogroup is said to be abelian if the triangle
Q
7 \
where T'(qi,q2) = (<7г,<71) for q2 £ Q, is homotopy commutative.
If Q and O' are H cogroups with comultiplications v and v', respectively,
a continuous map /j: Q Q' is called a homomorphism if the square
QAQvQ
Q'^Q'vQ'
is homotopy commutative.	i
The proofs of the following theorems are dual to the proofs of the ;
corresponding statements about H groups (see theorems 1.5.4, 1.5.5, 1.5.6, 
and 1.5.7) and are omitted.
I theorem A pointed space having the same homotopy type as an ,
H cogroup is itself an H cogroup in such a way that the homotopy equiva-
lence is a homomorphism. 
2	theorem If Qis an H cogroup, ttq is a covariant functor from the homo-
topy category of pointed spaces with values in the category of groups and
homomorphisms. If Q is an abelian H cogroup, this functor takes values in the
category of abelian groups. 
3	theorem If Q is a pointed space such that ttq takes values in the cate- ’
gory of gro ups, then Q is an H cogroup (abelian if ttq takes values in the cate-;
gory of abelian groups). Furthermore, the gro up structure on 'Tq(X) is identical |
with that determined by the H cogroup structure of Q as in theorem 2. и •
4	theorem If fi: Q Q' is a map between H cogroups, then fi# fCj
a natural transformation from ttq’ to ttq in the category of groups if and oiihi:
if fi is a homomorphism. 
We describe an example of an II cogroup dual to the loop-space example
of an H group. Let Z be a pointed topological space with base point Zq. The)
s|jC, () SUSPENSION
suspension of Z, denoted by SZ, is defined to be the quotient space of Z x I
in which (Z X 0) U (zo X I) U (Z X 1) has been identified to a single point.
This is sometimes called the reduced suspension in the literature, the term
‘suspension” being used for the suspension in the category of spaces (no base
points). The latter is defined to be the quotient space of Z X I in which Z X 0 is
identified to one point and Z X 1 is identified to another point.
If (z,t) £ Z X I, we use [z,t] to denote the corresponding point of SZ
under the quotient map Z x I SZ. Then [z,0] = [z0,t] = [z',1] for allz,
£ Z and t£l. The point [zo,0] £ SZ is also denoted by z0, and SZ is
a pointed space with base point z0. If f. Z s Z', then Sf. SZ SZ' is defined
j by S/[z,t] = lf(z), t]- Thus S is a covariant functor from the category of
pointed spaces and continuous maps. To show that it is a covariant functor
to the category of H cogroups and homomorphisms, we define a comultiplication
v. SZ-> SZv SZ
by the formula
v(\z m _ f([^,2t], z0)	0 < t < %
(Lz,tJ) _	2f _ ц) y2 < f < p
and illustrate it in the diagram (where the dotted lines are collapsed to one
point).
The map v provides SZ with the structure of an H cogroup such that if
Д Z —> Z', then S/ is a homomorphism. This can be verified directly or
deduced from properties of loop spaces already established. We follow the
latter course.
The functors Я and S defined from the category of pointed spaces and con-
tinuous maps to itself are examples of adjoint functors. This means that for
pointed spaces Z and Y there is an equivalence
hom (SZ,Y) ~ hom (Z,S2Y)
where both sides are interpreted as the set of morphisms in the category of
pointed spaces and continuous maps. This equivalence results from theorem 2.8
in the Introduction, and if g: Z SY, the corresponding g': SZ —> Y is de-
fined by g'[z,t] — g(z)(f) for z £ Z and t £ I. It is obvious that if h : Y —> Y',
then (flh ° g)' = h° g', and if f: Z' Z, then (g ° f)' = g' ° Sf. Therefore
the equivalence g g' comes from a natural equivalence from the functor
hoin (S • , •) to the functor hom (• , S2 •).
42	HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP, ]
This natural equivalence passes to morphisms in the homotopy category
of pointed spaces. For pointed spaces a homotopy G: Z X / Y must map!
Zo X I into !/o- Therefore it defines a map F: Z X I/zo X I Y. Because!
S(Z X I/Zo X I) can be identified with SZ X I/zo X /by the homeomorphismj
[(z,t), t'J ([z,t'J, t) z £ Z; t, t' £ 1	j
it follows that homotopies F: Z X I/zo X 7 —> £2Y correspond bijectively to
homotopies F': SZ X I/zq X I —> Y. Therefore the equivalence above givej
rise to an equivalence
[SZ;Y] ~ [Z;£2Y]	!
f
such that if the maps g: Z —> £2 Y and g': SZ —> Y are related by g'[z,t] = g(2)(t).l
then [g'| corresponds to [g]. Hence there is a natural equivalence from thet
functor [S- ; • ] to the functor [ • ; £2 • ].	I
It follows from these remarks that for a fixed pointed space Z the functor!
77sz is naturally equivalent to the composite functor tiz ° £2. Here £2 is)
regarded as a covariant functor to the homotopy category of H groups and;,
homomorphisms. Then the composite ttz ° £2 takes values in the category ol]
groups and homomorphisms. By theorem 3, SZ is an H cogroup, and the map^
v: SZ —> SZ v SZ defined above is the one which is the comultiplication in the!
H cogroup SZ (or is homotopic to it). In similar fashion, if f: Z —> Z'.f
the natural transformation (Sf )# from ti^z' to -ttsz corresponds to the natural
transformation f# from the composite -ttz’ ° £2 to the composite ttz ° £2. Because
the latter is a natural transformation in the category of groups, so is (Sf)#,
and by theorem 4, Sf is a homomorphism of the H cogroup SZ to the
H cogroup SZ'.	!
These statements are summarized as follows.	<
5 theorem The suspension functor S is a covariant functor from t/i^
category of pointed spaces and maps to the category of H cogroups and con- •
tinuous homomorphisms. 	!
The functor S also preserves homotopies. That is, if fa, fa: Z Z' aft;
homotopic by a homotopy fa, then Sfa, Sfa are homotopic by a homotopy Sj^
which is a continuous homomorphism for each t £ I.	I
We now show that for n > 1 the sphere Sn is homeomorphic to a sus-
pension, and thus obtain an interesting family of H cogroups. The correspond-
ing functors are known as the homotopy group functors and are particular^
important.
6 lemma For n > 0, S(Sn) is homeomorphic to S!1+1.	1
proof Let po = (1,0, ... ,0) be the base point of Sn. We regard R"11
imbedded in R),+2 as the set of points in R"12 whose (n + 2)nd coordinate
is 0. Then S” is imbedded as an equator in S’i+1.	I
Sn = {z E Ril+21 |k|| = 1 and Zn+2 = 0}	I
and En+1 is also imbedded in En+2:	t
-eC> 6 suspension	43
E«+i = {z E R«+2| ||z|| < 1 and z,l+2 = 0}
Let H+ and H be the two closed hemispheres of S’"11 defined by the equator
Then
H+ = {z E S”+4z„+2 >0} and = {z E S«+i|zn+2 < 0}
and S'1+1 = H+ U H_ and Sn = H+ П H. Furthermore, the projection map
jpi i2 _> R«+1 defines projection maps p+: H+ En+1 and p: H_ En+1,
which are homeomorphisms. A map f: S(S") Sn+1 is defined by
rr _ (p-~l(2tz + (1 - 2t)p0)	o < t < 14
JL ’J “ [p^((2 - 2t)z + (2t - l)p0)	% < t < 1
and is verified to be a homeomorphism f: S(Sn)	S”+1. 
For n > 1 the nth homotopy group functor irn is the covariant functor
on the homotopy category of pointed spaces defined by тгп = тт8п. It follows
from theorems 6 and 5 that these functors take values in the categoiy of
groups and homomorphisms.
In the last two sections of this chapter we give another definition of тг ,
and study it in more detail. In Chapter 7 we return to the study of the higher
homotopy groups w„.
The following necessary and sufficient condition for a map Sn —> X to
represent the trivial element of ^„(X) is an immediate consequence of
theorem 1.3.12.
7 theorem A map a: Sn X represents the trivial element of rrn(X) for
n > 1 if and only if a can be continuously extended over En+1. 
Before leaving this section let us consider the interplay between two
possible group structures on the set [X; Y] for particular pointed spaces X and
1' (for example, if X is an H cogroup and Y is an H group, this set can
be given a group structure in two ways). It is a fact that under rather general
circumstances two laws of composition on hom (X,Y) in a category are equal,
aid we establish this result.
3 theorem Let X and Y be objects in a category arid let * and *' be two
lines of composition in hom (X,Y) such that
(a)	* and *' have a common two-sided identity element
(b)	* and *' are mutually distributive
Then * and *' are equal, and each is commutative and associative.
PROOF
f A>
Statement (a) means there is a map fa: X Y such that for any
f*fa=fa*f=f = f*'fa=fa*'f
Statement (b) means that for fa, fa, gi, g2: X —» Y
(fi * fa) *' (gi * gz) = (fi *’ gi) * (fa *'
HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP, j
 г. <7 THE FUNDAMENTAL GROUPOID
SgC< I
45
THE FlINBAMENTAL «ROUPOI»
5
44
Iff, g: X Y, then
f *g = (f*'fo) * (Jo *'g) = (f *fo) *'(fo * g) =f *'g
and g*f = (fo *'g) * (f*'fd) = (fo*f) *'(g*fo) =f *'g
Therefore f*g=f*' g = g*f. For associativity we have
(f * g) * h = (f * g) *' (f0 * h) = (f *'f0) * (g *' fi) = f * (g * /г)
» corollary If P is an H space and Q is any II cogroup, then [Q;P] (s |
an abelian group and the group structure is defined by the multiplication map|
in P.
proof This follows on observing that the two laws of composition defined )
in [Q;F] by using the comultiplication in Q or the multiplication in P satisfy^
the hypotheses of theorem 8. 	j>
Note that if P is just an H space (but not an H group), the law of cona-|
position in [X;P] defined by the multiplication in P is in general not a group)	gaSy consequences	of general properties of categories.	"
structure on [X;P], However, if X is an H cogroup (for instance, a suspension),^
it follows from corollary 9 that this law of composition is a group structure oifl	1 ^'ie ,e at^on	between objects A and В of a groupoid defined by the con-
[X;P], and in this case the resulting group structure on [X;P] is the same iitii	didon hom (A,B)	0 is an equivalence relation. 
matter what multiplication map P is given (so long as it is an H space).
IO corollary If P is an II space, irn(P) is abelian for all n > 1------.
group structure in riT.„(P) is defined by the multiplication map in P.  j
For a double suspension S(SZ) whose points are re;
[[z,t],t'], with z G
set of maps S(SZ) X. Iff g: S(SZ) X, we define
(f*g)tM,d = [g[fe2f_lu1
This section concerns paths in a topological space. This leads to another
description (in Sec. 1.8) of the first homotopy group tti, introduced in Sec. 1.6.
\Ve shall have occasion to define a number of homotopies between paths in a
topological space. These homotopies are generalizations (to nonclosed paths)
of those used in Sec. 1.5 to prove that a loop space is an H group and are de-
fined by the same formulas (except that the t and t' arguments are interchanged),
ft is clear that this repetition of formulas could have been eliminated by
proving a suitably general result about path spaces instead of merely consid-
( ei-jng loop spaces in Sec. 1.5. However, each usage has its own value, and
it is hoped that the repetition may be an aid to understanding the formulas.
I A groupoid is a small category in which every morphism is an equiva-
| lence. We list without proof a number of facts about groupoids which are
_______, ,	presented in the form|
C Z and t, t’ G L there are two laws of composition in the!
- -	f
The equivalence classes of this equivalence relation are called the com-
1 and the] ponents of the groupoid. The groupoid is said to be connected if it has just
one component.
2	For any object A of a groupoid, the law of composition which sends
fi g: A —> A to f ° g: A —> A is a group operation in hom (A,A). 
3	There is a covariant f unctor from any groupoid to the category of groups
and homomorphisms which assigns to an object A the group hom (A,A) and
to a morphism f.A -^B the homomorphism
hp. hom (A,A) —» hom (B,B)
defined by hf(g) = f ° g ° f~x for g: A A. 
Because any morphism f: A —> В in a groupoid is an equivalence,
hp hom (A,A)	hom (B,B) is an isomorphism. The following statement
describes the collection of isomorphisms obtained by taking all possible mor-
phisms f: A —> B.
1 If A and В are in the same component of a groupoid, the collection
of isomorphisms {hf | f: A —> B} is a conjugacy class of isomorphisms
hom (A,A^) -э- hom (B,B). 
» Let F be a covariant functor from one groupoid G to another (S'. Then F
maps each component of G into some component of G', and there is a natural
transformation F* (A) from the covariant functor hom^ (A,A) on G to the co-
variant functor home' (F(A), F(A)) on G defined by
F*(A)(f)=F(f):F(A)^F(A) f.A^A -
0
y2
and
0
(f *' g)[M> И = (gffiot'1- i]
The corresponding operations in [S(SZ);X] satisfy the hypotheses of theorem Ц
Therefore they are equal, and [S(SZ);X] is an abelian group. In particular, we?
have the following corollary.	|
11 corollary For n > 2, rrn is a functor to the category of abelid»[
groups. 	f
A similar argument can be applied to the loop space ЙР, where P is itself
an II space. There is a multiplication map in QP, because it is a loop space,,
and another multiplication obtained from the original multiplication in P. The
corresponding laws of composition in [Х;ЙР] satisfy theorem 8. Therefore il
follows that if P is an H space, iraP is a contravariant functor to the category
of abelian groups.	|
47
46	HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP. | <SEC< 1 THE FUNDAMENTAL GROUPOID
With these general remarks about groupoids out of the way, we proceed and fllustrated in the diagram
to define the fundamental groupoid. A path co in a topological space is defined!
to be a continuous map co: I X [note that the path is the map, not just th(!
image set co(f)]. The origin of the path is the point co(0), and the end of th?i
path is the point co(l). We also say that co is a path from co(0) to co(l). A close®
path, or loop, at x0 6 X is a path co such that co(O) = xo = co(l). If co and
are paths in X such that end co = orig co', there is a product path co * co' in J
defined by the formula
.	.	(co(2t)	0 < t < Fa
(co * co )(t) -	_ у) y2 < t < !
Then orig (co * co') = orig co and end (co * co') = end co'.
We should like to form a category whose objects are the points of
Then F * F: co * co' ~ cox * col rel I. 
1 theorem For each topological space X there is a category ^(X) whose
objects are the points of X, whose morphisms from xy to x0 are the path
classes with xy as origin and xy as end, and whose composite is the product of
path classes.
X, whose morphisms from xy to Xo are the paths from x0 to xy, and with ВД. proof To prove the	existence	of identity morphisms, let	ex:	I	X be the
composite defined to be the product path. With these definitions, neithei constant map of I to x for any x £ X. We show that [e J	=	1If co	is a path
axiom of a category is satisfied. That is, there need not be an identity morj xvitli co(l) = x, we must prove that co * ex ~ co (with a similar property for
phism for each point, and it is generally not true that the associative law fo| paths with origin at	x). Such	a homotopy F: I x I	X	is	defined by
product paths holds [that is, co * (co' * co") is usually different from (co * co') * co"]} formula
A category can be obtained, however, if the morphisms are defined not to Ц1
the paths themselves, but instead, homotopy classes of paths.	v
Two paths co and co' in X are briefly said to be homotopic, denoted
by co ~ co', if they are homotopic relative to I. Thus a necessary condition'
that co ~ co' is that co(0) = co'(0) and co(l) = co'(l). For any Xo, xy 6 X th4
relation co co is an equivalence relation in the set of paths from Xq to pictured in the diagram
The resulting equivalence classes are called path classes, and if co is a path inj
X, the path class containing it is denoted by [co]. Since two paths in the same!
path class have the same origin and the same end, we can speak of the origin?
and the end of a path class.	j
We shall construct a category whose objects are the points of X and|
whose morphisms from xy to x0 are the path classes with x0 as origin and у
as end. The following lemma shows that the path class of the product of tW(|
paths depends only on the path classes of the factors, and it will be used to
define the composite in the category.
6 lemma Let [co] and [co'] be path classes in X with end [co] = orig [co'].
There is a well-defined path class [co] * [co'] = [co * co'] with orig ([co] * [co ]) a
orig [co] and end ([co] * [co']) = end [co'].
F(f,t') =
coi
0 < t < f + 1
“ “	2
^-«.1
lA similar homotopy shows that if co(0) = x, then ex * co ~ co.
To prove the associativity of the composite of morphisms, let co, co', and
и be paths such that end co = orig co' and end co' = orig co". We must prove
that (co * co') * co" ~ co * (co' * co"). Such a homotopy G: I X 1 X is defined
[by the formula
proof To prove that co *£ coi and co' ~ col imply co * co' ~ cop * coi, ld>
F: I X Г —> X be a homotopy relative to I from co to coi and let F': I X I —> J
be a homotopy relative to 1 from co' to col. A homotopy F * F: I X I -> Xi
defined by the formula
G(t,t') =
col
t'+ 1
4
„	(F&t,f)	0 < t < Fa
(F * F)(t/) - [F,(2t _ у Fa < t < 1
co'(4t - t’ - 1)	'
,,/4t — 2 — f'\ f + 2
co I----------------I ---------!---
\	2 - tr /	4
4
HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP,
49
48
and pictured in the diagram
О o'
co co
i
The category ^(X) is called the category of path classes of X, or	A th_connected ig connectecL
fundamental groupoid of X, the latter because of the following theorem. {
8 theorem 9(X) is a groupoid.	f
proof Given a path co in X, let co-1: I —> X be the path defined by co 4
co(l — f). To prove that [co1] = [co] :l in vP(X), we must show that co * co1 ~
[and also that co-1 * co ~ еш(1), which follows, however, from the first honid
topy, because co = (co-1)-1]. Such a homotopy H: I X I —> X is defined E
the formula
co(O)
co(2t — t’)
w(2 - 2t - f)
n(t,e) =
co(O)
and pictured in the diagram
0<t<|
1^1
2 ~ ~ 2
i _ £ < t < i
2 ~ -
Iliis completes the construction of the fundamental groupoid. The coj
ponents of the fundamental groupoid are called path components of X. It _____
clear that Xo, and xx are in the same path component of X if and only if then \
is a path co in X from x0 to Xi. X is said to be path connected if its fundament 12 theorem If X is a contractible space, then rrx and w0 are naturally
groupoid is connected. The following is an alternate characterization of Й
path components.
9 THEOREM
subspaces of X.
ggc. 1 the fundamental groupoid
pjcOOF Let A be a path component of X and let co be a path in X such that
w(()) E A. We show that co is a path in A. For each t £ I define a path
W(. J X by co((t') = afttT) for P E I. Then co( is a path in X from co(0)
to w(t)- Therefore co(t) is in the same path component of X as x0, namely A.
Since this is so for every t £ I, co is a path in A.
A is path connected because if x0, x, £ A there is a path co in X from
r0 to Xi. By the above result, co is a path in A. Therefore any two points of A
can be joined by a path in A, and A is path connected. Since any path in X
that starts in A stays in A, A is a maximal path-connected subset of X. 
proof If co is a path in X, then co(l), being a continuous image of the con-
nected space I, is connected. Therefore co(0) and co(l) lie in the same compo-
nent of X. If X is path connected, any two points of X lie in the same
component, and X is connected. 
The converse of lemma 10 is false, as is shown by the following example.
1 j example Let X be the subspace of R2 defined by
X = ((x,y) E R2 | x > 0, у = sin — or x = 0, — 1 < у < 1}
x
Then X is connected, but not path connected.
 Given a map f: X Y, there is a covariant functor f# from ?P(X) to ?P(Y)
J. which sends an object x of vP(X) to the object f(x) of and the morphism
j [«] of to the morphism f#[a>] = [f ° co] of °P(Y). The functorial proper-
j ties off# are easily verified. From the first part of statement 5, or by direct
j verification, it follows that f maps each path component of X into some path
/ component of Y. Therefore there is a covariant functor ir0 from the category
j of topological spaces and maps to the category of sets and functions such that
15To(X) equals the set of path components of X, and
^o(f) = f#: 77o(X) TTo(Y)
I maps the path component of x in X to the path component of fix) in Y. If
’ fo ~ fl, then for any r £ X there is a path ых in Y from fo(x) to fi(x) de-
fined by = F(x,t) for t E I. Therefore fo(x) and fi(x) belong to the same
path component of Y, and fo# = f-##. It follows that Wq can be regarded as a
1 covariant functor from the homotopy category to the category of sets and
T (unctions. This functor characterizes the functor irx for a contractible space X
as follows.
equivalent functors on the homotopy category.
proof If X and X' have the same homotopy type, then rrrx and tTx’ are
The path components of X are the maximal path-connect naturally equivalent. It follows from corollary 1.3.11 that if P is a one-point
фасе, wAis naturally equivalent to rrP. It therefore suffices to prove that rrP
50
HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP, |
sgC. 8 THE FUNDAMENTAL GROUP
51
is naturally equivalent to 7*0. 77o(P) consists of the single path component f
and a natural transformation
f: rrP-^ 77O	j
is defined by i^[/] = f#(P) for [/'] £ [P;X]. Because Xp is in, one-to-one correJ
spondence with X in such a way that homotopies P X I —» X correspond to!
paths I —> X, it follows that f is a natural equivalence. E	j
The functor 77o is closely related to the functor Ho of example 1.2.6. Ы
fact, for spaces X whose components and path components coincide, H(1 jJ
the composite of тго with the covariant functor which assigns to every set thei
abelian group generated by that set. In particular, 77O could have been use^
to obtain the results of Sec. 1.2 that were obtained by using Ho.	I
SB ТИЕ fundamental group
By choosing a fixed point xp € X and considering the path classes in X with!
Xo as origin and end, a group called the fundamental group is obtained. We)
show now that this group is naturally equivalent to the first homotopy group!
77i, defined in Sec. 1.6. The section closes with a calculation of the fundamental^
group of the circle.	!
Let X be a topological space and let xp € X. The fundamental group o/i
X based at xo, denoted by 77(X,xp), is defined to be the group of path classed
with x'o as origin and end. It follows from theorem 1.7.8 and statement 1.7v2;
that this is a group, and if f: (X,Xq) —> (T</o), then/# is a homomorphism front]
77(X,Xo) to tt(Y,j/o)- If, fi f'- (A,xp) —> Cfil/o) are homotopic, then	j
f# = f#- w(X,x0)	ir(Y,y0).	।
Therefore, we have the following theorem.
1	theorem There is a covariant functor from the homotopy category ofi
pointed spaces to the category of groups which assigns to a pointed space its’
fundamental group and to a map f the homomorphism f#. 	;
We show that the fundamental group functor 77 is naturally equivalent toj
77i, defined in Sec. 1.6. Let X: I —> S(S°) be defined by X(t) = [ — l,t], where)
S° consists of the two points — 1 and 1 and 1 is its basepoint.	Then X induces^
a bijection X# between the homotopy classes of maps (S(S°),	1)	(X,xp)	and,
the path classes of closed paths in X at xp defined by	f
A#[g] = [gA] g: (S(S<>), 1)	(X,x0)	|
From the definition of the law of composition in [S(S°);X] and in 77(X,xp), XT
is seen to be a group isomorphism. Given a map fi (X,xp) —> (Y,t/o), X# com-1
mutes with f#. By lemma 1.6.6, S(S°) is homeomorphic to S1.	[
2	theorem The map X* is a natural equivalence of the first homotopy
group functor 77i with the fundamental group functor 77.
It will sometimes be convenient to regard the elements of 77(X,x'o) as
homotopy classes of maps (Sx,po) —» (X,xp), rather than as path classes.
Because any closed path at xp (and any homotopy between such paths)
must lie in the path component A of X containing xp, it follows that 77(A,x0)
тг(Х,хр). Hence the fundamental group can give information only about the
path component of X containing xp. From general groupoid considerations
(see statements 1.7.3 and 1.7.4), if [<;] is a path class in X from xp to Xi, then
fiwis an isomorphism from 77(X,xx) to 77(X,x0).
3	theorem The f undamental groups of a path-connected space based at
different points are isomorphic by an isomorphism determined up to
conjugacy. 
Even though the fundamental groups based at different points of a path-
connected space are isomorphic, we cannot identify them, because the iso-
morphism between them is not unique. If the fundamental group at some
point (and hence all points) is abelian, the isomorphism is unique. In general,
the fundamental group need not be abelian; however, the following conse-
quence of theorem 2 and corollary 1.6.10 is a general result about the com-
mutativity of fundamental groups.
4	theorem The fundamental group of a path-connected H space is
abelian, and if co and cd are closed paths at the base point, then
[co] * [co'] = [fi ° (co,co')]
where p is the multiplication map in the H space. 
A space X is said to be n-connected for n > 0 if every continuous map
f;Sfc X for к < n has a continuous extension over Ek+1. A l-c<_ oected
space is also said to be simply connected. Note that if 0 < m < г, an
«.-connected space is m-connected. It follows from theorem 1.6.7 that a space
X is n-connected if and only if it is path connected and 77/c(X,x) is trivial for
every base point x E X and 1 < к < n. From corollary 1.3.13 we have the
following result.
5	lemma A contractible space is n-connected for every n > 0. 
Note that a space is О-connected if and only if it is path connected, and
a space is simply connected if and only if it is path connected, and 77(X,xp) = 0
for some (and hence all) points Xo € X.
From theorem 1 we know that- two pointed spaces having the same
homotopy type as pointed spaces have isomorphic fundamental groups. To
prove a similar result for two path-connected spaces which have the same
52	HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP. I
homotopy type as spaces (no base-point condition) we need some preliminary
results.
6	lemma Let	X and let «о, Po, and Pi be the paths in X |
defined by restricting h to the edges of I x I [that is, aft) — h(i,t) and \
Pft) = h(t,i)]. Then (exo * Pi) * («i-1 * Po-1) fe a closed path in X at /1(0,0) j
which represents the trivial element of 7*(X, /1(0,0)).
proof Let ад, a), fig, and fi] be the paths in I X I defined by aft) = (i,t)
and fi'ft) = (t,i). Then (a'o * Pi) * (ai-1 * j66-1) is a closed path in I X I at
(0,0) and h maps this closed path into («о * Pi) * («1 1 * Po1)- Since I X 1 is
a convex subset of R2, it is contractible, and by lemma 5, it is simply con- j
nected. Therefore	i
(«о * Pi) * (al”1 * Po1) e(o,o)
and
(a0 * fif) * (a”1 * Pg-1) = h ° ((a'g * Pi) * (a'r1 * PtT1))
— h ° 6(0,0) = 6ft(o,o) 
7	theorem Let f: (X,xg) —» (X,y0) and g: (X,x'o) (Y,yi) be homotopic as
maps of X to Y. Then there is a path co in Y from y0 to yi such that	I
f# = о g#: тг(Х,х0) ir(Y,yg)	|
I
proof Let F: X X f —> Y be a homotopy from / to g and let co: I Ybe
defined by co(t) = F(x0,t). Then co is a path in Y from y0 to y±. If co' is any
closed path in X at Xg, let h: I X I Y be defined by h(t,F) = F(co'(t), /').
Then /i(0,F) = F(xg,F) = co(F), h(t,l) — g^'(f), h(l,F) = co(F), and h(t,O) = i
fofit). By lemma 6 we have	’
(co * geo') * (co”1 * (/co')”1) eVo
This implies [co] ° g^fco'] ° [co]1 = /#[co'], or (hM ° g#)[co'] = /#[co']. Since [co']
is an arbitrary element of rr(X,xg), h^ ° g# = /#. 	i
1
8	theorem Two path-connected spaces with the same homotopy type i
have isomorphic fundamental groups.	j
!
proof Let /: X —> Y be a homotopy equivalence with homotopy inverse ]
g: Y X. Let x0 € X and set t/0 = f(Xo), Xi = g('/o)> and t/i = /(xi). Let [
/0: (X,x'o) —> (Y,yo) and/i: (X,X|) —» (Y,yi) be maps defined by/(that is, /0 and
fi are both equal to / but are regarded as maps of pairs), and let g': (Y,t/o)	[
(X,xi) be defined by g. Then g' 0 /0: (X,xg) —> (X,xi) is homotopic, as a map of j
X to X, to 1(х,Жо): (fi,xg) C (X,x0), and/i 0 g': (Y,t/0) —> (X'/i) is homotopic, as j
a map of Y to Y, to l(v,i/0): (¥,Уо) C (Y,i/0). It follows from theorem 7 i
that there are paths co in X from Xi to Xo and co' in Y from yi to yg such that [
Ь[И] = (g' ° fo)# = g# ° fo# and h^ = (fi о g')# = fi# о g'# J
Шс. 8 THE FUNDAMENTAL GROUP	53
'Therefore we have a commutative diagram
77(X,XO)	77(X,X1)
”(У,Уо)	<ЪУ1)
g# is an epimorphism because is, and it is a monomorphism because /г^-]
is. Therefore g# is an isomorphism. 
We close with an example of a space with a nontrivial fundamental
group. For this purpose we compute 7r(S1 *,p0) following a method used
by Tucker1, where S1 = {eil’} and p0 = 1.
The exponential map ex: R —» S1 is defined by ex(t) = e24rit. Then ex is
continuous, ex(t] + t2) = ex(ti) ex(t2) (where the right-hand side is multiplica-
tion of complex numbers), and ex(ti) = ex(t2) if and only if ti — t2 is an
integer. It follows that ex| ( — Fa, Fa) is a homeomorphism of the open interval
(—й,й) onto S1 — {e7ri}. We let
Ig: S1 - {e^} -> (-Fa,^)
be the inverse of ex | (— Fa, Va).
A subset X C R“ will be called starlike from a point x'o € X if, whenever
x С X, the closed line segment [x'o,x] from x0 to x lies in X.
9	lemma Let X be compact and starlike from Xo € X. Given any contin-
uous map f: X S1 and any t0 G R such that ex(to) = f(Xo), there exists a
continuous map f': X R such that/'(x'o) = to and ex(/'(x)) = _f(x) for all
x G X.
proof Clearly, we can translate X so that it is starlike from the origin; hence
there is no loss of generality in assuming Xo = 0. Since X is compact, f is uni-
formly continuous and there exists e > 0 such that if ||x — x'|| < e, then
||f(x) — У(х')|| < 2 [that is, /(x) and f(x') are not antipodes in S1]. Since X is
bounded, there exists a positive integer n such that ||x||/n < e for all x G X.
Then for each 0 < / < n and all x X
(? + l)x _ /XII = ||x|| < e
n n ll n
\
It follows that the quotient f(fj + l)x/n)/ffx/n) is a point of S1 — (e7™). Let
gf. X —> S1 — {e’™} for 0 < j < n be the map defined by g/x) =
1 See A. W. Tucker, Some topological properties of disk and sphere, Proceedings of the
Canadian Mathematical Congress, 1945, pp. 285-309.
54
HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP. 1
/((/ + T)x/n)/f(jx/n). Then, for all x E X, we see that
f(x) = f(0)go(x)gi(x)    gn_x(x)
We define f: X —> R by
f'(x) = to + te(go(x)) + ?g(gl(x)) +  • • + ?g(gM-l(x))
f is the sum of n + 1 continuous functions from X to R, so it is continuous.
Clearly, /'(0) = to and ех(У'(х)) = /(x). 
10	lemma Let X be a connected space and let f, g': X R be maps such
that ex ° f' = ex ° g' andf '(x0) = g'(xo) for some x0 € X. Then f' = g'.
proof Let h = f — g': X —> R. Since ex ° f' = ex ° g', ex ° h is the con-
stant map of X to po. Therefore h is a continuous map of X to R, taking only
integral values. Because X is connected, h is constant, and sincy h(x(t) = 0,
h(x) = 0 for all x E X. 
Let a: I —> S1 be a closed path at p0. Because I is starlike from 0 and
«(()) = p0 = ex(0), it follows from lemma 9 that there exists a': I —> R such
that a'(O) = 0 and ex ° a' = a. By lemma 10, a' is uniquely characterized by
these properties. Because ex(a'(l)) = p0, it follows that a'(l) is an integer. We
define the degree of n by deg a = «'(I).
11	lemma Let a and (3 be homotopic closed paths in S1 at po. Then
deg a — deg j6.
proof Let F: I X I —> S1 be a homotopy relative to I from a to j8. Because
I X I is a starlike subset of R2 from (0,0), it follows from Iemma 9 that there
is a map F: I X I —> R such that F'(0,0) = 0 and ex ° F — F. Since F is a
homotopy relative to 1, F(O,t') = F(1,F) = p0 for all t £ I. Therefore F(f)f)
and F'(l,t') take on only integral values for all f € I. It follows that F(0f)
must be constant and F'(l,f) must be constant. Because F'(0,0) = 0,
F(f),t') = 0 for all F E I. Define a', /3': I R by a'(t) — F(t,0) and =
F(t,l). Then a'(0) = 0 and ex ° a' = a. Therefore deg a = <x'(l) = F'(l,0),
Similarly, j8'(0) = 0 and ex ° {3' = (3, so deg (3 = (3'(T) = F'(l, 1). Because
F'(1,F) is constant, F'(l,0) = F'(l,l) and deg a = deg {3. 
It follows that there is a well-defined function deg from 7*(S',po) to Z de-
fined by
deg [a] = deg a
where a is a closed path in S1 at po-
12	theorem The function deg is an isomorphism
deg: 7r(S1,p0) ~ Z
proof To prove that deg is a homomorphism, let a and {3 be two closed
paths in S1 at po and let a[3 be the closed path which is their pointwise
•SEC. 8 THE FUNDAMENTAL GROUP	55
product in the group multiplication of S1. We know from theorem 4 that
[л] * [/Я = [a/?L Let a', /Т: f -—> R be such that <x'(0) = 0, ex ° a’ = a,
)/(()) = 0, and ex ° /)' = Д Then a’ + /Т: I R is such that (a' + j6')(0) = 0
and ex ° (a' + (T) = afi. Therefore
deg ([a] * [Д]) = deg [аД] = (a' + Д')(1)
= deg a + deg = deg [a] + deg [/1]
showing that deg is a homomorphism.
The map deg is an epimorphism; for if n is an integer, there is a path a„
in R defined by a'n(f) = tn. Let an = ex ° a’n. Then clearly, deg [«„] = «,',(!) = n.
The map deg is a monomorphism; for if deg [a] = 0, there is a closed
path a' in R at 0 such that ex ° a' = a. Since R is simply connected (because
it is contractible, and by lemma 5), a' ~ eo. Then ex ° a' ePo. Therefore
a ~ ePo and [a] is the identity element of ^(Skpo)- 
The method we have used to compute 7r(S',po) will be generalized
in Chapter 2 to give relations between the fundamental group of a space and
the fundamental groups of its covering spaces.
It follows from theorem 2 that 7r(S1,po) [Sx,po; S1^]. Because S1 is a
topological group, the set [S1;S1] (with no base-point condition) is also a
group under pointwise multiplication of maps, and there is an obvious
homomorphism
y: [Si,p0; Si,p0] [ShS1]
13	lemma The homomorphism
У- [Shpo; Shpo] [ShS1]
is an isomorphism.
proof To show that у is an epimorphism, let/: S1 —> S1 and let/(po) =
for some 0 < 0 < 2тг. Define a homotopy F: S1 X I —» S1 by
F(z,t) = f(z)e~ite
Then F is a homotopy from f to a map /' such that f'(po) = po- Therefore
m>0 = if 1 = ifi
To show that у is a monomorphism, assume that f: (S1,p0) —> (S1,po) is
such that y[y i/J(, — [/] is trivial. Then f: S1 —> S1 is null homotopic. By
theorem 1.3.12, J is null homotopic relative to po- Therefore [y]M is trivial. 
It follows from theorem 12 and lemma 13 that [S^S1] Z. For each in-
teger n the map z zn from S1 to itself represents a homotopy class corre-
sponding to n under the above isomorphism.
56	HOMOTOPY AND THE FUNDAMENTAL GROUP CHAP. I
EXERCISES
!
A CONTRACTIBLE SPACES	'
1 The cone over a topological space X with vertex v is defined to be the mapping cyl- j
inder of the constant map X —> v. Prove that X is contractible if and only if it is p
a retract of any cone over X.	j
2 Prove that Sn is a retract of En+1 if and only if Sn is contractible.
3 If CX is a cone over X, prove that (CX,X) has the homotopy extension property with
respect to any space.
4 Prove that a space Y is contractible if and only if, given a pair (X,A) having the
homotopy extension property with respect to Y, any map A —> Y can be extended over X. !
5 Let Y be the comb space of example 1.3.9 and let yo be the point (0,1) g Y. Let Y' >
be another copy of Y, with corresponding point y'o. Let X be the space obtained by form- |
ing the disjoint union of Y and Y' and identifying y0 with y'D. Prove that X is n-connected £
for all n but not contractible. (Hint: Any deformation of X in itself must be a homotopy |
relative to yo-)
В ADJUNCTION SPACES	[
1	Let A be a subspace of a space X and let f: A —> Y be a continuous map. The ad- |
junction space of X to Y by/is defined to be the quotient space of the disjoint union of (
X and Y by the identifications x g A equals fix) g Y for all x g A. Prove that if X and Y
are normal spaces and A is closed in X, then Z is a normal space.
2	A space X is said to be binormal if X X I is a normal space. If X is a binormal space,
Y is a normal space, and /: X Y is continuous, prove that the mapping cylinder of/is j
a normal space.	(
3	Given a continuous map /: A —> Y, where A is a subspace of a space X, prove that f
can be extended over X if and only if Y is a retract of the adjunction space of X j
toYby/.
4	Let Z be the adjunction space of X to Y by a map f: A - - > У. Prove that (Z,Y) has I
the homotopy extension property with respect to a space W if and only if (X,A) has the t
homotopy extension property with respect to W.
<2 ABSOLUTE RETRACTS AND ABSOLUTE NEIGHBORHOOD RETRACTS
A space Y is said to be an absolute retract (or absolute neighborhood retract) if, given a
normal space X, closed subset А С X, and a continuous map f: A —> Y, then / can be
extended over X (or / can be extended over some neighborhood of A in X).	.
1 Prove that a normal space Y is an absolute retract (or absolute neighborhood retract) {
if and only if, whenever Y is imbedded as a closed subset of a normal space Z, then Y is )•
a retract of Z (or a retract of some neighborhood of Y in Z).	j
2 Prove that the product of arbitrarily many absolute retracts (or finitely many |
absolute neighborhood retracts) is itself an absolute retract (or absolute neighborhood f
retract).	|
3 Prove that R” is an absolute retract for all n.	I
exercises	57
4 Prove that a retract of an- absolute retract is an absolute retract and that a retract of
some open subset of an absolute neighborhood retract is an absolute neighborhood
retract.
5 Prove that En is an absolute retract and S'1 is an absolute neighborhood retract for
all n.
6 Prove that a binormal absolute neighborhood retract is locally contractible (that is,
every neighborhood U of a point x contains a neighborhood V of x deformable to x in U).
"3 Prove that a binormal absolute neighborhood retract is an absolute retract if and
only if it is contractible.
D HOMOTOPY EXTENSION PROPERTY
1 Let A be a closed subset of a normal space X, let /: X —> Y be continuous (where Y
is arbitrary), and let G: A X I -> Ybe a homotopy of /1 A. If there exists a homotopy
G: U X I Y of/| U which extends G,.where Uis an open neighborhood of A, show
that there exists a homotopy F: X X I —> Y of /which extends G.
2 Borsuk’s homotopy extension theorem. Let A be a closed subspace of a binormal
space X. Then (X,A) has the homotopy extension property with respect to any absolute
neighborhood retract Y.
3 Let A be a closed subset of a binormal space X and assume that the subspace
AxJUXxOCXxfis an absolute neighborhood retract. Then (X,A) has the
homotopy extension property with respect to any space Y.
4 Let A be a closed subset of X and В a subset of Y. Assume that (X,A) has
the homotopy extension property with respect to В and that (X X 1, X X I U A X 1)
has the homotopy extension property with respect to Y. Prove that if /: (X,A) —> (Y,B) is
homotopic (as a map of pairs) to a map which sends all of X to B, then it is homotopic
relative to A to such a map.
E COEIBRATIONS
1 Prove that any cofibration is an injective function.
2 Prove that a composite of cofibrations is a cofibration.
3 For a subspace A of X prove that the inclusion map А С X is a cofibration if and
only ifX X 0 U A x I is a retract of X X I-
4 If A is a subspace of a Hausdorff space X, prove that if А С X is a cofibration, then
A is closed in X.
5 Assume that X is the union of closed subsets Xi and Xz and let A be a subset of X
such that Xi П Xj C A. Prove that if А П Xj C Xi and А Г) Xg C Xg are cofibrations,
so is А С X.
в Let A be a closed subspace of a space X. Prove that the following are equivalent:1
(а)	А С X is a cofibration.
(b)	There is a deformation D: X X I —> X rel A [that is, D(x,0) = x and D(a,t) — a
for x g X, о £ A, and t g I] and a map <p: X I such that A = <p“1(l) and
WW] x 1) C A.
1 If X is normal, the equivalence of (a) and (c) is proved in G. S. Young, A condition for the
absolute homotopy extension property, American Mathematical Monthly, vol. 71, pp. 896-897,
1964.
58
HOMOTOPY AND THE FUNDAMENTAL GBOUP CHAP. J
(c) There is a neighborhood U of A deformable in X to A rel A [that is, there is a
homotopy H: U X I —> X such that H(x,O) = x, H(a,f) = a, and H(x,l) 6 A for
x 6 U, a 6 A, and f E I] and a map : X —> I such that A =	1(1) and <p(x) = 0
if x ex -u.
7 If А С X and В С Y are cofibrations with A and В closed in X and Y, respectively,
prove that AxBCXxBUAxY and X X В U A x Y С X X Y are cofibrations.
F LOCAL SYSTEMS1
1	A local system on a space X is a covariant functor from the fundamental groupoid of
X to some category. For any category t? show that there is a category of local systems on
X with values in (?. (Two local systems on X are said to be equivalent if they are
equivalent objects in this category.)
2 Given a map /: X —> Y, show that f induces a covariant functor from the category of
local systems on Y with values in S to the category of local systems on X with values in (?,
3 If A is an object of a category (?, let Aut A be the group of self-equivalences of A
in C. If <p: A В is an equivalence in <2, then show that q>: Aut A Aut В defined by
<p(a) = <p ° a ° <p-1 is an isomorphism of groups.
4 If Г is a local system on X and xo 6 X, show that Г induces a homomorphism
Г,„: tt(X,x0) Aut Г(х0)
5 If X is path connected, prove that two local systems Г and Г' on X with values in 6
are equivalent if and only if there is an equivalence <p: Г(х0) ~ Г'(хо), such that ° Гг||
is conjugate to Г(.„ in Aut Г'(х0).
6 If X is path connected, given an object A £ (? and a homomorphism a: тт(Х,хо) —>
Aut A, prove that there is a local system Г on X with values in 6’ such that Г(х0) = A
and Гт„ = a.
G THE FUNDAMENTAL GROUP
I Prove that the fundamental group functor commutes with direct products.
2	If co and o' are paths in X from Xo to Xi, prove that co ~ co' if and only if
co * co'1 ~ eXB.
3	Let a space X be the union of two open simply connected subsets U and V such that
U П V is nonempty and path connected. Prove that X is simply connected.
4	Prove that S" is simply connected for n > 2.
5	If there exists a space with a nonabelian fundamental group, prove that the “figure
eight” (that is, the union of two circles with a point in common) has a nonabelian funda-
mental group (cf. exercise 2.B.4).
6	Let f: I Rz be a continuously differentiable simple closed curve in the plane with
a nowhere-vanishing tangent vector [that is, /(0) — /(1), /'(0) — /'(I), and /'(t) 7^ 0 for
1 See N. E. Steenrod, Homology with local coefficients, Annals of Mathematics, vol. 41,
pp. 610-627, 1943.
exebcises
59
()</ <!] Let 6j: /	S'1 be the closed path defined by ro(t) = /'(t)/|| f'(t)II- Prove that
[w] is a generator of ^(S1).1
7 In Rz, let X be the space consisting of the union of the circles C,„ where C„
has center- (l/n,O) and radius 1/n for all positive integers n. In R3 (with R^ imbedded as
the plane хз = 0), let Y be the set of points on the closed line segments joining (0,0,1) to
X and let Y' be the reflection of Y through the origin of R3. Then Y and Y' are closed
simply connected subsets of R3 such that Y П Y' is a single point. Prove that Y U Y' is
not simply connected.2
|I SOME APPLICATIONS OF THE FUNDAMENTAL GROUP
1 Prove that S1 is not a retract of E2.
2 Prove that S1 and S” for n > 2 are not of the same homotopy type.
3 Prove that R2 and R” for n > 2 are not homeomorphic.
4 Let p(z) = z" + a„_i z'!-1 + •  • + a±z + Go be a polynomial of degree n, having
complex coefficients and leading coefficient 1, and let r/(z) — zn. For r > 0 let
CT = {r 6 R2 | ||x|| = r}. Prove that for r large enough, p | Cr and q | Cr are homotopic
maps of Cr into R2 — 0.
5 Fundamental theorem of algebra. Prove that every complex polynomial has a root.
(Hint: For any r > 0 the map г/1 C,: C,. —> R2 — 0 is not null homotopic because it in-
duces a nontrivial homomorphism of fundamental groups.)
1 See H. Hopf, Uber die Drehung der Tangenten und Sehnen ebener Kurven, Compositio
Mathematica, vol. 2, pp. 50-62, 1935. For generalizations see H. Whitney, On regular closed
curves in the plane, Compositio Mathematica, vol. 4, pp. 276-284, 1937, and S. Smale, Regular-
curves on Riemannian manifolds, Transactions of the American Mathematical Society, vol. 87,
pp. 495-512, 1958.
2 See H. R. Griffiths, The Fundamental group of two spaces with a common point, Quarterly
Journal of Mathematics, vol. 5, pp. 175-190, 1954.
CHAPTER TWO
COVMtING SPACES
ANU FIBBATIONS
THE THEORY OF COVERING SPACES IS IMPORTANT NOT ONLY IN TOPOLOGY, BUT
also in differential geometry, complex analysis, and Lie groups. The theory is
presented here because the fundamental group functor provides a faithful
representation of covering-space problems in terms of algebraic ones. This
justifies our special interest in the fundamental group functor.
This chapter contains the theory of covering spaces, as well as an intro-
duction to the related concepts of fiber bundle and fibration. These concepts
will be considered again later in other contexts. Here we adopt the view that
certain fibrations, namely, those having the property of unique path lifting,
are generalized covering spaces. Because of this, we shall consider these
fibrations in some detail.
Covering spaces are defined in Sec. 2.1, and fibrations are defined in
Sec. 2.2, where it is proved that every covering space is a fibration. Section 2.3
deals with relations between the fundamental groups of the total space and
base space of a fibration with unique path lifting, and Sec. 2.4 contains a solu-
tion of the lifting problem for such fibrations in terms of the fundamental
group functor.
61
62
COVERING SPACES AND FIBRATIONS CHAP. 2
The lifting theorem is applied in Sec. 2.5 to classify the covering spaces
of a connected locally path-connected space by means of subgroups of its
fundamental group. This entails the construction of a covering space starting
with the base space and a subgroup of its fundamental group. In Sec. 2.6 a
converse problem is considered. The base space is constructed, starting with
a covering space and a suitable group of transformations on it.
In Sec. 2.7 fiber bundles are introduced as natural generalizations of
covering spaces. The main result of the section is that local fibrations are
fibrations. This implies that a fiber bundle with paracompact base space is a
fibration. Section 2.8 considers properties of general fibrations and the con-
cept of fiber homotopy equivalence. These will be important in our later
study of homotopy theory.
1 COVERING PROJECTIONS
A covering projection is a continuous map that is a uniform local homeomor-
phism. This and related concepts are introduced in this section, along with
some examples and elementary properties.
Let p: X —> X be a continuous map. An open subset I/ C Xis said to be
evenly covered by p if p-1([7) is the disjoint union of open subsets of X each
of which is mapped homeomorphically onto U by p. If U is evenly covered
by p, it is clear that any open subset of U is also evenly covered by p. A con-
tinuous map p: X X is called a covering projection if each point r f X has
an open neighborhood evenly covered by p. X is called the covering space
and X the base space of the covering projection.
The following are examples of covering projections.
1	Any homeomorphism is a covering projection.
2	If X is the product of X with a discrete space, the projection X X
is a covering projection.
3	The map ex: В —> S1, defined by ex(t) = e2,irit, (considered in Sec 1.8) is
a covering projection.
4	For any positive integer n the map p: S1 —> S1, defined by p(z) = z”, is a
covering projection.
5	For any integer n > 1 the map p: Sn —> P'\ which identifies antipodal
points, is a covering projection.
6	If G is a topological group, H is a discrete subgroup of G, and G/H is
the space of left (or right) cosets, then the projection G -^> G/H is a covering
projection.
A continuous map f: Y —> X is called a local homeomorphism if each
point у E Y has an open neighborhood mapped homeomorphically by f onto
SEC. J COVERING PROJECTIONS	__-	63
an open subset of X. If this is so, each point of ¥ has arbitrarily small neigh-
borhoods with this property, and we have the following lemmas.
7	lemma A local homeomorphism is an open map. 
8	lemma A covering projection is a local homeomorphism.
proof Let p: X —> X be a covering projection and let x E X. Let U be an
open neighborhood of p(x) evenly covered by p. Then p-1(l7) is the disjoint
union of open sets, each mapped homeomorphically onto U by p. Let U be
that one of these open sets which contains x. Then U is an open neighbor-
hood of x such that p | U is a homeomorphism of U onto the open subset U
of X. 
A local homeomorphism need not be a covering projection, as shown by
the following example.
f> example Let p: (0,3) S1 be the restriction of the map ex: R S1 of
example 3 to the open interval (0,3). Because p is the restriction of a local
homeomorphism to an open subset, it is a local homeomorphism. It is also a
surjection, but it is not a covering projection because the complex number
1 E S1 has no neighborhood evenly covered by p.
The following is a consequence of lemmas 7 and 8 and the fact (immedi-
ate from the definition) that a covering projection is a surjection.
IO corollary A covering projection exhibits its base space as a quotient
space of its covering space. 
For locally connected spaces there is the following reduction of covering
projections to the components of the base space.
11	theorem If X is locally connected, a continuous map p: X X is a
covering projection if and only if for each component C of X the map
P I p'^': p1^ C
is a covering projection.
proof If p is a covering projection and C is a component of X, let x E C
and let U be an open neighborhood of x evenly covered by p. Let V be the
component of 17 containing x. Since X is locally connected, V is open in X,
and hence open in C. Clearly, V is evenly covered by p | p-1C. Therefore
p | p-1C is a covering projection.
Conversely, assume that the map p | p-1C: p-1C —» C is a covering pro-
jection for each component C of X. Let x E C and let U be an open neighbor-
hood of x in C evenly covered by p | p1 CJ. Since X is locally connected, C is
open in X. Therefore U is also open in X and is clearly evenly covered by p.
Hence p is a covering projection. 
In general, the representation of the inverse image of an evenly covered
open set as a disjoint union of open sets, each mapped homeomorphically, is
64
COVERING SPACES AND FIBRATIONS CHAP. 2
not unique (consider the case of an evenly covered discrete set); however, for
connected evenly covered open sets there is the following characterization of
these open subsets.
12	lemma Let U be an open connected subset of X which is evenly
covered by a continuous map p: X —» X. Then p maps each component of
p-^lfJ) homeomorphically onto U.
proof By assumption, p-1(I7) is the disjoint union of open subsets, each
mapped homeomorphically onto U by p. Since U is connected, each of these
open subsets is also connected. Because they are open and disjoint, each is
a component of U). B
13	corollary Consider a commutative triangle
Xi A X2
X
where X is locally connected and p^ and p2 are covering projections. If p is a
surjection, it is a covering projection.
proof If U is a connected open subset of X which is evenly covered by
and p2, it follows easily from lemma 12 that each component of p2-1(fl) is
evenly covered by p. 0
14 theorem If p: X —» X is a covering projection onto a locally connected
base space, then for any component C of X the map
p\C: Cp(C')
is a covering projection onto some component of X.
proof Let C be a component of X. We show that p( C) is a component of
X. p( C) is connected; to show that it is an open and closed subset of X, let x
be in the closure of p(C) and let U be an open connected neighborhood of x
evenly covered by p. Because U meets p(C), p^lfl) meets C. Therefore some
component U of p~\U) meets C. Since C is a component of X, VC C.
Then, by lemma 12, p(C) D p( 0) = U. Therefore the closure of p(C) is con-
tained in the interior of p(C), which implies that p(C) is open and closed. The
same argument shows that if x £ p( C) and U is an open connected neighbor-
hood of x in X evenly covered by p, then U C p(C) and (p | С)-1(17) is the
disjoint union of those components of p-1(L7) that meet C. It follows from
lemma 12 that U is evenly covered by p | C. Therefore p | С: C —» p( C) is a
covering projection. 0
The following example shows that the converse of theorem 14 is false.
15 example Let X = S1 X S1 x • • • be a countable product of I-spheres
and for n > 1 let Хй = В” X S1 X S1 X   • • Define pn: Xn —> X by
Я THE HOMOTOPY LIFTING PROPERTY	65
Sbv.
p„(#i, . . . ,tn, 21;22, . . .) = (ex(H), . . . ,ex(tn), zltz2, . . .)
Let X = V Хг, and define p: X —> X so that p | Xn — pn. The components of
X are the spaces Xn and the map p | Xn = pn: Xn —> X is a covering projec-
tion. However, p is not a covering projection, because no open subset of X is
evenly covered by p.
For later purposes we should Eke to have the analogues of theorems 11
and 14, in which “component” is replaced by “path component.” For this we
need the following definition: a topological space is said to be locally path
connected if the path components of open subsets are open. The following
are easy consequences of this definition.
1© Any open subset of a locally path-connected space is itself locally path
connected. H
17	A locally path-connected space is locally connected, и
IB In a locally path-connected space the components and path components
coincide. B
19	A connected locally path-connected space is path connected. B
From statements 17 and 18 we obtain the following extension of theorems
11 and 14.
20	theorem If X is locally path connected, a continuous map p: X —» X
is a coveting projection if and only if for each path component A of X
p | р~гА: р~гА A
is a covering projection. In this case, if A is any path component of X, then
p\A is a covering projection of A onto some path component of X. B
2 THE HOMOTOPY LIFTING PBOPEHTY
The homotopy lifting property is dual to the homotopy extension property. It
leads to the concept of fibration, which is dual to that of cofibration intro-
duced in Sec. 1.4. In this section we define the concept of fibration and prove
that a covering projection is a special kind of fibration. This special class of
fibrations will be regarded as generalized covering projections, and our subse-
quent study of covering projections will be based on a study of the more gen-
eral concept. At the end of the chapter we return to the general considera-
tion of fibrations.
We begin with an important problem of algebraic topology, called the
lifting problem, which is dual to the extension problem. Let p; E —» В and
Л X —> В be maps. The lifting problem for f is to determine whether there is
66
COVERING SPACES AND FIBRATIONS CHAP, 2
a continuous map /': X —» E such that f = p ° f—that is, whether the dotted
arrow in the diagram
E
x 4 в
corresponds to a continuous map making the diagram commutative. If there
is such a map/', then/ can be lifted to E, and we call/' a lifting, or lift, of/
In order that the lifting problem be a problem in the homotopy category, we
need an analogue of the homotopy extension property, called the homotopy
lifting property, defined as follows. A map p; E —» В is said to have the
homotopy lifting property with respect to a space X if, given maps /': X —» E
and F: X X I —> В such that F(x,O) = pf'ft) for x £ X, there is a map
F: X X I —> E such that F'(x,O) = /'(x) for x £ X and p ° F = F. If /' is re-
garded as a map of X X 0 to E, the existence of F is equivalent to the
existence of a map represented by the dotted arrow that makes the following
diagram commutative:
X X 0 4 E
I" / I’
xx/4 в
If p: E —» В has the homotopy lifting property with respect to X and
/о, /г: X —> В are homotopic, it is easy to see that /0 can be lifted to E if and
only if ft can be lifted to E. Hence, whether or not a map X —» В can
be lifted to E is a property of the homotopy class of the map. Thus the homo-
topy lifting property implies that the lifting problem for maps X —» В is
a problem in the homotopy category.
A map p: E —» В is called a fibration (or Hurewicz fiber space in the lit-
erature) if p has the homotopy lifting property with respect to every space. E
is called the total space and В the base space of the fibration. For
b £ B, p~ftb) is called the fiber over b.
If p: E —> В is a fibration, any path w in В such that w(0) £ p(E) can be
lifted to a path in E. In fact, w can be regarded as a homotopy w: P X I —* В
where P is a one-point space, and a point e0 £ E such that p(e0) = w(0)
corresponds to a map f.P —» E such that p/(P) = w(P,O). It follows from the
homotopy lifting property of p that there exists a path й in E such that
£(()) = e0 and p ° й = w. Then ы is a lifting of w.
1 example Let F be any space and let p: В X F —» В be the projection
to the first factor. Then p is a fibration, and for any b Q В the fiber over b is
homeomorphic to F.
To prove that a covering projection is a fibration, we first establish the
following unique-lifting property of covering projections for connected spaces.
SEC- 2 THE HOMOTOPY LIFTING PROPERTY
67
jg theorem Let p: X —» X be a covering projection and let f,g: Y X be
liftings of the same map (that is, p ° f = p ° g).If Y is connected and f agrees
with g for some point of Y, then f = g.
proof Let Y} = {y £ Y |/(y) = g(y)}. We show that Yj is open in Y.
If у € Yi, let U be an open neighborhood of pf(y) evenly covered by p and
let U be an open subset of X containing f(y) such that p maps U homeomor-
phically onto V. Then f~\ U) П g-1( U) is an open subset of Y containing у
and contained in Yj.
Let Y2 = (y £ YI f(y) 7^ g(y)}. We show that Y2 is also open in Y (if X
were assumed to be Hausdorff, this would follow from a general property of
Hausdorff spaces). Let у £ Y2 and let U be an open neighborhood of pf(y)
evenly covered by p. Since f(y) 7^ g(y), there are disjoint open subsets Ui and
(/2 of X such that /(y) £ Uv and g(y) £ U2 and p maps each of the sets
Cl and 02 homeomorphically onto U. Then /-1( Ui) П g'fL) is an open
subset of Y containing у and contained in Y2.
Since Y = Yr U Y2 and Yj and Y2 are disjoint open sets, it follows from
the connectedness of Y that either Yj = 0 or Yj = Y. By hypothesis,
Yr 7^ 0, so Y = Yi and f = g. B
We are now ready to prove that a covering projection has the homotopy
lifting property.
3	theorem A covering projection is a fibration.
proof Let p: X —» X be a covering projection and let f': Y —» X and
F: Y X I —» X be maps such that F(y,O) — pf'(y) for у £ Y. We show that
for each у £ Y there is an open neighborhood Nv of у in Y and a map
F'v: Nv X I —» X such that F'v(y',O) = f'(y') for y' £ Ny and pF'v = F\NV X I.
Assume that we have such neighborhoods Ny and maps F'v. If y" £NV П Ny',
then F'y I у" X I and F'V’ | у" X I are maps of the connected space у" X I
into X such that for t £ I
p°(Fv\ у" X I)(y",t) = F(y",t) = p ° (F’y> I у" x I)(y",t)
Because (F'y | у" X I)(y",O) = f'(y") - (F^ | у" x I)(y",O), it follows from
theorem 2 that F'v | у" x I = F'V' | у" x I- Since this is true for all
y" <=NV П Ny, it follows that F'y | (Ny A Nyf x I = F^ | (Nv A Nyf X I.
Hence there is a continuous map F: Y X I —> X such that F | Ny X I = F'v,
and F is a lifting of F such that F(y,Q) = f'(y) for у £ Y. Thus we have
reduced the theorem to the construction of the open neighborhoods Nv and
maps Fy.
It follows from the fact that p: X Xis a covering projection (and the
compactness of Г) that for each у £ Y there is an open neighborhood Ny of
у and a sequence 0 = t0 < h < • •  < tm — 1 of points of I such that for
i = I, . . . , m, F(NV X	is contained in some open subset of X
evenly covered by p. We show that there is a map F'y; Nv X I —> X with the
desired properties. It suffices to define maps
68
COVERING SPACES AND FIBRATIONS CHAP. 2
G4: Nv X	X i = 1, ... ,m
such that
p ° Gt = F | Nv X [h-i,h]
Gi(</',0) = f '(y')	y' E Nv
Gi_1(y',ti~1) = Gi(y',ti~f) у' E Ny, i = 2, . . . , m
because, given such maps Gj, there is a map Ff. Ny X I —> X such that
Fy | Ny X [й-i,tf] = Gt for i = 1, . . . , m. Then F'v has the desired properties.
The maps Gj are defined by induction on i. To define G1; let U be an
open subset of X evenly covered by p such that F(NV X [Ь>,#1]) C U. Let
{ Uj} be a collection of disjoint open subsets of X such that p1 (U) = U Ц
and p maps Ц- homeomorphically onto U for each j. Let V) = /,-1( Ц). Then
{V)} is a collection of disjoint open sets covering Nv, and G| is defined to be
the unique map such that for each j, Gi maps V; X [#o> t1] into Ц- to be a lift-
ing of F | Vj X [Wi]. This defines Gi.
Assume Gj_i defined for I < i < m. Let U' be an open subset of
X evenly covered by p such that F(Ny X	C U'. Let { if} be a collec-
tion of disjoint open subsets of X such that p-1( U') = U and p maps if
homeomorphically onto U' for each A. Let V^ = {// E Nv | G;i(y',t;i) E Ff}.
Then { Vh} is a collection of disjoint open sets covering Nv, and Gj is defined
to be the unique map such that for each k, Gj maps V/- X [h-i,ti] into if to
be a lifting of F | V/c X	This defines G{. в
A map p-. E —» В is said to have unique path lifting if, given paths w and
u>' in E such that p ° w — p ° w' and w(0) = a>'(0), then w = сУ. It follows
from theorem 2 that a covering projection has unique path lifting.
4	lemma If a °nap has unique path lifting, it has the unique-lifting
property for path-connected spaces.
proof Assume that p: E~> В has unique path lifting. Let У be path connected
and suppose that fi g: Y —» E are maps such that p ° f = p ° g and/(y0) = g(?/o)
for some yo E T. We must show f — g. Let у E Y and let w be a path in Y
from yo to y. Then f ° и and g ° и are paths in E that are liftings of the
same path in В and have the same origin. Because p has unique path lifting,
f ° и = g ° gj. Therefore
f(y) = (f ° w)(!) = (g ° w)(!) = ё(У) B
The following theorem characterizes fibrations with unique path lifting.
5	theorem A fibration has unique path lifting if and only if every fiber
has no nonconstant paths.
proof Assume that p: E —» В is a fibration with unique path lifting. Let <;
be a path in the fiber p'fb) and let gj' be the constant path in such
that gj'(O) = w(0). Then p ° gj = p ° gj', which implies gj = gj'. Hence gj is a,
constant path.
SEC. 2 THE HOMOTOPY LIFTING PROPERTY	69
Conversely, assume that p: E —> В is a fibration such that every fiber has
no nontrivial path and let w and be paths in E such that p ° w = p ° w'
and ы(0) = w'(0). For t £ I, let u't be the path in E defined by
_ W1 - 2t')f)	0 < f < ¥2
t[ 1 ~ U'((2f' - l)f) ¥2 < f < 1
Then w" is a path in E from w(f) to »:'(/), and p ° ы" is a closed path in В that
js homotopic relative to I to the constant path at pw(t). By the homotopy lift-
ing property of p, there is a map F': I X I —» E such that F'(t',0) — and
p maps OxIUIxlUlXfto the fiber p-1(pw(t)). Because p~'1(pco(t))
has no nonconstant paths, F' maps О X I, I X 1, and 1 X I to a single point.
It follows that F'(0,0) = F'(I,0). Therefore w"(0) = w"(I) and w(f) = co'(f). B
We have seen that a covering projection is a fibration with unique path
lifting. It will be shown in Sec. 2.4 that if the base space satisfies some mild
hypotheses, any fibration with unique path lifting is a covering projection.
One reason for studying fibrations with unique path lifting as generalized
covering projections is that the following two theorems are easily proved, but
both are false for covering projections.
6	theorem The composite of fibrations (with unique path lifting) is a
fibration (with unique path lifting). B
7	theorem The product of fibrations (with unique path lifting) is a
fibration (with unique path lifting), и
An example shows that theorem 6 is false for covering projections.
8	example Let X and X„, for n > 1, be a countable product of 1-spheres.
Let Xn = R” X X and define pn: Xn Xn by
pn(#i, . . . ,tn, -zi,z2, . . .) = (ex(#i), . . . ,ex(tn), z-^zz, . . .)
Then pn is a covering projection for n > I. It follows from theorem 2.1.11
that Vpn: VX„ —» VXn is a covering projection. Since VX„ is the product
of X and the set of positive integers, there is a covering projection VX?l —> X
(see example 2.1.2). The composite
vxn^ vx„^x
is not a covering projection (cf. example 2.1.15).
Similarly, theorem 7 is false for covering projections.
8	example For n > 1, let pn: Xn —» Xn be the covering projection
ex: R —> S1. Then
Xpn: XXn XXn
is not a covering projection.
It follows from theorem 6 that there is a category whose objects are
topological spaces and whose morphisms are fibrations with unique path
70
COVERING SPACES AND FIBRATIONS CHAP. 2
lifting. From theorem 7, this category has products. It is also easy to verify
that it has sums. We shall now describe a category, depending on a given
base space, which is of more use in studying covering projections or fibrations.
For a given space X there is a category whose objects are maps p: X —> X,
which are fibrations with unique path lifting, and whose morphisms are com-
mutative triangles
Xi 4 X2
Pl\
X
If pp Xj —> X is an indexed family of objects in this category, let p: V Xj —> X
be the map such that p | Xj = p,. Then p is also an object in the category and
is the sum of the collection {pj} in the category.
To show that this category also has products, given maps рр Xj —» X, let
X = {(Л)) e X X; I	for all /, /'}
and define p: X —» X by p((.q)) = pXx'j). If each pj is a fibration, so is p, and
if each p, has unique path lifting, so does p. Hence p is a product of (p/j in
the category of fibrations with unique path lifting. This map p is called the
fibered product of the maps {/)/} We consider it in more detail in Sec. 2.8.
There is a similar category whose objects are covering projections with
base space X and whose morphisms are commutative triangles. This category
has finite sums and finite products, but neither arbitrary sums nor arbitrary
products. In fact, for each n let
p.: R« X S1 X S1 X--------> S' X Si X •  
be defined by pn(ti, . . . ,tn, z>i,Z2, ...') = (e2’77il>, . . . ,e2vltn, z±, Z2, .  .),
as in example 8. Then the collection {pn} has neither a sum nor a product in
the category of covering projections with base space X.
3 RELATIONS WITH THE FUNDAMENTAL «ROUP
In a fibration with unique path lifting the fundamental group of the total
space is isomorphic to a subgroup of the fundamental group of the base
space. The corresponding subgroup of the fundamental group will lead to a
classification of fibrations with unique path lifting. In fact, we shall see in the I
next section that the fundamental group functor solves the lifting problem for j
fibrations with unique path lifting. The present section is devoted to consid-
eration of the relation between the fundamental groups of the total space and
the base space of a fibration with unique path lifting.
We begin with a localization property for fibrations which is an analogue
of theorem 2.1.14.	!
SEC. 3 RELATIONS WITH THE FUNDAMENTAL GROUP
71
1 lemma Let p: E В be a fibration. If A is any path component of E,
then pA is a path component of В and p\ A: A —» pA is a fibration.
proof Since pA is the continuous image of a path-connected space, it
is path connected. It is a maximal path-connected subset of B, for if co is a path
in В that begins in pA, there is a lifting co of co that begins in A. Since A
is a path component of E, &> is a path in A. Therefore gj = p ° a is a path in
pA. Hence pA is a maximal path-connected subset of В and, by theorem
1.7.9, a path component of B.
To show that p | A: A —» pA has the homotopy lifting property, let
f': Y —» A and F: У X I —> pA be maps such that F(t/,O) = pf'lij). Because p
is a fibration, there is a map F: Y x I —» E such that p ° F = F and
F(y>0) = /'(?/) Lor апУ У E F, F' must map у X I into the path component
of E containing F'(y,0). Therefore F(y X I) C A for all y, and F: Y X I A
is a lifting of F such that F(y,0) — f'(y). и
For locally path-connected spaces we have the following analogue of
theorem 2.1.20, which reduces the study of fibrations to the study of fibra-
tions with total space and base space path connected.
g theorem Let p: E —> В be a map. If E is locally path connected, p is
a fibration if and only if for each path component A of E, pA is a path com-
ponent of В and p fA: A pA is a fibration.
proof If p: E —» В is a fibration and A is a path component of E, it follows
from lemma 1 that pA is a path component of В and p | A: A —» pA is
a fibration.
To prove the converse, let f: Y —» E and F: Y x I —» В be such that
F(y,0) = f'(y)- Let {Aj} be the path components of E. Then {AJ are disjoint
open subsets of E. Let Vy = f^Aj). The collection { V/} is a disjoint open cov-
ering of Y. Therefore, to construct a map Ft Y x I —» E such that p ° F = F
and F(y,ty = f'(y), it suffices to construct maps F'p Vy x I —> E for all / such
that p ° F'j = F | Vj x I and Fy(y,O) = f '(?/,()).
Because F(y x I) is contained in the path component of В containing
F(y,O) = pf'b/f it follows from the fact that pAj is a path component of В
that F(Vy X I) C pAj for all j. Because p | Ay: Ay —» pAy is a fibration, there is
a map Fy: Vy X I —» Ay such that pF} = F\Vj x I and F}(y,b) = f'(y) for
у £ Vy. Therefore p has the homotopy lifting property. B
Since every path in a topological space lies in some path component of
the space, it is clear that theorem 2 remains valid if the term “fibration” is
replaced throughout by “fibration with unique path lifting.”
The main result on fibrations with unique path lifting is embodied in the
following statement.
3	lemma Let pt X —» X be a fibration with unique path lifting. If co and
<*>' are paths in X such that co(O) = 6J,(b) and P ° w ~ p ° d, then co ~ co'.
72
COVERING SPACES AND FIBRATIONS CHAP. 2
proof Let F: I X I —» X be a homotopy relative to I from p ° w to p ° w'
[that is, F(t,0) = pw(t) and F(f,I) = pw'(£), and F(O,t) — pa>(0) and F(l,t) =
pw(l)]. By the homotopy lifting property of fibrations, there is a map
F: I X I-> X such that F'(t,0) = w(t) and p ° F = F. Then F'(0 X I) and
F'(I X f) are contained in p 1(po>(0)) and	respectively. By
theorem 2.2.5, F'(0 X I) and F'(l X I) are single points. Hence F is a homo-
topy relative to I from co to some path u" such that co"(O) = co(O) and
p ° co" = p ° co'. Since co'(O) = co(O), it follows from the unique-path-lifting
property of p that co' = co" and F: <x> ~ <x>' rel I. и
It follows from lemma 3 that if p: X —» X is a fibration with unique path
lifting, then for any two objects and xy in the fundamental groupoid of X,
p# maps hom (x0,Xi) injectively into hom (p(x0),p(xi)). In particular, if
x0 — xi, we obtain the following theorem.
4	theorem Let p: X —» X be a fibration with unique path lifting. For
any x0 6 X the homomorphism.
p#: rr(X,x0) -» 7r(X,x0)
is a monomorphism, в
This last result provides the basis for the reduction of problems concern-
ing fibrations with unique path lifting to problems about the fundamental
group. In order that the fundamental group be really representative of the
space in question, we assume that the spaces involved are path connected. It
follows from theorem 2 that this is no loss of generality for locally path-
connected spaces.
5	lemma Let p: X X be a fibration with unique path lifting and
assume that X is a nonempty path-connected space. If x(J, xy £ X there is a
path и in Xfrom p(x'o) to p(xy) such that
p#rfX,Xo) = h^p#7fX,xfi
Conversely, given a path и in Xfrom p(x0) to xy, there is a point xq £ p~*(xi)
such that
hiJPttrtX’Zi) = P#ir(X,Xo)
proof For the first part, let co be a path in X from x0 to xy. Then тг(Ххо) =
/г[й]7г(ХХу). Therefore
p#77(X,X0) = /l[poS]p#77(X,^l)
and so p ° co will do as the path from p(x0) to р(лу).
Conversely, given a path co in X from p(*o) to xy, let й be a path in X
such that co(O) = x0 and рй — co. If xy = co(I), then
hMp#ir(X,xf) = p#(/i[c]77(X,ay)) = p#rr(X,x0) в
This easily implies the following result.
SEC. 3 RELATIONS WITH THE FUNDAMENTAL CROUP	73
G theorem Let p: X —» X be a fibration with unique path lifting and
assume that X is a nonempty path-connected space. For x'o C pX the collec-
tion {р#тт(Х,Хо) | xo € P-1(hj)} Is a conjugacy class in tt(X,x0). If w is a path
in pX from Xo to xq, then h^ maps the conjugacy class in tt(X,Xi) to the con-
jugacy class in tt(X,Xo). 0
Let p: X —» X be a fibration and let w be a path in X beginning at x0.
Define a map Fa: p-1(xo) X I X by FB(x,f) = w(t) and let i: p-1(x0) С X.
Then pi(x) = FM(x,0) for x £ p-1(x0). It follows from the homotopy lifting
property of p that there exists a map GB: p-1(x0) X I —» X such that
GB(x,0) = i(x) = x and p ° GB = FB.
Suppose now that p has unique path lifting. We prove that the map
x—> GB(x,I) of p-1(x0) to p-1(<о(1)) depends only on the path class of w.
If u,' ~ w and G'.-: p-1(xo) X I X is a map such that GB'(x,O) = x and
p о GB = FBs then for any x £ pH(xo), let w and w' be the paths in X defined
by w(t) = Ga(x,t) and w(t) = GB'(x,t). Then & and Д' begin at x and
p°w = cj~w' = p°w'
It follows from lemma 3 that Д ~ Д'. Then GM(x,l) = GB'(x,l) for every
x £ p"1(x0). Therefore there is a well-defined continuous map
fkT P^Hty) P~4^(l))
defined by /|B|(x) = GB(x,I), where GB is as above. It is clear that if w(I) =
w'(0), then = fM ° f[Mj.
7	theorem Let p: X —» X be a fibration with unique path lifting. There
is a contravariant functor from the fundamental groupoid of X to the cate-
gory of topological spaces and maps which assigns to x £ X the fiber over x
and to [w] the function flaJ. a
The fact that /[Bj is a homeomorphism for every [w] leads to the follow-
ing corollary.
8	corollary If p: X —» X is a fibration with unique path lifting and X
is path connected, then any two fibers are homeomorphic, a
If X is path connected and p: X —> X is a fibration with unique path lift-
ing, the number of sheets of p (or the multiplicity of p) is defined to be the
cardinal number of p^1(x) (which is independent of x £ X, by corollary 8).
For a path-connected total space, the multiplicity is determined by the con-
jugacy class as follows.
9	theorem Let p: X —» X be a fibration with unique path lifting and
assume X and X to be nonempty path-connected spaces. If Xq С X, the nd-
tiplidty of p is the index of р#гт(Х,х0) in гт(Х,р(х0У).
proof By theorem 7, 7т(Х,р(х0)) acts as a group of transformations on the right
on p~4p(xo)) by x ° [w] = f[M](«) for x e p-1(p(x0)). If Xi, x2 e p~4p(xof), let
w be a path in X from xq to x2. Then [p ° с] C 77(X,p(x0)) and Xi ° [pw] = x2.
74	COVERING SPACES AND FIBRATIONS CHAP. 2
Therefore 77(Х,р(х0)) acts transitively on p-1(p(x0)). The isotropy group of x0
[that is, the subgroup of 7r(X,p(x0)) leaving x0 fixed] is clearly equal to р#тт(Х,х0').
From general considerations1 there is a bijection between the set of right
cosets of р#тт(Х,х0) in я-(Х,р(х0)) and p-1(p(xo)). B
10	example For n > 2 the covering p-. Sn —» Pn of example 2.1.5 has
multiplicity 2. Because S’1 is simply connected, тг(Р1) Z2 for n > 2.
A fibration p: X —» X with unique path lifting is said to be regular
if, given any closed path и in X, either every lifting of и is closed or none is
closed.
11	theorem Let p: X —» X be a fibration with unique path lifting, p is
regular if and only if р#гт(Х,х0) = p^lXpif) whenever p(xf} = p(x,).
proof Assume that p is regular and let <5 be a closed path in X at x0. Then <5 is a
closed lifting of p&>. Therefore there is a closed lifting of рй at хь It fol-
lows that p#[w] = [p<5] = pff[wj. Therefore p#7r(X,x0) C p#(X,xf). Since they
are conjugate subgroups of тг(Х,р(х0)), they are equal.
Conversely, if p#7r(X,x0) = p#rr(X,Xi) whenever p(x0) = let w be a
closed path in X at plfio) having a closed lifting <5 at x0. Then
[w] = p#[w] C p#7t(X,^o) = p#w(X,«i)
Therefore there is a closed path wj in X at Xi such that ры± ~ w. If <5j is a
lifting of w such that <5i(0) — Xi, then by the unique-path-lifting property of
p, <51 = <5'1. Therefore <5j is a closed lifting of w at xy and p is regular. B
In case X is a nonempty path-connected space, theorems 6 and II give
the following result.
12	theorem Let p\ X —» X be a fibration with unique path lifting and
assume that X is a nonempty path-connected space. Then p is regular if and
only if for some x0 C Xo, p#7r(X,x‘o) is a normal subgroup of w(X,p(x0)). 
4k THE LIFTING PROItLEM
In this section we show that the fundamental group functor solves the lifting
problem for fibrations with unique path lifting. As a consequence of this, the
fundamental group functor provides a classification of covering projections,
which is discussed in the next section.
Our first result is that any map of a contractible space to the base space
of a fibration can be lifted.
1	lemma Let p: E —> В be a fibration. Any map of a contractible space
to В whose image is contained in p(E) can be lifted to E.
1 Whenever a group G acts transitively on the right on a set S there is induced a bijection
between the set of right cosets of the isotropy group (of any s € S) in G and the set S.
SEC- 4 THE lifting problem
75
proof Let Ybe contractible andlet/: Y В be amap such that/( Y) C p(E).
Because Y is contractible, f is homotopic to a constant map of Y to some
point of /(Y). /(Y) C p(E), so this constant map can be lifted to E. The
homotopy lifting property then implies that f can be lifted to E. a
Because we use the fundamental group functor, it will prove technically
simpler to consider the lifting problem for spaces with base points.
2	lemma Let p: (Х,Жо) (X,xo) be a fibration with unique path lifting.
If yo is a strong deformation retract of Y, any map (Y,y0) —» (X,xo) can be
lifted to a map (Y,y0)	(X,x0).
proof Let f: (Y,y0) —> (X,x0) be a map. f is homotopic relative to y0 to the
constant map Y x0. The constant map can be lifted to the constant map
У i'o. By the homotopy lifting property, f can be lifted to a map f':Y—>X
such that f' is homotopic to the constant map Y x0 by a homotopy which
maps y0 X J to ptyxo)- Because p-1(x0) has no nonconstant path by theorem
2.2.5, /'(yo) = xo- H
We shall apply theorem 2 to a contractible space in order to lift certain
quotient spaces of the contractible space. The usual way to represent a
space as the quotient space of a contractible space is to show it is a quotient
space of its path space. Given y0 E Y, the path space P(Y,y0) is the space of
continuous maps w: (1,0) —> (Y,yo) topologized by the compact-open topology.
There is a function <p: P(Y,y0) —> Y defined by <p(w) = w(l). If U is an open
set in Y,
= (1;U> = {co E P(Y,y0) I «(1) £ U}
is an open set in P(Y,y0). Therefore <p is continuous.
3	lemma The constant path at у о is a strong deformation retract of the
path space P(Y,y0).
proof A strong deformation retraction F: P(Y,y0) x f —> P(Y,y0) to the con-
stant path at y0 is defined by
F(w,t)(t') = W((l - t)t') w £ P(Y,y0); t, f E I и
We have shown that у is a continuous map of the contractible path space
P(Y,y0) to Y. If Y is path connected, <p is clearly surjective. If Y is also locally
path connected, the following theorem shows that <p is a quotient projection.
4	theorem A connected locally path-connected space Y is the quotient
space of its path space P(Y,y0) by the map <p.
proof We know that <p is continuous, and because a connected locally path-
connected space is path connected, it is surjective. To complete the proof it suf-
fices to show that <p is an open map. Let w E P(Y,y0) andlet W = П 1<гьп<К>;17г)
be a neighborhood of w, where Ki is compact in I and is open in Y. We
enumerate the K’s so that for some 0 < к < n, 1 E Ki Г1 • • • П Kk and

76	COVERING SPACES AND FIBRATIONS CHAP. 2
I £ Kk+1 U •  • U Kn. Because co(l) £ Ur П • • • П Uk, there is a path-
connected neighborhood V of co(l) contained in C7i П • •  П C7r. Choose
0 < t' < 1 such that [#', 1] Л (T<A4 i U • •  U Kn) = 0 and co([t',l]) С V.
To prove that <p(W) 2D V, which completes the proof, let if £ V and let
co' be a path in V from co(t') to y'. Define co: I Y by
co(f)	0 < t < t'
,(t - f \	. . . . ,
CO H----,1	f < t < 1
\1 - t) “ ~
For i > k, = ы(К;) C Uf. For i < k,
u>(Ki) = u>(Ki П [0,t']) U со(К; П [t',I]) C co(K{) U co'(I) C Ut U V = Ut
Therefore co £ W and y(co) = y'. Hence y( W) 22) V. и
We can put these results together to obtain the following result, called
the lifting theorem.
5 theorem Let pi (X,x0) —» (X,xf) be a fibration with unique path lift-
ing. Let Y be a connected locally path-connected space. A necessary and
sufficient condition that a map ft (Y,y0) —» (X,%o) have a lifting (Y,y0) (X,xo)
is that in tt(X,Xo)
f#rr{Y,y0) С р#7г(Х,х0)
proof If /': (Y,y0) (ХДо) is a lifting of f, then f = p ° f' and
f#^(Xyo) = P#f ИУ>!/о) c P#”(X>Xo)
which shows that the condition is necessary.
We now prove that the condition is sufficient. It follows from lemmas 3
and 2 that if coo is the constant path at y0, the composite
(P(Y,tJO), wo) Л (Y,yf) Л (X,x0)
can be lifted to a map f: (PfYp/o), coo) —» (X,x0)- We show that if/#7r(Y,yo) <22
P#tt(X,Xo) and if co, co' C P(Y,y0) are such that qftY) = yftY), then /(co) = flfi').
Let co and CY be the paths in P{Y,yf} from coo to co and co', respectively,
defined by co(f)(t') = co(tt') and co'(t)(t') = co'(tt'). Then/° co and/° co' are
paths in X from x() to /(co) and /(co'), respectively, such that
p ° / ° co = / ° <p ° co = / ° co and p ° f ° co' = / ° co'
Because co * co'1 is a closed path in Y at у о and Y,y0) С p#w(X,Xo), there
is a closed path co in X at x0 such that (/ ° co) * (/° co')-1 ~ p ° co. Then
p ° (/° co) = /° co ~ (p ° co) * (/° co') = p ° (co *(/° co'))
By lemma 2.3.3, /° co ~ co «• ( /° co'). In particular’, the endpoint of / ° co,
which is /(co), equals the endpoint of / ° co', which is /(co').
It follows that there is a function /': (Y,y0) —» (X,x0) such that /' ° <p = f.
SEC. 4 THE LIFTING PROBLEM
77
arid using theorem 4, we see that f' is continuous. Because
p°f'°<p = p°f = f°<p
and <p is surjective, p ° f' = f. Therefore f' is a lifting of f. 
Let p: E —» В be a fibration. A section of p is a map s: В —» E such that
p о s — IB (thus a section is a right inverse of p). It follows easily from the
homotopy lifting property that there is a section of p if and only if [p] has a
right inverse in the homotopy category. Because a section is a lifting of the
identity map В С B, the following is an immediate consequence of theorem 5.
G corollary Let p: (X,ito) —» (X,xq) be a fibration with unique path lift-
ing. If X is a connected locally path-connected space, there is a section
(X,x"o)	(X^o) of p if and only if р#тт(Х,Х0) = тт(Х,х0). в
7 corollary Let p: X —» Xbe a fibration with unique path lifting. If R
is a nonempty path-connected space and X is connected and locally path
connected, then p is a homeomorphism if and only if for some x0 £ X,
р#чт(Х,Хо) = ^(X,p(x0)).
proof If p is a homeomorphism, р#тг(Х,Хо) = 7T'(X,p(ito)). Conversely, if
р#7г(Х,Хо) = ^(X,p(x0)), then by theorem 2.3.9, p is a bijection. By corollary 6,
it has a continuous right inverse. Therefore p is a homeomorphism. B
If p: X —> X is a covering projection and X is path connected, a neces-
sary and sufficient condition that p be a homeomorphism is that р#тт(Х,х‘о) =
?r(X,p(xo)) for some Xo £ X. This condition on the fundamental groups implies
that p is a bijection, and by lemmas 2.1.8 and 2.1.7, p is open; hence for cov-
ering projections corollary 7 is valid without the assumption that X be locally
path connected. This is definitely false for fibrations with unique path lifting
if X is not locally path connected, because p need not be open. The following
example shows this.
в example Let X be the subspace of R2 defined to be the union of the
four sets
Ai =	{(x,y)	|	x	=	0, —2	<	у < 1}
A2 =	{(x,y)	j	0	<	x < 1,	у	- -2}
A3 =	{(x,y)	|	x	=	I, —2	<	у < 0}
A4 =	{(x,y)	I	0	<	x < 1,	у	= sin 2tt/x}
illustrated in the diagram
A;
(1,-2)
(0,-2)
78
COVERING SPACES AND FIBRATIONS CHAP. 2
Let X be the half-open interval [0,4) and define p: X —» X to map [0,1] linearly
onto Ai, [1,2] linearly onto A2> [2,3] linearly onto A3, and [3,4) homeomor-
phically onto A4 by the map t —> (t — 3, sin(27r/(t — 3))). Then X and X are
path connected and p: X —> X is a fibration with unique path lifting. However,
p is not a homeomorphism, although X and X are both simply connected.
For locally path-connected spaces the lifting theorem provides the fol-
lowing criterion for determining whether an open path-connected subset of
the base space is evenly covered by a fibration.	'
9 lemma Let p: X —+ Xbe a fibration with unique path lifting. Assume
that X and X are locally path connected and let U be an open connected sub-
set of X. Then U is evenly covered by p if and only if every lifting to X of a
closed path in U is a closed path.
proof If U is evenly covered by p and й is a path in p-1((7), then й is a
path in some component U of pH((7). By lemma 2.1.12, p maps U homeo-
morphically onto U. Therefore, if p 0 й is a closed path in U, й is a closed
path in U. Hence the condition is necessary.
It is also sufficient, because if x0 E U and x0 E p-1(x0), the hypothesis
that every lifting of a closed path in U at x0 is a closed path in X implies that
in w(X,Xo)
г^7г(и,.г'о) С р#тт(Х,^о) where i: (U,xf) C (X,x’o)
By theorem 5, there is a lifting i^0: (I7,xo) —» (X,*o) of i. The collection
{i'n0(U) I Xo € p“1(x0)} consists of path-connected sets which, by lemma 2.2.4,
are disjoint. We show that their union equals p“1(C7). If x £ p^fU), let w be
a path in U from p(x) to xo and let й be a lifting of w such that й(0) = x. Then
й(1) E pfixfi, and therefore й is a path in г ~(i )([/) Hence x E and
{iga(U) I Xo E Р^Цхо)} is a partition of p~1(U) into path-connected sets. Since
p ~'( (7) is open and X is locally path connected, i'gfU) is open in X for each
Xo E /’ '(А'о)- Clearly, p is a homeomorphism of i$0(U) onto U for each
x0 E /’ ’ (x'o), and U is evenly covered by p. и
A space X is said to be semilocally L-connected if every point Xo E X has
a neighborhood N such that 7r(X,x0) —» тг(Х,х0) is trivial.
10 theorem Every fibration with unique path lifting whose base space is
locally path connected and semilocally l-connected and whose total space is
locally path connected is a covering projection.
proof It follows from lemma 9 and the definition of semilocally l-connected
space that each point of the base space has an open neighborhood evenly
covered by the fibration, в
s£C. 5 THE CLASSIFICATION OF COVERING PROJECTIONS
79
5 THE CLASSIFICATION OF COVERING PROJECTIONS
This section contains a classification of covering projections over a connected
locally path-connected base space. It is based on the lifting theorem and re-
duces the problem of equivalence of covering projections to conjugacy of
their corresponding subgroups of the fundamental group of the base space.
A large part of the section is devoted to constructing a covering projection
corresponding to a given subgroup of the fundamental group of the base
space.
Let X be a connected space. The category of connected covering spaces
of X has objects which are covering projections p-. X X, where X is
connected, and morphisms which are commutative triangles
Xi Д X2
Pl\ / P2
X
If X is locally path connected and p: X —> X is an object of this category,
then, by lemma 2.1.8, p is a local homeomorphism and X is also locally path
connected. We show that in this case every morphism in this category is a
covering projection.
1 lemma In the category of connected covering spaces of a connected
locally path-connected space every morphism is itself a covering projection.
proof Consider a commutative triangle
Xi Л x2
Pl\ / P2
X
where pi and p2 are covering projections and X is locally path connected. It
follows from corollary 2.1.13 that J is a covering projection if it is surjective.
Because X2 is connected and locally path connected, it is path connected.
Let £i £ Xi and x2 £ X2 be arbitrary and let w2 be a path in X2 from f(xi)
to x2. Because pi is a fibration, there is a path й1 in Xj beginning at Xi such
that pi0 Wr = p2 0 w2. By the unique path lifting of p2, f 0 oh = w2. Therefore
f(«i(l)) = ЗД = x2
proving that Jis surjective, и
The next result determines when there is a morphism from one object to
another in the category of connected covering spaces of X.
2 theorem Let p±: Xi X and px X2 —» X be objects in the category
80
COVERING SPACES AND FIBRATIONS CHAP. 2
of connected coveting spaces of a connected locally path-connected space X.
The following are equivalent:
(a)	There Is a covering projection f: Xi —» X2 such that p2 ° f = p^.
(b)	For all Ж1 E Xi and x2 E X2 such that pi(xj) = p2(x2), pi#rr(Xi,xf) is
conjugate in ?r(X,pi(xi)) to a subgroup of p2#rr(X2,x2).
(c)	There exist xj E Хл and x2 E X2 such that pi(xf) = p2(x2) and
Pi#tt(Xi,Xi) is conjugate in 7r(X,pi(xi)) to a subgroup of р2#тт(Х2,х2).
proof (a) => (b) Given f: Xi —» X2 such that p2° f — pi, if Si E Xi and
x2 € X2 are such that pi(xi) = Рг(х2), then
р1#тт(Х1,Х!) = p2# ° f^X^xf) C p2#^(X2/(Si))
Because f(xf) and x2 lie in the same fiber of p2: X2 —» X, it follows from
theorem 2.3.6 that p2#^(X2,f(xi)) and р2#я-(Х2,х2) are conjugate in ?r(X,pi(xi)).
(b) => (c) The proof is trivial.
(c) => (a) Assume that ху E ^1 and x2 E X2 are such that pi(xi) = p2(x2)
and that pi#7r(Xi,Xi) is conjugate in 7r(X,pi(xi)) to a subgroup of р2#7т(Х2,х2).
By theorem 2.3.6, there is a point x2 E X2 such that p2(x'2) = p2(^2) and such
that
P1#77(X1,X1) C p2#rr{X2,x!2)
Because Xi is a connected locally path-connected space, the lifting theorem
implies the existence of a map f: (Xi,xi) —» (Х2,хг) such that p2 0 f = pi. a
3 corollary Two objects in the category of connected covering spaces
of a connected locally path-connected space X are equivalent if and only if
their fundamental groups (at some two points over the same point of X) map
to conjugate subgroups of the fundamental group of X (at this point), a
We give two examples.
4 Because every nontrivial subgroup of ^(S1) Z is infinite cyclic, by
corollary 3 every connected covering space X —» S1 is equivalent to
ex: R —» S1 or to the map S1 —» S1 sending z to x!! for some positive integer n.
5 For n > 2, w(Prt) Z2, and every connected covering space X —> Pn is
equivalent to the double covering S’1 —» Pn or to the trivial covering Pn С P”.
A universal covering space of a connected space X is an object p: X —» X
of the category of connected covering spaces of X such that for any object
p': X' —» X of this category there is a morphism
X Д X'
p\ fP'
X
in the category. The next result follows easily from theorem 2 and corollary 3.
6 corollary Two universal covering spaces of a connected locally path-
connected space are equivalent, a
SEC. 5 THE CLASSIFICATION OF COVERING PROJECTIONS	81
Another result also follows from theorem 2.
7 corollary A simply connected covering space of a connected locally
path-connected space X is a universal covering space of X. в
Having reduced the comparison of connected covering spaces of X to a
comparison of their corresponding subgroups of the fundamental group of X,
we shall determine which subgroups of the fundamental group correspond
to covering spaces. This necessitates the construction of covering spaces. Let
X be a space and let be an open covering of X. If x0 £ X, let	be
the subgroup of тт(Х,Ло) generated by homotopy classes of closed paths having
a representative of the form (w * w') * w-1, where w' is a closed path lying in
some element of and w is a path from x0 to w'(0). The following statements
are easily verified.
« If^ is an open covering of X that refines s’!, then rr^Xo) C 7r(L?l,x0). °
9 7r(^L,%o) is a normal subgroup of tt(X,x0). a
10 If w is a path in X, then h^rr(fil,u(l')'j =	a
The connection of the groups x0) with covering projections is
explained by the following result.
11	lemma Let p; X —» X be a covering projection and let be a covering
of X by open sets each evenly covered by p. For any Xo 6 X
77(Ч1ф(х0)) C p#7r(X,X0)
proof If w' is a closed path lying in some element of then, by lemma 2.4.9,
any lifting of w' is a closed path in X. Hence any path of the form (w * w') * w-1,
where ы' is a closed path lying in some element of Ql, can be lifted to
a closed path [namely, to (<5 * &') * w-1, where Д and &' are suitable liftings
of cj and cj', respectively]. Hence any element of 77(^1,p(xo)) has a representa-
tive which can be lifted to a closed path at x0. a
The following theorem characterizes those fibrations with unique path
lifting which are covering projections.
12	theorem Let p: X X be a fibration with unique path lifting, where
X and X are connected locally path-connected spaces. Then p is a covering
projection if and only if there is an open covering Ql of X and a point х0 £ X
such that
р(хоУ) С р#тт(Х,х0)
proof If p is a covering projection, the desired result follows from lemma 11.
Conversely, if there is such an open covering '-ll and point x0 £ X, it follows
from statements 9 and 10 that for any point x'o £ X, тт(%,р(х'о')') C p#rr(X,x'o).
Using lemma 2.4.9, it follows that every element of is evenly covered
bv n. и
82
COVERING SPACES AND FIBRATIONS CHAP. 2
Lemma 11 gives a necessary condition for a subgroup of tt(X,x0) to
correspond to a covering space. The next result proves that this necessary
condition is also sufficient.
13	theorem Let X be a connected locally path-connected space and let
x0 G X. Let H be a subgroup of and assume that there is an open cov-
ering QL of X such that tt(QL,Xo) С H. Then there is a covering projection
p: (X,xo) —» (X,x0) such that р#тт(Х,хо) = H.
proof Suppose such a covering projection exists, and suppose, moreover,
that the space X is path connected. The projection <p: (P(X,xo),wo) —> (Xpyf)
of the path space of (X,xo) can then be lifted to a map (P(X,xo),coo) (X,xo),
which is surjective. If co and co' are elements of P(X,x0), then <p’(u) = fp'lcT)
if and only if <p(co) = <p(a>') and [co * co'1] £ р№тт(Х,Хо) = H. Therefore, for
path-connected X there is a one-to-one correspondence between the points of
X and equivalence classes of P(X,x0) identifying co with co' if co(I) = co'(I)
and [co * co"1] G H (the group properties of H imply that this is an
equivalence relation). Hence it is natural to try to construct X by suitably
topologizing these equivalence classes of P(X,xo). We could start with the
compact-open topology on P(X,x0) and use the quotient topology on the set
of equivalence classes, but it seems no simpler than merely topologizing the
set of equivalence classes directly, as is done below.
We consider the set of all paths in X beginning at Xq. If co and co' are two
such paths, set co — co' if co(l) = co'(l) and [co * co"1] C H. This is an equivalence
relation, and the equivalence class of co will be denoted by (co). Let X be the
set of equivalence classes. There is a function p: X —» X such that p((co)) =
co(l). If U is an open subset of X and co is a path beginning at xo and ending
in U, (<x,U) will denote the subset of X consisting of all the equivalence
classes having a representative of the form co * co', where co' is a path in U
beginning at co(I).
We prove that the collection {(co, CT)} is a base for a topology on X. If
(co') G (co, CT), then co' — co * u" for some path co" lying in U. If co is any path
in U beginning at co'(I), then
co' * co .— (co * co") * Ы — co * (co" * co)
showing that (co',СТ) С (co,CT). Since co — co' * co"1, (co) G (co',CT). The same
argument shows that (co,CT) C (co',CT), and so (co,CT) = (co',CT). Therefore, if
co" G (co,СТ) Г* (w',17'), then (co", U Г} СТ') С (co,СТ) П (co',CT'), and so the
collection {(co,U)} is a base for a topology on X.
Let X be topologized by the topology having {(co,CT)} as a base. Then p
is continuous; for if p((co)) G U, then p((co,CT)) C U. p is also open, because
p((co, CT)) clearly equals the path component of CT containing co(I), and this is
open because X is locally path connected.
Let 9l be an open covering of X such that rr{fli,xf} С H and let V be an
open path-connected subset of X contained in some element of 91. We show
that V is evenly covered by p, which will imply that p is a covering projection.
SEC. 5 THE CLASSIFICATION OF COVERING PROJECTIONS
83
If (co) E p-1(V), then (<o,V) C p-^V). The sets {(w,V) | (w> E p^(V)}
are open and their union equals p'1(V). If (co,V) Fl (co',V) =£ 0, let
(co") E (co,V) О (co'.V). Then (co",V) = (co,V> and (co",V) = (co',V).
Hence the sets {(co,V) | (co) E P-1(V)} are either identical or disjoint. To
prove that V is evenly covered by p, it suffices to show that p maps each set
(co,V) bijectively to V (because p has already been shown to be continuous
and open). If x E V, let co' be a path in V from co(I) to x. Then (co * u>") £ (co, V)
and p((co * co'» = x, showing that p is surjective. Assume p(co * co» =
p(u * co2). Then cor(l) = сог(1), so (co * co,) * (co * co2)'1 is a closed path in X
at x0. Also,
[(<0 * cor) * (co * 0:2) 1] — [(co * (cor * <02 1)) * co 1]
Since coj * CJ2- 1 is a path in V and V is contained in some element of 17,
[(co * (Wi * w2-1)) * co '] E ”’(‘?l,Xo) С H. Therefore co * cor ~ co * co2 and
(co * co» = (co * co2), showing that p is injective.
We have shown that p: X X is a covering projection. Let Xq — (coo),
where coq is the constant path in X at Xq. It remains only to verify that
р#тг(Х,Хо) — H. For this we need an explicit expression for the lift of a path
in X that begins at Xo. Let co be a path in X beginning at x0, and for t E I, de-
fine a path cof in X beginning at x0 by co((t') = co(tt'). Let co: I —> X be defined
by co(/) = (co». We prove that co is continuous. If co(to) E (co',CT), then
pG>(to) = co(f0) E U and (co',IT) = (co(t0),(7). Let N be any open interval in I
containing t0 such that co(N) C U. If t E N, then cot ~ cofo * cotO)f, where
Dft) f(t') = co(to + t'(t — t0)). Therefore, for t E N
&(t) = (co» = (co(to) * cof(b» E (co(to),I7> = (co',[7>
and so co is continuous. Furthermore, pco(t) = cof(l) = co(t). Hence co is a lift
of co beginning at cb(O) = x0 and ending at cd(l) = (co).
If [co] E H, then co — coo and (co) = x'o. Therefore the lift co of co con-
structed above is a closed path in X at x0, proving that И C р#тг(Х,Жо). On
the other hand, if co' is a closed path in X at x0 and pco' = co, let co be
the path in X constructed above. Since co is a lift of co beginning at x0,
it follows from the unique path lifting of p that co = co'. Therefore co(I) =
co'(l) = Xq. Since co(l) = (co), co — coo, showing that p#?r(X,Xo) С H. a
A space X is semilocally I-connected (defined in Sec. 2.4) if and only if
there is an open covering Ql of X such that w(Qt,x0) = 0. Hence we have the
following result.
14	corollary A connected locally path-connected space X has a simply
connected covering space if and only if X is semilocally 1-connected. B
From corollaries 14 and 6 and theorem 2 we obtain the next result.
15	corollary Any universal covering space of a connected locally path-
connected semilocally 1-connected space is simply connected. B
84
COVERING SPACES AND FIBRATIONS CHAP. 2
Not every connected locally path-connected space has a universal cover-
ing space. We give two examples.
1 ® An infinite product of 1-spheres has no universal covering space.
17 Let X be the subspace of R2 equal to the union of the circumferences of
circles Cn, with n > 1, where Cn has center at (1/n, 0) and radius 1/n. Then
X is connected and locally path connected but has no universal covering
space.
It is possible for a connected locally path-connected space to have a uni-
versal covering space that is not simply connected. We present an example.
18 example Let Yj be the cone with base X equal to the space of example 17
[Yi can be visualized as the set of line segments in R3 joining the points of X to
the point (0,0,1)] and let //j be the point at which all the circles of X are tan-
gent. Let (Y2,y2) be another copy of (Y|,t/|). Let Z = Y| V Y2. Then Z
is connected and locally path connected but not simply connected (cf. exercise
LG.7, a closed path oscillating back and forth from Yi to Y2 around the
decreasing circles Cn is not null homotopic). However, Yi and Y2 are each closed
contractible subsets of Z. By the lifting theorem, each of them can be lifted
to any covering space of Z, so that y\ is lifted arbitrarily and y2 is lifted arbitrar-
ily. Therefore any covering projection with base Z has a section. It follows
that any connected covering space of Z is homeomorphic to Z.
In the category of fibrations with unique path lifting over a fixed path-
connected base space (and with path-connected total spaces) there is always
a universal object (that is, an object which has morphisms to any other object
in the category). We sketch a proof of this fact. Let X be a path-connected
space and let X(X) be the collection of topological spaces whose underlying
sets are cartesian products of X and the set of right cosets of some subgroup
of the fundamental group of X. It follows from theorem 2.3.9 that any fibra-
tion whose base space is X and total space is path connected is equivalent to
a fibration X X, where X g 'X(X). Since ?C(X) is a set, those fibrations
X —> X with unique path lifting, where X is a path-connected space in ft(X),
constitute a set. We may form the fibered product of this set (as in Sec. 2.2).
This fibered product is then the desired universal fibration with unique path
lifting.
If X is a connected locally path-connected space, it follows from theorem 13
that for any open covering of X there is a path-connected covering space
of X whose fundamental group is isomorphic to 7t(QL,Xo). This implies that if X
is a universal object in the category of path-connected fibrations over X with
unique path lifting, then л/ХДо) is isomorphic to a subgroup of Glk?J w(Ql,xo).
In particular, if w(Ql,x0) = 0, then X has a simply connected fibration with
unique path lifting that is a universal object in the category. Thus the spaces
in examples 16 and 17 both have universal fibrations with unique path lifting
that are simply connected. The space Z of example 18 is its own universal
fibration with unique path lifting.
SEC. 6 COVERING TRANSFORMATIONS
85
© COVERING TRANSFORMATIONS
In this section we consider a problem inverse to the one of the last section, in
which we constructed covering projections with given base space; we ask for
covering projections with given covering space. On any regular covering
space we prove that there is a group of covering transformations. The cover-
ing projection is then equivalent to the projection of the covering space onto
the space of orbits of the group of covering transformations.
Let p: X —> X be a fibration with unique path lifting. It is clear that
there is a group of self-equivalences of this fibration (a self-equivalence is a
homeomorphism f: X —> X such that p ° f = p). We denote this group by
G(X | X). In case p: X X is a covering projection, G(X | X) is also called the
group of covering transformations of p. In general, there is a close analogy of
G(X | X) with the group of automorphisms of an extension field leaving a
subfield pointwise fixed.
If X is path connected, it follows from lemma 2.2.4 that two self-
equivalences of p: X —> X that agree at one point are identical. Hence we
have the following lemma.
1	lemma Let p: X —> X be a fibration with unique path lifting. If X is
path connected and x0 € X, then the function f-^ flxo) is an injection of
G(X | X) into the fiber of p over p(xf). ®
Theorem 2.3.9 established a bijection from the set of right cosets of
р#тг(Х,Хо) in тг(Х,р(хо)) to the fiber of p over p(xo)- Combining the inverse of
this bijection with the function of lemma 1 yields an injection i^from G(X | X)
to the set of right cosets of p#w(X,x0) in w(X,p(xo)). f is defined explicitly as
follows. For any f E G(X | X) let & be a path in X from x0 to /(x0). Then p ° & is
a closed path in X at p(xf), and the right coset (p*n-(X,x0)) [p ° «] is independ-
ent of the choice of &. The function f assigns to f this right coset.
Given x0 E X, let X(p#w(X,Xo)) be the normalizer of р#7т(Х,х0) in w(X,p(x0)).
Thus X(p#w(X,Xo)) is the subgroup of тт(Х,р(хо)) consisting of elements
[w] E тг(Х,р(хо)) such that р#тг(Х,х0) is invariant under conjugation, by [и].
Х(р#тт(ХДо)) is the largest subgroup of тт-(Х,р(хо)) containing p#vr(X,x0) as a
nonnal subgroup.
2	theorem Let p: X X be a fibration with unique path lifting. Let X
be path connected and let xq E X. Then f is a monomorphism of G(X | X) to
the quotient group Nfp^^^^/p^ir^xo). If X is also locally path connected,
f is an isomorphism.
eroof We already know that f is an injection. We show that f is a function
from G(X | X) to the set of right cosets of p#v(X,x0) by elements of Х(р#7т(Х,х0)).
86
COVERING SPACES AND FIBRATIONS CHAP. 2
If co is a path in X from x0 to fyxo), there is a commutative square
^(X,*o)	^X,f(x0))
^X,p(x0))	7г(Х,р(х0))
Since/: (X,x0) —> (X,/(x0)) is a homeomorphism,
У>(Х,«о) =
and since p#f# = p#,
hipo^^p^rfXflxffi ~ h[i>‘c\}p^-fdXr(XyX(^ = h[p.,igfp#'n'(X,Xo)
= p^M4T(X,f(x0)) = р#тт(Х,х0)
Hence [p ° co] € N(p#4r(X,Xo)). Because fy/) is equal to the right coset
(p#w(X,Xo)) [p ° G>], / is an injection of G(X | X) into the set of right cosets of
p#w(X,Xo) by elements of N(p#Tr{X,x0)).
We now verify that / is an homomorphism. If fy, fy £ G(X | X) let co । and
co2 be paths in X from x0 to fy(x0) and fyfyo), respectively. Then fy ° co2
is a path fromfy(x0) to fyfy(xo), and co3 * (fy ° co2) is a path from x0 tofyfy(x0).
Therefore fyfyfy) is the right coset
(p#fyX,x0))[(p ° cox) * (p ° fy ° «2)] = (p#<X,x0))[p 0 wi] * [p ° «2]
and this equals fyfy)fyfy).
Finally, we show that if X is locally path connected, / is an epimorphism
to the set of right cosets of р#7т(Х,х0) in N(pv7T(X,%o)). Assume that [co] £
w(X,p(xo)) belongs to 7V(p#fyX,Xo)). Let co be a lifting of co ending at x0 and
let x — co(O). Then
p#fyX,x0) = 7гГЕ5](р#тт(Х,Хо)) = pfyfycjfyXfyo)) = p#fyX,x)
Because X is connected and locally path connected, the lifting theorem
implies the existence of maps f: (X,x'o) —> (X,x) and g: (X,x) —> (X,x0) such
that p ° f = p and p ° g = p. From the unique-lifting property (lemma 2.2.4),
it follows that f ° g = lx and g ° f — fy. Therefore f £ G(X | X) and fy/)
equals the right coset (p#7z(X,Xo))[co]-1. 
Combining theorem 2 with theorem 2.3.12, we have the following
corollary.
3	corollary Let p: X —> X be a fibration with unique path lifting. If X
is connected and locally path connected and x0 £ X, then p is regular if and
only if G(X | X) is transitive on each fiber of p, in which case
fy G(X | X) 77(X,p(xo))/p#77(X,xo) 
SEC. 6 COVERING TRANSFORMATIONS
87
If X is simply connected, any fibration p: X Xis regular, and we also
have the next result.
4	corollary Let p: X X be a fibration with unique path lifting,
where X is simply connected, locally path connected, and nonempty. Then
the group of self-equivalences of p is isomorphic to the fundamental group of
X. 
If p: X X is a regular covering projection and X is connected and
locally path connected, then X is homeomorphic to the space of orbits of
G(X | X) (an orbit of a group of transformations G acting on a set S is
an equivalence class of S with respect to the equivalence relation .s'i ~ s2 if
there is g € G such that g.si = sf). We are interested in the converse problem
—that is, in knowing what conditions on a group G of homeomorphisms of a
topological space Y will ensure that the projection of Y onto the space of
orbits Y/G is a regular covering projection whose group of covering trans-
formations is equal to G.
A group G of homeomorphisms of a topological space Y is said to be dis-
continuous if the orbits of G in Y are discrete subsets of Y. G is properly
discontinuous if for у £ Y there is an open neighborhood U of у in Y such
that if g, g' E G and gU meets g'U, then g = g'. G acts without fixed points
if the only element of G having fixed points is the identity element. The
following are clear.
5	A properly discontinuous group of homeomorphisms is discontinuous
and acts without fixed points. 
6	A finite group of homeomorphisms acting without fixed points on a
Hausdorff space is properly discontinuous. 
If G is the group of covering transformations of a covering projection,
then a simple verification shows that G is properly discontinuous. We now
show that any properly discontinuous group of homeomorphisms defines a
covering projection.
7	theorem Let G be a properly discontinuous group of homeomorphisms
of a space Y. Then the projection of Y to the orbit space Y/G is a covering
projection. If Y is connected, this covering projection is regular and G is its
group of covering transformations.
proof Let p: Y Y/G be the projection. Then p is continuous. It is
an open map, for if U is an open set in Y, then p-1(p(G)) = U {gG | g £ G)
is open in Y, and therefore pU is open in Y/G. Let U be an open subset of Y
such that whenever gU meets g'U, then g = g'. We show that p(U) is evenly
covered by p. The hypothesis on U ensures that (gU | g C G} is a disjoint col-
lection of open sets whose union is p-1(p(G)). It suffices to prove that p | gU
is a bijection from gU to p(G). If у E U, then p(gy) = p(y), so p(gU') — pfU).
If P(g!/r) = p(g?/2), with yb y2 E U, there is g' E G such that g(/i = g'gi/2-
88
COVERING SPACES AND FIBRATIONS CHAP. 2
Therefore gU meets g'gU, and g = g'g. Hence g7 = ly and gf/i = gy2. We
have proved that p is a homeomorphism of gU onto p(U). Since G is properly
discontinuous, the sets p(C7) evenly covered by p constitute an open covering
of Y/G.
Because p(gy) — p(y), we see that G is contained in the group of
covering transformations of p. Since G is transitive on the fibers of p, it
follows from theorem 2.2.2 that if Y is connected, G equals the group of cov-
ering transformations. Since the group of covering transformations is transi-
tive on each fiber, the covering projection is regular. 
Я corollary Let G be a properly discontinuous group of homeomorphisms
of a simply connected space Y. Then the fundamental group of the orbit
space Y/G is isomorphic to G.
proof By theorem 7, G is the group of covering transformations of the reg-
ular covering projection p: Y--> Y/G. By theorem 2, f is a monomorphism of
G into the fundamental group of Y/G. Because G is transitive on the fibers of
p, f is an isomorphism. 
f> example Let S3 = {(zo,zi) 6 C2 | |zo|2 + |zi|2 = 1} and let p and q be
relatively prime integers. Define h: S3 —-> S3 by
h(zo,zi) = (e2^/J’zo,e2';r9i/p2i)
Then h is a homeomorphism of S3 with period p (that is, h? = 1), and Zp acts
on S3 by
n(zo,zf) = hn(zo,zf)
where n denotes the residue class of the integer n modulo p. In this way Zp
acts without fixed points on S3. The orbit space of this action of Zp on S3 is
called a lens space and is denoted by L(p,q). By statement 6 and corollary 8,
the fundamental group of L(p,q) is isomorphic to Zp.
1® example Let S2?t+1 = {(2o,zi, . . . ,zn) C C’t+1 | S |z.j|2 = 1} and let
qi,    T[n be integers relatively prime to p. Define 7г:	11 —> S2?l+1 by
h(zo,zi, . . . ,zf) — (e2vi/Pz^,e2vlq‘/pz^ . . . te2iriqn/pz^
Then, as in example 9, h determines an action of Zp on S2n+1 without fixed
points; the orbit space is called a generalized lens space and is denoted by
Lf),q\, . . . ,q?t). Its fundamental group is isomorphic to Zp.
It is possible to use theorem 7 to show that the projection Y—> Y/G is
a regular fibration with unique path lifting even when it may not be a cover-
ing projection. Note that if G acts on Y without fixed points, so does any sub-
group of G, and if G' is a normal subgroup of Y, then G/ G' acts without fixed
points on Y/G'.
11 theorem Let G be a group of homeomorphisms acting without fixed
points on a path-connected space Y and assume that there is a decreasing
sequence of subgroups
FIBER BUNDLES
89
G - Go D Gi D • • • □ G„ □ Gn+1 D • • •
such that
i (a) AG» = {Ip}
!	(b) Gre+i is a normal subgroup of G„ for n > О
(	(c) Gn/G„+3 is a properly discontinuous group of homeomorphisms on
। Y/Gn+i and the projection Y Y/G„ is a closed map for n > О
(d) Any orbit of Y under Gn for n > О is compact
! Then the projection p: Y Y/G is a regular fibration with unique path
j lifting whose group of self-equivalences is G.
; proof Since Y/G„ = (Y/G„+i)/(Gn/G„+i), it follows from (c) and theorem
7 that the projection
j	pn+i'- Y/Gn+±—» Y/Gn
! is a regular covering projection for n > 0. Let
i	Y = {(У») £ x (Y/G„) I pn+i(yn+i) = yn for n > 0}
and define p: Y Y/G by p((i/H)) = yo- It is easy to verify that p is a fibra-
' tion with unique path lifting (it is the fibered product of the maps {;>/})
i For n > 0 there is a continuous closed projection map <pn: Y —-> Y/Gn
j such that p,i+i ° <pn+i = <pn- Therefore there is a continuous closed map
, <p: Y —> Y defined by <p(y) = (<p,i(q)) and such that p ° <p — p. To prove that <p
I is a homeomorphism, it suffices to show that it is a bijection. If cp(y) = <p(y')>
 then for n > 0 there is gn £ Gn such that у = gny'. Then gny' — gmy' for all
' in and n, and because G acts without fixed points, gm = gn for all m and n.
j Therefore g„ C Gm for all in, and by (a), g„ = Ip. It follows that у = у', and
i hence that <p is injective.
 И (Уп) C Y, then <pm 1yn is an orbit of Y under Gn, By (d), cpn 1y,l is
compact. Since
<Pn ^y?i = <Рп+1Рп+1Уп —> фи+ГУп+1
the collection {<ря-1Уи} consists of compact sets having the finite-intersection
I property. Therefore Pl	0. If у £ П	then <p(y) = (y„),
1 showing that <p is surjective.
We have shown that <p: Y—Y is a homeomorphism. Therefore p: Y-^Y/G
' is a fibration with unique path lifting. Since each element of G is a self-
* equivalence of p, the group of self-equivalences of p is transitive on each
fiber. By corollary 3, p is a regular fibration and G is the group of self-
; equivalences of p. 
7 FIBER BUNDLES
A covering space is locally the product of its base space and a discrete space.
This is generalized by the concept of fiber bundle, defined in this section,
because the total space of a fiber bundle is locally the product of its base
90	COVERING SPACES AND FIBRATIONS CHAP. 2
space and its fiber. The main result is that the bundle projection of a fiber
bundle is a fibration.1
A fiber bundle £ = (E,B,F,p) consists of a total space E, a base space B,
a fiber F, and a bundle projection p: E ^ В such that there exists an open (
covering {17} of В and, for each U E {17}, a homeomorphism cpp: U X F i
p-1(U) such that the composite	I
UXF p-\U) и	|
is the projection to the first factor. Thus the bundle projection p: E ^> В and [
the projection В X F В are locally equivalent. The fiber over b £ В is de- I
fined to equal p-1(b), and we note that F is homeomorphic to p~4b) for every I
Usually there is also given a structure group G for the bundle consisting ।
of homeomorphisms of F, and we define this concept next.
Let G be a group of homeomorphisms of F. Given a space F and a col- j
lection Ф = {<p} of homeomorphisms <p: F —> F, define <pg: F F for <p E Ф •
and g E Gby <pg(i/) = <p(gy) for у E F. The collection Ф is called a G struo j
ture on F if	j
(a)	Given t/ C Ф and g EG, then <pg E Ф	1
(b)	Given <pi, <g2 E Ф, there is g E G such that <pi = <p2g	I
Condition (a) implies that G acts on the right on Ф, and condition (b) implies
that this action of G is transitive on Ф. A fiber bundle (E,B,F,p) is said to have
structure group G if each fiber р~ЦЬ) has a G structure Ф(Ь) such that there
exists an open covering {U} of В and, for each U E {U}, a homeomorphism }
Фи: U X F —> p-1(U) such that for b E U, the map F —> // ’(b) sending x to }
<pu(b,x) is in Ф(Ь). It is clear that a given fiber bundle can always be given the |
structure of a fiber bundle with structure group the group of all homeomor- (
phisms of F. It is also clear that a given fiber bundle can sometimes be given ?
the structure of a fiber bundle with two different structure groups of homeo- i
morphisms of F.	[
An n-plane bundle, or real vector bundle, is a fiber bundle whose fiber is |
RM and whose structure group is the general linear group GE(Rn), which con- j
sists of all linear automorphisms of R”. A complex n-plane bundle, or complex (
vector bundle, is a fiber bundle whose fiber is C" and whose structure group I,
is GL(Cn).
We give some examples.
1	For spaces В and F the product bundle is the fiber bundle (В X F, B, F, p),
where p: В X F —» В is projection to the first factor (it has the trivial group
as structure group).	j
2	Given that p: X X is a covering projection and X is a connected
space, if xo E X, then (X,X,p~ r(xo),pj. is a fiber bundle (and if X is path
connected, it can be given the structure of a fiber bundle with structure group
1 For the general theory of fiber bundles see N. E. Steenrod, The Topology of Fibre Bundles,
Princeton University Press, Princeton, N.J., 1951.
sEC. 7 FIBER BUNDLES	QI
tt(X,xo), where w(X,xo) acts on p-1(xo) by [co] x = xfoj]1, with the right-hand
side as in the proof of theorem 2.3.9).
3 Given that M is a differentiable n-manifold and T(M) is the set of all tan-
gent vectors to M, there is a fiber bundle (T(M),M,R”,p), where p: T(M) M
assigns to each tangent vector its origin. This is called the tangent bundle
and is denoted by r(M). Because it can be given the structure group GL(R«),
it is an n-plane bundle, and if M is a complex manifold of complex dimension
m, then t(M ) is a complex ni-plane bundle.
4 Given that H is a closed subgroup of a Lie group G and that G/H is the
quotient space of left cosets and p; G G/H the projection, then (G,G/H,H,p)
is a fiber bundle (having structure group H acting on itself by left translation).
5 Represent Sre as the union of two closed hemispheres E’L and EJ whose
intersection is an equatorial sphere S"1 and let G be a group of homeo-
morphisms of a space F. Given a map <p: S''1 --> G, let Ev be the space
obtained from (E’L X F)v {E’f X F) by identifying (x,t/) E E’l X F with
(х,<р(х)г/) E f " X F for x E S'"1 and у E F. These identifications are compat-
ible with the projections EL X F-^ E’L and E? X F —> Ef. Therefore there
is a map pv: E,, —> S" such that each of the composites
Е» X F-> f,,A S» and	f« x F-^ fiA S«
is projection to the first factor. Then (E^SnJqp,/) is a fiber bundle (having
structure group G) which is said to be defined by the characteristic map <p.
0 Let ВЯ(С) be the n-dimensional complex projective space coordinatized
by homogeneous coordinates. If z0,	. . . , zn E C are not all zero, let
[zb^t, • • • ,%n] E Fn(C) be that point of Fn(C) having homogeneous coordi-
nates Zo, z±, ... , Zn. Regard S2n+1 as the set {(zo,zi, . . . ,z„) E C’i+1 |
S Ы2 = 1} and define p: S2«+1F„(C) by p(z0^i, . . . ,zn) = [zo,zr, . . . ,z„].
If Ui C F,j(C) is the subset of points having a nonzero ith homogeneous
coordinate, it is easy to see that // 1(U;) is homeomorphic to U.; X S1. There-
fore there is a fiber bundle (S2?t+1,F?!(C),S1,p) (having structure group S1 acting
on itself by left translation), and this is called the Hopf bundle.
7	If Q is the division ring of quaternions, there is a similar map
p- S4?t+3	F„(Q) and a quatemionic Hopf bundle (S4n+3,Fn(Q),S3,p) (having
structure group S3 acting on itself by left translation).
The structure group will not be important for our purposes. Thus we
define an n-sphere bundle to be a fiber bundle whose fiber is S” [usually it is
also required that it have as structure group the orthogonal group 0(n -|- 1) of
all isometries in GE(Rn+1)]. If £ is an n-sphere bundle, we shall denote its
total space by Ef. The mapping cylinder of the bundle projection Ef ^ В is
the total space Ej of a fiber bundle (Ej,B,En+1,pj), where pg Ef-^ В is the re-
traction of the mapping cylinder to В (and pc | Eg. Er ^ В is the original
bundle projection).
92	COVERING SPACES AND FIBRATIONS CHAP. %
If £ = (E,B,R’t+1,p) is an (n + l)-plane bundle having structure group
0(n + 1), it is possible to introduce a norm in each fiber p-1(h). The subsets
E' С E of all elements in E having unit norm is the total space of an n-sphere
bundle (E', B, Sn, p | E') called the unit n-sphere bundle of £. If the base space
В of an (n + l)-plane bundle is a paracompact Hausdorff space, the bundle
can always be given 0(n 4- 1) as structure group, hi particular, there is a unit
tangent bundle of a paracompact differentiable manifold.
Two fiber bundles (E!,B,F,pi) and (E2,B,F,p2) with the same fiber and
same base are said to be equivalent if there is a homeomorphism h: Ei E2
such that p2 0 h = p±. If they both have structure group G, they are equiva-
lent over G if there is a homeomorphism h as above, with the additional
property that if <p E Ф1(Ь), then h 0 <p £ Ф2(Ь) for b £ B. A fiber bundle is
said to be trivial if it is equivalent to the product bundle of example 1 (or,
equivalently, if it can be given the trivial group as structure group).
In view of example 2, fiber bundles are related to covering spaces in
much the same way that fibrations are related to fibrations with unique path
lifting. The rest of this section is devoted to a proof of the fact that in a fiba-
bundle (E,B,F,p) whose base space В is a paracompact Hausdorff space the
map p is a fibration.
A map p: E В is called a local fibration if there is an open covering
{17} of В such that p |	р'-Ци') —> 17 is a fibration for every U £ {17}.
It is clear that a fibration is a local fibration1 and that any bundle projection
is a local fibration.
Given a map p: E B, we define a subspace В С E X B1 by
В = {(e,co) € E X Bi I w(0) = p(e)}
There is a map p: E1 —> В defined by p(c7) = (co(O), p ° &) for <0: I —-> E,
A lifting function for p is a map
Л: В Ei
which is a right inverse of p. Thus a lifting function assigns to each point
e E E and path co in В starting at p(e) a path X(e,co) in £ starting at e that is a
lift of co. The relation between lifting functions and fibrations is contained in
the following theorem.
8	theorem A map p: E ^ B is a fibration if and only if there exists d
lifting function for p.
proof The proof involves repeated use of theorem 2.8 in the Introduction.
If p is a fibration, let f': В --> E and F: В X I В be defined by /'(e,co) = e
1 Our proof of the converse for paracompact Hausdorff spaces В can be found in W. Hurewicz,
On the concept of fibre space, Proceedings of the National Academy of Sciences, U.S.A., vol
41, pp. 956-961 (1955). Another proof can be found in W. Huebsch, On the covering homo-
topy theorem, Annals of Mathematics, vol. 61, pp. 555-563 (1955). Generalizations and
related questions are treated in A. Dold, Partitions of unity in the theory of fibrations, Annals
of Mathematics, vol. 78, pp. 223-255 (1963).
«SEC* 7 FIBER BUNDLES	03
an<l F((e,co), t) = w(0* Then
F((e,w), 0) = w(0) = p(e) = (p° f')(e,u)
gy the homotopy lifting property of p, there is a map F': В X I —> E such
that F'((e,co), 0) = /'(e,co) = e and p ° F' = F. F' defines a lifting function X
for p by X(e,w)(t) = F'((e,w), t).
Conversely, if X is a lifting function for p, let/': X-^E and F: X X l—> В
be such that F(x,0) = pf'(x)- Let g: X —> B1 be defined by g(x)(t) = F(x,t).
There is a map F': X X I -* E such that F'(x,t) = X(/'(x),g(x))(t). Because
F'(x,0) = f(x) and p ° F' = F, p has the homotopy lifting property. 
Let p: E ^ В and let W be a subset of B1. Let W be defined by
W = {(e,co,s) £ E X W X 11 w(s) = p(e)}
An extended lifting function over W is a map
Л: W-> E1
such that p(A(e,co,s)(t)) = co(t) and A(e,oj,.s)(.s) = e. Thus an extended lifting
function is a function which lifts paths to paths that pass through a given
point of E at a given parameter value. It is reasonable to expect the following
relation between the existence of lifting functions and extended lifting
functions.
9	lemma A map p: E —> В has a lifting function if and only if there is
an extended lifting function over B1.
proof If A is an extended lifting function over B1, a lifting function X for p is
defined by X(e,co) = A(e,cj,0).
To prove the converse, given a path co in B, let co5 and co® be the paths in
В defined by
,л f <4s — t) 0 < t < s
“*> = L(0)	„<,<1
^s(t} _	+ f) 0 < t < 1 - s
W - U(l) 1 - s < t < 1
The maps (co,s) —> cos and (co,s) —-> co® are continuous maps В1 X I —» B1.
Given a lifting function X: Ё —> E1 for p, we define an extended lifting func-
tion Л over B1 by
л (P ,, eV A - RtGUsXs - t) o < t < s
A(e,^s)(t) - [X(e^t	s < t < 1 
The main step in proving that a local fibration is a fibration is the fitting
together of extended lifting functions over various open subsets of B1. For this
we need an additional concept. A covering {W} of a space X is said to be
numerable if it is locally finite and if for each W there is a function
fw: X [0,1] such that W = {x £ X | fw(x) =£ 0}.
94	COVERING SPACES AND FIBRATIONS CHAP. 2
10	lemma Let p: E —» В be a map. If there is a numerable covering {
of B1 such that for each j there is an extended lifting function over Wj, then:
there is a lifting function for p.
proof Let the indexing set be J = {j} and for each / Ict f;: В1 I be a map
such that Wj = {co £ B11	7^ 0}. For any subset a C J let Wa = Ujea Wj
and define fa: B1 —> R by
f<№) = 'Zj^fjfS)
(this is a finite sum and is continuous because {W/} is locally finite). Then
/„(co) > 0 for co £ B1 and
Wa= {со£В'|Дсо)^0}
We define Ba = {(e,co) £ В | co £ W„).
Consider the set of pairs (a,X„), where a C J and X„: Ba E1 is a lifting
function over B„ [that is, X„(e,co)(O) = e and pX„(e,co)(t) = co(t)]. We define a
partial order < in this set by (а,Л„) < (ДЛ^) if a C /> and X„(e,co) = X₽(e,co)
whenever (e,co) £ Ba and/„(co) = /g(co) [so if (e,co) £ Ba and X„(e,co) 7^ X^(e,co),
then co £ Wj for some /' £ (3 — a].
To prove that every simply ordered subset {ft;,A,,/} has an upper bound,
let /> = U a.j. We shall define \p: Bp —> E1 so that («г,Л„,-) < (ДЛ^) for alii.
Let U be any open subset of Wp meeting only finitely many Wj with / £
say Wj,, . . . , Wjr (Wp can be covered by such sets U). Choose i so that
/1, . . . , jr all belong to «». Then if a.; C afc, fai | U = fak | U. Because
(«iAai) < («fcAac), it follows that XQj(e,co) = Xn7(e,co) for (e,co) £ Bai, with
co £ U. Therefore there exists a map \p: Bp E1 such that X^(e,co) = X„{(e,co)
for a, sufficiently large. We now show that (л;,Х„;) < (/3,Xp). If (e,co) £ Baj
and X„,(e,co) 7^ X^(e,co), there exists n/; such that (a;,Xaj) < (ofoXJ and
XKj(e,co) 7^= \.,(e,co). This implies co £ Wj for some / £ «л- — Therefore
co £ Wj for some j £ [3 — a», hence (ft;,\,,-) <
By Zorn’s lemma, there is a maximal element (ft,X„). To complete the
proof we need only show that a = J. If a 7^= J, let jo £ J — a and let
/3 = a U {/0}. Define g: Wp -> R by g(co) = fa(v)/fp(u). Then 0 < g(co) < 1,
g(co) 7^ 0 <=4> co £ Wa, and g(co) 7^ 1 co £ W;o. Define /t: Bj0 —> E by
_ f^(e,co)(g(co))	g(co) 7^ 0
~ [e	g(co) = 0
Then p is continuous. Let Л be an extended lifting function over Wj„
define \p: Bp --> E1 by
X (e <	- f A“(e’w)(f)	0 < t < g(co), g(co) 7^ 0
X } ~ lA(M(e,co), co, g(co))(t) g(co) < t < 1, g(co) 1
Then hp is a well-defined lifting function over Wp. Moreover, for (e,co) £ Ba,
if XQ(e,co) X^(e,co), then g(co) 7^ 1 and co £ W,o. Since jo £ /3 — a, this
means that (а,Ла) < (J3,hp), contradicting the maximality of (а,Ла). 
and
95
' rr 1 FIBER BUNDLES
SLU* 
In case p has unique path lifting, lemma 10 would hold for any open
Covering {W)} of B1 such that there is a lifting function over Wj for each j
(because the uniqueness of lifted paths enables the extended liftings to be
amalgamated to a lifting for p). This was used in the proof of the theorem
that a covering projection is a fibration (theorem 2.2.3), which was valid with-
out any assumption on the base space.
11	lemma Given a map p: E —» В and subsets и1г . .
that there is an extended lifting function over Ufi, U21,  
the subset of B' defined by
Uh of В such
Ukf let W be
W = (w € B'l	—к-Я) C Ut for i = 1, . . . , к
I	\L к к J/
Then there is an extended lifting function over W.
proof Let A; be an extended lifting function over Uf for i = 1, . . . , k.
Given a path w £ W, let co.; be the path equal to w on [(/ — l)/k, i/k] and
constant on [0, (i — Л )//<| and on [i/k, 1]. Given (e,co,s) £ W such that
(n. — l)/k < s < n/k, define ei (/ E for i = 1, . . . , к inductively so that
(n — 1 \
к )
Cn—1 — A?;.(f,C0?1,S
An extended lifting function Л over W is defined by
(7 — 1 \
e;_l,C0.;,——----)(t)
к /
i—^<t< 4~,i = 1, ... ,k 
к	к
We are now ready for the main result on the passage from a local fibra-
tion to a fibration.
12	theorem Given a map p: E —> В and a numerable covering s’I of В such
that for U £ p I p-1(f7): p;1 (t.7) —> U is a fibration, then p is a fibration.
proof Let QJ = { 17/} and for к > 1, given a set of indices /1, . . . , y:, let
Wwz... ik be the subset of Br defined by
W.hj2... jk = L € В11 w (ГЦЛ 41) C f • • • > k]
v	\L К /С J/	J
It is then clear that the collection {Wjj.,...} (with к varying) is an open
covering of B1, and by lemma 11, each set Wjrj2... jk has an extended lifting
function. For к fixed the collection {Wj,j2...} is locally finite. In fact, if
96	COVERING SPACES AND FIBRATIONS CHAP. 2
co £ B1, for each i = 1, . . . , к there is a neighborhood of co([(i — !)//<, i/k]j
meeting only finitely many Uj. Then П1<$£й <[(i — l)/k, i/k], V,-> is a neigh,
borhood of co meeting only finitely many { W)j,_,.
For each / let fi: В —> I be a continuous map such that fj(b) 7^ 0 if and
only if b £ Uj. Define fj,... jk: B1 —> 1 by
fh = inflow I < t <y> i = 1, • • • >
к	I К	К	J
Then fjA... jk(ff) 0 if and only if co £ Wj±... jk.
Unfortunately, the collection {Wj^... jk} (all k) is not locally finite,
otherwise the proof would be complete by lemma 10. This difficulty is circum-
vented by modifying the sets Wyij!!... jk. Since for fixed m the collection
{ W)j... with к < m is locally finite, the sum of the functions fi1... jk with
к < m is a continuous real-valued function g.m on B1. Define
f'h mf(sup(0,^ • • -jm - mgm), 1)
Then f'}1 ,,.]m:B'^I and we define W/x... }m = {co £ B11 ff .	=£ 0}.
Clearly,	C Wh... jm-, therefore there is an extended lifting function
over Wf ... jm. To complete the proof, it follows from lemma 10 that we need
only verify that { WJj ...jk} (with к varying) is a locally finite covering of Bl.
For co С B1, let m be the smallest integer such that	7^ 0 for
some /ь • • • , jm. Then gw(co) = 0 and/J,.. ,;m(co) = Д. ..;„>) 7^= 0. There-
fore co £ Wj1... jm, proving that { Wf is a covering of B1. To show that it
is locally finite, assume N chosen so that N > m and fj1...	> 1/N. Then
gv(co) > 1/N and Ag/V(oo) > 1. Hence Ag.y(oj') > 1 for all co' in some neighbor-
hood V of co. Therefore all functions /h ... jk with к > N vanish on V. But this
means that the corresponding set Wj1 jk is disjoint from V. Since the collec-
tion { W)j...;, } with к </ N is locally finite, the collection { Wh ...} (all k) is
locally finite. 
The fact that any open covering of a paracompact Hausdorff space has a
numerable refinement, leads to our next theorem.
13	theorem If В is a paracompact Hausdorff space, a map p: E В is a
fibration if and only if it is a local fibration. 
A bundle projection is a local fibration. Therefore, we have the following
corollary.
14	corollary If (E,B,F,p) is a fiber bundle with base space В paracom-
pact and Hausdorff, then p is a fibration. 
Ж FIBRATIONS
This section contains a general discussion of fibrations. We establish a relation
between cofibrations and fibrations which allows the construction of fibrations
from cofibrations by means of function spaces. We also prove that every map
is equivalent, up to homotopy, to a map that is a fibration (this dualizes a
SEC. 8 FIBRATIONS
97
similar result concerning cofibrations). The section contains definitions of the
concepts of fiber homotopy type and induced fibration and a proof of the
result that homotopic maps induce fiber-homotopy-equivalent fibrations.
We begin with an analogue of theorem 2.7.8 for cofibrations. Given a
map /: X' —-> X, let X be the quotient space of the sum (X' X I) v (X X 0),
obtained by identifying (x',0) £ X' X I with (J(x'),0) E X X 0 for all x' £ X'.
We use [x',t] and [x,0] to denote the points of X corresponding to (x',t) EX' X I
and (x,0) £ X X 0, respectively. Then [x',0] = [/(x'),0]. There is a map
i: X-> X X I
defined by
i[x',t] = (f(x'),t) x' E X', t E I
i[x,o] = (x,o) x e x
A retracting function for f is a map
p: X X I X
which is a left inverse of i. In case fis an inclusion map, so is i, and a retracting
function for / is a retraction of X X I to the subspace X' X I U X X 0.
I theorem A map f. X' —» X is a cofibration if and only if there exists a
retracting function for f.
proof If f is a cofibration, let g: X —> X and G: X' X I —> X be the maps
defined by g(x) = [x,0] and G(x',t) = [x',t]. Because
G(x',0) = [x',0] = [f (x'),0] = gf'(x)
it follows from the fact that/is a cofibration that there exists a map p: X X I—> X
such that p(x,0) = g(x) and p(/(x'),t) = G(x',t). Then p is a retracting function
forf.
Conversely, given maps g: X Y and G: X' X I —> Y such that
G(x',0) = gf(x') for x' E X', define
G: X ^ Y
by G[x',t] = G(x',t) and G[x,0] = g(x). If p: X X I —» X is a retracting func-
tion for f, the map F = G ° p: X X I —* Y has the properties F(x,0) = g(x)
and F(/(x'),t) = G(x',t), showing that fis a cofibration. 
This leads to the following construction of fibrations from cofibrations.
2 theorem Let f: X’ X be a cofibration, where X' and X are locally
compact Hausdorff spaces, and let Y be any space. Then the map p: YY —> Yr
defined by p(g) = g ° f is a fibration.
proof Let p: X X I —> X be a retracting function for f (which exists by
theorem 1). Then p defines a map
p': YY ^ Yvx'
such that p'(g) = g ° p for g: X —> Y. Because X' and X are locally compact
98
COVERING SPACES AND FIBRATIONS CHAP. 2
Hausdorff spaces, so is X, and by theorem 2.9 in the introduction, Y-Yx/ ~ (Yxy
and
Y* ~ {(g,G) e Yx x (Y^y | g ° / = G(0)}
Therefore p' corresponds to a lifting function for p: YxYx', and by
theorem 2.7.8, p is a fibration. 
3 corollary For any space Y let p: Y1 —> Y x Y be the map p(ui) =
(oj(()),oj(1)) for co: I —> Y. Then p is a fibration.
proof Because I X I U I X 0 is a retract of I X I, the inclusion map I С I
is a cofibration [equivalently, the pair (I,I) has the homotopy extension prop-
erty with respect to any space]. The result follows from theorem 2 and the
observation that Y1 is homeomorphic to Y X Y under the map g —> (g(O),g(l))
for g: t Y. 
Let f:B'—>B and p: E В be maps and let E' be the subset of В' X E
defined by
E'= {(b',e) £ В'x E \ f (b'j = p(e)}
E'is called the fibered product of B' and E (more precisely, the fibered product
of /and p; cf. Sec. 2.2). Note that there are maps p'-. E' —-> B' and/'; E' —> E
defined by p'(b',e) = b’ and f'(b',e) = e. E' and the maps p' and f' are
characterized as the product of f. B' --> B and p: E В in the category
whose objects are continuous maps with range В and whose morphisms are
commutative triangles
Xi Д x2
в
The following properties are easily verified.
4	If p is injective (or surjective), so is p'. 
5	Ifp-. В X F В is the trivial fibration, then p': E' B' is equivalent to
the trivial fibration B' x F —» B'. 
в Ifp is a fibration (with unique path lifting), so is p'. 
7	Ifp is a fibration, f can be lifted to E if and only if p' has a section. 
Note that since the fibered product is symmetric in В and E (or rather,
in f and p), there is a similar set of statements where p and p' are replaced by
/ and/'.
If p: E —> В is a fibration (or covering projection) and/: В' В is a map,
then, by property 6 (or property 5), p': E' —-> B' is a fibration (or covering
projection) and is called the fibration induced from p by / (or covering pro-
jection induced from p by/). If £ = (E,B,F,p) is a fiber bundle and/: В' В
is a map, it follows from property 5 that there is a fiber bundle (E',B',F,p').
This is called the fiber bundle induced from £ by / and is denoted by /* c. In
the case of an inclusion map i: В' С В we use E | B' to denote the fibered
SEC, 8 FIBRATIONS	99
product of B' and E, and if £ is a fiber bundle with base space В, £ | B’ will
denote the fiber bundle with base space B' induced by i. Observe that c | B'
is equivalent to (p-1(B'), B', F, p | p“1(B')).
0 corollary For any space У and point yo € Y, let p: P(Y,y0) —> Y be
fpe map sending each path starting at yo to its endpoint. Then p is a fibra-
tion whose fiber over у о is the loop space S2Y.
proof Let /: Y —» Y X Y be defined by /((/) = (y0,i/) and let p: Y1 У X Y
be the fibration of corollary 3. The fibration induced by f is equivalent to the
map p- P(Y,y0) —> Y, where p(co) = co(l), and /> '(?/o) the fiber over y0, is by
definition, the loop space SY. 
ft follows from corollary 3 that the map p': Y1^ Y defined by ;/(<y) = cj(O)
[or by p'lfi) = co(l)| is a fibration, because it is the composite of fibrations
Y1 Y X Y Y. If p: E В is any map and p': В1 В is the fibration
defined by	= cj(O), then the fibered product of E and B1 is just the
space В used to define the concept of lifting function for p.
These remarks about fibered products and induced fibrations have ana-
logues for cofibrations. Given maps fi: X —> Xi and fi: X —> X2, the cofibered
sum of Xi and X2 is the quotient space X' of Xi v X2 obtained by identifying
jj(x) with fi(x) for all x F X. There are maps ip Xj —> X' and i2: X2 X', and
these characterize X' as the sum of /1 and fi in the category whose objects
are maps with domain X and whose morphisms are commutative triangles. If
fi: X —> Xi is a cofibration, so is i2: X2X!, and this is called the co fibration
induced from fi by fi.
The map ho". X' —> X' X I defined by hofifi = (x',0) is a cofibration for
any space X', and if f: X' —-> X is any map, the cofibered sum of X' X I and X
is just the space X used to define the concept of retracting function for fi
Let p: E —> В be a fibration. Maps fo, fi: X-^E are said to be fiber
homotopic, denoted by fi ~ ft, if there is a homotopy F: fi ~ fi such that
pF(x,t) = pfifx) for x Q X and t F I (in which case p ° fi = p °fi). This is an
equivalence relation in the set of maps X —> E. The equivalence classes are
denoted by [X;E]P, and if fi X —> E, [f]p denotes its fiber homotopy class.
The concept of fiber homotopy is dual to the concept of relative homotopy.
We use induced fibrations to prove that any map is, up to homotopy
equivalence, a fibration. Let fi X —> Y and let p': Y1 —> Y be the fibration de-
fined by p'(co) = w(0). Let p: Pf —» X be the fibration induced from p' by/.
It is called the mapping path fibration of f and is dual to the mapping cylinder.
There is a section s: X Pf of p defined by s(x) = (х,ыдЖ)), where is the
constant path in Y at fix). There is also a map p”: PfY defined by
p"(x,w) = cc(l). We then have the following dual of theorem 1.4.12.
9 theorem Given a map fi X Y, there is a commutative diagram
X Л Pf
f\ / p"
Y
100
COVERING SPACES AND FIBRATIONS CHAP. 2
such that
(a)	lPr~s°p
(b)	p" is a fibration
proof The triangle is commutative by the definition of the maps involved.
(a)	Define F: Pf x I —> Pf by F((x,«), t) — (x,oj|Z), where oj|((/') ~
co((l — t)f). Then F is a fiber homotopy from 1P/ to s ° p.
(b)	Let g: W Pf and G: W X I —> У be such that G(w,O) = p"g(w)
for w £ W. Then there exist maps g': W—> X and g":	Y1 such that
g"(w)(O) = /g'(w) and g(w) = (g'(rv),gf(w)) for w £ W. We define a lifting
G': W X I —> Pf °f G beginning with g by G'(w,t) = (g'(w), g(w,t)), where
g(w,t) £ Yr is defined by
0(... bit’} - fg"(w)(2f/(2 ~ f))	o < 2t' < 2 - t < 2, w e W
+ t-2)	1 < 2 - f < 2f < 2, w(=W
Since p" has the homotopy lifting property, it is a fibration. 
It follows that the fibration p": Pf Y is equivalent (by means of
s: X Pf and p: Pf —> X) in the homotopy category of maps with range Y to
the original map fi X ^> Y. In replacing f by an equivalent fibration, we
replaced X by a space Pf of the same homotopy type, whereas in Sec. 1.4,
when f was replaced by an equivalent cofibration, the space Y was replaced
by a space Zf of the same homotopy type.
Two fibrations p±: Et В and p2: E2 -^ В are said to be fiber homotopy
equivalent (or to have the same fiber homotopy type) if there exist maps
fi Ei E2 and g: E2 —> Ei such that g ° f ~ 1P] and f ° g ~ I/;., (in which
case /and g both preserve fibers in the sense that p2 ° f = p± and pi ° g = p2).
Each of the maps / and g is called a fiber homotopy equivalence. The rest of
this section is concerned with fiber homotopy equivalence.
We begin with the following result concerning liftings of homotopic maps.
IO theorem Let p: E —» В be a fibration and let Fo, Fi: X X I—> E be
maps. Given homotopies H: p ° Fo ~ p ° Fi and G: fi, | X X 0 — Fi | X X 0
such that H(x,O,t) = pG(x,O,t), there is a lifting H': X X I X 1 E of H
which is a homotopy from Fo to Fj and is an extension of G.
proof Let A = (I X 0) U (О X I) U (I X 1) С I X I and define
/: X X A —» E by
/(x,t,0) = F0(x,t)
/(x,0,t) = G(x,0,t)
/(x,t,l) = Fi(x,t)
Then H\XxA=p°f. Because there is a homeomorphism of I X I with
itself taking A onto I X 0, there is a homeomorphism of X X I X I with
itself taking X X A onto X X I X 0. It follows from the homotopy lifting
property of p that there is a lifting H': X X I X I —> E of H such that
H' | X X A = f. 
SEC-8 fibrations	101
Taking H and G to be constant homotopies, we obtain the following
corollary.
j 1 corollary Let p: E В be a fibration and let Fo, Fi: X X I E be
lifting8 of the same map such that Fo | X X 0 = F| | X x 0. Then Fo ~ Ft
relXxO. 
Let p: F —» В be a fibration and let co: / —> be a path in its base space.
By the homotopy lifting property of p, there exists a map F: 1(oj(())) X I —> E
such that pF(x,t) = co(t) and F(x,0) = x for x £ p~1(u>(0)') and t £ I. Let
f p-1(co(O)) —> p-1(co(l)) be the map/(x) — F(x,l). It follows from theorem 10
that if co co' are homotopic paths in В and if F, F': p-1(co(O)) X I —> E are
such that pF(x,t) = co(t), pF'(x,t) = co'(t), and F(x,0) = x = F'(x,0) for
x E P-1(w(0)) and t E I, then the maps fi fi: p-1(co(O)) —> p^1(co(l)) defined by
fix) = F(x,l) andf'(x) = F'(x,l) are homotopic. Hence there is a well-defined
homotopy class [f ] E [p-1(co(O));p-1(co(l))] corresponding to a path class [co]
in B. We let h[co] = [f ].
The following is the form theorem 2.3.7 takes for an arbitrary fibration.
12	theorem Let p: E В be a fibration. There is a contravariant functor
from the fundamental groupoid of В to the homotopy category which assigns
tob E В the fiber over b and to a path class [co] the homotopy class h[co],
proof If cob is the constant path at b, let F: ;/ *(/>) X 1 —> E be the map
F(x,t) = x. The corresponding map f: p^lb) —> p t(b) defined by fix) = F(x,l)
is the identity map. Hence
7i[cob] = [l„-i(ft)|
showing that h preserves identities.
Let co and co' be paths in В such that co(l) = co'(O). Given a map
F: p-1(co(O)) X I —> E such that F(x,0) = x and pF(x,t) = co(t) for x E p~1(co(O))
and t E I, and given F': p-1(co(l)) X I E such that F'(x',0) = x' and
pF'(x',t) = co'(t) for x' E	and t E I, let f: p-fitfiO)) —> p-1(co'(O)) be
defined by fix) = F(x,l) and let F": p-1(co(O)) X I E be defined by
F(x,2t)	0 < t < %, x E p'(co(O))
F'(f(x), 2t - 1)	% < t < 1, x E p-1(co(O))
Then pF"(x,t) = (co * co')(t) and F"(x,0) = x for x E p-1(<o(0)) and t E I- Let
f: p^'(0))	be defined by/'(x') = F'(x',l). Then F"(x,l) = fi(fix))
for x E p-1(co(O)), which shows that
h[co * co'] = /г[со'] * /г[со]
Therefore h is a contravariant functor. 
This yields the following analogue of corollary 2.3.8 for an arbitrary
fibration.
13	corollary If p: E —> В is a fibration with a path-connected base space,
any two fibers have the same homotopy type. 
F"(x,t) = I
102
COVERING SPACES AND FIBRATIONS CHAP, 2
The following result asserts that homotopic maps induce fiber-homotopy-
equivalent fibrations.
14	theorem Let p: E В be a fibration and let fo, fi: X В be homo-
topic. The fibrations induced from p by f0 and by f are fiber homotopy
equivalent.
proof Let po'- Eo X and pi: Ei —> X be the fibrations induced from p by
fo and fi, respectively, and let/L Eo E and/1: Ei —> E be the correspond-
ing maps such that p ° fo = f0 ° po and p ° fi = fi ° pr. Given a homotopy
F: X X I —> В from fi, to fi, there are maps Fo: Eo X I —> E and
Fi: Ei X I E such that p ° F'o = F ° (p0 X Ir) and p ° F\ = F ° (pi X 1/)
and Fo | Eo X 0 = f 6 and Fl | Et X 1 = fi- Let g0: Eo -> Ei and gy. Ei -> Eo
be the maps defined by the property Fo(x,l) = /igo(x) for x E Eo and
Fi(y,O) = f ogi(i/) for у £ Ei. Then
p ° Eo ° (gi X Ir) = F ° (p0 X Ir) ° (gi X Ir) = F ° (px X Ir) 
and	F’o ° (gi X b) | Ei X 0 = Fl | Ei X 0
It follows from theorem 10 that Fi ~ Fo ° (gi x Ir)- In a similar fashion
Fo ~ Fl ° (go X Ir). This implies that gogi 1Я1 and gig0 ~ l£o. 
Clearly, a constant map induces a trivial fibration, and we have the
following result.
15	corollary If p: E —» В is a fibration and В is contractible, then p is
fiber homotopy equivalent to the trivial fibration В X P1 (bo) В for any
bQ E B. 
Let В be a space which is the join of some space У with S°. Then
В = C_Y U C+Y, where C_Yand C+Yare cones over Yand C_Y Г) C+Y = Y.
Let yo E Y and let p: E —> В be a fibration with fiber Fo = p^1(yo)- It follows
from corollary 15 that there are fiber homotopy equivalencesELY X Fo—>
p-1(C_Y) and g+: px(C+ Y) —> C+Y X Fo. A clutching function p: Y X Fo Fo
for p is a function p defined by the equation
g+/-(l/x) = (y> М(ух)) у E Y, z E Fo
where/-: C_Y X Fo —> p H(C Y) and g+: p-1(C+Y) -» C+Y x Fo are fiber
homotopy equivalences. If C_Y and C+Y are contractible to y0 relative to yo,
it follows from theorem 10 that /_ and g+ can be chosen so that z f(yo,z)
is homotopic to the map Fo C p-1(C-Y) and z —-> g+(z) is homotopic to the
map z —-> (г/оХ) of Fo to C+ У X Fo- In this case the clutching function p cor-
responding to /_ and g+ has the property that the map z —> p(z/oX) L homo-
topic to the identity map Fo C Fo.
Let Ev be the fiber bundle over S” defined by a characteristic map <p:
S”1 G, as in example 2.7.5 (where G is a group of homeomorphisms of the
fiber F). Then Ел = C-S”-1 and Ef — C+S"-1, and it is easy to verify that
/_ and g+ can be chosen so that the corresponding clutching function
p: S?!-1 X F--> Fis the map p(x,z) = <p(x)z.
exercises
103
EXERCISES
Д LOCAL CONNECTEDNESS
j Prove that a space X is locally path connected if and only if for any neighborhood
U of x in X there exists a neighborhood V of x such that every pair of points in V can be
joined by a path in U.
2 If X is a space, let X denote the set X retopologized by the topology generated by
path components of open sets of X. Prove that X is locally path connected and that the
identity map of X is a continuous function j: X —> X having the property that for any
locally path-connected space Y a function f:Y—>X is continuous if and only if
j ° /: Y —> X is continuous.
3 For any space X let X and X—> X be as in exercise 2. Prove that	tt(X,xo) ~ tt(X,Xo).
В COVEBING SPACES
I Let X be the union of two closed simply connected and locally path-connected sub-
sets A and В such that А П В consists of a single point. Prove that if p: X -a X is a non-
empty path-connected fibration with unique path lifting, then p is a homeomorphism.
2 Let X = {(x,i/) g R2 | x or у an integer) and let
X- S'vS1 = {(z1;~2) € S1 x S11 zt = 1 or z2 = 1}
Prove that the map p: X—> X such that p(x,y) = (e2™^2"^'11} is a covering projection.
3 With p: X —> X as in exercise 2 above, let Y С X be defined by
Y = {(x,y) g X| 0 < x < 1, 0 < ^ < 1}
Prove that Y is a retract of X and that (p | Y)# maps a generator of tt(Y) to the commu-
tator of the two elements of тг(Х) corresponding to the two circles of X.
4 Prove that wjS1 v S1) is nonabelian.
C the covebing space ex: R —» S1
1	For an arbitrary space X prove that a map /: X S1 can be lifted to a map
/: X—> R such that f = ex ° / if and only if fis null homotopic.
2	Let X be a connected locally path-connected space with base point x0 g X. Prove
that the map
[X,xo; S»,1] Hom (тт(Х,х0), tt(S1,1))
which assigns to [/] the homomorphism
/#: tt(X,Xo) -> ’’’(S1,!)
is a monomorphism (the set of homotopy classes being a group by virtue of the group
structure on S1).
3	Prove that any two maps from a simply connected locally path-connected space to S1
are homotopic.
4 Prove that any map of the real projective space P" for n > 2 to S1 is null homotopic.
104	COVERING SPACES AND FIBRATIONS CHAP, g
5 Prove that there is no map/: S’1 —> S1 for n > 2 such that /( — x) — —fix).
6 Borsuk-Ulam theorem. Prove that if/: Sz —> Rz is a map such that/( — re) = —fix),
then there exists a point x'o € S2 such tliat /jh'o) = 0.
U COVERING SPACES OF TOPOLOGICAL GROUPS
1 Let H be a subgroup of a topological group and let G/H be the homogeneous space
of right cosets. Prove that the projection G G/H is a covering projection if and only
if H is discrete.
2 Prove that a connected locally path-connected covering space of a topological group
can be given a group structure that makes it a topological group and makes the projec-
tion map a homomorphism.
A local homomorphism </ from one topological group G to another G' is a contin-
uous map from some neighborhood U of e in G to G' such that if gb gz, gigz € V,
then <p(gigz) = <p(gi)<p(gz)- A local isomorphism from G to G' is a homeomorphism <p from
some neighborhood U of e to some neighborhood U' of e' such that </, and are both
local homomorphisms (in which case G and G' are said to be locally isomorphic).
3 Prove that a continuous homomorphism y : G —> G' between connected topological
groups is a covering projection if and only if there exists a neighborhood U of e in Q
such that <p | U is a local isomorphism from G to G’.
4 Let <p be a local homomorphism from a connected topological group G to a topological
group G' defined on a connected neighborhood U of e in G. Let G be the subgroup
of G X G' generated by the graph of </ (that is, generated by {(g,g') £ G X G' | g' — y(g),
g £ U}). G is topologized by taking as a base for neighborhoods of (e,e') the graph of
<p | N as N varies over neighborhoods of e in U. Prove that G is a connected topological
group, the projection pi: G —> G is a covering projection, and the projection pz: G—>G'
is continuous.
5 Prove that two connected locally path-connected topological groups are locally iso-
morphic if and only if there is a topological group which is a covering space of each of
them.
6 If G is a simply connected locally path-connected topological group and <p is a local
homomorphism from G to a topological group G', prove that there is a continuous homo-
morphism <p': G —> G' which agrees with <p on some neighborhood of e in G.
E FIBRATIONS
I If p: E В is a fibration, prove that p(E) is a union of path components of B.
2 If a fibration has path-connected base and some fiber is path connected, prove that
its total space is also path connected.
3 Let p: E В be a fibration and let X be a locally compact Hausdorff space. Define
p': Ex —> B-r by p'(g) = p ° g for g: X —> E. Prove that p' is a fibration.
4 Let p: E -^> В be a fibration and let bo £ p(E), F = P'W’ '	X be a space
regar ded as a subset of some cone CX. Prove that the map
p#: [CX,X; E,F) -> [CX,X; B,b0]
is a bijection.
5 Let p: E —> В be a fibration and let e0 £ E, b0 = p(e0), and F = p’(b<)). If В is
simply connected, prove that fiF,e0) —> ir(E,e0) is an epimorphism.
j ? 6XBBCBES	105
Let p: E —> В be a fibration and let <’o £ E and bo = p(eo)- If p-1(bo) is simply con-
nected. prove that	*
P#= fiE,e0) ~ rr(B,b0)
Let p: E В be a fibration and bo £ p(F). If E is simply connected, prove that
llieie is a bijection between тт(В,Ьо) and the set of path components of p~ /Ifi).
i
$
t
I
!
CHAPTER THREE!
POLYHEDRA
IN CHAPTER TWO THE FUNDAMENTAL GROUP FUNCTOR WAS USED TO CLASSIFY
covering spaces. We now consider the problem of computing the fundamental
group of a specific space. We shall show that the fundamental groups of many
spaces (the class of polyhedra) can be described by means of generators and
relations.
; A polyhedron is a topological space which admits a triangulation by a
i simplicial complex. Thus we start with a study of the category of simplicial
'complexes. A simplicial complex consists of an abstract scheme of vertices and
•simplexes (each simplex being a finite set of vertices). Associated to such a
simplicial complex is a topological space built by piecing together convex cells
. with identifications prescribed by the abstract scheme. Since the topological
properties of these spaces are determined by the abstract scheme, the study
। of simplicial complexes and polyhedra is often called combinatorial topology.
A compact polyhedron admits a triangulation by a finite simplicial com-
plex. Thus these spaces are effectively described in finite terms and serve as a
useful class of spaces for questions involving computability of functors.
Sections 3.1 and 3.2 are devoted to definitions and elementary topological
107
Sl'.C
J SIMPLICIAL COMPLEXES
109
108	POLYHEDRA СНДр, g
properties of polyhedra. Section 3.3 introduces the concept of subdivision
a simplicial complex, and it is shown that a compact polyhedron admits arbi-
trarily fine triangulations. This result is used in Sec. 3.4 to prove the simplicial:
approximation theorem, which asserts that continuous maps from compact 1
polyhedra to arbitrary polyhedra can be approximated by simplicial maps.
The technique of simplicial approximation is used in Sec. 3.5 to prove
that the set of homotopy classes of continuous maps from a compact polyhedron
to an arbitrary polyhedron can be described abstractly in terms of triangula-
tions of the polyhedra. In Sec. 3.6 this result provides an abstract description f
of the fundamental group of a polyhedron as the edge-path group of a triam f
gulation, which is used in Sec. 3.7 to obtain a system of generators and rela-!
tions for the fundamental group of a polyhedron. It is also shown in Sec. 3,7 *!
that the fundamental group functor provides a faithful representation of the ;
homotopy category of connected one-dimensional polyhedra. Section 3.8 con*
sists of applications of the results on the fundamental group, some examples
of polyhedra, and a description of the fundamental group of an arbitrary
surface.
I
SIMPLICIAL COMPLEXES
This section contains definitions of the category of simplicial complexes and
of covariant functors from this category to the category of topological !
spaces.	1
A simplicial complex К consists of a set {«} of vertices and a set {«} of •
finite nonempty subsets of {«} called simplexes such that	?
(a)	Any set consisting of exactly one vertex is a simplex.	j
(b)	Any nonempty subset of a simplex is a simplex.	j
A	simplex s containing exactly q + 1 vertices is called a q-simplex.	We	1
also say that the dimension of s is q and write dim s = q. If s'	C	s,	then	s' is	j
called a face of s (a proper face if s' =^= s), and if s' is a p-simplex, it is called
a p-face of s. If s is a q-simplex, then s is the only q-face of s, and a face s'
of s is a proper face if and only if dim s' < q. It is clear that any simplex has
only a finite number of faces. Because any face of a face of s is itself a face
of s, the simplexes of К are partially ordered by the face relation (written ’
s' < s if s' is a face of s).	f
It follows from condition («) that the O-simplexes of К correspond bijec- '
tively to the vertices of K. It follows from condition (b) that any simplex is
determined by its О-faces. Therefore К can be regarded as equal to the set of
its simplexes, and we shall identify a vertex of К with the О-simplex corre-
sponding to it.
We list some examples.

The empty set of simplexes is a simplicial complex denoted by 0.
For any set A the set of all finite nonempty subsets of A is a simplicial
2
complex.
3	И s is a simplex of a simplicial complex K, the set of all faces of s is a
simplicial complex denoted by s.
4	If s is a simplex of a simplicial complex K, the set of all proper faces of s
is a simplicial complex denoted by s.
5	If К is a simplicial complex, its q-dimensional skeleton Кч is defined to
fie the simplicial complex consisting of all p-simplexes of К for p < q.
G Given a set X and a collection = { W } of subsets of X, the nerve of L7lf,
denoted by /<(l’M), is the simplicial complex whose simplexes are finite non-
empty subsets of Lflf with nonempty intersection. Thus the vertices of Kjflfi)
are the nonempty elements of W.
7	If Ki and K2 are simplicial complexes, their join Kj * K2 is the simplicial
complex defined by
Ki * K2 = Kj v K2 U {«j v s2 | sj E Ki, s2 E K2}
Thus the set of vertices of Ki * K2 is the set sum of the set of vertices of /<i
and the set of vertices of K2.
в There is a simplicial complex whose set of vertices is Z and whose set of
Simplexes is
{{n} | n E Z} U {{n, n + 1} | n E Z}
9 For n > 1 regard Zn as partially ordered by the ordering of its coordi-
nates (that is, given x, x' E Zn, then x < x' if for the ith coordinates
г,- < Xi in Z). There is a simplicial complex whose set of vertices is Zn and
whose simplexes are finite nonempty totally ordered subsets {%0, . . . ,x®}
of Z" (that is, x° < x1 < • • • < x®) such that for all 1 < i < n, x^ — x,0 = 0
or 1.
If К is a simplicial complex, its dimension, denoted by dim K, is defined
 to equal — 1 if К is empty, to equal n if К contains an «-simplex but no (n + 1)-
simplex, and to equal oo if К contains «-simplexes for all n > 0. Thus
dim К — sup {dim s | s E К). К is said to be finite if it contains only a finite
number of simplexes. If К is finite, then dim К <E oo; however, if dim К <E oo,
К need not be finite (example 8 is an infinite simplicial complex whose
dimension is 1).
A simplicial map <p: K\ K2 is a function <p from the vertices of Ki to
the vertices of K2 such that for any simplex s E Ki its image <p(s) is a simplex
of K2. For any К there is an identify simplicial map fy: К К corresponding
- 1 SIMPLICIAL COMPLEXES	Ц1
jgC. *
for only a finite set of vertices).
(h) For any a, 2Pei:«(r) = 1.
jf X = we define |K| = 0.
The real number a(y) is called the nth barycentric coordinate of a. There
js a metric d on |K| defined by
d(a,(3) = VS„eJC[a(o) - ffvff
md the topology on |K| defined by this metric is called the metric topology.
The set |K| with the metric topology is denoted by |К|Й.
We shall define another topology on |K|. For s E К the closed simplex |s|
js defined by
|s| = {a £ \K\ I a(o) 0=> о E s)
If s is a (/-simplex, |s| is in one-to-one correspondence with the set
E R'(+1 I 0 < Xi < 1, S xi = 1). Furthermore, the metric topology on |К|Й
induces on |s| a topology that makes it a topological space |.sj,/ homeomorphic
to the above compact convex subset of R"11. If si, s2 E K, then clearly
S1 П s2 is either empty (in which case |«i| П |s2| = 0) or a face of «i and of
4'2 (in which case |«i П s2| = |«i| G |s2|). Therefore, in either case |.S'i|d П |s2|d
is a closed set in |«1|й and in |s2|d, and the topology induced on this intersec-
tion from |si|d equals the topology induced on it from |s2|d. It follows from
theorem 2.5 in the Introduction that there is a topology on |K| coherent with
{|s|d I« E K). This topology will be called the coherent topology. The space
of K, also denoted by |K|, is the set |K| with the coherent topology. (What we
call here the coherent topology is known in the literature as the weak topology.)
Note that |s| = |s|d; we shall also use |s| to denote the space |s|.
Because a subset A C |K| is closed (or open) in the coherent topology if
and only if А П |s| is closed (or open) in |s| for every s Ek, we have the fol-
lowing theorem and its corollary.
15	theorem A function f: |K| X, where X is a topological space, is con-
tinuous in the coherent topology if and only iff) |s|: |s| X is continuous
for every s £ K. 
16	corollary A function f: |K| X is continuous in the coherent topol-
V&l if anc^ only if f\ |K®|: |K®| Xis continuous for every <7 > 0. 
It follows from theorem 15 that the identity map of the set |K| is a con-
tinuous map |K| —> |К|Й. Note that L С К => |L| C |K| and |L|d is a closed
subset of |К|Й (which implies that |L| is a closed subset of |K|). Furthermore,
is a collection of subcomplexes of K, then U |L;j = | U Lj\ and
П |L;| = |П L;|.
The coherent topology has the following property.
17	theorem For any simplicial complex K, its space |K| is a normal
llnusdorff space.
110	POLYHEDRA CHAP, 3
to the identity vertex map. Given simplicial maps /<1	/<2 —> K3, the corn,
posite simplicial map f 0 <p: Ki K3 corresponds to the composite vertex map,
Therefore there is a category of simplicial complexes and simplicial maps.
A subcomplex L of a simplicial complex K, denoted by L С K, is a sub-
set of К (that is, s £ L => s £ K) that is a simplicial complex. It is clear that
a subset L of К is a subcomplex if and only if any simplex in К that is a face
of a simplex of L is a simplex of L. If L С K, there is a simplicial inclusion i
map i: L С K.	j
A subcomplex L С К is said to be full if each simplex of К having all its ।
vertices in L itself belongs to L. There is a subcomplex N of К consisting of ;
all simplexes of К with no vertex in L. Clearly, N is the largest subcomplex i
of К disjoint from L. If s = {u0,Pi, • • • >vq} is any simplex of K, then either j
no vertex of s is in L (in which case s £ TV), or every vertex belongs to L (in j
which case, if L is full, s £ L), or the vertices can be enumerated so that i
и,. E L if i < p and Vi L if i > p, where 0 < p < q. In the latter case, I
s = s' U s", where s' = {о0, . . . ,vp} is in L, if L is full, and s" = {up+i, . . . ,(;rJ (
is in N. Therefore we have the following result.	:
10	lemma If L is a full subcomplex of К and N is the largest subcomplex
of К disjoint from L, any simplex of К is either in N, or in L, or of the form •
s' U s" for some s' € L and s" E TV. 	(
There is a category of simplicial pairs (K,L) (that is, К is a simplicial j
complex and L is a subcomplex, possibly empty) and simplicial maps <p; '
(Ki,Li) —> (K2,L2) (that is, ф is a simplicial map Ki —> K2 such that <p(Lf) C Ifp •
The category of simplicial complexes is a full subcategory of the category of i
simplicial pairs. There is also a category of pointed simplicial complexes К |
(that is, К is a simplicial complex together with a distinguished base vertex)
and simplicial maps preserving base vertices which is a full subcategory of the >
category of simplicial pairs. Following are some examples.	!
11	For any q the (/-dimensional skeleton is a subcomplex of K, and if j
p <(/,/<(' is a subcomplex of Kq.	I
i
12	For any s E К there are subcomplexes s C s С K.	1'
13	If is a family of subcomplexes of K, then DL, and UL; are alsg I
subcomplexes of K.	i
14	Given that А С X, ^llf = { W) is a collection of subsets of X, and Ka(*®) 1
is the collection of finite nonempty subsets of whose intersection meets A j
in a nonempty subset, then Кл(*№) is a subcomplex of the nerve K(v¥). j’
We now define a covariant functor from the category of simplicial cofflj •
plexes and simplicial maps to the category of topological spaces and continuous
maps. Given a nonempty simplicial complex K, let |K| be the set of all func-
tions a from the set of vertices of К to I such that
(a) For any a, { v £ К | a(c) 7^ 0} is a simplex of К (in particular, a(v) ff О1 >
112	POLYHEDRA CHAP. 3
proof Because |К|Й is a Hausdorff space and i: |K| —> |К|Й is continuous
|K| is a Hausdorff space. To prove that |K| is normal it suffices to show that
if A is a closed subset of |K|, any continuous map/: A I can be continue
ously extended over |K|. By theorem 15, the existence of such an extension of
/ is equivalent to the existence of an indexed family of continuous maps;
{/s: |.sj —> 11 s £ K) such that	j
(a) If s' is a face of s, then /s | |s'| = /s-	
(b)/g | (A A |s|) = /1 (A A |s|)	'
The existence of the family { fs} is proved by induction on dim s. If s i$;
a О-simplex, |s| is a single point, and either |s| £ A, in which case we define ;
fs =f] |s|, or |sj £ A, in which case we define fi arbitrarily.	1
Let q > 0 and assume fi defined for all simplexes s with dim s < q to,
satisfy conditions (a) and (b). Given a (/-simplex s, define/': |s| U (A A |s|) —> J<
by the conditions	-	1
fi | |s'| = /S' s' a face of s
/' | (A A |s|) =f\ (A A |s|)	, j
Because {fi'}itins'<q satisfies conditions (a) and (b), f's is a continuous map of»',
the closed subset |s| U (A A |s|) of |s| to I. By the Tietze extension theorem, •
there exists a continuous extension fi: |s| -a> 1 of /s'. 	I
The same technique can be used to prove that |K| is perfectly normal '
(that is, every closed subset of |K| is the set of zeros of some continuous reals
valued function on |K|) and paracompact.	[
For s £ К the open simplex <s) C |K| is defined by	I
<s> = {a £ |K| | a(o)	E s]	}
Although a closed simplex is a closed set in |K|, an open simplex need not be '
open in |K|. However, the open simplex is an open subset of |s| because,-
(s) = |s| — |s|. Every point a £ |/<| belongs to a unique open simplex (namely,;
the open simplex <s), where s = {c £ К | a(o) =fi 0}). Therefore the open '
simplexes constitute a partition of |/<|.	,
If A is a nonempty subset of |K| that is contained in some closed simplex ,'
|s|, there is a unique smallest simplex s £ /( such that A C |s|. This smallest ?
simplex is called the earner of A in K. If A C <s), then the carrier of A isj
necessarily s. In particular any point a of |K| has as carrier the simplex s sucB
that a £ (s).
18 lemma Let A c |K|; then A contains a discrete subset (in the coherent
topology) that consists of exactly one point from each open simplex meeting A f
proof For each s £ К such that A A (s) =/= 0 let os £ A A (s) and let!
A' — {««} Because any closed simplex can contain at most a finite subset of
A’, it follows that every subset of A' is closed in the coherent topology and 4
is discrete. H
SEC. 1 SIMPLICIAL COMPLEXES	113
Because a compact subset of any topological space can contain no infinite
discrete set, we have the following result.
lf> corollary Every compact subset of |K| is contained in the union of a
finite number of open simplexes. 
A finite simplicial complex has a compact space. The converse follows
from corollary 19.
20 corollary A simplicial complex К is finite if and only if |K| is
compact. 
We establish the following analogue of theorem 15 for homotopies.
21 theorem A function F: |/<| X / X is continuous if and only if
F | (lsl X I)r |s| X i X is continuous for every s £ K.
proof Because |K| has the topology coherent with the collection of its closed
simplexes, and each closed simplex is a closed compact subset of |K], it follows
that |K| is compactly generated. By theorem 2.7 in the Introduction, |K| X I
is also compactly generated. It follows from corollary 19 that every compact
subset of |K| X I is contained in |L\ X 1 for some finite subcomplex L С K.
Therefore |K| X I has the topology coherent with the collection {|L| X 11
L С K, L finite}. It is clear that this topology is identical with the topology
coherent with {|s| X 11 s £ K} (because if L is finite, |L| X I has the topology
coherent with {|s| X 11 s £ L}). 
If <p: Kj -a is a simplicial map, then there is a continuous map
|ф|й:	-> |K2|d defined by
|<p|d(a)(o') =	a(v) v' £ K2
The same formula defines a continuous map |<p|: |Ki| |K2], and there is a
commutative square
|Kr| -a l/GIrf
l<rl| |l<d й
|K2| -+ |K2|d
An easy verification shows that | | and | |й are covariant functors from the
category of simplicial complexes to the category of topological spaces, and
|K| —> |К|Й is a natural transformation between them. These functors can also
be regarded as defined on the category of simplicial pairs to the category of
pairs of topological spaces.
A triangulation (K,f) of a topological space X consists of a simplicial
Complex К and a homeomorphism /: |K| X. If X has a triangulation, X is
called a polyhedron. Similarly a triangu lation ((K,L), f) of a pair (X,A) con-
sists of a simplicial pair (K,L) and a homeomorphism/: (|K|, |L|) -a (X,A). If
114
POLYHEDRA CHAP, 3
(X,A) has a triangulation, (X,A) is called a polyhedral pair. In general, a given
polyhedron will have triangulations and (K2J2), for which /<1 and /<2
are not isomorphic simplicial complexes.
Following are some examples.
22 For any n > 1, (En+1,Sn)is homeomorphic to (|s|,|s|), where s is an (n +
simplex. Therefore (E"+1,Sn) is a polyhedral pair.
23 Given that К is the simplicial complex of example 8 and f: |K| -h> R is
defined so that	« and /| |{n, n + 1}| is a homeomorphism of
|{n, n + 1}| onto the closed interval [n, n + I], then (K,f) is a triangulation
of R, and R is a polyhedron.
24 For n > 1, given that К is the simplicial complex of example 9 and
f: \K\	R'! is defined by the equation (/(a)), =	a(x)(x)«, then (KJ) is
a triangulation of R”, and R" is a polyhedron.
Given a vertex v E K, its star is defined by
st о = {a E |K| I a(v) ¥= 0}
Because a —> a(o) is a.continuous map from |К|Й to I, st v is open in \K\d, and
hence also in |K|. It is immediate from the definition that
a E st v <=> carrier a has r as vertex
<=> a E <s> where s has v as vertex
Therefore st v — U {(s) | и is vertex of s}.
25	lemma Let L С К and let v0, vi, ..., vq be vertices of K. Thein
vq, vi, . . . , Vg are vertices of a simplex of L if and only if
no.m9st vi П =L 0
proof If there is a simplex s £ L with vertices t>0, •  • , G;, then (s) C st iq
for every i, and (x) C |L|. Therefore Г) st о, П |L|	0. Conversely, if
П st с, П |L|	0, let a E П st vt Fi |L|. Then a(yf 0 for 0 < i < q, and
carrier a is a simplex s of L whose vertices include Vq, ... , vq. Then the set
. . . ,oQ} is a face of s and must belong to L, because L is a complex. •
This yields the following relation between К and the open covering of |K|
of vertex stars.
26	theorem Let = {st r | e E K}- The vertex map rp from К to K(^l) de-
fined by <p(p) = st v is a simplicial isomorphism <p: К Gr and for any
L C K,<p\L:L^z K|l](o71). »
2 UWEARITY IX SIMPLICIAL COMPLEXES
The linear structure in the set of all functions from any set to R defines lin-
earity in the space of a simplicial complex. This section is devoted to a study
SEC. 2 LINEARITY IN SIMPLICIAL COMPLEXES	115
of such linearity. We show that a closed simplex |s| is homeomorphic to the
cone with base |s|. This implies that a closed simplex can be parametrized by
“polar coordinates,” which are convenient for the construction of maps. We
use them to prove that a polyhedral pair- has the homotopy extension property
with respect to any space.
We also consider linear imbeddings in euclidean space of the space of a
simplicial complex; this entails a discussion of locally finite simplicial com-
plexes. Such complexes are characterized by the property that their spaces
are locally compact or the equivalent property that the coherent and metric
topologies coincide on their spaces.
Let К be a simplicial complex and let «i, . . . , ap be points of a closed
simplex |s|. Given real numbers t15 . . . , tp such that 0 < tt < 1 for
i= 1, . . . , p and such that St, = 1, the function a = Stjiq is again a
point of |s|. Therefore each closed simplex has a linear structure such that
convex combinations of its points are again points of the closed simplex.
Conversely, if a = has a simplex s as carrier (so that a E (s)), then
each a; E |s|. Therefore we have the following lemma.
1	lemma A convex combination of points of |K| is again a point of |K|
if and only if the points all lie in some closed simplex. 
We shall find it convenient to identify the vertices of К with their char-
acteristic functions. That is, if и is a vertex of K, we regard v as also being
the function from vertices v' £ К defined by
If a E |K|, then we can write a = ~S.Vf:Ka(v)v, the sum on the right being a
convex combination of points of |K|.
Let X be a topological space which is a subset of some real vector space.
We assume that X has a topology coherent with its intersections with finite-
dimensional subspaces each such intersection being topologized as a sub-
space of the finite-dimensional topological linear space in which it lies. For
example, X is euclidean space or X is the space of a simplicial complex.
A continuous map f: |K| X is said to be linear on К if it is linear in terms of
barycentric coordinates. That is, f is linear if for every a £ |K|, a(v)fv)
is a point of X and
fa) = 2ViKa(v)f(y)
It is then clear that a linear map is uniquely determined by the vertex map f0
from vertices of К to X such that fo(v) = fy). Conversely, a vertex mapfo
from vertices of К to X may be extended to a linear map f: |K| X if and
only if for every simplex s E К all convex combinations of elements in f(s)
lie in X.
116
POLYHEDRA CHAP. 3
If <p: Ki K2 is a simplicial map, then the definition of |<p| shows that
|<P|(«) = S a(u)|<p|(u)
Therefore |<p| is linear.
Let X be a topological space. The cone X * w with base X and vertex w
is defined to be the mapping cylinder of the constant map X w. The points
of X * w are parametrized by [x,t] with x £ X and t £ I, where x £ X is
identified with [x,0] and [x,l] is identified with w for all x £ X. Because w is
a strong deformation retract of X * w, a cone is contractible.
2	lemma For any simplex s of К the cone |s| * w is homeomorphic to |s|.
proof Choose a point w0 £ <s) and define a map f: |sj * w —> |s| by
У([а,/]) = tw0 + (1 — t)a. Then f is continuous (because the linear opera-
tions in |s| are continuous). To show that f is injective, assume /([a,t]) =
У([/?Х]) for a, ft £ |s| and t,tf £ I. Then
tw0 + (1 — t)a = fwo + (1 — tftft
Let s have vertices o0, t>i, . . . , vQ and suppose that a = 'SmiO-i, ft = 5Дог,
and w0 = 2 YiOj. Because a, ft £ |s|, there is / such that cq = 0 and there is к
such that ftjc = 0. Then
ty, - fy, + (1 - f'lftj and (t -	= (1 - tftftj
Because Yj =# 0, t > t7. Similarly, ty* + (1 — fta^ = t'y/: and so t7 > t. There-
fore t = t'. It follows then that (1 — t) a = (1 — f)ft, and if t =y= 1, a = ft.
Therefore either t = t’ and a = ft or t = t’ = 1. In either case [a,t] = [At7],
and / is injective.
We now show that У is surjective. Clearly, y([a,0]) = a andy([a,l]) =
Wo, and so У maps onto |s| and w0. To show that every point of (s) — w0 is
on a unique line segment from w0 to some point of |s|, let a £ <s), with
а =У= Wo, and suppose that a = Sa;Lj. Consider the function <p(t7) —
(I + t')« — Two. <p(0) = у £ and as t’ increases, the barycentric coordi-
nates of <р(Т) change continuously. Because a =f= w0, there is some i such that
аг Уг- Therefore
<p(^)(oi) = at - ^(yi. - aft
is a monotonically decreasing function of t'. By continuity, there exists a ,
unique t' > 0 such that <p(f')(oj) = 0. Hence there exists a t'o > 0 which is
the smallest f for which (/.(tlft/i'ft = 0 for any 0 < i < q. Then <p(io) £ Ml and
a =	+ у
1 + to -1 + to
shows that a = y([<p(to)> to/(l + #6)])> and У is surjective.
Because у is a continuous bijection from a compact space to a Hausdorff
space, it is a homeomorphism. 
SEC. 2 LINEARITY IN SIMPLICIAL COMPLEXES	117
The barycenter b(s) of the simplex s = {d0,oi, • • • ,vQ} is defined to be
the point
] £ Cj
Clearly, h(.v) E <s), and so the carrier of b(s) is s. By lemma 2, |s| is homeo-
morphic to |s| * w in such a way that w corresponds to b(s). If a E |s| and
t E I, Ле point ibis') + (1 — t)a of |s| will be parametrized by polar co-
ordinates [a,t], where [a,t] denotes the point of |s| * w corresponding to the
given point of |s|. Then [«,()] = a and [a,l] = b(s) for all a E |s|. We use
polar coordinates for the following homotopy.
3 lemma For any simplex s, |s| x 0 U |s| x I is a strong deformation re-
tract of |s| X I-
proof If s is a 0 simplex, |s| = 0 and we know the point |s| X 0 is a strong
deformation retract of the closed interval |s| X I- If dim s > 0, we define a
deformation retraction
F: |s| X I X I |s| X I
to |s| X 0 U |s| X I by the formula in polar coordinates
F([a,t], t', t") =
([«, (1 - t")t +	(1 - t")F)
([a, (1 - f')t], (1 - t")f +
t' < 2t
2t <f
and diagram it for the cases of a I simplex and a 2 simplex:
4 corollary For any subcomplex L С К the subspace |K| X 0 U |L| X I
is a strong deformation retract of |K| X I-
proof Let Xn = |K| X 0 U |K” U L\ X I for n > — 1. We first show that
for each n > 0 the space Xя”1 is a strong deformation retract of Xn. For each
«-simplex s E К — L let Fs: |s| X I X I |s| X I be a strong deformation
retraction of |s| X I to |s| X 0 U |s| X I (which exists, by lemma 3). For
•i > 0 define a map
F„: Xя X I Xя
118
POLYHEDRA CHAP. 3
by the conditions
Fn | |s| X I X I = Fs for an n-simplex s £ К — L
Fn(x,t) = x x E X'"', t £ I
Then Fn is well-defined and continuous (because for every simplex s the
restriction Fn | |s| X I X I is continuous), and F„ is a strong deformation
retraction of Xn to X”-1.
Let f„; Xn —> X””1 be the retraction defined by fn(x) = F„(x,l) for
x E X”. Let an = 1/n for n > 1, and define G„: Xn X I —> X" by induction
on n so that
G0(x,t) =
0 < f < 02
a2 < t < 1
and for n > 1
0 t < (^n+2
Gn(r,t) =
Fn(x, —-------
' fln+1 ^n-t-2
G„-i(f?!(x),t)
an+2 < t < On+r
On+i E / 'v 1
By induction on n, it is easily verified that Gn is a strong deformation retrac-
tion of Xя to X-1 such that Gn | Xя 1 x I = Gtl_i. Therefore there is a map
G: |K| X IX 1^ |K| X I
such that G | XяX I = Gn. Then G is a strong deformation retraction of
|K| X I to |K| X 0 U |L| X I- 
5	corollary A polyhedral pair has the homotopy extension property
with respect to any space.
proof It suffices to show that if L С K, then (|K|, |L|) has the homotopy
extension property with respect to any space Y. Given g: |K| —> Y and
G: \L\ X I—> Ysuch that G(a,0) = g(a) for a E |L|, let/: |/<| X 0 U |L| X > b
be defined by/(a,O) = g(a) for a E |K| and f(a,f) — G(a,t) for a E |L| and
t E I- Because |L| is closed in |K|, /is continuous. By corollary 4, |K| X 0 U
|L| X I is a retract of |K| X I- Therefore f can be extended to a continuous
map F: |K| X I-> F. Then F(a,0) = g(a) for a E \K| and F | \L\ X I = G. 
Let us now consider lineal’ imbeddings of |K| in euclidean space.
6	lemma A linear map f: |s| R” is an imbedding if and only if it maps
the vertex set of s to an affinely independent set in R”.
proof Let/(uj) = pi, where s = {c;}. We show that the set {p,} is affinely
dependent if and only if/is not injective, (p^) is affinely dependent if and
only if there exist a, not all zero such that Sc^p, = 0 and Sa, = 0. Assume
the points pi enumerated so that а, > 0 for i < f0 and а, < 0 for i > jo.
SEC. 2 LINEARITY IN SIMPLICIAL COMPLEXES
119
Then 5г,;оа.грг = Si>;o (— ai)pi. If a = 2^0 a, —	— cq, then
= 2;>j0(— «,/«)/),. It follows from the linearity of f that
f(Zi<j„(ai/a)vi) =	—	showing that/is not injective.
Conversely, if/is not injective, then/Sa;!:/ = JfSffivi), where ajo 7^ //„
for some j0. Then 5(a4 — Д4)р4 = 0 and 5(а4 — Д) = 0. Because a}o —	7^0,
the set {p, } is affinely dependent. 
A simplicial complex К is said to be locally finite if every vertex о of К
belongs to only finitely many simplexes of K.
7	lemma I/ К is locally finite, every point of |K|d has a neighborhood of
the form |L|d, where L is a finite subcomplex of K.
proof Let a £ |K|d. Then a £ st v for some vertex v of K. Because v is a
vertex of only finitely many simplexes {s,} of K, st v is contained in the com-
pact set U |«г|. Let L = {s £ К | s is a face of s, for some i). Then L is a finite
subcomplex of K, and a £ st v C |L| д. 
8	theorem For a simplicial complex K, the following are equivalent:
(a)	К is locally finite.
(b)	|K| is locally compact.
(c)	|K| |K|d is a homeomorphism.
(d)	jK| is metrizable.
(e)	|K| satisfies the first axiom of countability.
proof (a) => (b). By lemma 7, if a is a point of |K|d, there is a finite sub-
complex L С К such that a is in the interior of |L| d. Then a is in the interior
of IL\ in |K|. Therefore |L| is a compact neighborhood of a in |K|.
(b)	=> (c). To show that |KI —> |K|d is an open map, let 17 be an open sub-
set of |K| with compact closure U in |K|. It suffices to show that 17 is open in
|К|Й. Because U is compact, there is a finite subcomplex L С К such that
U C |L| (by corollary 3.1.19). Let Ki be the subcomplex of К defined by
Ki = {s £ К I |s| П 17= 0}
If s £ К — I<i, then |.sj П U is a nonempty open subset of |s|. Therefore
<s> П U =£ 0 and <s> П |L| 7^ 0. The fact that the open simplexes of К
form a partition of К implies that s £ L, and we have shown that К — f<i U L.
Now, |K|d — is an open subset of |K|d. Because L is finite, |L| |L|d is
a homeomorphism. Therefore 17 is open in |L|d, and so it is open in |L|d — |K!|d.
Because |L|d — |Ki|d — |K|d — |Ki|d, 17 is open in |K|d.
(c)	=> (d). Because |K|d is metrizable, if |K| and |K|d are homeomorphic,
then |K| is also metrizable.
(d)	=> (e). Every metrizable space satisfies the first axiom of countability.
(e)	==> (a). Assume that К is not locally finite and let v be a vertex of an
infinite set of simplexes {sj} 4-1,2,... of K. Assume that v has a countable base
of neighborhoods {I7i}i=i,2,... in |K|. Without loss of generality, we may
120
POLYHEDRA CHAP. 3
assume Ui D C7i+i for all г > 1. For each i, A U-i 0, because v, being
a vertex of s-i, is in the closure of Let cq E (s,) П U{. Then the sequence
{«,} has v as a limit point (because each U-i contains all cq with / > i), but in
the coherent topology the set {«,} is discrete, because it meets every closed
simplex |s| in a finite set. 
A realization of a simplicial complex К in RM is a linear imbedding of |/q
in Rn. The following theorem characterizes those complexes К which have
realizations in some euclidean space.
9	theorem If К has a realization in Rn, then К is countable and locally
finite, and dim К < n. Conversely, if К is countable and locally finite, and
dim К < n, then К has a realization as a closed subset in R2,l+1.
proof Let fi | К | —> R” be a linear imbedding. If К is uncountable, it follows
from lemma 3.1.18 that |K| contains an uncountable discrete set A'. Then
/(A') is an uncountable discrete subset of R’1, which is impossible because Rn
is separable. Therefore К is countable. Clearly is metrizable and, by
theorem 8, К is locally finite. It follows from lemma 6 and theorem 5.3 in the
Introduction that dim К < n.
To prove the converse statement, let {p,} be a sequence of points
in R2i,+1 such that
(«) Every set of 2n + 2 of the points p, is affinely independent.
(b) If C is any compact subset of R2?l+1, there exists j such that C is dis-
joint from the convex subset of R2n+1 generated by the set (pi | i > /}.
For example, let Hi D H2 A • • • be a decreasing sequence of closed half-
spaces of R2n+1 such that ГШ, = 0, and assuming p,- defined for i <T q,
inductively choose pQ to be a point of Hg not lying on any of the finite
number of affine varieties determined by 2n + 1 or fewer points of the set
{pi]l <i <j - 1}-)
Assume that К is countable and locally finite and dim К < n, and let
{vi}i-1,2,... be an enumeration of the vertices of K. Define fi |K| —> R2?i+1 to
be the linear map such that /(o,) = pi- Because of condition (o), it follows
that for any s С K, f\ |s| is a linear imbedding of |s| in |K|, and if s and
s' E K, then
>1 П |s'|)=>|) П>'|)
Therefore /is injective. Because of condition (b), if C is any compact subset of
R2n+i, there is / such that /”1(С) C U {st o.j | i < /}. Since К is locally finite,
this implies that /-1(С) C |L| for some finite subcomplex L С K. Therefore
/-1(C) is compact in |К[. If A is closed in |K| and C is compact in R2n+1, then
f(A) A C = f(A А /-!(С)) is closed in C [because A A /-1(C) is a closed sub-
set of the compact subset/'(C) of |K| and/|/1(C) is a homeomorphism of
SEC. 3 SUBDIVISION
121
y-'i(C) to f[f 1(C))]. Therefore / is a closed map and is a linear imbedding of
|K| as a closed subset in R2i,+1. 
3 SUBDIVISION
Our main interest in simplicial complexes is in the polyhedra they describe.
To study a polyhedron it is important to consider its different triangulations
and their interrelationships. This section is devoted to proving the existence
of “small” triangulations of a polyhedron, which are used in the next section
in proving that arbitrary continuous maps between polyhedra can be approxi-
mated by simplicial maps.
Let К be a simplicial complex. A subdivision of К is a simplicial complex
A' such that
(a)	The vertices of K' are points of |K|.
(b)	If s' is a simplex of K', there is some simplex s of К such that s' C |s|
(that is, s' is a finite nonempty subset of |s|).
(c)	The linear map |K'| —> |K| mapping each vertex of K' to the corre-
sponding point of |K| is a homeomorphism.
Note that conditions (a) and (b) assert that every simplex s' of K' has a
carrier s £ K. If K' is a subdivision of K, we identify |K'| and |K| by the linear
homeomorphism of condition (c). The following fact is immediate from the
definition.
1	Any subdivision of a subdivision of К is itself a subdivision of K. 
The next fact is also true (but somewhat more difficult to prove).
2	If K' and K" are subdivisions of K, there is a subdivision K'" of К that
is a subdivision of K’ and of K".
Thus, statements 1 and 2 assert that the subdivisions of К form a directed
set with respect to the partial ordering defined by the relation of subdivision.
3	lemma Let К and K' be simplicial complexes satisfying conditions (a)
and (b). If s £ К is the carrier of s! £ K', then (s') C (s).
pboof Let v'o, . . . , v.p be the vertices of s' and let o0, . . . , vq be the
vertices of the carrier s of s'. Because s' C |s|, for 0 < i < p, v£ = ~SMi}-Vj.
Because s is the smallest such simplex, for 0 < j < q there exists 0 < i < p
such that ац 0. Let ft £ (s'). Then
ft =	= S(S/h.ay)u;
and because fi > 0 for all i, > 0 for all j. Therefore ft £ (s) and
(s') C (s). 
122
POLYHEDRA CHAP, 3
4	theorem Let K' and К be simplicial complexes satisfying conditions
(a) and (b). Then K' is a subdivision of К if and only if for s £ К the set
((s') | s' £ K', (s') C (s)} is a finite partition of (s).
proof Assume that K' and К satisfy conditions (a) and (b) and the condition
that ((s') | s' £ K', (s') C (s)J is a finite partition of (s) for s £ K. Because
any simplex s £ К has only a finite number of faces, it follows that
A"(s) = (s' £ K' | there exists a face si of s such that (s') C (sj)}
is a finite subcomplex of K', and the linear map hs: |K'(s)| —> |s| that maps
each vertex of K'(s) to itself is a homeomorphism. Therefore there is a contin-
uous map g: |K| —» |K'| such that g | |s| = hsA for s £ K, which is an inverse
of the linear map h: |K'| —> |K|. Therefore h is a homeomorphism, and K' and К
satisfy condition (c).
Conversely, if K' is a subdivision of K, then (s' | s' £ /<'} is a partition of
|K'| — |K|. For s £ K, consider the sets (s') A (s) for s' £ K'. By lemma 3,
either (s') П (s) = 0 or (s') C (s). Therefore ((s') | s' £ K', (s') C (s)}
is a partition of (s). Because |s| is compact, it follows from corollary 3.1.19
that this set is a finite partition of (s). 
We use this result to show that any subdivision of К simultaneously
subdivides every subcomplex of K.
5	corollary Let K' be a subdivision of К and let L be a subcomplex
of K. There is a unique subcomplex L' of K' which is a subdivision of L.
proof If L' is a subcomplex of K' that is a subdivision of L, then
L' = (s' £ K' | (s') C |L|}, which proves the uniqueness of L'. To prove the
existence of L', we prove that (s' £ K'\ (s') C |L|) has the desired properties.
It is clear that this set is a subcomplex L' of K' and that L' and L satisfy con-
ditions («) and (b) above. We use theorem 4 to show that L' is a subdivision
of L. If s £ L, by theorem 4 the set ((s') | s' £ K', (s') C (s)} is a finite par-
tition of (s). By definition of L',
((s') | s' £ K', (s') C (s)} = {(s') | s' £ L', (s') C (s)}
Therefore, by theorem 4, L' is a subdivision of L. 
The subdivision L' of L in corollary 5 is called the subdivision of L
induced by K' and is denoted by K’ | L.
From the definition of subdivision two facts are immediate.
6	Iff: |K| X is linear on К and K’ is a subdivision of K, then f is also
linear on K'. 
7	If {{K,L), f) is a triangulation of(X,A) and K' is a subdivision ofK, then
((K',K' | L), f) is also a triangulation of X. 
For any simplicial complex we construct a particular subdivision, called
the barycentric subdivision. For this we need the following lemma, which
shows how to extend a subdivision of s to a subdivision of s for any simplex s.
SEC. 3 SUBDIVISION
123
8	lemma Let s be a simplex of some complex and let K' be a subdivision
of s. For any Wo E <s), K' * w0 is a subdivision of s.
proof In the statement of lemma 8, w0 is regarded as a simplicial complex
having a single vertex and K' * wq is the join defined in example 3.1.7. It is
clear that /<'* w0 satisfies requirements (a) and (b) for a subdivision of s. It
follows from lemma 3.2.2 that any point of |s| either equals iv0, belongs to |s|,
or belongs to a unique open simplex of the form (s' U {w0}>, where s' £ K'.
Therefore the open simplexes of \K' * w0| constitute a finite partition of |s|,
and by theorem 4, K' * wo is a subdivision of s. 
The subdivision of s obtained by applying lemma 8 is pictured below for
a 2-simplex s.
Д. Л A
s = triangle and	K' = pictured subdivision of	K’ * wq = pictured
its faces	the boundary of the triangle	triangles and their faces
We are now ready to prove the existence of the barycentric subdivision.
Let К be a simplicial complex. We define sd К to be the simplicial complex
whose vertices are the barycenters of the simplexes of К and whose simplexes
are finite nonempty collections of barycenters of simplexes which are totally
ordered by the face relation in K. Thus the simplexes of sd К are finite sets
{b(«o), . . . ,b(sff\ such that Sj-t is a face of s, for i — 1, . . . , q. We shall
always assume the vertices of a simplex of sd К to be enumerated in this order.
It is clear that sd К is a simplicial complex and that if L is a subcomplex
of K, then sd L is a subcomplex of sd K. Furthermore, if b(sQ) is the last
vertex of a simplex s' £ sd K, then s' C |sQ|, and since sQ is the carrier of b(sQ),
sQ is the carrier of s'. Therefore sd К and К satisfy conditions (a) and (b).
f> theorem sd К is a subdivision of K.
proof We show that sd К and К satisfy the hypotheses of theorem 4.
If s E K, then, by lemma 3 and the remarks above,
{s' E sd К | <s'> C <s>} = {s' E sd К | last vertex of s' = h(s)}
= {s' E sd s | <s'> C <s>)
Therefore we need only show that sd s is a subdivision of s for any s E K. We
do this by induction on dim s. If dim s = 0, sd s = s is a subdivision of s. For
q > 0, assume that sd si is a subdivision of sj for every simplex Sj with
dim si < q, and let s be a (/-simplex. By the inductive assumption, sd s is a
subdivision of s. The definition of the barycentric subdivision shows that
sd S = sd s * b(s). By lemma 8, this is a subdivision of s. 
124
POLYHEDRA CHAP. 3
The subdivision sd К is called the barycentric subdivision of K. The
iterated barycentric subdivisions sd" К are defined for n > 0 inductively,
so that
sd° К = К
sd’1 К = sd (sd,rl K) n > 1
10	lemma If L is a subcomplex of K, sd L is a full subcomplex of sd K.
proof Let {b(so), . . . ,b(sq)} be a simplex of sd К all of whose vertices
belong to sd L. Then Sj -i is a face of s,- for i = 1 . . . , q and each s,- £ L.
Therefore {b(so), . . . ,b(sg)} £ sd L. 
11	corollary Let (X,A) be a polyhedral pair. Then A is a strong deforma-
tion retract of some neighborhood of A in X.
proof Because of statement 7 and lemma 10, it suffices to consider the case
(X,A) = (|K|,|L|), where L is a full subcomplex of K. Let N be the largest
subcomplex of К disjoint from L. We prove that |L| is a strong deformation
retract of |K| — |1V|. If a £ |K| — |A|, then, by lemma 3.1.10, either a £ ]L]
or there exist vertices vq, . . . , vp £ L and vertices tyii, . . . , vq £ N,
with 0 < p and p + 1 < q, such that £ (to, . . . In the latter case,
a — 5o<i<eaiFj, with > 0, and we define a = So<i<p«i- Then 0 a 1
and we let nf = су/a for 0 < i < p and a" = a';/(l — d) for p + 1 < i < q.
Then a = aa' + (1 — where a' = So<j<p«iUi is in |L| and a" —
Sp+i<j<Qa"uj is in |N|. A strong deformation retraction F: (|KI — |A|) X I —>
|K| — |A| of |K| — l-N] to |L| is defined by
p/ a _	« € IH t € I
~ (fa' + (1 - t)a |K| - (|2V| U |L|), t £ 1
F is continuous because F | |L| X I is continuous, and for any simplex of К of
the form s' U s", where s' £ L and s" £ N, F | [|s' U s"| П (|K| — |7V|)] X I
is continuous. 
Let X be a polyhedron and let be an open covering of X. A triangula-
tion (K,f) of X is said to be finer than if for every vertex v £ К there is
U £ Ql such that /(st v) C U. A simplicial complex К is said to be finer than
an open covering of |K| if the triangulation (K,1|k|) of |K| is finer than <
(that is, for each vertex v £ К there is V £ such that st о C U). We show
that if is any open covering of a compact polyhedron, there are triangula-
tions finer than $1.
A metric on |K| is said to be linear on К ii it is induced from the norm
in R” by a realization of К in R?i. Any finite simplicial complex has linear
metrics, and if K' is any subdivision of K, a metric that is linear on |K| is also
linear on )K'|.
12	lemma Given a metric linear on an m-simplex s, then for any s' £ sd s
diam Is'l < ——— diam |s|
1 1 — m + 1	11
SEC. 3 SUBDIVISION
125
proof Let (p;1 0 < j < m} be points of Rn and assume that у is a convex
combination of {pj} (that is, у = St;p;, where St; = f and q > 0) and let
x £ R” Then
11-'' f/1)	1|я	= liSfXx- — P;)|| < St;||x Pjll
Therefore ||x — i/Ц < sup |(x — p;||. If x is also a convex combination of {pj},
then ||x - у|| < sup ||p4 - p;||.
Regard |.sj as imbedded linearly in Rn, with vertices po, p\, . . . , pm.
Then, by the above result, diam |s| < sup ||p4 — p/||, and if s' is a simplex of
sd s, diam |s'| < sup {||p' — p"|| | p’, p" £ s'}- Therefore we need only show
that if p' = (po + •  • + p<z)/(</ + 1) and p" = (p0 +   • + pr)/(r + 1),
where q < r, then ||p" — p"|| < [m/(m + 1)] sup ||p4 — р,-||. Again by the
result above,
lip' - p"|| < sup {||p{ - p"|| I 0 < i < q}
and also, for 0 < i < q,
Ups - p"|| = ||р| - —S p;|| < ~~ S ||p{ - p;||
I -Г JL 0<j<r	? -f- 1 0<5<r
< yyy sup llpi - Pill
Therefore
\\p' - p'l < ~~j- sup {||p{ - P;|| I 0 < г < q, 0 < j < r)
< —-— diam |s|
~ r + 1	11
Because r < m, r/(r + 1) < m/{m + 1) and diam |s'| < [m/(m + 1)] diam |s|. 
Given a metric on |K|, we define mesh of К by
mesh К = sup {diam |s| | s £ K}
13 corollary If К is an m-dimensional complex and |K| has a metric
linear on K, then
mesh (sd K) < ——— mesh К 
m + I
This gives us the important result toward which we have been heading.
14 theorem Let be an open covering of a compact polyhedron X. Then
X has triangulations finer than c?l.
eroof Let (K,f) be a triangulation of X. We shall show that there exists an
integer N such that if n > N, then (sd" K, f) is finer than QL Let |K| be pro-
vided with a metric linear on К and let e > 0 be a Lebesque number of the
open covering J1 °)I = {/-1G | U £ ^1} with respect to this metric [thus, if
126
POLYHEDBA CHAP. 3
A C |K| and diam A < e, then /(A) is contained in some element of Ql], Such
a number e > 0 exists because |K| is compact. Let m =. dim К and choose
N so that [m/(m + 1)]л mesh К < e/2 (such an N exists because lim^^
[m/(m + I)]71 = 0). If n > N, then, by corollary 13, mesh sd’1 К < e/2. If v'
is any vertex of sd" K, diam (st i;') < 2 mesh sd" К < e. Therefore /(st i;') is
contained in some element of 4/ and (sd" K, f) is finer than if n > A7. 
This last result is true even if X is not compact. More precisely, if (K,/)
is a triangulation of a polyhedron X and Ql is an open covering of X, there
exist subdivisions K' of К such that is finer than A.1 However, when X
is not compact K' cannot generally be chosen to be an iterated barycentric
subdivision of K, and so the proof for this case is more complicated than the
proof of theorem 14. We need only the form proven in theorem 14, however,
and so omit further consideration of the more general case.
4 SIMPLICIAL APPROXIMATION
A continuous map between the spaces of simplicial complexes can be suitably
approximated by simplicial maps. This section contains a definition and
characterization of the approximations and a proof of their existence for maps
of a compact polyhedron into any polyhedron. Finally, we apply the result
obtained to deduce some connectivity properties of spheres.
Let Ki and K2 be simplicial complexes and let f: \K11 —> |K2| be contin-
uous. A simplicial map <p; Kf —> K2 is called a simplicial approximation to/if
/a) € <«2> implies |<p|(a) E |sz| for a E |Ki| and s2 E K2. Note that if v is a
vertex of K± such that//) is a vertex of K2, then |<p|(o) = /(c). Therefore we
obtain the following result.
1	lemma Let f: |Kj| —> |K2| be a map and suppose that for some subcom-
plex Li C Ki, /| |Li] is induced by a simplicial map L± —> K2. If
<p: K± K2 is a simplicial approximation to f then |<p| | |Li| = /| |Li|. 
In particular, the only simplicial approximation to a map |<p|: |Ki| —> |K2|
induced by a simplicial map <p: f<i K2 is <p itself. One sense in which
a simplicial approximation is an approximation is the following.
2	lemma Let <p-. K\ K2 be a simplicial approximation to a map f:
]/<i| —> |K2| and let A C \Ki| be the subset of |Ki| on which |<p| andf agree.
Then |<p| ~ / rel A.
proof A homotopy relative to A from |<p| to / is defined by the equation
F(a,t) = t/(a) + (1 — *)(|<p[(a)) a E |Ki|, t E I
1 See theorem 35 in J. H. C. Whitehead, Simplicial spaces, nuclei, and m-groups, Proceedings of
the London Mathematical Society, vol. 45, pp. 243- 327 (1939).
SEC. 4 SIMPLICIAL APPROXIMATION
127
The right-hand side is well-defined, because if Да) £ (s2>, then |<p|(a) £ |a2|,
and so F(a,t) £ |s2| for t G I. The continuity of F is easily verified. Clearly, if
I a € A, then F(«,t) = /(a) for all t G I. Therefore F: |<p| ~ /rel A. в
•	The following theorem is a useful characterization of simplicial approxi-
j mations.
!	3 theorem A vertex map <p from Ki to K2 is a simplicial approximation
' to f- |Ki | —> |K2| if and only if for every vertex vCKi
*	/(st v) C st <p(n)
' proof Assume that <p is a simplicial approximation to f. Let a G st v and
suppose/(a) G <s2}. Then a(y) 0 and G |«г|- Because <p is simplicial,
'	|<p|(a)(<P(r)) 7^ 0. Therefore <p(n) is a vertex of |s2|, and f(a) £ st <p(y). Since
; this is so for eveiy a G st v, /(st с) C st <p(u).
I Conversely, assume that <p is a vertex map such that/(st v) C st <p(v) for
(	every vertex v G K±. We show that <p is a simplicial map. If {vf} are vertices
of a simplex of Ki, then Cl st vt 0 (by lemma 3.1.25) and
0 т^/(П st vi) С П /(st vi) С Л st <p(fj)
By lemma 3.1.25, {<p(vi)} are vertices of some simplex of K2. Therefore <p is
a simplicial map Kt —> K2.
To show that <p is a simplicial approidmation to /, assume a £ (si) and
.	/(a) G <s2> and let v be any vertex of si. Then a G st v and, by hypothesis,
	/(«) £	т(г)- Therefore <p(u) is a vertex of s2. This is so for every vertex о of
;	st. Because <p is simplicial, |<p|(|si|) C |s2|. Hence |<p|(a) G |s2|, and <p is a
• simplicial approximation to /. 
к
’ We are also interested in simplicial approximations <p: (Ki,Li) —> (K2,L2)
to maps /: (IFGjJ^LiJ) —> (|K2|,|L2|). The following corollary shows that any
simplicial approximation Ki K2 to a map /: (|Ki|,|Li|) —> (|K2|,|L2|) is
' automatically a simplicial approximation when regarded as a map of pairs.
[	4 corollary Let f: |K1| -7 |KZ| be a map such that /(|Li|) C |L2| for
i Lj С and L2 C K2 and let cp: K± K2 be a simplicial approximation to
f f Then <p | Li maps Lt to L2 and is a simplicial approximation to f | |Li|.
! proof By theorem 3, it suffices to show that if о is a vertex of Li, then <p(o)
j is a vertex of L2 such that
/(st v П |Li|) C st <p(o) П |L2|
Since <p is a simplicial approximation to / /(st v) C st <p(o), and if о is a vertex
of Li, then f(v) G <s2> for some s2 C L2 [because /(|Lr|) C |L2|]- Therefore
<p(v) is a vertex of L2 and
/(st u О |Li|) C /(st v) П |L2| C st <p(t>) П |L2| 
It follows from corollary 4 that any simplicial approximation to a map
128	POLYHEDRA CHAP. 3
/: (|Ki|,|Lr|) -a (|K2|,|L2|) is a simplicial map tp: (KuLf) -a (K2,L2). From
lemma 2, it follows that f ~ |g?| as a map of pairs.
5 corollary The composite of simplicial approximations to maps is a
simplicial approximation to the composite of the maps.
proof Let <p: Ki -» K2 be a simplicial approximation to /: |Ki| -a |K2| and
let /: K2 -a K3 be a simplicial approximation to g: |K2| —> |Кз|. Then,
by theorem 3, for a vertex v £ Ki
gf(st v) C g(st <p(n)) C St ftp(v)
and ftp: Ki Кз is thus a simplicial approximation to gf: [К^ -а |Кз| . 
Theorem 3 leads to the following necessary and sufficient condition for
the existence of a simplicial approximation to a map.
6	theorem A map f: |Ki| -a |K2| admits simplicial approximations
Kj K2 if and only if Kt is finer than the open covering {/-1(st v) | v is a
vertex of K2}.
proof By theorem 3, there exist simplicial approximations to /if and only
if for each vertex vt £ Kt there is a vertex v2 6 K2 such that st vt C /-1(st v2).
This is equivalent to the condition that Ki is finer than 
If К is a subdivision of K, then for vertices v' £ K' and v 6 К
v' 6 Six V <=> st/r v' C Sfo V
Combining this fact with theorem 3 yields the following corollary.
7	corollary Let K' be a subdivision of K. A vertex map tp from K' to К
is a simplicial approximation to the identity map |K'| C |K| if and only if
v' 6 st tp(v') for every vertex v' £ K'. 
In particular, if K' is a subdivision of K, there exist simplicial approxima-
tions К' К to the identity map |/<'| C |A|. Combining theorems 6 and
3.3.14 and corollary 4, we obtain the following simplicial-approximation
theorem.
8	theorem Let (Ki,Li) be a finite simplicial pair and letf:	—>
(IKzI^LzI) be a map. There exists an integer N such that if n > N there are
simplicial approximations (sd” Kj, sd" Lj) (K^Lf) to f. 
As remarked at the end of Sec. 3.3, theorem 3.3.14 is also valid for an
arbitrary polyhedron X. Therefore, if Ki is arbitrary and/: |/<i| -а |K2| is a
map, there exists a subdivision Kj of Kt and a simplicial approximation
K\ —> K2 to /: |K1| —» |f<2|. If Кг is not finite, however, Kf cannot generally
be taken to be an iterated barycentric subdivision of K±.
9	example If sis the complex consisting of all proper faces of a 2-simplexs,
then |.s| is homeomorphic to S1, and therefore [|s|;|s|] is an infinite set
SEC. 5 CONTIGUITY CLASSES	129
Because s is a finite complex, there are only a finite number of simplicial maps
j sd” s -a s for any n. Therefore for any n there exist maps |s|	|s| having no
| simplicial approximation sd" s -a s.
• Ю example Let s be as in example 9 and let its vertices be v0, tq, v2. De-
* fine/ |s| —» |s| to be the map linear on sd s such that
i
/(u0) = h{c0,m} f(fo) = b{V1,v2}	f(y2) = b{«2,«o}
! ftblvopV}}) = rn f(b{v1,v2}) = V2	f(b{vZ,V0}) = V0
j Then /	| lj|, but there is no simplicial approximation s -a s to /. There are
exactly six simplicial approximations <p: sd s -a s to /[<p is unique on b{v0,v^},
Ь(у1,с2], and b{v2,v0}, and <p(u0) = v0 or tq, tp(vf) = m or v2, and tp(v2~) = v2
or 00].
! As an application of the technique of simplicial approximation, we
deduce the following useful result.
I 11 theorem S" is (n — l)-connected for n > 1.
proof By theorem 1.6.7, it suffices to prove that if m n, any map
Sm -a S" is null homotopic. Let si be an (m 4- l)-simplex and s2 an (n + 1)-
simplex. Then Sm and Sn are homeomorphic, respectively, to J.st | and |s2|. By
theorem 8 and lemma 2, it suffices to show that if tp: sd’ .i| -a s2 is any
' simplicial map, then |<y| is null homotopic. Because dim (sd’ sf) = m fn, tp
1 maps sd’ into the m-dimensional skeleton of s2. Therefore there is some
a £ |s2| such that
|r/|(lsd‘ sr|) C |s2| - a
t Because |.<2| — a is homeomorphic to Sn minus a point, which is homeomor-
' phic to R", it is contractible. Therefore |<p| is null homotopic. 
1
In particular, we have the following result.
12 corollary For n > 1, S” is simply connected. 
Because S’" is locally path connected, corollary 12 and the lifting theorem
imply that any continuous map/: S’" —» S1 can be factored through the covering
j map ex: R -a S1. Since R is contractible, this implies the following corollary.
•' 13 corollary For n > 1 any continuous map S” -a S1 is null homotopic. 
, 5 CONTIGUITY CLASSES
j In the last section it was shown that any continuous map between the spaces
1 of simplicial complexes has simplicial approximations defined on sufficiently
. fine subdivisions of the domain complex. In general, simplicial approximations
i fo a given continuous map are not unique, and in this section we investigate
j this nonuniqueness.
130
POLYHEDRA CHAP. 3
We shall define an analogue of homotopy, called contiguity, in the cate-
gory of simplicial pairs and simplicial maps. Different simplicial approxima-
tions to the same continuous map will be shown to the contiguous. The main
result of the section is the existence of a bijection between the set of homo-
topy classes of continuous maps (from the space of a finite simplicial complex
to the space of an arbitrary complex) and the direct limit of a certain sequence of
contiguity classes of simplicial maps.
Let (Kp,Li) and (Kz,Lf) be simplicial pairs. Two simplicial maps <y,
<p':	are contiguous if, given a simplex s G /(, (or s f L1)>
<p(«s) u </(s^) is a simplex of Kz (or of L2). Obviously, this is a reflexive and
symmetric relation in the set of simplicial maps (K1;Li) -> (KZ,LZ), but
in general it is not transitive. There is, however, an equivalence relation,
denoted by <p ~ <p', in this set of simplicial maps that is defined by g— <p' if
and only if there exists a finite sequence <p0, <Pi,   . , <pn such that <p0 = <p
and <p„ = <p' and such that <p{_i and <p,- are contiguous for i = 1, 2, . . . , n.
The corresponding equivalence classes are called contiguity classes, and the
set of contiguity classes of simplicial maps from (Ki,Li) to (K2,L2) is denoted
by [Ki,Li; K2,L2]. If <p: (/<i,L|) —> (KZ,LZ) is a simplicial map, its contiguity
class is denoted by [<p].
We shall see that contiguity classes are algebraic analogues of homotopy
classes. We begin by showing that contiguity classes can be composed.
1	lemma Composites of contiguous simplicial maps are contiguous.
proof Assume that <p, <p': (/<i,L|) —> (K2,L2) are contiguous and f, ip';
(KZ,LZ) —> (K3,L3) are contiguous. If s is a simplex of /<1 (or Li), <p(s) U <p'(s)
is a simplex of K2 (or L2). Therefore
’/'(?(») U <p'(s)) U <p'(<p(s) U <p'(s))
is a simplex of K3 (or L3). This implies that the subset ftp(s) U f'tp'(s) is
a simplex of K3 (or L3) and that ftp, f'tp'- (Ki,Lf) —> (K3,L3) are contig-
uous. 
It follows easily from lemma 1 that if <p — <p' and f — f', then
jp<p ~ ftp' — f'<p'- Therefore there is a well-defined composite of contiguity
classes
И 0 [<p] = [<p<p]
for (Ki,Li) (KZ,L2) -Л (K3,I3). Thus there is a contiguity category whose
objects are simplicial pairs and whose morphisms are contiguity classes |
of simplicial pairs. There are full subcategories of the contiguity category
determined by the pairs (K, 0) or by the pointed simplicial complexes. 1
2	lemma Contiguous simplicial maps which agree on a subcomplex 
define continuous maps which are homotopic relative to the space of the
subcomplex.
CgEC- 5 CONTIGUITY CLASSES	131
proof Assume that <p, <p': (Ki,Li) —> (Кг,Ь2) are contiguous and agree on
g c K. Define a homotopy F: (|Ki| X I, |Li| X’l) —> (|K2|,|L2|) rel |L| from
|ф| to |<p'| by the equation
F(a,t) - (1 — t)(|<p|(a)) + t(|<p'|(a))	« € |^i|, t € 1 
Since homotopy is an equivalence relation, if <p — <p', then |<p| ~ |<p'|.
Therefore we have the following result.
3	corollary There is a covariant functor from the contiguity category
of simplicial pairs to the homotopy category of topological pairs which
assigns to (K,L) the pair (|K|,|L|) and to [<p] the homotopy class [|<p|]. 
The next result considers different simplicial approximations to the same
continuous map.
4	lemma Two simplicial approximations (Ki,Lj) —> (A2,L2) to the same
continuous map are contiguous.
proof Let <p> <p/; (-Ki,Li) —> (I<2,L2) be simplicial approximations to f:
—> (|K2|,{L2|) and let {oj} be a simplex of Kj. Then П st yt 0,
and by theorem 3.4.3,
0 #=ДП st Vi) C n/(sl v^ С П (st qfvi) A st <р'(г»))
Therefore (ф(г;,-)} U {<р'(о»)} a simplex of K2. If {г:;} is a simplex of Lj, a
similar argument shows that (<р(ч)} U {<p'(oj)} is a simplex of L2. Therefore
(p and <p' are contiguous. 
Since it was necessary to subdivide in order to obtain simplicial approxi-
mations to arbitrary continuous maps, we should also expect to subdivide to
make contiguity classes correspond to homotopy classes. An example will
illustrate the relation between homotopy and contiguity.
5 example Let s be a 2-simplex with vertices v0, v±, v2 and let /<1 = K2 = s.
Any vertex map from K\ to K2 is a simplicial map. Therefore there are
exactly 27 simplicial maps Kt —> K2. Of these 27, there are 21 which map
into a proper subcomplex of K2, and these constitute one contiguity class. Of
the remaining 6, each is the only element of its contiguity class, the 3 even
permutations of the vertices defining homotopic continuous maps correspond-
ing to one generator of the group
[|£r|;|K2|]	~ Z
and the 3 odd permutations corresponding to the other generator of this
group. Therefore [Ki;K2] consists of 7 elements, and the image
[Kl;K2] [(MIM
consists of 3 elements.
This example shows that simplicial maps which define homotopic con-
tinuous maps need not be in the same contiguity class. The next result shows
132	POLYHEDRA CHAP. 3
that a finite simplicial complex can be subdivided so that homotopic simplicial
maps from it to some other complex can be simplicially approximated on the
subdivision by maps in the same contiguity class; it is the analogue for
homotopy of the simplicial-approximation theorem.
6 theorem Letfif': (|Kt|,|Li|) -а (|K2|,|L2|) be homotopic, where Кг is
finite. Then there exists N such that f and f' have simplicial approximations
<p, <p': (sd^ Kb sdN Lf)(K2,L2)
respectively, in the same contiguity class.
proof Let F: (|Ki| X I, |Li| Xl)-> (|K2|,|L2|) be a homotopy from/to/'.
Because |Ki| is compact, there exists a sequence 0 = to < t± < • • • < tn = f
of points of I such that for a E |Ki| and i = 1, 2, . . . , n there is a vertex
t; E K2 such that F(«,E_|) and F(a,t4) both belong to st v. Let/: (|Kt|,|Li|) —>
(|K2|,|L2|) be defined by /(a) = F(a,t{). Then f = fo and f' = fn, and for
i = 1,2, . . . , n the set
= {/-1(st v) П v) | v E K2}
is an open covering of |Ki|. Let N be chosen large enough so that sdA' Kx is
finer than ^li, 712, . . . , (which is possible, by theorem 3.3.14). For
i = 1, 2, . . . , n let <pj be a vertex map from sdA Ki to K2 such that
/(st v) U fj_i(st v) C st <pi(v)
for each vertex v E Kj (such a vertex map exists because sdA’ K| is finer
than 9l-j). By theorem 3.4.3,
<pj: (sd« K1; sdwLr) (K2,L2)
is a simplicial approximation to fi and to fi-±. Because c/.,- and c/-;+i are
simplicial approximations to fi, it follows from lemma 4 that and are
contiguous for i = 1, 2, . . . , n — 1. Therefore <pj ~ <p„, and also <pi is a
simplicial approximation to fo = f and <pn is a simplicial approximation to
fn =f'- •
Unlike the simplicial-approximation theorem, this last result is definitely
false if Kr is not a finite simplicial complex. That is, given homotopic maps
fi f': |Ki| —> |K2|, there need not be a subdivision K't of Ki such that/and f
have simplicial approximations K'i —> K2 in the same contiguity class.
7 example Let Ki = K2 equal the simplicial complex of example 3.1.8,
with space homeomorphic to R. Let <p: K, -a- K2 be the identity simplicial
map and <p': Ki —» K2 be the constant simplicial map sending every vertex of
Ki to the vertex 0 of K2. Since R is contractible, |<p| However, if K{ is
any subdivision of Kx, a simplicial approximation /: Kj —> K2 to |<p| must be?
surjective to the vertices of K2 and a simplicial approximation K'i —> K2
to |<p'| must be the constant map to 0. Since two contiguous maps K'i —> K$.
either both map onto a finite set of vertices or neither does, / and are not;
in the same contiguity class.
sf:( . 5 CONTIGUITY CLASSES	133
We show that if Ki is finite the set of homotopy classes of maps
[|Ki|,|bi|; lK2|,|L2|] is the direct limit of the set of contiguity classes
|	[sd” Ki, sd” Li; K2,L2]
I
i Note that simplicial approximations (sd Ki, sd Lf) —> (Ki,Lf to the identity
 map (|sd Ki|, |sd Li|) C (|Ki|,|Li|) exist, by corollary 3.4.7, and any two are'
I contiguous, by lemma 4. Because the composites of contiguous simplicial
•’ maps are contiguous by lemma 1, there is a well-defined map
i	sd: [K^Ln K2,L2] [sd K1; sd Li; K2,L2]
defined by
sd[<p] = [<pA]
i where A: (sd Ki, sd Lf) (K|,L|) is any simplicial approximation to the.
identity (|sdKi|, |sd Li|) C (|Ki|,|2Li|) and <p: (Kr,Li) (K&Lf) is an arbitrary
i simplicial map. By iteration there is obtained a sequence
------> [sd” Ki, sd” Lt; K2,L2]	[sd”+1 Kt, sd”+i K2,L2]	  •
which begins with [Ki,Lr; K2,L2] on the left and extends indefinitely on the
right. The direct limit linn, {[sd” Ki, sd” Lr; K2,L2]} is a functor of two argu-
 ments contravariant in (Kr,Lr) and covariant m (K2,L2). For finite Ki this
| functor is naturally equivalent to the functor [|Kr|,|Lx|; ]K2],|L2]].
1 « theorem If is a finite simplicial complex, there is a natural
equivalence
lim_ {[sd” Къ sd” Li; K2,L2]} ~ [IKrIJLrl; |K2|,|L2|]
I proof A function from the direct limit to [|FGL|,|K.1|; |K2|,|L2|] consists of a
sequence of functions
I	fn: [sd” Ki, sd” Li; K2,L2] -+ [|Kl|,|Li|; |K2|,|L2|]
for n > 0 such that fi, = fn+1 ° sd for n > 0. Such a sequence f„ is defined
by/n[<p] = [|<p|] for <p: (sd" Ki, sd” Lf) (K2,L2), because if
A„: (sd"+1 Ki, sd”+1 Lf) (sd” Ki, sd” Lf)
I is a simplicial approximation to the identity map
’	(|sd”+iKi|, |sd”+1Li|) C (|sd”Ki|, |sd”Li|)
i then, by lemma 3.4.2, |A„| ~ 1, and
5	fn+i sd[<p] = [|<pA„|]	[|<p|] = /n[<p]
< The sequence {/„} defines a natural transformation
j	/: lim_ {[sd” Ki, sd” Lr, K2,L2]} [IKxlJLrl; |K2|,|L2|]
and we show that / is a bijection.
It follows easily from the simplicial-approximation theorem that {/?l}
satisfies (a) of theorem 1.3 of the Introduction; for if g: (|Kr|,|Lr|)	(|K21,|L2|)
134	POLYHEDRA CHAP. 3
is a map and <p: (sdw K±, sd” L|) —> (K2,L2) is a simplicial approximation to g(
then |<p| ~ g, and
f»[«p] = [|<p|] = и
To show that {/„} satisfies (b) of theorem 1.3 of the Introduction, assume j
<p, <p': (sdB Kb sd” Lt) —> (KZ,LZ)	j
are such that |<p| ~ |<p'|. By theorem 6, there exists m > n such that |<p| and >
|<p'| have simplicial approximations	j
ip': (sd™ Ki, sd™ Lx) -a (KZ,LZ)
in the same contiguity class. Let
Am,„: (sd™ Ki, sd»’ Lt) -a (sd” Kb sd” Lt)	(
be the composite XWIi„ = X„ Xn+1   • Xm~i. Then Xm,n is a simplicial approxi-
mation to the identity map, and because <p is a simplicial approximation ’
to |<p|, <pXW!,B is also a simplicial approximation to |<p|, by corollary 3.4.5. By ‘
lemma 4, <pABljB is contiguous to ip. Similarly, (p'A/B;B is contiguous to ip'. Since
ip and ip' are in the same contiguity class, so are <pAM,B and <p'Am;„. This means
that sd™~”[<p] = sd™**’l[<p/] in [sd™ Ki, sd™ Li; K2,L2]. 	I
For finite Ki the last result furnishes an algebraic description of the set j
[|Ki|,|Li|; |K2|,|L2|]. As an application, note that if K2 is a countable complex, ,
there are only a countable number of simplicial maps (sd" Ki, sd” Li) -> •
(K2,L2) for n > 0. Therefore [sdB Kls sd” Li; K2,L2] is countable for n > 0, :
Because the direct limit of a sequence of countable sets is countable, we
obtain the following result.
9 corollary Let (ХД) and (Y,B) be polyhedral pairs with X compact
and Y the space of a co untable complex. Then [X,A; Y,B] is a countable set. 
6 THE ЕМЕ РАТП GROUPOID
[
It was shown in the last section that for finite Kb [|Ki[;[K2|] is describable as, J
a limit in which Ki is subdivided but K2 is not. In particular, for any simplicial;
complex К the set of path classes of |K| from v0 to vi is determined by the j
simplicial structure of K. This is made explicit in the present section, where :
we define a simplicial analogue of the fundamental groupoid of a space. In
the next section the fundamental group of a polyhedron is presented in terms [
of generators and relations.	j
An edge of a simplicial complex К is an ordered pair of vertices
which belong to some simplex of K. The first vertex v is called the origin of-1
the edge, and the second vertex v' is called the end of the edge. An edge path
f of К is a finite nonempty sequence e, e2  •  er of edges of К such that end
SRC. 6 THE EDGE-PATH GROUPOID	135
= orig ej+i for i = 1, . . . , r — 1. We define orig f = orig ei and end f =
end er. A closed edge path at vo € К is an edge path f such that orig f =
n0 = end f. If fi and f 2 are edge paths of К such that end f । = orig f2, we
define the product edge path f if2 to be the edge path consisting of the
sequence of edges of fi followed by the sequence of edges of f2. Then
orig fif2 = orig fi and end fxf2 = end f2. It is clear that if end fi = orig f2
and end f2 = orig f3, then fi(f2f3) = (fifzjfs- The product of edge paths thus
satisfies associativity; however, there are no left or right identity elements for
the product. To obtain a category (as was done for paths in a topological
space) it is necessary to define an equivalence relation in the set of edge paths
of K.
Two edge paths f and f' in К are simply equivalent if there exist vertices
V, v', and v" of К belonging to some simplex of К such that the unordered
pair {f,f'} equals one of the following:
The unordered pair {(0,13"), (v,v')(v',v")}
The unordered pair {f 1(0,0"), fi(o,o')(o',o")} for some edge path fi in К
with end fi = о
The unordered pair {(o,o")f2, (o,o')(o',o")f2) for some edge path f2 in К
with orig f2 = 0"
The unordered pair {fi(o,o")f2, fi(o,o')(o',o")f2} for some edge paths
fi and f2 in К with end fi = о and orig f2 — v".
Two edge paths f and f' will be said to be equivalent, denoted by f — f',
if there is a finite sequence of edge paths fo, fi, . . . , fB such that f — f0
and f' = fn, and f{_i and f,- are simply equivalent for г = 1, ... , n. This
is an equivalence relation, and the following statements are easily verified.
I f ~ f' implies that £ and f' have the same origin and the same end. 
2 fi ~ fl, f2 ~ Г2 and end fx = orig f2 imply fif2 ~ flf'2. 
3 If orig f = 01 and end f = o2, then (oi,oi)f f — f(o2,o2). 
If f is an edge path, [f] denotes its equivalence class. It follows from
statement 1 that orig [f] and end [f] are well-defined (by orig [f] = orig f
and end [f] = end f). From statement 2 there is a well-defined composite
[ft]0 If2] ~ [fif2] defined if end fi = orig f2. We then have the following
simplicial analogue of theorem 1.7.7.
•I theorem There is a category S(K) whose objects are the vertices of К
and whose morphisms from O| to v0 are the equivalence classes [f] with
brig [f] = vo and end [f] = cj and whose composite is [fj 0 [f2].
Proof The existence of identity morphisms follows from statement 3, and
file associativity of the composite is a consequence of the associativity of the
product of edge paths. 
We now show that £(K) is a groupoid. If e = (v,v') is an edge of K, we
136
POLYHEDRA CHAP. J
define e 1 = (t:',r;), and if f = e\ •  • er is an edge path in K, we define
G 1 — er-1 • •  ei-1. The following statements are then easily verified.
5	= f. 
6 orig f-1 = end f and end	= orig f. 
7	fi — &> implies fr1 ~ ЙГ1- 
8 If orig f = ui and end f = v2, then ff-1 ~ (ui,ui) and f-1f — (v2,v2~). 
It follows that in £(K), [f1] = IfJ h ail<^ so is a groupoid. This
groupoid is called the edge-path groupoid of K. If vq is a vertex of K, the
operation [f] ° [fz] in the set of elements of &(K) with origin and end at v0 is
a group denoted by E(K,vo) and is called the edge-path group of К with base
vertex v0.
To compare S(K) [and E(K,vf)] with ^(|К?|) [and тт(|К|,оо)], for r > llet
IT denote the subdivision of I into r equal subintervals; that is, Ir is the
simplicial complex
r [ [ 1 1 I	. • J* 1 I . [ [ 'I ' 1 I 1 I -|	. 1
Ir = {{— 1 0 < г < r} U {{-----------, — 1 1 < t < r 1
Ц r JI ~ J Ц r r JI ~ ~ J
Given an edge path f = ei   • er in К with r edges, let <p?: Ir —> К be the sim-
plicial map defined by
/ i \	forig ei+i	0	< i < r	— 1
<Pt I— I =	1		,	,
\r /	[end e4	1	< i < r
Then |ф<|: I —> |K| is a path in |K|, and it is easily seen that the following
statements hold.
9 orig |<pf| = orig f and end |<p?| = end f. 
Ю f f' implies [<pf [ ~ |<pf'| rel I. 
Il If end fi = orig f2, then |<pW2| ~ |<ph| * |<pf2| rel I. 
It follows that there is a natural transformation p from S(K) to ?P(|K|)
which assigns to v £ К the point v £ |K| and to a morphism [f] in S(K) the
morphism [|<Pj|] in ?P(|K|). We shall prove that for vertices oo, oi 6 K, pis a
bijection
p: homs (ui,uo) ~ honig, (ui,uo)
This can be obtained from theorem 3.5.8, but there is also a direct proof
(which seems no longer than a proof based on theorem 3.5.8).
12 lemma For any vq, Vi £ К the function
p: hom6 (t>i,t>o) —> hom,P (t>i,i>o)
is surjective.	,__
proof Given a path oj: / —> |K| from v0 to Vi, because I = |Ii|, it follows
Jgc. 6 THE EDGE-PATH GROUPOID	137
from theorem 3.4.8 that there is a simplicial map
<p: sd'" /| —> /<
which is a simplicial approximation to w. Since sdB L = I2n, there is an edge
patli f = ei   • e2n defined by ег = (<p((i — l)/2’!), <p(i/2n)) such that <p =
for this f. Because <p(0) = w(0) and <p( 1) = oj( 1), it follows from lemma 3.4.2
that |<p| w rel I. Therefore [w] = [|<p|] = [|<pf|] = p[f], 
We shall need some preliminary lemmas before showing that p is injective.
13 lemma For any simplex s two edge paths in s with the same origin and
the same end are equivalent.
proof It suffices to prove that if f is any edge path in s from orig f = v and
end f = o', then f is equivalent to the edge path consisting of the single edge
(o,o'). This is proved by induction on the number of edges of f. 
14 lemma Let f and be edge paths in K, each with r edges, such that
фр Ф?':	К are contiguous. Then f — f.
proof Let f = ei - -  er, where о» = (u-i,o»), and let = el • • • of, where
di = (oUi,o|). For 0 < i < r let ё; = (this is an edge of К because
and <pp are contiguous). From the definition of equivalence
f ~ Oi0i0i~10202 •• • cr\er
Because and are contiguous, for each 1 < i < r there is some simplex
Si of К such that e{, e|, ё»_1, and 0,- all are edges of Sj. By lemma 13, 0|Oi — el
and OjliOjOj — е{ for 2 < i < ?• — 1, and effer ~ eh Therefore
О1ё1ёГ1е2ё2 •  • eT-\er ~ 0102   • of = f 
15 lemma Let £ = ei •   er be an edge path of К and let X: I2r -a Irbe a
simplicial approximation to the identity map |I2r| C |Ir|. Then = <p^:
l2r К for some if — el  •  02r and f — £'.
Proof Let Oj = (ui-i,Fi) for 0 < i < r. Then 024-1024 = (14.|,С;)(ад) for
a vertex £»• which equals either or u». By lemma 13, 024-10'2» ~ 0» and
Г ~ ? • 
We are now ready for the main result on the edge-path groupoid.
16 theorem For vertices vq, V i £ К the f unction
p: homg (n,r0) homg> (t>i,t>0)
is a bijection.
proof In view of lemma 12, it only remains to prove that p is injective.
Assume that f and f' are edge paths from v0 to vi such that |<p?| ~ [Wj'| rel L
By juxtaposing trivial edges (vt,vd) at the end of f or f' sufficiently often, we
can replace f and f' by equivalent edge paths having an equal number of
POLYHEDRA CHAP. 3
138
edges. Hence there is no loss of generality in assuming f and both to have f
edges. Then <pf, <p^: Ir-^> К are such that |<p?| ~ |<Prl rel 1. By theorem 3.5.6
there exists m such that if A: sdm Ir —> Ir is a simplicial approximation to the
identity, then <p?X and c/.j-X are in the same contiguity class. Now <p?X =
and y.j'X = for edge paths and in K. By lemma 15 (and induction i
on m), f ~ fi and f' — fi. From lemma 14 it follows that f । — fl. Therefore !
f ~ r 	;
If <p: Ki —> K2 is a simplicial map, there is a covariant functor >
<p#: S(Ki) —> &(K2) defined by	I
<P#[f] = [<p(f)]
where, if f = (u0,Ui)(ui,d2)  (ur_i,ur), then <p(f) = (<p(u0),<p(ui))
i),c/,(i:T)). It is trivial to verify that commutativity holds in the square
fe(Ki) —£(K2)
L?(|K1|) ^<3>(|K2|)	,
Therefore we have the following result.
j
I f corollary On the category of pointed simplicial complexes К with !
base vertex Vq, p is a natural equivalence of the covariant f unctor E(K,vq} [
with the covariant functor tt(|K|,i;o). 	i
Note that &(K) is determined by the 2-skeleton of K; that is, the edge
paths of К are determined by pairs of vertices of К which are vertices of a ’
simplex, and the equivalences between edge paths are determined by triples
of vertices which are vertices of a simplex. Hence &(K2) ~ &(K).	!
t
18 corollary For any pointed simplicial complex (K,vf), the inclusion j
map К2 С К induces an isomorphism	vJ
77(|K2|,Co)	77(|K|,Oo) 	I
If s is a simplex of K, any two of its vertices belong to the same com- |
ponent of S(K). Therefore the components {Z^} of &(K) define a partition
of К into subcomplexes {Kj}, called the components of K, defined by '
Kj = {s £ К | s has some vertex in /^}. К is said to be connected if it contains i
exactly one component.
19 theorem If {Kj} are the components of K, then {|Kj|} are the path
components of |K|.	j-
proof If о is a vertex of K, then st v is path connected and so belongs to; ’
the same path component of |K| as v. It follows from theorem 16 that two vertices7
of К are in the same component of °P(|K|) if and only if they are in the same-
component of £(K). Therefore, if {E3} is the set of components of &(K), the
sl.(. 7 graphs	139
path components of |K| are the sets {//} defined by Pj = U{st v | v 6 Ej}.
Clearly, Pj - |K;|. 
7 GliAPMS
\Ve show how a system of generators and relations for the edge-path group
E(K,«o) can be determined. This provides a method for finding generators
and relations of the fundamental group of a polyhedron. Since every edge
path of К is an edge path of the one-dimensional skeleton of K, we start with
a description of the edge path group of a one-dimensional complex.
A one-dimensional simplicial complex is called a graph. A tree is defined
to be a simply connected graph.
| lemma A graph is a tree if and only if it is contractible.
proof Since a contractible space is simply connected, a contractible graph
is a tree. Conversely, let К be a tree and let ao be a point of |K|. We shall de-
fine a homotopy F: |K| X Z -> |K| from the identity map 1 of |K| to the con-
stant map c of |K| to ao- Since |K| is path connected, for each vertex v of К
there is a path	in	|K| from v to	ao. We	now define F on v X I by F(u,t) =
wv(t'). For every	1-simplex s of	К the	map F is defined	on the subset
(|s| X 0) U (|s| X	1)	U (|s| X I) of	|s| X I-	Since |K| is simply	connected and
(|s| X I, (|s| X 0)	U	(|s| X 1) U (|s| X I))	is homeomorphic	to (E2,SX), it
follows that F can be extended over |s| X I- In this way we obtain a map
F: |K| X I —> |K| whose restriction to each |s| X I is continuous. Then F is
continuous and F: 1 ~ c. 
2 lemma Let К be a connected simplicial complex. Then К contains a
maximal tree, and any maximal tree contains all the vertices of K.
proof The collection of trees contained in К is partially ordered by inclu-
sion. Let {Kj} be a simply ordered set of trees in К and let T = U Kj. We
show that T is a tree. Since Kj is one-dimensional, T is one-dimensional.
Since {Kj} is a simply ordered set of trees, it follows that any finite subcom-
plex of T is contained in some Kj. To show that T is simply connected, let f:
.S’ —> |T|, where i = 0 or 1. Then/(S’) is compact and is therefore contained
in |Kj| for some j. Since |K;| is simply connected, the map/: S’ |/<;| can be
extended to a map/': E’+1	|/</| C |T|, and |T| is simply connected.
As a result, every simply ordered set of trees in К has a tree as upper
bound. Zorn’s lemma can be applied to yield a maximal tree in K. If T is a
maximal tree of К and does not contain all the vertices of K, it follows from
the connectedness of К that there is a 1-simplex {/1,1:2} £ К with t‘ i 6 7' and
c2 4 T. Let 7} = T U {{/2}, {v\,v2}}. Since vi is a strong deformation re-
tract of |{t>p,t?2}|, |Г | is a strong deformation retract of |Tp|. Therefore |Tp| is
contractible, and so 7} is a tree strictly larger than T, contradicting the max-
irnality of T. 
140
тетхивокА. chap. 3
It follows from lemma 2 that if К is a connected complex and Г is
a maximal tree in K, then К — T consists of simplexes of dimension > 1.
Because | T | is contractible, any edge path in К is determined by its part in
К — T, as we shall see below. This motivates the following definition.
Let T be a maximal tree of the connected complex K. Let G be the
group generated by the edges of К with the relations
(o) If (0,0') is an edge of T, then (v,v') = 1.
(b) If v, v', and v" are vertices of a simplex of K, then (VjV'jly' ,v") = (y,v").
3 theorem With the notation above, E(K,Vq) G.
proof Since T is connected and contains the vertices of K, for each vertex
v of К there is an edge path in T such that orig = t;0 and end — v. If
is an edge of K, the edge path	is a closed edge path in К at
vQ. Therefore there is a homomorphism a of the free group F generated by
the edges of К into E(K,v0) defined by a(v,v') =
We show that a can be factored through G. If (y,v') is an edge of T, then
is a cl°se<3 edge path in T. Because T is simply connected,
= 1 and a sends relations of type (a) into 1. If v, v' and v" are
vertices of a simplex of K, then
0 [WM = [«iwXWX'-1]
= [ЦЛ '“4
Therefore a((v,v')(v',v"')) = a(v,v"), and so there is a homomorphism
a'-. G —> E(K,Uo) such that a'(v,v') = a(y,v') =
To prove that a' is an isomorphism we construct an inverse /3': E(K,vq)	Q
as follows. For each closed edge path f = e±  • • er let /1(f) = er •   er,
where the right-hand side is interpreted as a product in G. If f and J'
are simply equivalent, then because of the relations of type (b), /1(f) = jG(f').
Therefore there is a homomorphism (3'-. E(K,v0) G such that j6'[f] =	,
We show that a' and /3' are inverses of each other. Given an edge path
f = (u0,ui)(ui,u2) •   (vnv0), then = [Г], where
£ — ^*vo(UO,Ul)^*v1 1{.vi(Fi,O2)£v2 1   • £ vr(vr,Vq)£v0 1
~ ^(UO,U1)(U1,«2) • • • (Vo)k1
Since is a closed edge path in the simply connected complex T, G(1 ~ 1
and — f. Therefore a'(3’ is the identity on E(K,Vq).
Consider /3'a'(v,v') = /3^v')(v,v'')/3(^v^1'). Since	and are paths in
T, they are products of edges in T. Hence (3^v) and ') are both equal to j
1 by relations of type (a). Therefore /3'a'(v,v') = (0,0'), and since {(0,0')} :
generate G, f3'a’ = 1 on G. 	;
In case К is finite, there are only a finite number of edges of K, and G |
is finitely generated. Similarly, there are only a finite number of relations of
type (a) or (b). G is thereby represented as the quotient group of a finitely4,
generated free group by a finitely generated subgroup. Hence we have the
following corollary.
,jgC. 7 GRAPHS	141
,ц corollary If К is a finite connected simplicial complex, then E(K,u0)
js finitely presented. 
5	corollary If К is a connected graph, E(K,vo) is a free group, and if T
is a maximal tree in K, the number of generators of E(K,vo) is in one-to-one
correspondence with the 1-simplexes of К — T.
proof Consider the group G. Because of relations of type (a), we need only
consider edges of К not in T. Every such edge e corresponds to an order of
the vertices of some 1-simplex of К — T. The relations of type (b) imply that
the oppositely ordered edge equals e1 in G. Therefore the group G is gener-
ated by edges one for each 1-simplex of К — T. There are no relations
on these generators of G, for if v, v', and v" are vertices of a simplex of K,
then at least two of them are equal. If v = v' or v' = v", the corresponding
relation of type (b) is the trivial relation 1(и',о") = (t>',t>") or (i;,i;')l = (t;,i;').
If v — o", the corresponding relation is (i;,t;')(i;',i;) = 1, which, in terms of
our generators, becomes ее-1 = 1. 
6	example Let J — {/} be a set and let X be the pointed space which is
the sum (in the category of pointed spaces) of pointed 1-spheres {/У
Each Sj1 can be triangulated by s,, where s,; is a 2-simplex Sj = {vpvfv") and
t>j corresponds to the base point of Sfi. Then X can be triangulated by the
complex К with vertices
{o} U
and 1-simplexes
{{M)W u	U
Let T be the maximal tree in К such that К — T consists of the collection
By corollary 5, E(K,v) is a free group on generators in one-to-one
correspondence with J. Therefore there is an isomorphism of tt(X,xo), where
io corresponds to v, with the free group generated by J.
7	example Let X be the complement in R2 of a set of p disjoint closed
disks or points. Then X has the same homotopy type as the sum of p pointed
1-spheres. Therefore the fundamental group of X is a free group with p
generators.
For connected graphs the fundamental group functor is a faithful repre-
sentation of the category of their underlying spaces and homotopy classes by
means of groups and homomorphisms. This is summarized in the following
theorem.
8	theorem Let /<, and К-г be connected graphs and let vq be a vertex of
Ki. Then
(a)	Any continuous map f: \K\| —> |K2| is homotopic to a continuous
map f: |Kx| —> |K2| such that /'(«o) is a vertex of K2.
(b)	If v'o is any vertex of K2 and h:	—> 7t(|K2|,uo) is an
arbitrary homomorphism, there exists a continuous map f: (|/<i|,i:o) —>
142
POLYHEDRA CHAP. 3
(|K2|,fo) such that h — fa.
(c)	Let v'o and if be vertices of K2 and assume that fa, fa: |Kp| —> |K2j
are maps such thatfa(v0) = v'o and /2(o0) = fa'- Then fa ~ fa if and only
if there is a path w in |K2| from v'o to vg such that the following triangle
is commutative:
I Ki |>«o)
'•/ v-=
w(IK21,Uo) ^(\K2\,v'o)
proof Since K2 is connected, it is path connected, and (a) follows from the
fact that the pair (|Kp|,oo) has the homotopy extension property with respect
to |K2| (by corollary 3.2.5),
To prove (b), let T be a maximal tree in Ki- Let {/;) be the simplexes of
Kp — T and for each j let e, = fafa) be an edge whose vertices are the
vertices of Sj in some order. For each vertex v in Kp there is an edge path
in T from Vo to v. For each j let
Wp =
Then [wj £ 7т(|Кр|,по)> and by corollaries 5 and 3.6.17, the set {oj;} is a sys-
tem of free generators of тт(|Кр|,оо)- For each /' let w) be a closed path in |Кг|
at v'o such that Ь[соД = [ыД. We define a continuous map
f: (|Kp|,u0)	(|K2|,o&)
by/(|7)) = v'o, and for each j we define/] |sy| by
fltv' + (1 - t)Vp) = co)(t)
where the points of |.Sj| are written in the form tuj + (1 — t)Vj for t £ I.
f is continuous because its restriction to | T | and to each |.s;| is contiiju-
ous. Clearly, fa[coj] = [юД = Ь[юД; therefore fa = h.
To prove (c), note that if fa ~ fa, there is a path w in |K2| from
Co to v'o such that, by theorem 1.8.7, fa# = h#:f2#. Conversely, if fa# =
h^fa#, let T be a maximal tree in Kp and for each vertex v of Kp
let fa be an edge path in T from o0 to v. We shall define F: |Kp| X I -> |K2|,‘
a homotopy from fa to fa, in several stages. First we set F(x,0) = fa(x) and !
F(x,l) = fa(x) for x £ |Kp|. Then F has been defined on (|Kp| X 0) U (|Kp| X 1)<
If о is a vertex of Kp, we define F(o,t) = ((/i(|ftX x|) * co) * /2|fJ)(l) for t £l.
Then F(o,0) = fafv) and F(o,l) = fa(v), and F is thus defined on |Kp°| X I to
be consistent with its previous definition on (|Kp| X 0) U (|Kp[ X 1)- И only j
remains to extend F over |.s| x I for each 1-simplex s £ Kp. Let v and v' be
the vertices of s in some order. Then |.sj X I is a square with the following^
product, arbitrarily associated, as boundary
SEC- 8 EXAMPLES AND APPLICATIONS	143
(|(u,o')| X 0) * (o' X I) * (|(u',n)| X 1) * (г X I)-1
F can be extended over |s | X I if and only if F maps this product into a null
homotopic path of |K2|. By the definition of F, the above path is sent into a
path homotopic to the following product associated arbitrarily
ДКо.о')! *	* 6J *f2|(o',o)| * (f2|fWX| * W-1 *
^/i|(u,o')| *	* (w *fz(|^| * |(o',o)| *	* W-1)
*»~1I */i(M * IMI * M
ЕЛ(П
Therefore Fcan be extended over |s| X I, and the resulting map F: |Kr| X I~^
|XZ| will be continuous, because for each closed simplex |.s| of /<i its restric-
tion to |s| X I is continuous. Then F: Д ~ /2. 
It follows from theorem 8b that if f: (|/<i|,l;0) —> (|-K2|,f6) induces an iso-
morphism f#: 7t(|Ki|,o0) ~ 7r(|K2|,ob), then there is a continuous map g:
(|K2|,nb) —» (|Ki|,no) such that g-~ = (J#)-1. By theorem 8c, it follows that
gf— 1Щ and/g ~ 1щ. Hence we have the next result.
f) corollary Let K\ and K2 be connected graphs with o0 a vertex of f<i
and «о a vertex of K2. A continuous map f: (|Ki|,o0) —» (|K2|,t>b) is a homo-
top ij equivalence if and only if f induces an isomorphism f#: tt(|Ki|,Oo) ~
й'(|К2|,Оо)- 
The step-by-step extension procedure used to construct the homotopy F
to prove theorem 8c is a standard method for constructing continuous maps
on the space of a complex. The map is constructed on one skeleton at a time
and extended over the next skeleton.
8 EXAMPLES AXB APPLICATIONS
This section contains assorted results concerning the fundamental group. We
begin with some applications to the theory of free groups; in particular, we
show that any subgroup of a free group is free. Next we consider the effect
on the fundamental group of attaching 2-cells to a space. We use the result
obtained to prove that any group is isomorphic to the fundamental group of
some space. Finally, we describe how the fundamental group of a surface can
be represented by means of generators and relations.
If К is a simplicial complex and a E |K| has carrier s (that is, a E («}),
then for any subdivision K' of s the simplicial complex K' * a is a subdivision
of S (by lemma 3.3.8). It follows that a modified barycentric subdivision of К
can be constructed whose vertices are a and the barycenters of simplexes of
f other than s. Therefore there is a subdivision of К having a as a vertex,
and we have the following result.
144	POLYHEDRA CHAP. 3
I lemma If a £ |K|, there is a subdivision K' of К having a as a
vertex. 
2 theorem A polyhedron is locally contractible.
proof In view of lemma 1, it suffices to prove that if о is a vertex of
a simplicial complex K, every neighborhood U of v in |7<| contains a neighbor-
hood V of v deformable in U to v. Let U be a neighborhood of v and let
A = st v. Define F: A X I |K| by
F(a,t) = tv + (1 — t)a
Then F is a deformation of A in |К| to the point v, and F(y X T) = v £ U,
Therefore there is some neighborhood V of v in A such that F(V X I) С V,
Because A — st v is open in |К|, V is a neighborhood of v in |K|. Since
F | V X I is a deformation of V in U to v, |К| is locally contractible. 
It follows from theorem 2 that the theory of covering projections applies
to polyhedra, and corresponding to any subgroup of the fundamental group
of a polyhedron there is a covering projection. We show that any covering
projection of a polyhedron corresponds to a simplicial map.
3 theorem Let p: X —> X be a covering projection, where X is a poly-
hedron. Then X is a polyhedron, of the same dimension as X, in such a way
that p corresponds to a simplicial map.
proof Assume that p: X |K| is a covering projection. For any simplex
s £ К the closed simplex |.sj is simply connected. It follows from the lifting
theorem that the inclusion map |s| C |K| can be lifted to a map |s| —> X, and
it follows from the unique lifting theorem that two such liftings are either
identical or have disjoint images. Hence there are as many liftings of |.sj as
sheets of X over |s|.
Define a simplicial complex К to have the collection {p-1(«) | v is a
vertex of !<} as vertex set and to have simplexes {s}, where s = {vq,  . . ,6a}
is a simplex of К if and only if there is a simplex s = {t'o> • • • ,vq} in К and
a lifting fi: |s| —> X of |s| such that = «5, for 0 < i < q [in which case
s = p(s) and fs are both unique]. Then К is a simplicial complex and has the
same dimension as K. If Si is a face of s, then p(sf) is a face of p(s) and
/s | |p(«i)| = fsi- Therefore the collection {ffii! defines a continuous map
f: |K| —> X such that
/(SajOi) = fs(Zaip(vi')') StqOj £ |s|
Let <p: К —> К be the simplicial map <p(o) = p(o). Then there is a commuta-
tive triangle
|K| 4 x
|£|
fS EXAMPLES AND APPLICATIONS	145
To complete the proof it suffices to prove that (K,f) is a triangulation of
fi (that is, that f is a homeomorphism). If о is a vertex of K, then st v,
|eing contractible, is evenly covered by p. For v £ let be the
component of p ](st «) containing v. Then p | O',- is a homeomorphism of Uj
onto st v. By the definition of К and <p, |<p| | st v is a homeomorphism of st v
i ()П[о st v for v £ p-1(«). From the commutativity of the above triangle,
i (| st v is a homeomorphism of st v onto U» for v £ p-1(«). Since |<p|-1(st v) =
j (J {st v | v £ /)1(с)}, у | |<p|-1(st v) is a homeomorphism of |<p|1(st v) onto
! p-i(st v). Since this is so for every vertex v of K, fis a homeomorphism of |K|
I pntoX. "
i The following corollary is an interesting application of these results.
: I corollary Any subgroup of a free group is free.
j proof Let F be a free group. It follows from example 3.7.6 that there is a
i polyhedron (in fact, a wedge of 1-spheres) X with base point .гр such that
i rfX,Xo) ~ F. Let F be any subgroup of F. Under the above isomorphism F
i’ corresponds to some subgroup H C fiX,xf). Let p: X —> X be a covering pro-
jection such that X is path connected, p(x0) = x0> and p#w(X,Xo) — H. By
theorem 3, X is homeomorphic to the space of a connected graph. By corol-
lary 3.7.5, tt(X,^o) is a free group. 
If К is a finite connected graph, it follows from corollary 3.1.5 that
! E(K,«o) is a free group on 1 - t»o + «1 generators, where no is the number
of vertices of К and is the number of 1-simplexes of K. If p: X —> |jK| is a
i covering projection of multiplicity m, the number of «/-simplexes in the corre-
! spending triangulation (K,f) of X (given by theorem 3) equals mnq, where nq
• is the number of «/-simplexes of K. Therefore the method used to prove
' corollary 4 also yields the following result.
 5 corollary Let F be a free group on n generators and let F be a sub-
‘ group of F of index in. Then F is a free group on 1 — m + mn generators. 
We now investigate the effect on the fundamental group of the process
of attaching cells. Let A be a closed subset of a space X. X is said to be
obtained from A by adjoining n-cells {ef}, where n > 0, if
' (a) For each j, ef is a closed subset of X.
(h) If ef = ef A A, then for j =^= j', ef — ef is disjoint from e/” — ef1.
(с) X has the topology coherent with {_A,ef}.
(d) For each j there is a map
fi: (E«,S”-i) (ef,ef)
such thaty(E”) = ef,fi maps En — S’1^1 homeomorphically onto ef — ef,
and every compact subset of ef — ef is closed in ef.
Note that if n — 0, X is the topological sum of A and a discrete space.
A map fi: (En,Sn~l) (ef,ef) satisfying condition (d) above is called a
146
POLYHEDRA CHAP. 3
characteristic map for e/", and/-1 S”-1: S”-1 —> A is called an attaching map,
for е/. X is characterized by A and the collection {/• | S”-1} of attaching
maps. Given A and an indexed collection of maps {g(-: S"-1	A}, there is
space X obtained from A by attaching n-cells {(:,-"} by the maps g,-. X js>
defined as the quotient space of the topological sum V Ef v A, where
Ejn = En for each j, by the identifications z 6 S/" 1 equals g/z) £ A. Then the
inclusion map (Efl,Sjn~1) С (V Ej" v A, V S/'1 v A) followed by the projec-
tion to (X,A) is a characteristic map /: (E/hS/1-1) —> (X,A) for an n-cell
ef1 = Жи)-
Following are two examples.
6 If К is a simplicial complex, |K4| is obtained from IK^1! by adjoining
q-cells {|s| | s is a (/-simplex of K}.
7 For i = 1, 2, or 4 let F{ be R, C, or Q, respectively, and for q > 0 let
Pq(Ft) be the real, complex, or quaternionic projective space of dimension q,
Pq(Fi) is imbedded in PQ+i(Fj) by the map [t0, b, • • • ,tq] [t0, h, • • . ,/,0]
for tj € Fi. Then Pe+i(F,-) is obtained from Pq(Fi) by adjoining a single (q + l)p
cell. If E<a+r>1 is identified with the space {(to,tn  • • ,/) € F;<;H | S|t,|2 < 1],
then a characteristic map /: (E(9+1)’,S(<?+1),_1) —> (PQ+1(Fj),PQ(Fj)) for this
single cell is defined by the equation
/(t0,tl, . . . ,tq) = [t0,tl,  . . ,tq, 1 - S |t,-|2]
8 lemma Let X be obtained from A by adjoining n-cells for n > 2. Then
for any point Xo € A the inclusion map i: (A,Xq) C (X,x(i) induces an
epimorphism
i#: -n(A,x0)	w(X,x0)
proof Let X be obtained from A by adjoining the n-cells {&/'}, and for each
/' let г/,- € e," — ef1 and let В,- be a neighborhood of t/j in e,” — e.j“ homeomor-
phic to En. Let w: (1,1) —> (X,x0) be a closed path at Xq. We show that w is
homotopic to a path in G = X — {///};. By the compactness of I, we can sub-
divide I by points 0 = to < ti < • • • < tn = 1 such that for 0 < i < n either
w[ti,ti+i] C U or w[ti,ti+1] C B, for some /'. If w[b,b+i] U w[ti+i,b+2] C
we can omit the point t.;+i from the subdivision of I to obtain another subdi-
vision of I with the same property. Continuing in this way we can obtain a
subdivision such that if w[ti,tj+i] C Bj, then neither w[b-i,b] nor wfb+bA+z]
is contained in Bj. It follows that w(tj) 7^ yy and w(h+1) Уз- For each such i,
because Bj — </,- is path connected and Bj is simply connected, w | [/;,/-;+i | is
homotopic rel } to a path contained in Bj — г/,-. Since altogether there
are only a finite number of such subintervals of I, и ~ u', where w'(J) C U.:S,
Because S’1-1 is a strong deformation retract of En minus a point, it follows
that is a strong deformation retract of e/! — г/,. Therefore A is a strong defor-
mation retract of U and w' w", where w"(f) C A. Then i#[w"] — [w]. -—
^gC. 8 EXAMPLES AND APPLICATIONS	147
corollary For all </>(), P„(C) and Pn(Q) are simply connected.
proof Because P0(C) and Po(Q) are each one-point spaces, the result follows
by induction on q, using lemma 8 and the fact that Pe+i(C) is obtained from
pe(C) by adjoining a 2(q + l)-cell and Pe+i(Q) is obtained from Pq(Q) by
adjoining a 4(q + l)-cell. 
We want to compute the kernel of i# for the case n = 2. Given any map
g: S1 —> A, where A is path connected, and given a point xp £ A, a normal
subgroup of я-(А,х0) is determined as follows. If g(po) = *i and w is a path in
p from xo to %i, then h[w]g#(w(S1,po)) is a cyclic subgroup of я-(А,х0), and for a
different choice of w we obtain a conjugate subgroup in w(A,Xo). Therefore
the normal subgroup of vr(A,Xo) generated by h[(J]g#(v7(S1,po)) is independent
of the choice of the path w. Similar statements apply to a collection of maps
(gy: S1 —A}. There is a well-defined normal subgroup of тг(А,Хо) determined
by these maps.
10 theorem Let Abe a connected polyhedron and let X be obtained from
A by attaching 2-cells to A by maps {gp S1 A}. If N is the normal sub-
group of тг(А,Хо) determined by the maps {g/}, then
i#: я(А,х0)	я(Х,х0)
is an epimorphism with kernel N.
proof By lemma 8, is a surjection. Let p: A A be a covering projection
such that A is path connected, p(xo) = xo, and p#(7r(A,x0)) = N. Because N is
normal in vr(A,Xo), p is a regular covering projection. Because N is the subgroup
determined by the maps {g;}, each map g; lifts to a map gp S1 A. Let X
be the space obtained from A by attaching 2-cells for all the lifted maps {g/}
and extend p to a map p': X —> X such that p' maps each 2-cell of X homeo-
morphically onto its corresponding 2-cell of X. Then p' is easily seen to be a
covering projection.
We know from the definition of N that i#(7V) — 1. Assume that
[w] £ тг(А,Хо) is in the kernel of i#. Let & be any lifting of w in A such that
w(0) = Xo- Then & is a lifting of w in X. Because w is null homotopic in X,
& is a closed path in X. Therefore <5 is a closed path in A, and so
[w] = p#[w] 6 N 
Note that for the proof of corollary 9 it was not necessary that A be a
connected polyhedron. It would have been sufficient to assume A path
connected, locally path connected, and semilocally 1-connected.
11 corollary For any group G there is a space X with w(X,x0) ~ G.
Proof Bepresent G as the quotient group of a free group F and a normal
subgroup N. There is a polyhedron A such that tt(A,Xo) ~ F (in fact, as in
example 3.7.6, A can be taken to be a wedge of 1-spheres). For each X £ N
POLYHEDRA CHAP. 3
148
let gx: (S1,jD0) —> (A,x0) be a map such that [gx] corresponds to X under the
isomorphism я-(А,х0) ~ F. Let X be the space obtained from A by attaching
2-cells by the maps {g/}- By theorem 10, there is an isomorphism 5т(Х,го) ~ Q. *
We now specialize to the case of a surface. These are the spaces of finite
two-dimensional pseudomanifolds without boundary. An n-dimensional f
pseudomanifold without boundary (or absolute n-circuit) is a simplicial com- I
plex К such that	i
(я) Every simplex of К is a face of some n-simplex of K.
(b) Every (n — l)-simplex of К is the face of exactly two n-simplexes of K,
(c) If .S' and s' are n-simplexes of K, there is a finite sequence
$ = sls «2, • • • , s-m = s' of n-simplexes of К such that s, and s.i+1 have
an (n — l)-face in common for 1 < i <( m.
We define a surface to be the space of a finite two-dimensional pseudo-
manifold without boundary. By scissors-and-paste techniques it can be shown1 j
that every surface can be represented in a normal form consisting of a polygon <
in the plane with suitable identifications of its edges. These fall into classes,
those with h > 0 handles and those with к crosscaps. The surface with
0 handles is the polygon with identifications of its edges pictured as
a
Surface with 0 handles
Topologically it is homeomorphic to the 2-sphere S2. For h > 0 the surface
with h handles is pictured as
The surface with one handle is topologically the torus.
1 See S. Lefschetz, Introduction to Topology, Princeton University Press, Princeton, N.J., 1949,
and H. Seifert and W. ThrelfaH, Lehrbuch der Topologie, B. G. Teubner, Verlagsgesellscliaft, \
Leipzig, 1934.
gxEBCiSES	149
For к > 1, the surface with к crosscaps is pictured as
Surface with к crosscaps
The surface with one crosscap is topologically the real projective plane P2,
and the surface with two crosscaps is topologically the Klein bottle.
The normal form represents a surface with h > 1 handles as a wedge of
2// 1-spheres with a single 2-cell attached by a suitable map. If A is the wedge
of 2h 1-spheres, then тг(А) is a free group on 2h generators, which generators
we denote by a, and b;, where 1 < i < n. If X is the'surface with h handles,
X is obtained from A by attaching a single 2-cell to A by a map g: S1 —> A
such that g# maps a generator of w(S1) to the element a^bya^b^1   
aftbhOii^bir1 € tt(A). Theorem 10 then provides a description of •nfX) in
terms of generators and relations. Similar remarks apply to a surface with
к > 1 crosscaps. The result is summarized below.
12 The fundamental group of a surface is
(o') Trivial for the surface with no handles.
(b) A group with generators «i, bi, ... , ah, bjt and the single relation
flibifli-1bi-1  • • ajlbtlair1bit~1 = 1 for a surface with h > 1 handles.
(c) A group with generators ci, c%, . . . , c^ and the single relation
Ci2C22 • • • Cfc2 = 1 for a surface with к > 1 crosscaps. 
EXE18CISES
A TOPOLOGICAL PROPERTIES OF POLYHEDRA
I Prove that a compact polyhedron is an absolute neighborhood retract. (Hint: Assume
X = |I<| and let К be a subcomplex of a simplex s. Use induction on the number of sim-
plexes in s — К and the fact that a retract of an open subset of an absolute neighborhood
retract is an absolute neighborhood retract.)
2	Give an example of a space X and closed subset А С X such that A and X are both
polyhedra but (X,A) is not a polyhedral pair.
3	Prove that an open subset of a compact polyhedron is a polyhedron. [Hint: Since
|K| — U is a Gs, there exists a sequence of open subsets V; of |K| such that
П V; = |K| — U. By induction on n, construct a sequence of subdivisions K„ and sub-
complexes Ln C Kn such that (a) Kn is finer than the covering {U,Vn}, (b) Ln is the
largest subcomplex of Kn such that |Ln| C U, and (c) Kn+i is a subdivision of K„ contain-
ing Ln as subcomplex. Then L' = U Ln is a simplicial complex such that |L| = |K| — U.]
i50	POLYHEDRA CHAP. 3 'J
4	Let У be an n-connected space and К be a simplicial complex. Prove that any con- 
tinuous map |I<| У is homotopic to a map which sends |Kn\ to a single point. If
fo, fi- (|K|,|K"|) —> (У,!/о) are homotopic, prove that they are homotopic relative to
5	Let У be a space which is n-connected for every n and let (X,A) be a polyhedral pair. :
Prove that two maps X У which agree on A are homotopic relative to A.
6	Prove that a polyhedron is contractible if and only if it is n-connected for every (i’
If it has finite dimension in, it is contractible if and only if it is m-connected.
В EXAMPLES
1	Prove that Pn is a polyhedron for all n.
2	Let К be the simplicial complex consisting of vertices «i, v2, . . . , vp and simplexes <
{t>i,t>2}> {t’zd’s}, • • • , {«р-i,fp}, and and let P be the simplicial complex wi^
0 and 1 as vertices and {0,1} as 1-simplex. Then К * I is a simplicial complex with
vertices t>i, . . . , vp, 0, and 1. If q is an integer relatively prime to p and o, is defined
for all integers i to be equal to Vj if i = / mod p, then let X be the space obtained from
|I< * I| by identifying the 2-simplex {ui,ui+i,0} linearly with the 2-simplex {oi+Q,ui+g+1,l}
for all i. Prove that X is homeomorphic to the lens space L(p,q) and that X is a polyhedron,
3	Prove that the generalized lens space L(p, q±, . . . ,qn) is a polyhedron.
4	If X and У are polyhedra and one of them is locally compact, prove that X *-Y and
X X У are also polyhedra.
C PSEUDOMANIFOLDS
A simplicial complex is said to be homogeneously n-dimensional if every simplex is a face
of some n-simplex of the complex. An n-dimensional pseudomanifold is a simplicial*
complex К such that
(a)	К is homogeneously n-dimensional.
(b)	Every (n — l)-simplex of К is the face of at most two n-simplexes of K.
(c)	If s and s' are n-simplexes of K, there is a finite sequence s = sb s2, . . . , sm = s'
of n-simplexes of К such that and s;+i have an (n — l)-face in common (or
1 < i < in.
The boundary of an n-dimensional pseudomanifold K, denoted by K, is defined to be
the subcomplex of К generated by the set of (n — l)-simplexes which are faces of exactjy
one n-simplex of K. (If К is empty, then К is an n-dimensional pseudomanifold without
boundary, as defined in Sec. 3.8.)
1	Prove that an n-simplex is an n-dimensional pseudomanifold whose boundary, as a
pseudomanifold, is s.
2	If К is a pseudomanifold and L is a subdivision of K, prove that L is a pseudo-
manifold and L — L | K.
3	If К is a finite 1-dimensional pseudomanifold, prove that К is either empty or con-
sists of exactly two vertices.
4	Give an example of an n-dimensional pseudomanifold К such that К is neither
empty nor an (n — l)-dimensional pseudomanifold.
D SIMPLICIAL MAPS
In the first four exercises К will be a finite n-dimensional pseudomanifold, where n > 0,
(JXUICISES
qth nonempty boundary К, K' will be a simplicial subdivision of K, and <p: К' —> К will
|jc a simplicial map such that </. | K’ maps K' to К and is a simplicial approximation to
{be identity map |R"| C |I<|. Furthermore, s"1 will be a fixed (n — l)-simplex of К and
jii will be the unique n-simplex of К having s" 1 as a face.
। y’or each n-simplex s' of K' let n(s') be the number of (n — l)-faces of s' mapped
onto s"1 by Ф- r>rove that a(s') = lif and only if <[. maps s' onto sn and that a(s') — 0
,or2 Otherwise.
2 Prove that the number of n-simplexes of K' mapped onto s" by <p has the same parity
as the number of (n — l)-simplexes of K' mapped onto s'1”1 by <p. [Hint: They both have
the same parity as S a(s'), the summation being over all n-simplexes s' of K'J
Л Sperner lemma. Prove that the number of n-simplexes of K' mapped onto sn by <p is
odd' (Hint: Use induction on n.)
,| prove that |I<| is not a retract of |K|.
S Brouwer fixed-point theorem. Prove that every continuous map of E” to itself has a
fixed point.
E simplicial mapping cylinders
Let <p: К —> L be a simplicial map between simplicial complexes whose vertex sets are
disjoint. We assume that the vertices of К are simply ordered. The simplicial mapping
cylinder M of <p is the simplicial complex whose vertex set is the union of the vertex sets
of К and L and whose simplexes ar e the simplexes of К and of L and all subsets of sets
of the form {to, . . . ,vk, <p(r'7.), . . . ,<p(vp)}, where {vo>Vb •  • >vp} is a simplex of К
arid Vo < hr < • • • < vp in the simple ordering of the vertices of K.
1	Prove that the inclusion maps i: К С M and j: L С M are simplicial maps. If
p:M—?L is defined by p(v) = <p(v) for v a vertex of К and p(v') = v' for o' a vertex of L,
then prove that p is a simplicial map such that <p = p ° i and p ° j = I/,.
2	If К is finite, prove that / ° p and lj/ are contiguous.
.‘I Prove that |M| is homeomorphic to the mapping cylinder of the continuous map
W |K|	|L|.
F EDGE-PATH GROUPS
I Prove that if К is a simplicial complex, there is a one-to-one correspondence between
equivalence classes of local systems on |I<| with values in G and natural equivalence
classes of covariant functors from the edge-path groupoid of К to (?.
2 Van Kampen’s theorem for simplicial complexes.1 Let К be a connected simplicial
complex with connected subcomplexes Li and L2 such that Li П L2 is connected and
К = Li U L2. Let Co be a vertex of Li Я L2 and let ii: (Lj Я L2, v0) C (Li,v0) and
1'2: (Li Я L2, Vo) L (L2,Vo). Prove that E(K,Vo) is isomorphic to the quotient gr oup of the
free product of E(Li,v0) with L'(I.2,i:()) by the normal subgroup generated by the set
((MW ° (Mfr1) I и e E(L1 n l2> «0)}
3 If G is a finitely presented group, prove that there is a finite connected two-dimen-
sional simplicial complex К whose edge-path group is isomorphic to G.
1 For the topological case see P. Olum, Non-abelian cohomology and Van Kampen’s theorem,
Annals of Mathematics, vol. 68, pp. 658-668, 1958.
152	POLYHEDRA CHAP, 3 T
•1 Let X be a space with base point л’о € X. Prove that there exists a polyhedron у I
with base point y0 £ Y, and a continuous map f: (Y,y0) —> (X,x0) such that
/*: 77(Y,i/o) ~ w(X,x0)-
<S NERVES OF COVERINGS
If 9l = {U } in an open covering of a space X and K(Ql) is its nerve, a canonical map
f: X |K(d?l)| is a continuous map such that /1(st U) C U for every U g <?l.	'
I If 91 is a locally finite open covering X, prove that there is a one-to-one correspond- ;
ence between canonical maps X —> |K(Ql)| and partitions of unity subordinate to 9l,	:
2 If 91 is a locally finite open covering of X, prove that any two canonical maps
X —> |K(9l)| are homotopic.	;
If and T are open coverings of X, with Ta refinement of *?l, a canonical projection from '
T to Ql is a function <p which assigns to each V g T an element <p(V) g V such that
v c <p(V).	;
3 Prove that a canonical projection from Tto 01 defines a simplicial map K(Y~) —> R'(9() ‘
and any two canonical projections from T to 9l define contiguous simplicial maps 1
K(<Y)	K(<?l).
4 If tp: K(T) —> is a canonical projection and/: X |K(T)| is a canonical map; [
prove that the composite |<p| ° /: X —> |K(0l)| is a canonical map.
5 Let X be a paracompact space and let g: X —> |I<| be a continuous map (where К is 
a simplicial complex). Prove that there exists a locally finite open covering 01 of X and a
simpficial map K(0l) К such that for any canonical map /: X |К(91)| the com- |
posite |<p| ° /is homotopic to g. [Hint: Choose “51 to be any locally finite open refinement >
of the open covering {g1(st o) | v a vertex of K}, and for U g Ql choose rp(U) a vertex }
of К such that U C g-1(st <p(U)).]	j
в Let X be a compact Hausdorff space and let К be a simplicial complex. Prove that !
there is a bijection	;
linu {[K(9l);K]} S [X;|I<|]
where the direct limit is with respect to the family of finite open coverings of X directed ।
by refinement with maps induced by canonical projections and the bijection is induced i
by canonical maps.
H DIMENSION THEORY	j
A topological space X is said to have dimension < n, abbreviated dim X < n, if every 1
open covering of X has an open refinement whose nerve is a simplicial complex of '
dimension < ft. If dim X < n but dim X 4 n — 1, then X is said to have dimension n,
denoted by dim X = n. If dim А' <р1 for any n, we write dim X = co.	;
1	If A is	a closed subset of X, prove that dim A	< dim X.
2	If К is	a finite simplicial complex with dim К	< n, prove	that dim |K|	<	n.
3	If s is	an n-simplex, prove that dim |s| = n.	(Hint: Let	be the open	covering of
|s|	of stars	of the vertices of s and assume that there is a refinement T	of	9|	such that
dim КГ-1') < n — 1. Let K' be a subdivision of s finer than cil. There are simplicial maps '
К' --> K(Y) s whose composite X is a simplicial approximation to the identity map |
|K'| c |4)	— 
I.XBRCISES	1S3
,j Let X be a paracompact space with dim X < n. Prove that any map X S'", with
, n, is null homotopic.
5 Let X be a compact metric space and let C be the space of maps /: X R2,,+ I
topologized by the metric
d(/,g) = sup {||/(4 - g(x)|| | x g X)
prove that C is a complete metric space, and if
Cm — (feC\ diam / /z) < ifor all z g R2,,+I}
then show that Cm is an open subset of C for every positive integer m and A Cm is the
set of homeomorphisms of X into R2n+1.
6 If X is a compact metric space of dimension < n, prove that Cm is a dense subset
of C for every positive integer m. [Hint: Let 9l be a finite open covering of X by sets of
diameter <Z 1/m such that dim f\(9l) < n and let h: |K(9t)| —> R2n+1 a realization of
K(9l). If/: X -» |K(9L)| is any canonical map, then h °/ g C,„. Given g: X-^ R2"+I and
given e > 0, show that it is possible to choose Ql and h as above, so that d(h ° /, g) < t-.j
7 If X is a compact metric space of dimension < n, prove that X can be embedded in
R2»+r (in fact, the set of homeomorphisms of X into R2"11 is dense in C).
CHAPTER FOUR
HOMOLOGY
THIS CHAPTER INTRODUCES THE CONCEPT OF HOMOLOGY THEORY, WHICH IS OF
fundamental importance in algebraic topology. A homology theory involves a
sequence of covariant functors H?i to the category of abelian groups, and we
shall define homology theories on two categories-the singular homology theory
on the category of topological pairs and the simplicial homology theory on the
category of simplicial pairs. The former is topologically invariant by definition
and is formally easier to work with, while the latter is easier to visualize
geometrically and by definition is effectively computable for finite simplicial
I complexes. The two theories are related by the basic result that the singular
homology of a polyhedron is isomorphic to the simplicial homology of any of
its triangulating simplicial complexes.
The functor Hn measures the number of “n-dimensional holes” in the
space (or simplicial complex), in the sense that the n-sphere Sn has exactly one
’ n-dimensional hole and no m-dimensional holes if m =^n. A O-dimensional
hole is a pair of points in different path components, and so Ho measures
path connectedness. The functors Hn measure higher dimensional connected-
ness, and some of the applications of homology are to prove higher dimensional
155
156
HOMOLOGY CHAP, 4
analogues of results obtainable in low dimensions by using connectedness
considerations.
Sections 4.1 and 4.2 are devoted to the definition of the category of chain
complexes and to an appropriate concept of homotopy in this category.
Homology theory is introduced as a sequence of covariant functors naturally
defined from the category of chain complexes to the category of abelian groups.
Simplicial homology theory is defined by means of a covariant functor
from the category of simplicial complexes to the category of chain complexes.
We study it in detail in Sec. 4.3, where it is shown that two different defini-
tions (one based on oriented simplexes, the other on ordered simplexes) are
isomorphic. In similar fashion, singular- homology theory is defined via a
covariant functor from the category of topological spaces to the category of
chain complexes. Its basic properties ar e considered in Sec. 4.4, where it is
shown that “small” singular simplexes suffice to define singular homology.
Section 4.5 introduces the concept of exact sequence. All the homology
functors Hn occur together in the exact sequences of homology, and it is for
this reason that we consider all these functors simultaneously, rather than one
at a time. Section 4.6 is devoted to the exact Mayer-Vietoris sequences con-
necting the homology of the union of two spaces (or simplicial complexes),
the homology of the spaces, and the homology of their intersection. We use
these to prove the isomorphism of the simplicial homology groups of a simplicial
complex with the singular homology groups of its corresponding space;
Section 4.7 contains some applications of homology theory. We prove (
that euclidean spaces of different dimensions are not homeomorphic. We also |
prove the Brouwer fixed-point theorem and the more general Lefschetz fixed- 
point theorem. Finally, we prove Brouwer’s generalization of the Jordan curve
theorem (that an (n — l)-sphere imbedded in S" separates S" into two com-
ponents), and we establish the invariance of domain. Section 4.8 contains a
discussion of the axiomatic characterization of homology given by Eilenberg
and Steenrod, as well as some related concepts.
1 CHAIN COMPLEXES	'
This section introduces the category of chain complexes and chain maps and
the homology functor on this category. We also define covariant functors from
the category of simplicial complexes and from the category of topological -
spaces to the category of chain complexes. The composites of these and the i
homology functor define homology functors on the category of simplicial I
complexes and on the category of topological spaces.	I
A differential group C consists of an abelian group C and an endomor- ;
phism 3: С C such that 88 = 0. The endomorphism 8 is called the differ- |
ential, or boundary operator of C. There is a category whose objects are '
differential groups and whose morphisms are homomorphisms соштнбй^
with the differentials.
SEC. 1 CHAIN COMPLEXES	157
For a differential group C there is a subgroup of cycles Z(C) — ker 8 and
a subgroup of boundaries B(C) = ini 8. Because 88 = 0, B(C) C Z(C). The
homology group H(C) is defined to be the quotient group
H(Q = Z(C)/B(C)
The elements of ЩС) are called homology classes. If z is a cycle, its homology
class in H(G) is denoted by {z}. Two cycles Zi and z2 are homologous, denoted
by Zi ~ z2, if their difference is a boundary, that is, if {^i} = {z2}.
If т: С C is a homomorphism of differential groups commuting with
the differentials, then т maps cycles of C to cycles of C and boundaries of C
to boundaries of C'. Therefore т induces a homomorphism
уДСНВД
such that t* [z] = {r(z)} for z g Z(C). Because (tit2)^. = ri,,. r2:J. , there is a
covariant functor from the category of differential groups to the category of
groups which assigns to a differential group C its homology group H(C) and
to a homomorphism т its induced homomorphism .
A graded group C = {Cq} consists of a collection of abelian groups Cq
indexed by the integers. Elements of Cq are said to have degree q. A homo-
morphism т: C —> C of degree d from one graded group to another consists of a
collection г = {rQ: Cq--> Cq+a} of homomorphisms indexed by the integers. We
shall omit the subscript in rq where there is no likelihood of confusion. It is
obvious that the composite of homomorphisms of degrees d and d' is a homo-
morphism of degree d + d', and that there thus is a category of graded
groups and homomorphisms (with each homomorphism having some degree).
It has a subcategory of graded groups and homomorphisms of fixed degree 0,
Because the sum of two homomorphisms from C to C of degree 0 is again a
homomorphism from C to C of degree 0, hom (C,C') is an abelian group
[hom (C,CZ) being the set of morphisms in the category whose morphisms are
homomorphisms of degree 0].
A differential graded group (sometimes abbreviated to DG group) is a
graded group that has a differential compatible with the graded structure
(that is, the differential is of degree r for some r). A chain complex is a differ-
ential graded group in which the differential is of degree — 1. Thus a chain
complex C consists of a sequence of abelian groups Cq and homomorphisms
8q". Cq —>
indexed by the integers such that the composite
is the trivial homomorphism. The elements of Cq are called q-chains of the
complex'. Most of the chain complexes we consider will have the additional
property that Cq = 0 for q < 0. Such a complex is said to be nonnegative.
A free chain complex is a chain complex in which Cq is a free abelian group
for every q.
158
HOMOLOGY CHAP. 4
For a DG group the group of cycles Z(C) is a graded group consisting of
the collection {Zq(C) — ker <;q}, and the group of boundaries B(C) is a graded
group consisting of {BQ(C) = im Эв+1}. The homology group H(C) is a graded
group consisting of {Hq(C) = Zq(C)/Bq(C)}.
A chain map 7: СC' (also called a chain transformation) between
chain complexes is a homomorphism of degree 0 commuting with the differ-
entials. Thus 7 is a collection {rq: Cq Cq] such that commutativity holds in
each square
Cq h C^
т4
cq % cq^
It is clear that there is a category of chain complexes whose objects are chain
complexes and whose morphisms are chain maps. It is also clear that if
C and C' are two objects in this category, hom (C,C) is an abelian group.
If т: C —> C' is a chain map, its induced homomorphism
is the homomorphism of degree 0 such that (7i.)q{z] = {'o/2')} f°r z € Zq(C).
The following theorem is easily verified.
I theorem There is a covariant functor from the category of chain com-
plexes to the category of graded groups and homomorphisms of degree 0
which assigns to a chain complex C its homology group 1.1(C) and to a chain
map г its induced homomorphism 7*. For any two chain complexes the map
т —> is a homomorphism from hom (C,G) to hom (H(C),H(C')). •
A subcomplex C' of a chain complex C, denoted by С С C, is a chain
complex such that Cq C Cq and Ъ'д — (’Q | C' for all q. There is then an inclu-
sion map г: С' С C consisting of the collection of inclusion maps {Cq C Cq).
There is also a quotient chain complex C/C = {Cq/Cq} with boundary oper-
ator induced from that of C by passing to the quotient. The collection of pro-
jections {Cq —> Cq/Cq] is the projection chain map C C/C'.
To describe a covariant functor from the category of simplicial complexes
to the category of free chain complexes, let К be a simplicial complex. An
oriented q-simplex of К is a «/-simplex s £ К together with an equivalence
class of total orderings of the vertices of s, two orderings being equivalent if
they differ by an even permutation of the vertices. If v0, m, . . . , vq are the
vertices of s, then	• • • ,vq] denotes the oriented «/-simplex of К
consisting of the simplex s together with the equivalence class of the ordering
</ t>i <f • •  </ vq of its vertices.
For q < 0 there are no oriented «/-simplexes. For every vertex v of К
there is a unique oriented О-simplex [o], and to every «/-simplex, with q > 1,<;
there correspond exactly two oriented «/-simplexes. Let Cq(K) be the abelian
group generated by the oriented «/-simplexes u® with the relations + Стг® = ®
SEC. 1 CHAIN COMPLEXES
159
if op‘ and огв are different oriented «/-simplexes corresponding to the same
q-simplex of K. Then C0(jK) = 0 for q < 0, and for q > 0 Cq(K) is a free
abelian group with rank equal to the number of (/-simplexes of K. If К is
empty, Cq(K) = 0 for all q.
We define homomorphisms й(;: Cq(jK) —> Ce_!(K) for q > 1 by defining
them on the generators by
Id) 39[«o,W, • • • ,«e] =	(— l)i[i.(l,i;1, . . . ,Vj, . . .
0<i<q
where	. ,oe] denotes the oriented (q — l)-simplex obtained
by omitting v,. If + o2q — 0 in Cq(K), then it is easily verified that
0g(oiQ) + 3e(^2Q) = 0 in Сд-|(/<). Therefore dQ extends to a homomorphism
from CQ(K) to C(;1 (/<). For q < 0 we define to be the trivial homomorphism
from Cq(K) to C(ri(7<). It is not difficult to show that 3Q3Q+i = 0 for all (/.
Therefore there is a free nonnegative chain complex C(K) = {CQ(K),0Q},
which is called the oriented chain complex of K. Its homology group, denoted
by H(K), is a graded group {Hq(K) = Hq(C(Kf)}, called the oriented homology
group of K. Hq(K) is called the qth oriented homology group of K.
If К is realized in some euclidean space, the oriented (/-simplexes of К
are «/-simplexes of К together with orientations, in the sense of linear algebra,
of the affine varieties spanned by them. The boundary of an oriented (/-simplex
is the sum of its oriented (q — l)-faces, with each face oriented by the orien-
tation compatible with that of the q-simplex, as shown in the diagrams.
Vs

An oriented q-cycle z of К is a “closed” collection of oriented q-simplexes,
with each (q — l)-simplex lying in the boundary of z the same number of
times with each orientation. Hq(jK) is the group of equivalence classes of these
q-cycles, two cycles being equivalent if their difference is a boundary. Thus
corresponds intuitively to the group generated by the q-dimensional
“holes” in |jK|.
It is convenient to add more generators and more relations to the chain
groups Cq(jK). If Vo, «i, . . . , vq are vertices (not necessarily distinct) of
some simplex of K, we define [t?o,iq, . . . ,rj £ Cq(K) to be 0 if the vertices
are not distinct and to be the oriented q-simplex as defined above if they are
distinct. Then equation (a) remains correct for these added generators (that is,
if the vertices	. . ,vq are not all distinct, the left-hand side of equa-
tion (a) is 0 and the right-hand side can also be verified to be 0).
160
HOMOLOGY CHAP. 4
If <p: Ki —> K2 is a simplicial map, there is an associated chain map
C(<p): C(jKj) —> C(K2) defined by
(b)	C(<p)([«0,«l, • • • ,t>g]) = [<p(t>o)>	g)]
Note that if v0, vlt . . . , vq are distinct vertices of some simplex of Кг, then
<p(«o), <p(«i),  - • , <p(oe) are vertices of some simplex of K2 but are not
necessarily distinct. Therefore the right-hand side of equation (b) above would
not be defined unless we had defined [o0,«i, . . . ,oe] as an element of Cq,
whether or not the terms t>j are distinct. It is easy to verify that C(<p) is
a chain map.
2 theorem There is a covariant functor C from the category of simplicial
complexes to the category of chain complexes which assigns to К its oriented
chain complex C(K). 
The composite of the functor C and the homology functor is a covariant
functor, called the oriented homology functor, from the category of simplicial
complexes to the category of graded groups. To a simplicial complex К
it assigns the graded group H(K) = {Hq(K) = Hq(C(Kf)}, and to a simplicial
map <p:	—> K2 it assigns the homomorphism : ЩК1} H(K2) of degree
0 induced by C(</): C(7<-|) —> C(7<2). If L is a subcomplex of K, and i: L С K,
then C(f): C(L') C(K) is a monomorphism by means of which we identify
C(L) with a subcomplex of C(f<).
We next describe the singular chain functor from the category of
topological spaces to the category of chain complexes. Let po, pi, p2, ... be
an infinite sequence of different elements fixed once and for all. For q > 0
let Д« be the space of the simplicial complex consisting of all nonempty sub-
sets of {po.pi, • • • ,pQ} (therefore Д« is the closed simplex |po,p1; • • • ,р9\).
For q > 0 and 0 < i < q + 1 let
4+i: Д« -> Д«+1
be the linear map defined by the vertex map
4+i(P>) = f p’
t Рз+i
Then е^+1(Д«) is the closed simplex |p0,pi, • • • ,pi,    ,pq\ in Д®+1 oppo-
site the vertex and direct computation shows that
3 tf 0 < j < i < q + 1, then 4+24+1 = 4+г4+1- 
Let X be a topological space. For q > 0 a singular q-simplex о of X is
defined to be a continuous map
о: № -> X
For q > 0 and 0 < i < q the ith face of o, denoted by a®, is defined to be
the singular (q — l)-simplex of X which is the composite	.__
u® o ° eql: Д'?-1 —> Д'? —> X
gEC, 1 CHAIN COMPLEXES
161
jt follows from statement 3 that
4	If q > i and 0 < j <i < q, then (o®)® = (a®)(i-1>. 
The singular chain complex of X, denoted by Д(Х), is defined to be the
free nonnegative chain complex Д(Х) = {Дв(Х),Эв), where Д9(Х) is the free
abelian group generated by the singular (/-simplexes of X for q > 0 [and
Д(;(Х) = 0 for q <Д 0], and for q > 1, is defined by the equation
ae(a) = S (-1)%®
0<i<q
This is a chain complex because 3Q3e+i = 0 is an immediate consequence
of statement 4. If X is empty, Д9(Х) = 0 for all q.
If f: XY is continuous, there is a chain map
Д(У): Д(Х) Д(У)
defined by Л( / )(<т) = f ° a for a singular (/-simplex о: Д'? X. Then Д(/) is
a chain map, and we have the following result.
5	theorem There is a covariant functor Д from the category of topologi-
cal spaces to the category of chain complexes which assigns to X its singular
chain complex Д(Х). 
The composite of the functor Д and the homology functor is a covariant
functor, called the singular homology functor, from the category of topologi-
cal spaces to the category of graded groups. To a space X it assigns the
graded group H(X) = {HQ(X) = Н9(Д(Х))} and to a map /: X —> Y it assigns
the homomorphism
f*:H(X)^H(Y)
of degree 0 induced by Д(/): Д(Х) Д(У). Hq(X) is called the qth singular
homology group of X. If A is a subspace of X and i: А С X, then the map
A(i): Д(А) —> Д(Х) is a monomorphism by means of which we identify Д(А)
with a subcomplex of Д(Х).
The category of chain complexes has arbitrary sums and products of
indexed collections. That is, if {O}jej is an indexed collection of chain com-
plexes, there is a sum chain complex (J) G and a product chain complex X G
in which (@O)Q = @Gq and (X G)q = X Gq for every q. It follows that
Ш)О) = ©Z0(O) and Bq(@G) = @Bq(G) and Ze(X О) = X ZQ(O)
and BQ(X G) = X Be(O) for all q. Therefore H(© O) = ©) H(O) and
B(X О) = X H(O).
6	theorem On. the category of chain complexes the homology functor
commutes with sums and with products. 
The category of chain complexes also has direct and inverse limits (whose
. (/th chain groups are appropriate limits of the qth chain groups of the factors).
162
HOMOLOGY CHAP. 4 j SEC. 2 CHAIN HOMOTOPY
7	theorem The homology functor commutes with direct limits.
proof Let {С“,т„0} be a direct system of chain complexes and let {C,ia} be
the direct limit of this system (that is, ia: Ca —> C, and if a < ft, then
ia = ip ° rft1: Ca —> CP C). Then {H(Ca),rKg} is a direct system of graded
groups, and we show that {/7(Cj,j,,,.} is the direct limit of this system.
We show that I.Зя of the Introduction is satisfied. Let {z} C Hq(C). Then
z = i„c,j for some ca E (C'%. Since
0 = dqZ — dqiaca = ftCf'c"
there is ft with a < fi such that rftTy/’c" — 0. Then 7aPca is a cycle of (C₽)e
and ip7aPca = i„c" = z. Therefore ip*	= {z}.
We show that 1.3b of the Introduction is also satisfied. Because we are
dealing with the direct limit of groups, it suffices to show that if {z'1} £ Н^Са)
is in the kernel of ia*, then there is у with « < у such that {z'1} is in the
kernel of rj*. If ia*{za} = 0, then iaza — SQ+1c for some c £ CQ+1. Because
c = ipcP for some Д we have iaza — ip3^+1cP. Choose / so that a, ft < /,
Then iy'(Tay'za — 7pY'd$+icP) = 0. Therefore there is у with у' < у such
that Tyi'(TaY'z“ — TpV'S^+iC^) = 0. Then 7 rtf = df+ilrprcP), so 7 J* {za} = 0. 
(
r
I
i
i
It is false that the homology functor commutes with inverse limits. We 1
present an example to show this.
8	example For any integer n > 1 let Cn be the chain complex with 1
(Ся)е = 0 if q 0 or 1 and (C,^	(C„)o equal to Z —> Z, where the homo-
morphism is multiplication by 2. For each n let r'“: Cn+1 Cn be the chain
map which is multiplication by 3 on each chain group, and for n < m define
r»”1: Cm Cn to be the composite rnm = тити+1 • • • т™”1. Then {Cn,rnm} is
an inverse system whose inverse limit is the trivial chain complex. Therefore ।
Я(;(Ит< {С1,,т?,г"}) = 0 for all q. However, H0(Cn) — Z2 for all n and ।
Н0(Сда) H0(Cn) for all n < m. Therefore linm {НО(СИ), тЛ) Z2.	;
2 CHAIN HOMOTOPY	' I
This section deals with homotopy in the category of chain complexes. For
free chain complexes we prove that contractibility is equivalent to triviality I
of all the homology groups. This leads to discussion of a method for construct- 7
ing chain maps and homotopies by a general procedure known as the method
of acyclic models. The section closes with a definition of mapping cone of a |
chain map and its relation to the chain map.	|
Let г, г': С C be chain maps. A chain homotopy D from т to r', de-
noted by D: т ~ t', is a homomorphism D — {DQ} from C to C' of degree 1
such that for all q
d'q+lPq +	— 7q — 7 q: Cq > Cq
163
If there is a chain homotopy from т to T, we say that т is chain homotopic to
t' and write т ~ r'. It is trivial that chain homotopy is an equivalence rela-
tion in the set of chain maps from C to C'. The corresponding equivalence
classes are denoted by [C;C'], and if т: C -s> C' is a chain map, its equivalence
class is denoted by [т].
I lemma The composites of chain-homotopic chain maps are chain
homotopic.
proof Assume D: т ~ r', where т, t': C C, and D: f ~ f', where
т, т': С C". Then
tD + Dr': С -z С C"
is of degree 1 and is a chain homotopy from rr to r'r'. 
It follows that there is a category whose objects are chain complexes and
whose morphisms are chain homotopy classes. A chain map г: С C is
called a chain equivalence if [r] is an equivalence in the homotopy category
of chain complexes. If there is a chain equivalence from C to C', we say that
C and C' are chain equivalent.
2 theorem If т, т': C C are chain homotopic, then
r* = 7^. H(C) ЩС')
proof Assume D: т ~ r'. For any z C. ZQ(C)
d^+1Dq(z) - 7q(z) - r'q(z)
showing that rq(z) ~ 7q(z) and t* [z] = 7* {z}. 
A chain contraction of a chain complex C is a homotopy from the iden-
tity chain map 1c to the zero chain map 0g of C to itself. If there is a chain
contraction of С, C is said to be chain contractible. C is said to be acyclic if
H(C) = 0 (that is, HQ(C) — 0 for all q).
3 corollary A contractible chain complex is acyclic.
proof Assume that C is a chain complex such that 1c ~ ()<. By theorem 2,
'	z/' '	”	and (0c)* = Oh(0. Therefore
— 0. 
with Cq = 0 for q 0, 1, 2 and
(Io)* ~ (Oc)*- However, (1с)* =
= Ощсу, which can happen only if H(C)
The converse of corollary 3 is false.
4 example Let C be the chain complex ч _____________________v
with C2 Ci Co equal to Z ^-> Z Z2, where a(n) = 2n,	= 0, and
ft(2m + 1) = 1. Then C is acyclic but not contractible. In fact, if D: lc — Cfr
were a chain contraction of C, then the homomorphism ft would have a right
inverse Do: Z2 Z, but any homomorphism Z2 Z is trivial.
If C is assumed to be a free chain complex, there is a converse of
corollary 3.
164
HOMOLOGY CHAP. 4
5 theorem A free chain complex is acyclic if and only if it is contractible.
proof We show that if C is an acyclic free chain complex, it is contractible.
For each q the map dg is an epimorphism of CQ to B0_i(C) = ZQ_i(C),i.
Because C'Q. | is free, so is ZQ~i(C), and there is a homomorphism
Sq-Г- Zq-l(C) —> Cq
which is a right inverse of dg. Then lc — sQ~i3Q maps Cg to ZQ(C), and we
define {Dg} by
= sp(lq — Sq—ldq)- Cg > Cg+±
Then
3g+rDg + Dq—idq = lcq ' Sg_jdq + Sg—l(lq — Sg^2^q—l)9g = 1(7Q
which shows that {DQ} is a chain contraction of C. 
The method of proof of theorem 5 is a standard one used to construct
chain maps and homotopies from a free chain complex to an acyclic chain
complex. We now extend it to obtain a general method of constructing chain
maps and chain homotopies, called the method of acyclic models. Repeated
application of this method will be made in subsequent discussions. We con-
sider a special version of the method of acyclic models which suffices for our
applications.1
A category with models consists of a category (? and a set 'Dll of objects
of C called models. Given a covariant functor G from a category with
models 9)1 to the category of abelian groups, a basis for G is an indexed col-
lection (gj £ G(Mj)}jeJ, where Mj £ 911 such that for any object X of C the
indexed collection
{G(/ )(&)he J,M>om Wj ,X)
is a basis for G(X). If G has a basis, it is called a free functor on 6 with models 9)1.
In this case, if h £ hom (X,Y), then G(/i) maps each basis element of G(X)
to some basis element of G(Y). Hence G is the composite of the covariant ,
functor which assigns to X the set {G(f )(&•) | / € J, f £ hom with the ’
covariant functor of example 1.2.2, which assigns to every set the free abelian
group generated by it.
Let G be a covariant functor from a category (? with models 911 to the
category of chain complexes. G is said to be free if Gg is a free functor to the
category of abelian groups.
6 example Let К be a simplicial complex and let (?(K) be the category i
defined by the partially ordered set of subcomplexes of К (as in example 1.1.11).	:
Let 91c(K) = {s | s £ K) be models for (?(K). We show that the covariant <
functor C which assigns to each subcomplex of К its oriented chain complex .
is a free nonnegative functor on £(K) with models 9ll(K) to the category of |
1A general treatment can be found in S. Eilenberg and S. Mac Lane, Acyclic models, Ainerb\
can Journal of Mathematics, vol. 79, pp. 189-199 (1953).
SEC. 2 CHAIN HOMOTOPY
165
chain complexes. For each model s of dimension q choose once and for all an
oriented «/-simplex o(s) which generates CQ(s). Then the indexed collection
I dim s — q}SeK is a basis for CQ. Hence Cq is free with models 91l(K).
f 7 example Let & be the category of topological spaces with models
|	= {Д'7 | q > 0} and let Д be the singular chain functor. Then Д is free and
! nonnegative on 6 with models 911. In fact, if £Q: Де С Де, then the singleton
{	{£0 € Д9(Д®)} is a basis for Aq.
Let G be a covariant functor on a category 6 to the category of chain
।	complexes. Then there are covariant functors Hq(G), for all q, from £ to the
i’	category of abelian groups that assign to an object X the group Hq(G(Xf). If
,	£ is a category with models 911, a functor G from G to the category of chain
i	complexes is said to be acyclic in positive dimensions if JIq(G(M)) = 0
I	for q > 0 and M £ 911. We now establish the main result dealing with
, the construction of chain maps and homotopies.
)	8 theorem Let G he a category with models 911 and let G and G be co-
1	variant functors from 6 to the category of chain complexes such that G is free
' and nonnegative and G is acyclic in positive dimensions. Then
‘	(a) Any natural transformation He,(G) H0(G') is induced by a natural
f chain map t: G-^ G'.
I (b) Two natural chain maps r, t': G —> G inducing the same natural
f transformation Ho(G) —> Ho(G') are naturally chain homotopic.
proof For every object X of G we must define a chain map r(X): G(X) —>
G'(X) [or a chain homotopy D(X): r(X) ~ r'(X)] such that if h: X —> У is a
morphism in G, then
;	r(Y)G(h) = G'(/i)t(X) [or L>(Y)G(7r) = G'(/i)E»(X)]
For q > 0 let (gj £ Gq(Mj)}jeJq be a basis for Gq, where Mj £ 911 for
, each j £ Jq. Then GQ(X) has the basis
j It follows that Tq(X) [or Dq(X)] is determined by the collection {Tq(Mj)(gj))jejQ
j and the equation
j (a)	T9(^)(^niiGQ(/ij)(gj)) = S?rjjGq(yij-)TQ(Mj){gj)
; or by the collection fDq(Mj)gj].jejq and the equation
(b)	Dq(X)^nijGq(fij)(gj)') = ^n^G'^f^D^M^g;)
j We shall define rq(X) by induction on q so that
J (c)	aTq(x) = tq_!(x)3
i and define DfX) by induction on q so that
i W	dDq(X) = rq(X) - r'(X) - Dq^X)d
166	HOMOLOGY CHAP. 4
Having defined 7j [or /Д-J for i < q, where q > 0, it suffices to define
for 1 € A so that
(e)	3tq(W(S) = Tq-itM^Sgj)	j
and to define DQ(Mj)(gj) for / £ JQ so that
(/)	- T'q(Mj)(gj) - Dq^M^)
since rq(X) [and Dg(X)] are then determined by equation (a) [or by (b)]. It will
then be true that те(Х) [and DQ(X)] are natural and will satisfy equation (c)	!
[and (d)].	•
Given a natural transformation tp; Uq(G) H^G'), the inductive definition
of т proceeds as follows. For q = 0 we define To(Mj)(^) for j C Jo to be any ele-	j
ment of G'o(M}) such that {ro(Afy)(g,-)} =	We use equation (o) to !
define 70(X) for all X. Then, for g € G0(X), {70(X)(g)} = ф(Х){д). Ь I
particular, for j £ Jx, To(Mj)(dgj) is a boundary in G'0(Mj). Hence we can define
Ti(Afi;)(gj) £ Gl(Mj) so that 37X(Mj)(g) = 7o(M7-)(8g;). We then use equation (a) J
to define 7X(X) for all X. Assuming n defined for i < q, where q > 1, so that /
equation (c) is satisfied, we observe that the right-hand side of equation (e) is
a cycle of Gq_i(Mf. Because q > 1, Hg_x(G'(My)) — 0, and we define
to satisfy equation (<e). We next define rg(X) for all X to satisfy equation (o).	'
This completes the definition of r.
Given т, r': G —> G' such that т and t' induce the same natural transfer- j
mation H0(G) —> Ip(G'), we define D0(Mj)(gj) for / £ Jo to be any element of
G'i (Л/;) whose boundary equals TofMj)(gf) — To(Afy)(g)). Then D0(X) is defined
for all X by equation (b). Assuming D, defined for i < q, where q > 0, so that
equation (d) is satisfied, we observe that the right-hand side of equation (/)
is a cycle of G'(Mj). Because q > 0, HQ(G'(M7)) = 0, and this cycle is a	(
boundary. We define DQ(Mj)(g) £ Gg+x(Mj) to satisfy equation (/), use equa-	[
tion (fi) to define Dq(X) for all X, and complete the definition of D. 	*
The last result provides another proof of theorem 5 for nonnegative com- ।
plexes. In fact, if C is a free nonnegative chain complex, let Q be the category
consisting of one object X and one morphism H and let C be regarded aS a
covariant functor on & with model {X}. Then C is a free nonnegative functor,
and if C is an acyclic chain complex, the functor C is acyclic in positive
dimensions. In this case, because Ip and 0c are chain transformations of С I,
inducing the same homomorphism of Iki(C) = 0, it follows from theorem 8
that Lc — Qc, and C is contractible.
There is a useful algebraic object (related to the mapping cylinder of (
Sec. 1.4) which we now describe. Let т: С C' be a chain map. The mapping
cone of t is the chain complex C = {CQ,dQ} defined by CQ = CQ_X © Cq and f
Эд(с,с') = ( —39_x(c), 7(c) + 8'e(c')) c € C9 i, c' € C'Q	|
The following result is trivial to verify.
9	lemma C is a chain complex, and if C and C' are free chain complexes,
so is C. 
SEC. 3 THE HOMOLOGY OF SIMPLICIAL COMPLEXES	167
The next theorem is the main reason for introducing mapping cones.
10	theorem A chain map is a chain equivalence if and only if its map-
ping cone is chain contractible.
proof Assume that т: С C' is a chain equivalence. There exist т': С C
and D: C —> C and D': С' —> C' such that D: t't ~ 1c and D': tt' Io.
Define D: С C by D(c,c') = (cx,c2), where
a = D(c) + t'D't(c) — t'tD(c) -j- t'(c')
c2 = D'tD(c) — D'D't(c) — D'^c')
A straightforward computation shows that D is a chain contraction of C.
Conversely, assume that D is a chain contraction of C. Define т': C' —> C
and D: С C and D': G —> C' by the equations
(t'(c'), -D'(c')) = D(0,c')
(D(c), •) = D(c,0)
Direct verification shows 7' to be a chain map and D: t't 1c and D': tt' ~
1(7, so г is a chain equivalence. 
Combining this with theorem 5 and lemma 9 yields the following result.
II	corollary A chain map between free chain complexes is a chain
equivalence if and only if its mapping cone is acyclic. 
3 THE HOMOLO6Y OF SIMPLICIAL COMPLEXES
This section begins with a discussion of augmented chain complexes and
their reduced homology groups. Next we define the ordered chain complex of
a simplicial complex and prove that it is chain equivalent to the oriented
chain complex. We use this result to show that simplicial maps in the same
contiguity class induce chain-homotopic chain maps. We also compute IIo(K)
in terms of the components of K. At the end of the section the relative
homology groups and the Euler characteristic of a simplicial pair are defined.
In the category of nonempty simplicial complexes any simplicial complex
P consisting of a single vertex is a terminal object. If К is a nonempty sim-
plicial complex, the functorial map К —> P has a right inverse. Therefore the
induced homology map H(K) —> H(P) has a right inverse. Because Hq(P) = 0
if <7 =A 0 and Ho(P) 7Z Z, it follows that there is an epimorphism Hq(K) —> Z.
Since Hq(K) — Ce(K)/'diCi(K), there is an epimorphism e: C’o(K) —> Z such
that еЭх = 0. Similarly, in the category of nonempty topological spaces X any
One-point space is a terminal object. The same kind of considerations yield an
epimorphism e: Ao(X) —> Z such that еЭх = 0. This motivates the following
definition of augmentation.
An augmentation (over Z) of a chain complex C is an epimorphism
js: Cq Z such that еЭх: Cx Co Z is trivial. An augmented chain complex
168
HOMOLOGY CHAP, 4
is a nonnegative chain complex C with augmentation. An augmentation e of
a chain complex can be regarded as an epimorphic chain map of C to the
chain complex (also denoted by Z) whose only nontrivial chain group is Z in
degree 0. For this chain complex Z, it is clear that He(Z) = 0 for q 0 and
that Ho(Z) = Z. Therefore e induces an epimorphism e* : Ho(Q Z. Hence
an augmented chain complex has a nontrivial homology group in degree 0.
The oriented chain complex C(K) of a nonempty simplicial complex К is
augmented by the homomorphism e: C0(K) —> Z defined by e([u]) = 1 for
every vertex v of K. The singular chain complex A(X) of a nonempty space X
is augmented by the homomorphism e: До(Х) —> Z defined by e(<r) = 1 for
every singular 0-simplex of X.
A chain map т: С C between augmented chain complexes preserves
augmentation if e' ° т = e: Сц —> Z. Note that т preserves augmentation if
and only if r.,. does—that is, if and only if e(. ° ts. =	: Ho(C) —> Z. There is
a category of augmented chain complexes and chain maps preserving
augmentation. A chain homotopy in this category is any chain homotopy
between chain maps in the category.
We want to divide out the functorial nontrivial part of H0(C)
of an augmented chain complex C. The reduced chain complex C of
an augmented chain complex C is defined to be the chain complex defined
by Cq = Cq if q 0, Co = her e, and	[note that 9i( C,) C Cq
because e3i = 0]. Thus C is the kernel of the chain map e: C Z. If т: C—+ C
is a chain map preserving augmentation, т induces a chain map C —> C'
between their reduced chain complexes. The homology group H(C) is called
the reduced homology group of C and is denoted by H(C). For a nonempty
simplicial complex К we define H(K) = H(C(K)), and for a nonempty topo-
logical space X we define H(X) = /7(Д(Х)). Because the chain complex of an
empty simplicial complex or an empty topological space has no augmentation,
the reduced groups are not defined in this case. For that reason some of the
arguments, which otherwise involve the reduced groups, require a special
remark in the case of empty complexes or spaces.
Clearly, there is an inclusion chain map С С C.	'
1	lemma If C is an augmented chain complex, then
1 ~ [h0(Q Ф Z q = 0
proof Because Z is a free group, Co ~ Co ® Z. Then ZQ(C) = ZQ(C) if
q ф (j, Zo(C)	Zo(C) © Z, and Bq(C) = Bq(C) for all q. 	!
It is clear that if т: C C' is an augmentation-preserving chain map, the (
isomorphism of lemma 1 commutes with t* . It is also obvious that if C is a
free augmented chain complex, C is a free chain complex.	1
It follows from lemma 1 that if C is an augmented chain complex,
H0(C) ф 0. Hence an augmented chain complex is never acyclic. The most
that can be hoped for is that C will be acyclic.
sgc. 3 THE HOMOLOGY OF SIMPLICIAL COMPLEXES	769
2	lemma If C is an augmented chain complex, C is chain contractible if
and only if the augmentation e is a chain equivalence of C with the chain
complex Z.
proof Let C be the mapping cone of the chain map e: C Z. Then
Co = Z and CQ = CQ_i if q > 0, and Эх = e and = — 3Q_i for q > 1.
By theorem 4.2.10, e is a chain equivalence if and only if C is chain
contractible.
We show that C is chain contractible if and only if C is chain contractible.
If D: С —> C is a chain contraction of C, define D: С C by Dr/- \ —
— Dq I CQ_i. Then D is a chain contraction of C. Conversely, if D is a chain
contraction of C, define D: С C so that Do: Z Co is a right inverse of
e: Co —> Z, D±: Co C'i is 0 on Dq(Z) and equal to —Do on Co, and for
q > 1, Dq: Cql —> Cq is equal to — if;. Then D is a chain contraction of C. B
Let C be a category with models A functor G' from 6 to the category
of augmented chain complexes (and chain maps preserving augmentation) is
said to be acyclic if G'(M) is acyclic for M £ 'DIL For augmented chain com-
plexes there is the following form of the acyclic-model theorem.
3	theorem Let C be a category with models C?1L and let G and G' be
covariant functors from C to the category of augmented chain complexes
such that G is free and G' is acyclic. There exist natural chain maps preserving
augmentation from G to G', and any two are naturally chain homotopic.
proof Let {gj E Go(Mj)}jejo be a basis for Go. By lemma 1, e': JhfG'lMf) ~ Z,
and there is a unique Zj E Ho(G'(Mj)) such that e'(z7-) = s(gj). A natural trans-
formation Ho(G) —> Ho(G') is defined by sending {SnjyGo(/y)(gj)} E Ho(G(Xf)
to 2inyGb(fij)Zj E H0(G'(X)) for j E Jo and f-;; E hom (Mj,X) (where X is any
object of t), and this is the unique natural transformation Hq(G) —> Ho(G')
commuting with augmentation. The theorem now follows from theorem
4,2.8. 
With the hypotheses of theorem 3 there is a unique natural transforma-
tion from H(G) to H(G') commuting with augmentation. It is the homomor-
phism induced by any natural chain map preserving augmentation from G to G'.
4	corollary Let G and G' be free and acyclic covariant f unctors from
a category Q with models to the category of augmented chain complexes.
Then G and G' are naturally chain equivalent; in fact, any natural chain
map preserving augmentation from G to G' is a natural chain equivalence.
proof Let r: G —> G' be a natural chain map preserving augmentation
(which exists, by theorem 3). Also by theorem 3, there is a natural chain map
T: G' G preserving augmentation and there are natural chain homotopies
D: / ° t ~ 1G and D': т ° r' ~ 1^. 
We are ultimately interested in comparing the chain complex C(K) of a
simplicial complex К with the singular chain complex A(|K|) of the space of K.
170
HOMOLOGY CHAP. 4
For this purpose we introduce a chain complex A(/<) intermediate between
them. Let К be a simplicial complex. An ordered q-simplex of К is a sequence
v0, oi, . . . , vq of q + 1 vertices of К which belong to some simplex of /<.
We use (ro,L|, . . . ,oe) to denote the ordered q-simplex consisting of the
sequence t>0, m, . . . , vq of vertices. For q < 0 there are no ordered q-sim-
plexes. An ordered О-simplex (o) is the same as the oriented O-simplex [«].
An ordered 1-simplex (o,t/) is the same as an edge of K.
We define a free nonnegative chain complex, called the ordered chain
complex of K, by A(/<) = (Aq(K),dq), where Aq(K) is the free abelian group
generated by the ordered q-simplexes of К [and Aq(K) = 0 if q < 0] and
is defined by the equation
3Q(«0,01, • • • ,Vq) =	2 (—1У(о0>  • • Л,   • ,Vq)
0<i<Q
Then A(K) is a chain complex, and if К is nonempty, A(K) is augmented by
the augmentation e(o) = 1 for any vertex v of K. If <p: K\ K2 is a simplicial
map, there is an augmentation-preserving chain map
j(-( . 3 THE HOMOLOGY OF SIMPLICIAL COMPLEXES	171
chain equivalence. Define a homomorphism t: Z A0(K * w) by t(1) = (w)
alld regard it as a chain map r: Z —> A(K * w). Then e 0 т = lz. To show that
1a(K*w) — T ° e, define a chain homotopy D: 1д(к»гс) ~ т ° e by the equation
{	D(o0,vi, • •  ,vq) = (и>,ад, • •  ,vq) 
< Because a q-simplex is the join of a (q — l)-face with the opposite vertex,
1 4ye have the next result.
[ f corollary For any simplex s £ K, A(s) and C(s) are acyclic. 
; (f theorem Far any simplicial complex К the natural chain map
p: A(K) C(K) is a chain equivalence.
> proof If К is empty, A(K) — C(K) and p is the identity, so the result is true in
i jhis case. If К is nonempty, it follows from corollary 7 that A and C are free
acyclic functors on (3[K) with models ?4l(K) = [s | s £ K). By corollary 4, p is
i a natural chain equivalence of A with C on C?(/<). In particular, p: A(K) —> C(K)
is a chain equivalence. 
A(<P): A(Kp) Д(К2)
such that A(<p)(fo,t>i,  •  ,vq) = (ф(<--о), ф('т)> •  • ,<p(uQ)). Therefore we have
the following theorem.
5 theorem There is a covariant functor A from the category of nonempty
simplicial complexes to the category of free augmented chain complexes
which assigns to К the ordered chain complex A(K). 
If L is a subcomplex of К and i: L С K, then A(i): A(L) —> A(/<) is
a monomorphism by means of which we identify A(L) with a subcomplex of
A(K). If <2(/<) is the category defined by the partially ordered set of subcom-
plexes of К and 91t(X) = {s | x £ K}, then A is a free functor on F(K) with
models 91l(K).
For any simplicial complex К there is a surjective chain map (preserving
augmentation if К is nonempty)	x
p: A(K) C(K)	!
such that p(oo,Ui, . . . ,vq) = [n0,Oi . . . ,oj. Then p is a natural transfer- ।
mation from A to C on the category of simplicial complexes. We shall show
that it is a chain equivalence for every simplicial complex. The following:
theorem will be used to show that A and C are acyclic functors on <£(/<) with
models 9K(/<).
6 theorem Let К be a simplicial complex and let w be the simplicial <.
complex consisting of a single vertex. Then A(K * w) and C(K * w) are chain I
contractible.	’ I
proof Since the proofs are analogous, we give the details only in the ordered
complex. According to lemma 2, it suffices to prove that e: A(K * io) —> Z is a
1	The next result is that the functors A and C convert contiguity of sim-
{ plicial maps into chain homotopy of chain maps. This result could also be
proved by the method of acyclic models.
j 9 theorem Let cp, cp': Kt K2 be in the same contiguity class. Then
h(cp), A(q'): A(Kf) —> A(K2) are chain homotopic, and in similar fashion
, C(cp), C(<p'): C(Ki) —> C(K2) are chain homotopic.
proof Because chain homotopy is an equivalence relation, it suffices to prove
: the theorem for the case that <p and cp' are contiguous. An explicit chain
} homotopy D: A(cp) ~ A(q') is defined by the formula
‘ D(O0>fl, • • • ,Aj) = 2 ( — Ш>), . . . ,(p'(Vi), <p(Ui), . . . ,<p(vq))
0<?<q
: That C(q) and C(q') are chain homotopic follows from the fact that A(<p) and
Д(ф') are chain homotopic and from theorem 8. 
IО theorem Tire homology groups of a complex are the direct sums of the
I homology groups of its components.
proof If {Kf} are the components of K, then (ffClKfi = C(K). The result
[ follows from theorem 4.1.6. 
If {Ka} is the collection of finite subcomplexes of К directed by inclu-
sion, then C(K) ~ lirn , {C(/<„)}. From theorem 4.1.7 we have the next result.
11	theorem The homology groups of a simplicial complex are isomorphic
to the direct limit of the homology groups of its finite subcomplexes. 
We are now ready to compute Ho(K).
12	lemma If К is a nonempty connected simplicial complex, then Ho(K) = 0.
[ SEC' 4 SINGULAR HOMOLOGY	jyg
Eliminating p(ZQ(C)), we have
I	p(Cq) = p(HQ(C)) 4- p(Bq(C)) + p(Bq_fiC))
I Multiplying this equation by ( — l)e and summing the resulting equations over
q yields the result. 
j If H(K,L) is finitely generated, its Euler characteristic, called the Eider
j characteristic of (K,L), is denoted by x(K,L).
; 15 corollary If К	L is finite and if aq equals the number of q-simplexes
J ofK — L, then
[	x(K,L) = 2(-l)</aQ
! proof If К — L is finite, Cq(K)/Cq(L) is a free group of rank aq. The result
; follows from theorem 14. 
1
4 SINGULAR HOMOLOGY
HOMOLOGY CHAP. 4
proof Let t?o be a fixed vertex of K. For any vertex v of К there is an edge
path 6162 •  • er of К with origin at vq and end at v. Then ei + 62 +  • • + et
is a 1-chain cv £ Ai(/<) such that dcv = v — t>o- Since e(2mvv) = Sn„, we see
that if Snro is any О-chain of До(К), then ^nv = 0 and
S(^nvC-v) — 2^0 — 2nrCo — SjluC
Therefore Н0(Д(К)) = 0, and by theorem 8, Ho(K) = 0. 
13	corollary For any simplicial complex K, H0(K) is a free group whose
rank equals the number of nonempty components of K.
proof If К is empty, H0(K) = 0, and the result is valid in this case. If К is
nonempty and connected, it follows from lemmas 12 and 1 that Ho(K) ~ Z.
The general result then follows from theorem 10. 
If L is a subcomplex of K, there is a relative oriented homology group
H(K,L) = {Hq(K,L) = Hq(C(K)/C(L))} of К modulo L. If L is empty,
H(K,0) = H(K) is called the absolute oriented homology group of K. Sim-
ilarly, there is a relative ordered homology group H(&(K)/£fiL)) of К modulo
L that generalizes the absolute ordered homology group H(Д(К),Д( 0)). The
relative homology groups H(K,L) and Н(Д(К),Д(Ь)) are covariant functors
from the category of simplicial pairs to the category of graded groups.
If Hq(K,L) is finitely generated (which will necessarily be true if If- L
contains only finitely many simplexes), it follows from the structure theorem;
(theorem 4.14 in the Introduction) that Hq(K,L) is the direct sum of a frees
group and a finite number of finite cyclic groups Zni © Zn, © •   © Z„(,
where 1ц divides Uj+i for i = 1, . . . , к — 1. The rank p(HQ(K,L)) is called:
the qth Betti number of (K,L), and the numbers пъ n2, . . . , nk are called,
the qth torsion coefficients of (K,L). The Qth Betti number and the Qth
torsion coefficients characterize Hq(K,L) up to isomorphism.
A graded group C is said to be finitely generated if Cq is finitely gener-
ated for all Q and Cq = 0 except for a finite set of integers Q. It is obvious that
if C is a finitely generated chain complex, H(C) is a finitely generated graded:
group. Given a finitely generated graded group C, its Euler characteristic
(also called the Euler-Poincare characteristic), denoted by x(C), is defined by
X(Q = 2(-1)«p(Cq)
14	theorem Let C be a finitely generated chain complex. Then
x(Q = x(H(Q)
proof By definition, ZQ(C) C Cq and the quotient group Cq/Zq(C) 7Z Bq_fiCf
By theorem 4.12 in the Introduction,
p(cq) = P(ZQ(C)) + p(BQ_r(q)
Similarly, Hq(C) = ZQ(C)/BQ(C), and again by theorem 4.12 of the Introduction^
p(ZQ(Q) = p(HQ(C)) + p(Be(C))
In this section we define a natural transformation from the ordered chain
1" complex to the singular chain complex of its space. This will be shown
I, in Sec. 4.6 to be a chain equivalence for every simplicial complex K. We also
• give a proof, based on acyclic models, that homotopic continuous maps in-
duce chain-homotopic chain maps on the singular chain complexes. There is
then a computation of H0(X) in terms of the path components of X. The final
; result is that the subcomplex of the singular chain complex generated by
j small singular simplexes is chain equivalent to the whole singular chain
' complex.1
Let К be a simplicial complex. Given an ordered Q-simplex (oo, Oi, . . . ,vq)
t of K> there is a singular Q-simplex in |K| which is the linear map Де -ч> |K|
j sending pi to vi for 0 < i < q. This imbeds Д(К) in Д(|К|), and we define an
! augmentation-preserving chain map
'	г: Д(К) Д(|К|)
to send (o0,Oi, . . . ,vq) to the linear singular simplex defined above. Then v
 is a natural chain map from the covariant functor Д( •) to the covariant
functor Д(| • j) on the category of simplicial complexes. It will be shown in
Sec, 4.6 that v is a natural chain equivalence. We prove now that it is a chain
equivalence for the complex s of an arbitrary simplex s.
) I lemma Let X be a star-shaped subset of some euclidean space. Then
the reduced singular complex of X is chain contractible.
'Our treatment is similar to that in S. Eilenberg, Singular homology theory, Annals of Mathe-
matics, vol. 45, pp. 407-447 (1944).
174
HOMOLOGY CHAP. 4
proof Without loss of generality, X may be assumed to be star-shaped from
the origin. We define a homomorphism r: Z Ao(X') with t(1) equal to the
singular simplex Д° —> X which is the constant map to 0. Then e ° r = lz. We
define a chain homotopy D: l\(X) —> Д(Х) from to т ° e. If <т: Д9 X is a
singular (/-simplex in X, let D(o): Дз"1-1 X be the singular {(/ + l)-simplex in
X defined by the equation
P(n)(tp0 + (1 - t)a) = (1 — t)v(a)
for a £ |pi, . . . ,pQ+i| and t £ I. If q > 0, then (D(a))<°> = a, and for
0 < i < q, (D(a))<i+1> = D(n®). If q = 0, then (Z?(a))<°) = <r and (D(a))(« = т(1).
Therefore
ЭГ> + ОЭ = 1Д(л-) — t ° e
and D:	~ т ° t. By lemma 4.3.2, Д(Х) is chain contractible. 
2 corollary For any simplex s the chain map v induces an isomorphism
of the crrdered homology group of s with the singular homology group o/~ |.sj.
proof Because v preserves augmentation, v induces a homomorphism i’*
from Н(Д(«)) to and under the isomorphism of lemma 4.3.1, v* =
v* © lz. By corollary 4.3.7, Н(Д(«)) = 0. By Iemma 1 and corollary 4.2.3,
H(|s|) = 0. Therefore v* is an isomorphism. 
We use lemma I to prove that if /о, fp. X—>У are homotopic, then
Д(/о), Д(/1): Д(Х) —>	are chain homotopic. We prove this first for the
maps h0, hi: X X X I, where ho(x) = (x,0) and hfx) = (x,l).
3	theorem The maps ho, hp. X—> X X I induce naturally chain-homotopic
chain maps
^h0) ~ A(hx): Д(Х) -а Д(Х x I)
proof Let Д'(Х) = Д(Х X I). Then Д and Д' are covariant functors from
the category of topological spaces to the category of augmented chain com-
plexes and Д(/г0) and Д(/гх) are natural chain maps preserving augmentation
from Д to Д'. Since Д is free with models {Д'7} and	x
Д'(Д«) = Д(Д? x I)
is acyclic, by lemma 1, it follows from theorem 4.3.3 that Д(Ь0) and A(hi) are
naturally chain homotopic. 
This special case implies the general result.
4	corollary If fo, fy. X У are homotopic, then
&(fo)	Д(Х) A(Y)
proof Let F-. X x I —> У be a homotopy from fo to /1. Then fo — Fho and
fi = Fhx. Therefore, using theorem 3,	___
Д(/о) = W(M	= Д(/1) 
SEC. 4 SINGULAR HOMOLOGY
175
Since Дв is path connected for every q, any singular simplex u: Д« —X
maps Де to some path component of X. Hence, if {Xj} is the set of path com-
ponents of X, then Д(Х) = (+)Д(ХД By theorem 4.1.6, we have the following
theorem.
5	theorem The singular homology group of a space is the direct sum of
the singular homology groups of its path components. 
Because Де is compact, every singular simplex о: Де —> X maps Д4 into
some compact subset of X. Hence, if {X,,} is the collection of compact sub-
sets of X directed by inclusion, then Д(Х) = limA(X„). By theorem 4.1.7, we
have our next result.
6	theorem The singular homology group of a space is isomorphic to the
direct limit of the singular homology groups of its compact subsets. 
We now compute the O-dimensional homology group of a space.
7	lemma If X is a nonempty path-connected topological space, then
Ho(X) = 0.
proof Let xo be a fixed point of X. For any point x £ X there is a path
from x'o to x. Because Д1 is homeomorphic to I, corresponds to a singular
1-simplex <тж: Д1 X such that <r.r(0) = x and = xo. A singular 0-simplex
in X is identified with a point of X. Therefore a 0-chain (that is, a 0-cycle) of
X is a sum where nx = 0 except for a finite set of x’s. Since е(2нжх) =
2пж, we see that if е(2пжх) — 0 [that is, if 2nxx £ До(Х)], then
9(Sna?o'a?) = 2пжх — (2n,.)x() = Sn/r
Therefore Йо(Х) = 0. 
8	corollary For any topological space X, IfaX) is a free group whose
rank equals the number of nonempty components of X.
proof If X is empty, IfaX') = 0, and the result is valid in this case. If X is
nonempty and path connected, it follows from lemmas 7 and 4.3.1 that
Ho(X) ~ Z. The general result now follows from theorem 5. 
If A is a subspace of X, there is a relative singular homology group
H(X,A) = (HQ(X,A) = Не(Д(Х)/Д(А))} of X modulo A. H(X, 0) = H(X) is
called the absolute singular homology group of X. The relative homology
(group is a covariant functor from the category of topological pairs to the
«category of graded groups. We show that this functor can be regarded as de-
fined on the homotopy category of pairs.
0 theorem If f0, fa (X,A) —> (Y,B) are homotopic, then
fa = fa: H(X,A)
proof Let (X X Г, A x I) (Y,B) be a homotopy from fo to fa Then
fo = Fho and Д = Fh±, where h0, hr: (X,A) (X X I, A X I) are defined by
176	HOMOLOGY CHAP. 4
h0(x) = (x,0) and h\(x) = (x,l). By theorem 3, there is a natural chain homo-
topy D: Д(й0) ~ where h0, hi: X X X I are maps defined by ho and
ht. Because D is natural, D(A(A)) С Д(А X F). For i = 0 or 1 there is a com-
mutative diagram
Д(А) С Д(Х) Д(Х)/Д(А)
Д(А X I) С Д(Х X I) —> Х(Х х 1)/Д(А X Г)
and a chain homotopy D: Д(7г0) Д(М is obtained by passing to the quo-
tient with D. By theorem 4.2.2,
ho* = hi*: ff(X,A) HfX X I, A X I)
Then
fo* ~ F* ho* = F* hi* = fi* 
If H,fX,A) is finitely generated, its rank is called the qth Betti number of
(X,A) and the orders of its finite cyclic summands given by the structure
theorem are called the qth torsion coefficients of (X,A). If H(X,A) is finitely
generated, its Euler characteristic is called the Euler characteristic of (X,A),
denoted by x(X,A).
The remainder of this section is directed toward a proof that the sub-
complex of the singular chain complex generated by small singular simplexes
is chain equivalent to the singular chain complex. We begin by defining
a subdivision chain map in singular theory. A singular simplex a: Aq Д'* is
said to be linear if ofSq-p,) = St,a(p,) for t; Q I with St, = 1. If a is linear,
so is <t(,) for 0 < i < q. Therefore the set of linear simplexes in Ди generates
a subcomplex Д'(ДП) С Д(ДП).
A linear simplex u in Ди is completely determined by the points a(pi). If
Xo, Xi, . . . , xq £ Д», we write (xq,xi, • • • Xq) to denote the linear simplex
tj:	A” such that a(p/) = xt. With this notation, it is clear that
Э(х’о, . . . , Xq) =	l/(x0, . . . ,Xj, . . . ,Xg)
Furthermore, the identity map f,: Д” С Д’1 is the linear simplex rz
(p0,pi, . . . ,pn).
Let bn be the barycenter of A" (that is, bn = S(l/(n + 1))р,. For q > 0
a homomorphism
fin: Д^(Д») —> Д;+1(Д»)
is defined by the formula
Дп(Хо> • • • Ад) = (hn,Xo, • • • ,Xq)
Let r: Z —» До(Д”) be defined by r(l) = (b.„). Direct computation shows that
10	fn: 1д-(лн) ~ т ° e 	X
SEC. 4 SINGULAR HOMOLOGY	177
For every topological space X we define an augmentation-preserving
chain map
sd: Д(Х) Д(Х)
 afld a chain defonnation
D: Д(Х) Д(Х)
from sd to 1 д(л-), both of which are functorial in X. That is, if f: -X У, there
 are commutative squares
l	Д(Х)	Д(Х)	Д(Х) Д Д(Х)
|	д(/)1 U(f> Д(/Ц |ду)
i	Д(У)	Д(У)	Д(У) Л Д(У)
। Both sd and D are defined on q-chains by induction on q. If c is a О-chain, we
define sd(c) = c and D(c) = 0. Assume sd and D defined on q-chains for
0 < q < n, where n > 1. We define sd and D on the universal singular
н-simplex An C An by the formulas
!	sdftn) = fn(sd a(4))
D(L) = fi„(sd - E>0(^))
: for any singular n-simplex а: Д” —> X we define
sd(a) = Afvjfsdffn))
D(a) = Д(а)№))
) Then sd and D have all the requisite properties.
I If X is a metric space and c = ^n^a is a singular q-chain of X, we define
<	mesh c = sup (diam а(Дв) |	0}
f 11 lemma Let An have a linear metric and let c be a linear q-chain of A“.
j Then
I	mesh (sd c) < —-— mesh c
(
f proof The proof is based on induction on q, using the inductive definition
J of sd. It suffices to show that if a = (xo,x1; . . . ,x9) is a linear q-simplex of Д”,
I then mesh (sd u) < (q/(q + 1)) mesh a. If b = S (l/(q + l))x», a computation
similar to that of lemma 3.3.12 shows that the distance from b to any convex
, iWiibination of the points xo, Xi, ... , xq is less than or at most equal to
(Ч/(</ + 1)) mesh (xo, . . . ,x9). Therefore
mesh (sd a) < sup	mesh a, mesh (sd 9u)
 By induction
178
HOMOLOGY CHAP. 4
mesh (sd Эст) < —-mesh Эн
—-— mesh о
Я + i
which yields the result. 
We next define augmentation-preserving chain maps
sd™: Д(Х) Д(Х)
for m > 0 by induction
sd° = 1Д(л")
and sd™ = sd(sdm x)
m > 1
Then, from lemma 11, we obtain the following result.
12 corollary Let Д” have a linear metric and let с £ Дд(Д”). Then
mesh (sd™ c) < [q/(q + l)]m mesh c 
Let 01 = {A} be a collection of subsets of a topological space X and let
Д(s’l) be the subcomplex of Д(Х) generated by singular (/-simplexes n: Де -a X
such that <т(Д9) C A for some A E Ol [if о(Д9) C A, then п^^Д9”1) C A, and
so Д(01) is a subcomplex of Д(Х/]. Because sd and D are natural, sdl/Xf/lf) С Д(0()
and С(Д(01)) С Д(01).
13 lemma Let Ol = [A] be such that X = U {int A |A £ Ol}. For any
singular q-simplex a of X there is m > 0 such that sd™ a £ Д(01).
proof Because X = U{int A | A £ Ol}, Д9 = Uj cr ^int A) | A £ Ol}. Let
Д9 be metrized by a linear metric and let X > 0 be a Lebesque number for
the open covering {n-1(int A) | A £ 01} of Де relative to this metric. Choose
m > 0 so that [q/(q + I)]®1 diam Де < X. By corollary 12, mesh (sd"! £?) < A,
Therefore every singular simplex of sd™ maps into fT '(int A) for some
A £ Ol. Then sd™ a = Д(п) sd™ is a chain in Д(01). 
We are now ready to prove the chain equivalence mentioned earlier.'
14 theorem Let 01 = {A} be such that X = U {int A | A £ 0l}. Then the;
inclusion map Д(01) С Д(Х) is a chain equivalence.
proof For each singular simplex n in X let ?n(n) be the smallest nonnegative
integer such that sdm^a £ Д(<?1). Such an integer m(<f) exists by lemma 13, and
it is clear that ?n(n) = 0 if and only if n £ Д(01). Furthermore, т(а(-г>) < m(of
for 0 < i < deg n.
Define D: Д(Х) Д(Х) by D(n) = 20<;<т(а)-1 D sd>(a). Then D(d) =• 0 
if and only if n € Д(01). Also
AD(c) = Ssd>+1(a) — Ssdi(a) — SDsd>(da)
=	— a — 20<j<m(a)-i	(— 1)’D s<T(a®)
DS(a) = Si(-l)1 So<j<m(a<i>biD sdl^Y)
SEC. 5 EXACTNESS
Therefore
179
о + SD(o) + D3(a) = Sj( — ])’'
, is in Д(ч*1). Define т: Д(Х) Д(чЧ) by t(<t) = a + 9D(a) + D9(o). Then r is a
' chain map preserving augmentation. Clearly, if i: Д(А) С Д(^1), then
г о i = 1aW and D: i ° т ~ 1A(X). Therefore [r] = [i]-1, and i is a chain
equivalence. H
) S EXACTNESS
< In this section we consider the relations among the homology groups of С', C,
 and C/C, where C' is a subcomplex of C. A concise way of summarizing
these relations is by means of the concept of exact sequence. The basic result
t is the existence of an exact sequence connecting the homology of С', C, and
' C/C.
A three-term sequence of abelian groups and homomorphisms
j	G' Л G A G"
is said to be exact at G if ker /? = im a. A sequence of abelian groups and
I homomorphisms indexed by integers (which may or may not terminate at
either or both ends)
Grtn+1 „ Ctn
n+l --->	1 —> • • •
is said to be an exact sequence if every three-term subsequence of consecu-
tive groups is exact at its middle group. Note that an exact sequence termin-
ating at one end with a trivial group can be extended indefinitely on that end
to an exact sequence by adjoining trivial groups and homomorphisms.
) A short exact sequence of abelian groups, written
I	0 G' A G A G" 0
1 is a five-term exact sequence whose end groups are trivial. In such a short
I exact sequence a is a monomorphism and fi is an epimorphism whose kernel
I is a(G'). Therefore a is an isomorphism of G' with the subgroup a(G') C G,
? and fi induces an isomorphism from the quotient group G/a(G') to G". The
group G is called an extension of G’ by G".
Given an exact sequence
let G’n = ker = im all+i. Then the given sequence gives rise to short exact
sequences
0 —» Gh —» Gn —» Gj,_i —» 0
for every Gn not on one or the other end of the original sequence, and the
180
HOMOLOGY CHAP. 4
composite Gn Gn-i Gn-i equals an.
A homomorphism у from one sequence {G.n —G,, |} to another
(Hn	with the same set of indices (that is, of the same length) is a
sequence {уи: G.n H.n} of homomorphisms such that the following diagram
is commutative:
G<W1 ж—т	an
n+l ---> '-Tn— 1 —* ’ ’ ’
4	I7-1
ЙА+1 rr Ih TT
?l+l	-Г1?1	*	1	>  • •
There is a category of exact sequences with the same set of indices. In par-
ticular, there is a category of short exact sequences, and also a category of
exact sequences (indexed by all the integers).
Note that a sequence of abelian groups and homomorphisms
JVi-i
• ’ • —>	LJl+i	> '-‘n	>	1 —> ' ' '
is a chain complex if and	only if im	3n+i C	ker	3n for all n. This	is half of the
condition of exactness at Cn. For a chain complex C, the group Hn(C) =
ker 3„/im 9,i+i is a measure of the nonexactness of the sequence at Cn. Thus
a chain complex is an exact sequence if and only if its graded homology
group is trivial. In any case, the fact that the homology group measures the
nonexactness of the chain complex suggests that there should be some rela-
tion between homology and exactness, and this is indeed so.
A short exact sequence of chain complexes, written
0 С' 4 C A C" -> 0
is a five-term sequence of chain complexes and chain maps such that for all q
there is a short exact sequence of abelian groups
0 C' i CQ Л C0
A homomorphism from one short exact sequence of chain complexes to an-
other consists of a commutative diagram of chain maps
0-^C'4cAc"-^0
о с 4 c A c" о
There is a category of short exact sequences of chain complexes and homo-
morphisms.
1 example Let C be a subcomplex of a chain complex C and let
i: С С C and /: С C/C be the inclusion and projection chain maps,
respectively. There is a short exact sequence of chain complexes
0 C4 C A C/C 0
SEC. 5 EXACTNESS
181
Given a subcomplex С' С C and a chain map т: C —> C such that т(С') С C,
there is a homomorphism
() ^ c4 C 4 C/C' 0
4 4 iT"
о c4 c 4 c/c' -> о
where т' = т | C and t" is induced from т by passing to the quotient.
2 example If C is an augmented chain complex, there is a short exact
sequence of chain complexes
О С -a C 4 Z 0
There is a covariant functor C from the category of simplicial pairs to
the category of short exact sequences of chain complexes which assigns to
(/<,/.) the short exact sequence
0 —> C(L) -> C(K) -> C(K)/C(L) -> 0
Similarly, there is a covariant functor Д from the category of topological pairs
to the category of short exact sequences of chain complexes which assigns to
(A',A) the short exact sequence
О Д(А) Д(Х) Д(Л)/Д(А)	0
There is also a covariant functor Д from the category of simplicial pairs to the
category of short exact sequences of chain complexes which assigns to (K,T)
the short exact sequence
О Д(Ь) Д(К) Д(К)/Д(Т)	0
, Then is a natural transformation from Д to C and v is a natural transforma-
tion from Д to Д( | • |) (both natural transformations in the category of short
exact sequences of chain complexes).
We define covariant functors H', H, and H" from the category of short
exact sequences of chain complexes
0 C' 4 C 4 C" -a 0
to the category of graded groups such that H'H, and H" map the above
sequence into H(C'), H(C), and H(C,r), respectively.
3 lemma On the category of short exact sequences of chain complexes
o—>c,4c4c"—»o
'there is a natural transformation 8*: H" -a H' such that if {z”} £ H(C"),
then a* (z”} = {u-iajs-v'} e я(С').
182
HOMOLOGY CHAP. 4
proof There is a commutative diagram
0 C'+i Cq+i	C"+i 0
4	4	I0"
0 C'q Л Cq AC" 0
4	4	i0"
0 C^l Cg-t C"_1 -> 0
in which each row is a short exact sequence of groups. If z" is a (/-cycle of C",
let c € Cq be such that /?((-) = г". Then
/8(3с) = d"ft(.c) = d"d' = 0
Therefore there is a unique d £ C'q^ such that a(c') = Sc. Then
а(Э'с') = Эа(с') = ЭЭс = 0
Because a is a monomorphism, Э'с' = 0. Hence d is a (q — l)-cycle of C.
We show that the homology class of c' in C depends only on the homol-
ogy class of z" in C", which will prove that there is a well-defined homomor-
phism {z"} = {d}. Let Ci £ CQ be such that /l(c'i) — z". Then there is
d” € Cg+1 such that /8(ci) = /8(c) + d"d". Choose d £ Ce+i such that
/8(d) = d". Then
/8(ci) = /8(c) + 9"/8(d) = /8(c 4- 9d)
Therefore there is a d' € C'q such that C| = c 4- dd 4- a(d'), and
3ci = de -I- r'a(d') = a(d) 4- a(d'd') = a(d 4- 8'd')
Hence a~x(9ci) = d 4- Cd' — d and {a“1(9ci)} = {a-1(9c)}, showing that
d* is well-defined.
To prove that d* is a natural transformation, assume given a commuta-
tive diagram of chain maps
0	C4 C 4 ( " ^ 0	j
4 4 h"	' |
0	C' ^ c 4 C" 0	i
where the horizontal rows are short exact sequences. Then
Tg.dg.ld'} = 1* (a^d/T^z") = {T'a~1dfi~1z"}
= (d-ira/S-^"} = (a^dp-^'z"} = d*TZ{z"} 
The natural transformation d* is called the connecting homomorphism
for homology because of its importance in the following exactness theorem, j
4 theorem There is a covariant functor from the category of short exact’ I
sequences of chain complexes to the category of exact sequences of groups
which assigns to a short exact sequence
SEC. 5 EXACTNESS	183
o c C C" 0
the sequence
 •  Hq(C') H^C)	\ Hq_i(C') ^ ...
proof The sequence of homology groups is functorial on short exact
sequences because 3.,; is a natural transformation. It only remains to verify
that it is an exact sequence. This entails a proof of exactness at Hq(C'), HQ(C),
and HQ(C"), each exactness requiring two inclusion relations. Therefore the
proof of exactness has six parts. We shall prove exactness at Hq(C") and leave
the other parts of the proof to the reader.
(a)	im /?* C ker 8*. Let {z} £ HQ(C). Then
Z*ft*(z} = 3*{ft(z)} = {a^dft~ift(z)} = {a-idz} = {«-1(0)} = 0
(b)	ker 3^. C im ft*. Let {z"} £ ker 3*. Then there is с C Cq such that
/?(<-;) = z!' and a'13(c) = B'(d') for some d' £ Cq. The difference c — a(d') £ Cq
is such that
8(c — «(</')) = Sc — a(d'd') = 0
Hence {c — a(d')} € Hq(C) and
(c ~ «(d')} = {ft(c) - /ta(d')} = (z"} 
Combining theorem 4 with example 2, we again obtain lemma 4.3.1. As
an example of the utility of exactness, note that the following corollary
is immediate from theorem 4.
5 corollary Given a short exact sequence of chain complexes
0 C C C" 0
(о) C' is acyclic if and only if ft* : 77(C) ~
(b) C is acyclic if and only if д* : H(C") ~ H(C').
(с) C" is acyclic if and only if a* : H(C') zz 77(C). 
In (b) above it should be noted that 8^. has degree — I. It follows from
sCorollary 5 that if two of the chain complexes С', C, and C" are acyclic, so is
the third.
6 corollary Given an exact sequence of abelian groups
1 and a subsequence
l	  •	* Lrn+1 ---> Lxn --». L»n-1 —* ’ ’ ’
(that is, G'n C Gn and ct'„ = a.„ | G'f), the subsequence is exact if and only if
( the quotient sequence
-----> g„/g;,	
' is exact.
184
HOMOLOGY CHAP. 4
pkooj Let C be the chain complex consisting of the original exact sequence
and let C' be the subcomplex consisting of the subsequence. Then the quo-
tient chain complex С/C is the quotient sequence. Because C is an exact se-
quence, C is acyclic, and 3*: Hg(C/C) ~ Hg-x(C). Therefore C is exact
[that is, II(C') = 0] if and only if C/C is exact [that is, ЩС/С) ~ 0]. “
7 theorem The direct limit of exact sequences is exact.
proof Each exact sequence is an acyclic chain complex. The direct limit is
also a chain complex, and it is acyclic, by theorem 4.1.7. Therefore the limit
sequence is exact. “
. This result is false if direct limit is replaced by inverse limit, because the
homology functor fails to commute with inverse limits.
Let К be a simplicial complex and let L C L С K. By the Noether iso-
morphism theorem, there is a short exact sequence of chain complexes
0 CtLzl/ClLJ Л C(K)/C(Lf) Л C(K)/C(L2)	0
By theorem 4, there is an exact sequence
HqlL2,Lx) Hg(K,Lf) Hg(K,L2) Hg_^L2,L±)
where q. is induced by i: (L2,Lf) C (K,Li),is induced by j:	C (K,L2),
and d# is the connecting homomorphism. This sequence is called the homology
sequence of the triple (K,L2,Lf). It is functorial on triples. If Li = 0, the re-
sulting exact sequence
 • • Д H^) Hg(K) Hg(K,L2)
is called the homology sequence of the pair (K,L2f It is functorial on pairs.
Because there is an inclusion map of the triple (K,L2,0) into the triple
(K,L2,Lfp the next result follows.
8	lemma The connecting homomorphism d*:	Hg~i(L2,Lf) of
the triple (K,L2,Li) is the composite
H'/K'Lz) Д Hg^(L2) H^LM
of the connecting homomorphism of the pair (K,L2) followed by the homo-,
morphism induced by k: (L2,0) C (L2,Lf. “
If L is a nonempty subcomplex of a simplicial complex, C(L) C C(K),
and by the Noether isomorphism theorem, C(K)/C(L) C(K)/C(L). There-
fore there is a short exact sequence of chain complexes
0 -+ C(L) C(K) Д C(K)/C(L)	0
The corresponding exact sequence
  • Д ВД Д Нд(К) Д Hg(K,L) Д Hg-dL) ‘ ‘
is called the reduced homology sequence of the pair {K,L). It is not defined^
if L = 0, because C(L) has no augmentation in this case.
SEC. 5 EXACTNESS
185
In the same way, there is a singular homology sequence of a triple
(X,A,B) and of a pair (X,A). If A is nonempty, there is also a reduced homology
sequence of (X,A). All these sequences are exact, and the analogue of lemma 8
is valid relating the connecting homomorphism of a triple to the connecting
homomorphism of a pair.
9	lemma Let s be an n-simplex. Then
proof CQ(s) = Cfs) if q y= n. Therefore [C(s)/C(s)]Q = 0 if q yfi n, and
[C(s)/C(s)]n = Z. »
Because H(s) = 0, by corollary 4.3.7, it follows from the exactness of the
reduced homology sequence of (s,s) that 8*: Hq(s,s)	for all q.
Therefore we have the next result.
1 © corollary If s is an n-simplex, then
= {°
q ¥= n - i
q — n — 1 
We conclude by proving the following five lemma (so named because of
the five-term exact sequences involved in its formulation).
J 1 lemma Given a commutative diagram of abelian groups and homomor-
phisms
m i" m “A c “A r-
vrg —-> Cr4 —> Стз —> Ct2 —Cri
4 4 "I ”1 I"
H5 H4 H3 H2 Hi
in which each row is exact and 71, y2, У4, and 75 are isomorphisms, then 73
is an isomorphism.
 proof The proof is straightforward. To show that 73 is a monomorphism,
assume 73(gs) = 0. Then 72a3(g3) = faytfgs) = 0. Therefore a3(g3) = 0.
Hence there is g4 £ G4 such that a4(g4) = g3. Then f^yfigf) = 0, and there
[' is /i5 £ H5 such that fifhf) = y4(g4). There is g5 £ G5 with 75(gs) = /15. Then
' Y4(as(g5)) = 74(g4)> and so g4 = a5(g5)- Then g3 = a4a5(gs) = 0.
To show that 73 is an epimorphism let /13 £ H3. There is g2 £ G2 such
that y2(g2) = ^з(Ьз). Then 7i«2(g2) = PzPfihf) = 0. Therefore a2(g2) = 0,
and there is g3 £ G3 such that a3(g3) = g2- Then j63(/i3 — 73(gs)) = 0, and
there is 7i4 £ H4 such that j64(/i4) = h3 — 73 (gs). Let g4 £ G4 be such that
j Y4(g4) = h4. Then g3 -|- a4(g4) £ G3 and 73(g3 + а4Ы) = Ys(g3) + Pfihfi =
h3. и
Note that to prove 73 a monomorphism we merely needed 72 and y4 to
' be monomorphisms and 75 to be an epimorphism, and to prove 73 an epimor-
phism we merely needed 72 and y4 to be epimorphisms and 71 to be a
HOMOLOGY CHAP. 4
186
monomorphism. This type of proof is called diagram chasing and will be
omitted in the future.
We shall have several occasions to use the five lemma. We mention the
following as a typical example. For any simplicial pair (K,L) the natural trans-
formation p from the ordered homology theory induces a homomorphism of
the corresponding exact sequences
•	 	HQ(A(K)) Hq(A(K)/A(L))	•
j,

•	  Hq(L) Hq(K) Hq(K,L)
By theorem 4.3.8, /r,;. is an isomorphism on the absolute groups. It follows
from the five lemma that it is also an isomorphism on the relative groups.
12 corollary For any simplicial pair (K,L) the natural transformation p
induces an isomorphism from the ordered homology sequence of (K,F) to the
oriented homology sequence of (K,L). “
6 MAYEB-VIKTOBIS SEQUENCES
There is an exact sequence which relates the homology of the union of two
sets to the homology of each of the sets and to the homology of their inter-
section. This sequence provides an inductive procedure for computing the
homology of spaces wliicli are built from pieces whose homology is known.
We shall define this exact sequence as well as its analogue involving relative
homology groups, and use them to prove that the natural transformation v
from A(K) to A(|K|) is a chain equivalence for any simplicial complex K.
Let Ki and K2 be subcomplexes of a simplicial complex K. Then
Ki П I<2 and Ki U K2 are subcomplexes of K, and C(Kf), C(K2) C C(K).
Clearly OKt П K2) = C(Kf) Cl C(K2) and C(Ki) + C(K2) = C(Ki U K2),
Let ii: Ki Г) K2 C K±, i2. Ki С K2 C K2, jp Ki C K± U K2, and'
/2: K-2 C Ki U K2. Then we have a short exact sequence of chain complexes
0 C(Ki n K2) Л C(K!) ® C(K2) C(Ki U /<2) -a 0
where i(c) = (C(ii)c, — C(i2)c) and /(съсг) = C(/i)ci + C(j2)c2. The corre-
sponding exact sequence of homology groups
... ^ Hq(Ki n K2) Hq(Kf) ® HfKf) H^Kr U K2)
CK2) ...
is called the Mayer-Vietoris sequence of the subcomplexes /<1 and K2. The
homomorphisms i* and f. in the Mayer-Vietoris sequence are described by
means of homomorphisms induced by inclusion maps by
i*z — (iuz, — i2*z)	and j* (zi,z2) = /1*24 + /2*2:2
SEC. 6 MAYEB-VIETORIS SEQUENCES	187
for z € H(Kr n K2), Z1 e and z2 € H(K2).
If Ki П I<2 7^ 0 > there is a commutative diagram of abelian groups and
homomorphisms
0 -+ C0(Ki A K2) 4 C0(Ki) © C0(K2) 4 CO(K1 U K2) -> 0
4	£®Ч	4
0 —>	z 4 z © z 4 z 0
where a(ri) = (n, — n) and P(n,m) = n + m. Since the rows are exact and the
vertical homomorphisms are epimorphisms, it follows from corollary 4.5.6 that
there is an exact sequence of the kernels
0 CO(K1 A K2) 4 CO(K1) © Co(K2) 4 Co(Ki U K2B 0
and so there is a short exact sequence of chain complexes
0 -» C(K1 A K2) 4 C(Ki) © C(K2) C(Kr и K2) -+ о
The corresponding exact sequence of reduced homology groups
... 4 a K2) 4 HQ(K1) © Hg(K2) Hg(Kr и к2) 4 ...
is called the redqced Mayer-Vietoris sequence of Ki and K2.
If and(7<2,L2) are simplicial pairs in K, there is also a short exact
sequence
0 C(Li A I,2p C(Li) © C(L2) -> C(Li L L2) -M)
which is a subsequence of the short exact sequence
0 -» C(K1 A K2)	C(K1) © C(K2) C(K1 U K2)	0
It follows from corollary 4.5.6 that the quotient sequence is a short exact
sequence of chain complexes
0^ C(K1 A K2)/C(L1 A L2)	C(K1)/C(L1) © C(K2)/C(L2)
C(Kj U K2)/C(Li U L,b 0
The corresponding exact sequence of homology groups
... 4 Hq(Ki A K2, Lr n L2) 4 Hg(KM © Hg(K2,L2) J\
Hg{Kr U K2, Lr U L2) 4 ...
is called the relative Mayer-Vietoris sequence of (Ki,Li) and (K2,L2).
The relative Mayer-Vietoris sequence specializes to the exact sequence of
a triple or a pair. In fact, given a triple (KdLidLf), the relative Mayer-Vietoris
sequence of (K,L2) and (Lr,Lr) is easily seen to be the homology sequence of
the triple (K,Li,L2) as defined in Sec. 4.5. In case L2 = 0, the relative
Mayer-Vietoris sequence of (K, 0) and (Li,Lr) is the homology sequence of
the pah\(f<,Lt).
An inclusion map^(Ki,Li) C	is called an excision map if
— Li — K2 — L2. The exactness of the Mayer-Vietoris sequence is closely
188
HOMOLOGY CHAP, 4
. 1
sm —
x
f(x) =
0
related (in fact, equivalent) to the following excision property.
I theorem Any excision map between simplicial pairs induces an iso-
morphism on homology.
proof If (Ki,Li) C	is an excision map, then K2 = Ki U L2 and
Li = Kj IT L2. By the Noether isomorphism theorem,
C(K1)/C(L!)	[C(K1) 4- C(L2)]/C(L2) = C(K2)/C(L2) »
For the ordered chain complex it is still true that if Kj and K2 are sub-
complexes of some simplicial complex, then A(Kr U K2) = A(/< i) + Д(К2).
Therefore all the above results remain valid if the oriented homology is
replaced throughout by the ordered homology.
An inclusion map (Xi,Ai) C (^2,A2) between topological pairs is called
an excision map i£ Xj — Aj — X2 — A2. It is not true that every excision
map induces an isomorphism of the singular homology groups. Neither is it
true that there is an exact Mayer-Vietoris sequence of any two subsets Xi and
X2 of a topological space.
2 example Let f: R —» R be defined by
x > 0
x < 0
and let X+ = {(x,y) € R2 | у > fix) or x = 0, 11/| < 1} and X_ = {(x,y) € R21
У < f(x) от л: == (), ]г/| < 1}. Then X+ and X_ are closed path-connected sub-
sets of R2 such that X+ U X_ = R2 and X+ П X. consists of two path com-
ponents. Therefore there is no homomorphism H-fX-j U X2) —> H0(Xj Pl X2)
which will make the sequence
H1(X1 U X2) -+ H0(Xj Г) X2) -+ Ho(Xf) ® H0(X2)
exact at H0(Xj A X2) [the ends are both trivial, but H0(Xj П X2) yA 0].
We can, however, develop a Mayer-Vietoris sequence in singular lionipj-
ogy for certain subsets Xj and X2 of a tc •mlogical space. Let Xi and X2 be
subsets of some space. {Xi,X2} is said to be an excisive couple of subsets if
the inclusion chain map A(Xr) + A(X2) C A(Xi U X2) induces an isomor-
phism of homology. Our next result follows from theorem 4.4.14.
3 theorem If Xj U X2 = intA1L!A2 Xj U int^u^ X2, then [Xi,X2} is an
excisive couple. “
In particular, if А С X, then {X,A} is always an excisive couple. The
relation between an excisive couple {Xi,X2} and excision maps is expressed
as follows.
4 theorem {Xi,X2} is an excisive couple if and only if the excision map
(Xi,Xi IT X2) C (Xi U X2,X2) induces an isomorphism of singular homology:.
SEC. 6 MAYEB-VTETOBIS SEQUENCES
189
pboof We have a commutative diagram of chain maps induced by inclusions
A(^f)/A(X1 П X2) A(Xi U X2)/A(X2)
[A(Xi) + Д(Х2)]/Д(Х2)
where / is the excision map /: (Xi, Xi Г) X2) C (Xi U X2, X2). By the Noether
isomorphism theorem, i is an isomorphism; therefore = 1*1* is an isomor-
phism if and only if i* is an isomorphism. Using the exactness of the homology
sequence of a pair and the five lemma, ii. is an isomorphism if and only if the
inclusion map A(Xx) + Д(Х2) C A(Xi U X2) induces an isomorphism of homol-
ogy, which is by definition equivalent to the condition that {Xi,X2} be an
excisive couple. “
This yields the following excision property for singular theory.
5 cobollaby Let U С А С X be such that V C int A. Then the excision
map (X — U, A — U) C (X,A) induces an isomorphism of singular homology.
pboof The hypothesis U C int A implies int (X — U) D X — U ZD X — int A.
By theorem 3, {A, X — U} is an excisive couple, and the result follows from
this and from theorem 4. “
For any subsets Xi and X2 of a space, A(Xi Г) X2) = Д(Хр) Г) Д(Х2), and
there is a short exact sequence of singular chain complexes
0 Д(Х1 П X2) 4 A(Xi) © Д(Х2) 4 Д(Х1) + Д(Х2)	0
This yields an exact sequence
• • • 4 HQ(Xi n X2) 4 HQ(Xi) © HQ(X2) 4 HB(A(Xi) + Д(Х2)) 4
Hq-l(Xi Г) X2) —> . • 
If {Xi,X2} is an excisive couple, the group HB(A(Xi) + Д(Х2)) can be replaced
jby the group HQ(Xj U X2), and the resulting exact sequence is
... 4 HQ(Xi n x2) 4 He(Xi) © hb(x2)4 u x2)4
HQ_i(Xi n X2)	...
’ where i* (z) = (fr*z, — i2*z) and ^(zi,z2) =	4- /2*z2 for z G H(Xi П X2),
’ Zi E H(Xi), and z2 E H(X2). This is the Mayer- Vietoris sequence of singular
theory of an excisive couple {Xi,X2}. Similarly, if Xi Г' Х2 i 0, there is a
reduced Mayer-Vietoris sequence of {Xi,X2}.
If (Xi,Ai) and (X2,A2) are pairs in a space X, we say that {(Xi,Ai), (X2,A2)}
is an excisive couple of pairs if {Xi,X2} and {Ai,A2} are both excisive couples
of subsets. In this case it follows from the five lemma that the map induced
by inclusion
[A(Xi) + A(X2)]/[A(Ai) + A(A2)] [A(Xi) U A(X2)]/[A(Ai) U A(A2)]
induces an isomorphism of homology. Hence, if {(Xi,Ai), (X2,A2)} is an
190
HOMOLOGY CHAP. 4
excisive couple of pairs, there is an exact sequence
... He(Xi n X2, Ai Г) A2)	Ф HQ(X2,A2)
Hg(X! U X2, Ai U A2)	.
called the relative Mayer-Vietoris sequence of {(X-^A-f, (X2,A2)}.
The relative Mayer-Vietoris sequence specializes to the exact sequence
of a triple (or a pair). In fact, given a triple (X,A,B), {(X,B), (A,A)} is always
an excisive couple of pairs, and the relative Mayer-Vietoris sequence of
{(X,B), (A,A)} is the homology sequence of the triple (X,A,B).
We use the Mayer-Vietoris sequence to compute the singular homology
of a sphere.
6 theorem For n > 0
H«(S-) = {»	9*»
4 (Z	q — n
proof Let p and p' be distinct points of S’1. Because Sn — p and S” — p' are
contractible (each being homeomorphic to Rn), H(Sn — p) = 0 = 17 (S’1 — p').
Since S’1 — p and Sn — p' are open subsets of S”, it follows from theorem 3
that {Sn — p, Sn — p'} is an excisive couple. From the exactness of the corre-
sponding Mayer-Vietoris sequence, it follows that
0*: HQ(S”) HQ^(S” - (p U p'))
Because S’! — (p U p') has the same homotopy type as Sn-1, there is an
isomorphism /7V_](S’' — (p U p')) ~ H9_i(S”~1), and the result follows by
induction and the trivial verification that for n = 0 the theorem is valid, в
We now show that a couple consisting of polyhedral subsets of a poly-
hedron is excisive.
7 lemma Let Ki and K2 be subcomplexes of a simplicial complex K.
Then {|Ki|,|K2|} is an excisive couple.
proof Let Vbe a neighborhood of |Ki Г) K2| in |Ki| having |/<i П I<2| as a
strong deformation retract (such a V exists, by corollary 3.3.11). There is a
commutative diagram
Г) K2|) He(|Ki|)	HedKil, |Ki Г) K2|) -A ...
-----> HQ(V)	HQ(|Kr|)	Нд(\Кг\,У)
Because i: |/<i Г) K2\ C Vis a homotopy equivalence, i*: H(\Ki Г) K2|) ~ H(V).
By the five lemma, fo: H(|Ki|, |Kr П K2|) ~ H(|I<i|,V).
Also, V U |K2| is a neighborhood of |K2| in |/<7 U K2| having |K2| as a
strong deformation retract. Therefore a similar proof shows that
/;: H(\Ki U K2|, |K2|) ~ H(|Ki U K2|, V U |K2|)
By theorem 4, {|/<i j, |K2|} is an excisive couple if and only if the excision
SEC. 6 MAYEB-VIETOBIS SEQUENCES
191
map (|Ki|, |Ki П K2I) C (|Z<i U K2I, |Кг|) induces an isomorphism of homol-
ogy. In view of the isomorphisms j* and j*, this will be so if and only if the
excision map (|7<i|, V) C (|Kt U /<2), V U |K2|) induces an isomorphism of
homology. Again by theorem 4, this is equivalent to the condition that
V U be an excisive couple. This is so by theorem 3, since
|K2| C int (V U |K2|) and |Ki| — |K2| C int |Ki|. и
8> theorem For any simplicial pair (K,L) the natural transformation v
induces an isomorphism of the ordered homology sequence of (K,L) onto the
singular homology sequence of (|K|,|L|).
proof It suffices to prove that for any simplicial complex K, v... : H(A(K))
H(|K|), because the theorem will follow from this and the five lemma. We
prove this first for finite simplicial complexes by induction on the number of
simplexes. If К contains one simplex, then К = s, where s is a О-simplex, and
the result follows from corollary 4.4.2.
Assume the result inductively for simplicial complexes with fewer than
m simplexes, where m > 1, and let К contain exactly m simplexes. Let s be a
simplex of К of maximum dimension and let L be the subcomplex of К con-
sisting of all simplexes other than s. Then К = L U s and s = L Г) s. Because
L has exactly m — 1 simplexes, v* is an isomorphism H(A(L)) ~ H(|L|)
and an isomorphism H(A(s))	By corollary 4.4.2, v*-. H(A(s))	H(|s|).
By the exactness of the ordered Mayer-Vietoris sequence of L and л and the
Mayer-Vietoris sequence of singular theory for |L| and |s| (which exists, by
lemma 7), it follows from the five lemma that v* : H(A(K)) ~ H(\K|).
For infinite simplicial complexes К let {Ka} be the family of finite sub-
complexes of К directed by inclusion. It follows from theorem 4.3.11 that
H(A(K)) s: lirn , H(A(Ka)) and from theorem 4.4.6 that H(|K|) ~ lim_. H(|Ka|).
The theorem now holds for К because v* is natural. “
We show next that for free chain complexes a chain map is a chain
equivalence if and only if it induces an isomorphism in homology. First we
establish an exact sequence containing the homomorphism induced by a
chain map.
О lemma Let т: C —> C be a chain map and let C be the mapping cone
of t. There is an exact sequence
•  	HQ+1(C) HQ(C) HQ(C') HQ(C)   •
proof Let a: С' C be the chain map defined by a(c) = (0,c). Then
a imbeds C' as a subcomplex of C and the quotient complex С/ C is such that
(C/C')Q ~ the boundary operator of C/C' corresponds to the negative
of the boundary operator of C under this isomorphism. The desired exact
sequence is then obtained from the exact homology sequence of the short exact
sequence of chain complexes
() _^ С' Д C —> C/C' 0
192
HOMOLOG'S CHAP, 4
by replacing Hq(C/C) by HQ_i(C) and verifying that the connecting homo-
morphism Э*: HQ+i(C/C') -> Hg(C) corresponds to r,..: HQ(C) -> Hq(C'). и
I ® theorem If C and C' are free chain complexes, a chain map т: С C'
is a chain equivalence if and only if r,:. : H(C) ~	“
proof By corollary 4.2.11, r is a chain equivalence if and only if C is acyclic.
By lemma 9 and corollary 4.5.5, C is acyclic if and only if т* : H(C) ~ и
Because Д(К)/Д(Ь) and Д(|К|)/Д(|Ь|) are free chain complexes, we have
the following result.
11 corollary For any simplicial pair (K,L), v is a chain equivalence of
&(K)/&(L) with Д(|К|)/Д(|Ь|). s
If tp: Ki f<2 is a simplicial map, there is a commutative diagram
H(Kr)	Н(Д(Кг))	Н(|Кг|)
H(K2)	Н(Д(К2))	H(|K2|)
In particular, if K' is a subdivision of К and tp: К' —> К is a simplicial approx-
imation to the identity |K'| C |K|, then
|<p| ~ l|x| and |<p|* = 1Л(И)
From the commutativity of the above diagram we obtain our next result.
12 theorem Let K' he a subdi vision of К and let <p: K'	К be a simplicial
approximation to the identity map |K'| C |K|. Then
<P*: H(K') ~ H(K) ®
By theorem 10, C(<p): CfK!) —-» C(K) is a chain equivalence. It will
be useful to construct a chain map C(K) —> C(K') which is a chain homotopy
inverse of C(cp). If K' is a subdivision of K, an augmentation-preserving chain
map t: C(K) —> C(K') is called a subdivision chain map if t: C(L) C C(K' | L)
for every subcomplex L С К [ that is, if r is a natural chain map from C to
C(K' | •) on G(K)].
13 theorem If K' is a subdivision of K, there exist subdivision chain
maps t: C(K) C(K'). If <p: К' —> К is a simplicial approximation to
the identity |K'| C |K|, then т* = H(K) ~ ЩК').
proof If s is any simplex of K, then C(K' | s) is acyclic [because H(K' | s) ~
H(|.s|) = 0]. Hence, on the category G[K) of subcomplexes of К with models
C3H(K) = {.s | ,s С K}, the functor C is free and C(K' | •) is acyclic. It follows
from theorem 4.3.3 that there exist natural chain maps т from C to C(K' | •)
preserving augmentation.
If т is any subdivision chain map and tp: К' К is a simplicial approxi-
mation to the identity map |K'| С K, the composite
SEC. 7 SOME APPLICATIONS OF HOMOLOGY
193
C(g>)r: C(K) C(K)
js a natural chain map over G(K) from C to C preserving augmentation. Since
Cis free and.acyclic with models 91ЦК), it follows from theorem 4.3.3 that
С(ф)т 1 ода- Therefore т,:. = 1ида. Since, by theorem 12, <р* is an iso-
morphism, t* = (p* и
f SOME APPLICATIONS OF HOMOLOGY
In this section we use homology for some of the applications mentioned
earlier. We shall show that euclidean spaces of different dimensions are not
homeomorphic, and also that Sn is not a retract of En+1 (which is easily seen
to be equivalent to the Brouwer fixed-point theorem). This leads to the gen-
: eral consideration of fixed points of maps, and we prove the Lefschetz fixed-
point theorem. Finally, we shall consider separation properties of the sphere.
Proofs are given of Brouwer’s generalization of the Jordan curve theorem and
of the invariance of domain.
< 1 theorem If n m, S'" and Sm are not of the same homotopy type.
t proof By theorem 4.6.6, H,,(S?') 7^ 0 and Hn(Sm') = 0. “
2 corollary Ifn=f= m, Rn and R”1 are not homeomorphic.
. proof If R” and R’" were homeomorphic, their one-point compactifications
; S” and Sm would also be homeomorphic, in contradiction to theorem 1. “
( In corollary 2 both Rn and Rm are contractible. Therefore they have the
; .same homotopy type and cannot be distinguished by their homology groups.
) To distinguish them it was necessary to consider associated spaces having
nonisomorphic homology. We chose to consider their one-point compactifica-
j tions, but another proof could have been based on the fact that Rn minus a
I point has the same homotopy type as S"-1.
I These two results are applications of homology to the problem of dassi-
[ fying spaces up to topological equivalence. Our next application is to an
extension problem.
»3 lemma Let (X,A) be a pair such that A is a retract of X. Then
H(X) H(A) © H(X,A)
. proof Given i: А С X and /: (X, 0) C (X,A) and a retraction г: X —» A, then
) fi = 1л- Therefore г* i* = 1 н(л) and L,; is a monomorphism of H(A) onto a
‘ direct summand of H(X). The other summand is the kernel of r*. From the
 exactness of the homology sequence of (X,A)
[	• • 	Hq+1(X,A) He_i(A) Hq_!(X) 4 H^XA)
194	HOMOLOGY CHAP. 4
because ker i* = О, Э* is the trivial map. Therefore fi. is an epimorphism.
Since ker fi = im fi, fi induces an isomorphism of ker r* onto H(X,A). 
Note that lemma 3 is still valid if A is a weak retract of X.
4	corollary For n > 0, SB is not a retract of En+1.
proof By theorem 4.6.6, l/n(S?!)	0, but because En+1 is contractible
Йп(Еп+г) = 0. Therefore i/(S") is not isomorphic to a direct summand of
H(&‘+1). и
This implies the following Brouwer fixed-point theorem.
5	theorem For n > 0 every continuous map from En to itself has a
fixed point.
proof For n = 0 there is nothing to prove. For n > 0 let f: En En be
continuous. If f has no fixed point, define a map g: En —> Sn-1 by g(x) equal
to the unique point of S"1 on the ray from fix) to x, as shown in the figure.
Then g is a retraction from En to Sn \ in contradiction to corollary 4. “
We have, in fact, proved that corollary 4 implies theorem 5. The
converse is also true, for if r: En+1 Sn were a retraction, the map/: En+1 —> En+1
defined by fix) — — fix) would have no fixed points.
There is an interesting generalization of theorem 5 which contains a.
criterion for showing that a certain map from X to itself has a fixed point even if’
not every map of X to itself has fixed points. This generalization also illustrates;
another type of application of homology in that it is based on an algebraic
count of the number of fixed points, the algebraic count being formulated in»
homological terms. This type of application of homology occurs frequently.
Generally it involves a set of singularities of X of a certain type (for example,»
the set of fixed points of a map X —> X, the set of discontinuties of a function
X Y, the set of self-intersections of a local homeomorphism X —> R", etc.);
and measures the singular set by means of a homology class associated to it..
Let C be a finitely generated graded group and let h: С —> C be an en-
domorphism of C of degree 0. The Lefschetz number X(/i) is defined by the-
formula
X(h) = S(-l)«Tr(/iQ)
SEC. 7 SOME APPLICATIONS OF HOMOLOGY
195
where hg: Cq Ce is the endomorphism defined by h in degree q. The fol-
lowing Hopf trace formula equates the Lefschetz numbers of a chain map and
its induced homology homomorphism.
6	theorem Let C be a finitely generated chain complex and let т: С C
be a chain map. Then
Х(т) = A(t^)
proof The proof is similar to the proof of the corresponding statement
about the Euler characteristic (theorem 4.3.14), the Euler characteristic being
the Lefschetz number of the identity map, with theorem 4.13 of the Intro-
duction used in place of theorem 4.12. Details are left to the reader. “
Let fi X X be a map, where X has finitely generated homology. The
Lefschetz number off, denoted by X(/), is defined to be the Lefschetz number
of the homomorphism f* : H(X) H(X) induced by f. It counts the algebraic
number of fixed homology classes of fi.. The following Lefschetz fixed-point
theorem shows that Х(/) 0 is a sufficient condition for f to have a fixed point.
7	theorem Let Xbe a compact polyhedron and let f: X —» Y be a map.
If X(/)	0, then f has a fixed point.
proof We assume that/has no fixed point and prove X(/) = 0. Without
loss of generality, we may assume X = |L| for some finite simplicial complex L.
Because |L| is a compact metric space, if/has no fixed point, there is a > 0
such that d(a,/(a)) > a for all a £ |L|. Let К be a subdivision of L with mesh
К < a/3 and let K' be a subdivision of К for which there exists a simplicial
map <p: К' —> К which is a simplicial approximation to fi. |K|	|K|. Since
|<p|(ct) and f(a) belong to some simplex of K, d(\(p\(a),fiaf) < a/3 for a £ |K|.
If s is any simplex of K, |.sj is disjoint from |<p|(|s|), for if a £ |s| is equal
to |<р|(У) for f £ |s|, then
d(/W)) < d((3,a) + d(|<p|(j6)») < 2fl/3
in contradiction to the choice of a.
Let t: C(K) —> C(K') be a subdivision chain map (which exists, by
theorem 4.6.13). Then С(<р)т: C(/<) C(K) is a chain map. If a is an oriented
g-simplex on a Q-simplex s of K, then С(<р)т(о) is a Q-chain on the largest sub-
complex of К disjoint from s. Therefore С(<р)т(о) is a Q-chain having coefficient 0
on a. Since this is so for every a, all the coefficients summed in forming
Тг(С(<р)т)в are zero and Тг((С(<р)т)е) = 0 for all q, which implies Х(С(<р)т) = 0.
By theorem 6, X((C(q)t)>;. ) = 0. Let <p': К' К be a simplicial approximation
to the identity map \K'| C |K|. There is a commutative diagram
H(K) <	.- H(K')	—H(K)
H(A(K))	Н(ЩК'У)	H(A(K))
H(|K|)	H(|K|)	H(|K|)
196	HOMOLOGY CHAP. 4
from which it follows that
A(/#) =
By theorem 4.6.13,	= r* and Л(<р^ (<р* P) = Л^т*) = А([С(<р)т]^).
Therefore A(/) = 0. 
This yields the following generalization of the Brouwer fixed-point
theorem.
8	corollary Every continuous map from a compact contractible poly-
hydron to itself has a fixed point.
proof If X is contractible, H(X) = 0, and for any f: X X, A(/) = 1
[because f* is the identity map on H0(X) ~ Z]. “
This result is false for noncompact polyhedra. In fact, R is a contractible
polyhedron and any translation different from 1R fails to have a fixed point.
Given a continuous map fi S’1	S’1, the degree of f is the unique integer
deg/such that
M = (tegf)z
The following fact is. obvious.
О For any map fi Sn Sn, A(/) = 1 + (— l)n deg /. в
Since the antipodal map S’! Sn has no fixed points, the next result
follows from theorem 7 and statement 9.
1	® corollary The antipodal map of S” has degree (— l)’l+1. 151
11	corollary If n is even, there is no continuous map fi Sn —> Sn such
that x and fix) are orthogonal for all x E Sn.
proof Assume that such a map exists. Then a homotopy F: f ~ 1,s" is de-
fined by
p/„ f) = (1 ~	+ fa'
11(1 - t)f(x) + fall
This is well-defined, because the condition that x and /(x) be orthogonal
implies ||(1 — t)/(x) + fa||2 = (1 — t)2 + t2 p 0 for 0 < t < 1. Since /~ Is",
A(/) = X(ls") = 1 + (-I)*1 /0. Hence, by theorem 7, / must have a fixed. •
point, in contradiction to the orthogonality of x and fix) for all x. 
This last result is equivalent to the statement that an even-dimensional
sphere Sn has no continuous tangent vector field which is nonzero everywhere
on Sn. For odd n such vector fields do exist because the map /: S2”1-1	S2’’^1
defined by
/(Xi, . . . ,X2m) — (— Xg, Xi, . . . , X2m, X’2W. 1)
is continuous and has the property that x and f(x) are orthogonal for all x.
SEC. 7 SOME APPLICATIONS OF HOMOLOGY
197
Instead of considering vector fields, we consider one-parameter groups
of homeomorphisms. A flow on X is a continuous map
f: R X X -+ X
such that
(a)	i^(ti + t2, x) = ^(ti, i//(t2,x)) ti, t2 € R; x £ X
(b)	^(0,x) = x	x £ X
For t E R let i/v X X be defined by ^f(x) = i^(f,x). Then (a) and (b) imply
f_t = (ft)-1, and so ft is a homeomorphism of X for all t E R- A fixed point
of the flow is a point x0 E % such that ^(t,x0) = x0 for all t E R.
12	theorem If X is a compact polyhedron with x(X) 0, then any flow
on X has a fixed point.
proof Each ft is homotopic to Lf [by the homotopy F: X X I X defined
ЪуЕ(х,Т) = i^((l — t')/, x)]. Therefore
Kft) = A(lx) = X(A) 7^= 0
Hence, by theorem 7, each ft has fixed points. For n > 1 let An be the
closed subset of X consisting of the fixed points of V-t/2". Then A„+i C An,
and {An} is a decreasing sequence of nonempty closed subsets of the com-
pact space X. Let F = П An- Then F is nonempty, and any point of F is
fixed under ft. for all t of the form 1/2” for n > 1. This implies that each
point of F is fixed under ft for all dyadic rationale t = m/2n. Since the dyadic
rationale are dense in R, each point of F is fixed under ft for all t. и
We now turn our attention to separation properties of the sphere.
13	lemma If A C Sn is homeomorphic to Ik for 0 < к < n, then
H(Sn - A) = 0.
proof We prove this by induction on k. If к = 0, then A is a point and
Sn — A is homeomorphic to R". Therefore H(Sn — A) = 0.
Assume the result for к < m, where m > 1, and let A be homeomorphic
to Im. Regard A as being homeomorphic to В X I, where В is homeomorphic
to I™-1, by a homeomorphism h: В X I A. Let A' = Zi(B X [O,1/?]) and
A" = h(B x [1^,1]). Then A = A' U A" and А' Г) A" is homeomorphic to
В X h. By the inductive assumption, H(Sn — (А' Г) A")) = 0. Because
S” — A' and S’! — A" are open sets, they are excisive and from the exactness
of the corresponding reduced Mayer-Vietoris sequence
i*: HQ(S” - A) = H(/S” - A') © We(Sn - A")
If z E HQ(S?t — A) is nonzero, then either i'^z 0 in HQ(Sm — A') or i*z --A 0
in He(Sn - A"), where i': S’1 — A C S« - A' and i": Sn — A C Sn - A".
Assume i* z 0. We repeat the argument for A' and thus obtain a sequence
of sets
198
HOMOLOGY CHAP. 4
A D Ai Э A2D  • •
such that
(a)	The inclusion S” — A C S" — A; maps z into a nonzero element of
HQ(S« - A;).
(b)	Г1 At is homeomorphic to I”1-1
Because every compact subset of Stl — ПА, is contained in S'" — Aj for
some /, it follows from theorem 4.4.6 that Hq(Sn — П A;) ~ lim . {Hq(Sn — A3)}.
This is a contradiction because, by condition (a), the element z determines a
nonzero element of lim , {//(;(S" — Д)}, but by condition (b) and the inductive
assumption, HfSn — П Aj) = 0. и
14	corollary Let В be a subset of Sn which is homeomorphic to S* for
0 < к < n — 1. Then
(J - lb — Л. - J.
proof We use induction on k. If к = 0, then В consists of two points and
S’! — В has the same homotopy type as S’1-1. Therefore
f Z q = n — 1
If к > 1, set В = Ai U A2, where Ai and A2 are closed hemispheres of Sk
and assume the result valid for к — 1. Then Ai and A2 are homeomorphic to
Ik and Ai Г) A2 is homeomorphic to Sfc-1. Because Sn — Ai and S’! — A2 are
open, {Sn — Ai, S’1 — A2) is an excisive couple, and there is an exact reduced
Mayer-Vietoris sequence
Hq+1(S« - Ai) © HQ+i(Sn - A2) HQ+i(S» - (Ai П A2))
Hq(S” — B)He(S” - Ai) © Hq(S« - A2)
By lemma 13, the groups at the ends vanish. The result then follows from the
inductive assumption, и
For the special case of an (n — l)-sphere imbedded in Sn, we obtain the
following Jordan-Brouwer separation theorem.
15	theorem An (n — Ifsphere imbedded in Sn separates S” into two com-
ponents of which it is their common boundary.
proof If В C S® is homeomorphic to Sn-1, then Ho(Sn — B) Z, by corol-
lary 14. Therefore Sn — В consists of two path components. Since Sn — В is
an open subset of Sn, it is locally path connected and its path components V
and V, say, are its components.
Clearly, В contains the boundary of U and of V. To prove В C U Cl V,
let x E В and let N be a neighborhood of x in S’!. Let А С В П N be a subset
such that В — A, is homeomorphic to In-1. Then H(Sn — (B — A)) = 0, by
lemma 13, so Sn — (B — A) is path connected. If p £ U and q E V, there is a
SI;( i g AXIOMATIC CHARACTERIZATION OF HOMOLOGY	IBB
ipath w in S” — (B — A) from p to q. Because p and q are in different path
components of S’! — B, u> meets A. Therefore A contains a point of U and a
point of V. Hence N meets U and V, and x £ U (T V. B
A related result is the following Brouwer theorem on the invariance of
domain.
16	theorem If U and V are homeomorphic subsets of Sn and U is open in
gn then V is open in S’1.
proof Let h: U V be a homeomorphism and let h(x) = y. Let A be a
neighborhood of x in 17 that is homeomorphic to In and with boundary В
homeomorphic to S’1-1. Let A' = 7i(A) С V and let B' = h(B). By lemma 13,
gn _ A' is connected, and by theorem 15, Sn — B' has two components.
Because
Sn — B' = (S” - A') U (A' - B')
and Sn — A' and A' — B' are connected, they are the components of Sn — B'.
Therefore A' — B' is an open subset of Sn. Since у £ A' — В' С V and у was
arbitrary, V is open in Sn. “
0 AXIOMATIC CHARACTERIZATION OF HOMOLOGY
A simple set of axioms characterizing homology on the class of compact
polyhedral pairs has been given by Eilenberg and Steenrod1. This section
describes the axiom system and related concepts. For compact polyhedral
pairs, the axioms are categorical (that is, two theories satisfying them are
isomorphic). Thus the axioms are basic theorems from which other properties
:of homology theories can be deduced. In many cases, proofs based on the axioms
are simpler and more elegant than proofs which refer back to the original de-
finition of the homology theory.
To formulate the axioms it is usual to start with a suitable category of
topological pahs and maps (called “admissible categories” by Eilenberg and
Steenrod). We shall not define these categories. The category of all topologi-
cal pairs is such a category, and so are its full subcategories defined by the
! polyhedral pairs and defined by the compact polyhedral pairs. For our pur-
poses we shall always regard a homology theory as defined on the category of
all topological pairs, and we identify a space X with the pair (X, 0).
A homology theory H and 3 consists of
(a) A covariant functor H from the category of topological pairs and
maps to the category of graded abelian groups and homomorphisms of
degree 0 [that is, H(X,A) = [17f/(X,A)}|
1 See S. Eilenberg and N. E, Steenrod, “Foundations of Algebraic Topology,” Princeton Univer-
sity Press, Princeton, N.J., 1952,
200
HOMOLOGY CHAP; 4
(b) A natural transformation Э of degree —1 from the functor H on
(X,A) to the functor H on (A, 0) [that is, d(X,A) = {dQ(X,A): Hg(X,A)
H^A)}].
These satisfy the following axioms.
1	homotopy axiom If f0, fi’. (X,A)	(Y,B) are homotopic, then
H(J0) = H(ff): H(X,A) H(Y,B)
2	exactness axiom For any pair (X,A) with inclusion maps i: А С X and
j: X C (X,A) there is an exact sequence
... Hg(A) He(X) Hg(X,A) H^A)	.
3	excision axiom For any pair (X,A), if U is an open subset of X such
that U C int A, then the excision map j: (X — U, A — U) C (X,A) induces
an isomorphism
H(f)-. H(X - U, A - t7) ~ H(X,A)
4	dimension axiom On the full subcategory of one-point spaces, there is a
natural equivalence ofH with the constant functor; that is, ifP is a one-point
space, then
Obviously, the homotopy axiom is equivalent to the condition that the I
homology theory can be factored through the homotopy category of topologi- ]
cal pairs.
Singular homology theory is an example of a homology theory. In fact,
the homotopy axiom is a consequence of theorem 4.4.9, the exactness axiom ;
is a consequence of theorem 4.5.4, the excision axiom is a consequence |
of corollary 4.6.5, and the dimension axiom is a consequence of lemmas 4.4.1
and 4.3.1. Therefore, there exist homology theories.
Corresponding to any homology theory there are reduced groups defined
as follows. If X is a nonempty space, let с: X —» P be the unique map from
X to some one-point space P. The reduced group Й(Х) is defined to be the
kernel of the homomorphism
H(cf H(X) H(P)
Because c has a right inverse, so does H(c). Therefore
H(X) ~ H(X) © H(P)	!
and the reduced groups have properties similar to those of the reduced |
singular groups.
Given a triple В С А С X, let к: A C (A,B) and define Э(Х,А,В):
H(X,A) —> H(A,B) to be the composite
SEC. 8 AXIOMATIC CHARACTEBIZATION OF HOMOLOGY
201
S(X,A,B) = H(k)0(X,A): H(X,A) H(A) H(A,B)
§ theorem For any triple (X,A,B), with inclusion maps r. (A,B) C (X,B)
and j: (X,B) C (X,A), there is an exact sequence
. •  Hq(A,B) Hq(X,B) Hq(X,A)	Hq^A,B) •
proof The proof involves diagram chasing based on the exactness axiom 2.
We prove exactness at Hq(A,B) and leave the other parts of the proof to the
reader.
(a)	im 9q+i(X,A,B) C ker Hq(f). Н^г)дд+1(Х,А,В) is the composite
Hq+1(X,A)	H<j(A) Hq(A,B) Hq/X,B)
which also equals the composite
Hq+i(X,A) МЛ'-Л)> hq(A) —Hg(X) Hq(X,B)
where г': А С X and i": X C (X,B). By axiom 2, IIf/(i')0f/+i(X,A) = 0. There-
fore He(i)0e+i(X,A,B) = 0.
(b)	ker Hg(f) G im 3?+i(X,A,B). Let z E Hq(A,B) be such that Hq(fjz = 0.
Then 3Q(X,B)HQ(i)z = 0, and because 0Q(A,B) = 3Q(X,B)HQ(i), 0(;(A,B)z = 0.
By axiom 2, there is / E HQ(A) such that = z. Because the composite
Hq(A) Hq(X) HQ(X,B)
equals the composite He(i)HQ(k), it follows that
Hq(i")Hq(i')zf = Hq(i)Hq(ky = Hg(i)z = 0
By axiom 2, there is z" £ Hq(B) such that if f: В С X, then
Given В C A, then HQ(f) = HQ(i')HQ(/"). Therefore Hq(i'ff - Hq(ffd'} = 0.
Again by axiom 2, there is z C He+r(X,A) such that 0e+i(X,A)z = / - Hg(j")z".
Then, because Hg(k)Hq(j") = 0,
0Q+i(X,A,B)z = HQ(k)0Q+i(X,A)z = Hg(tyd - Hq(k)Hq(f'y = z
which shows that z is in im 0Q+i(X,A,B). 
The exact sequence of theorem 5 is called the homology sequence of the triple
(X,A,B). If В = 0, it reduces to the homology sequence of the pair (X,A).
Let H and 0 and IJ' and 0' be homology theories. A homomorphism from
H and 0 to H' and S' is a natural transformation h from H to II' commuting
with 0 and S'. That is, for every (X,A) there is a commutative diagram
H(X,A)	H(A)
H'(X,A)	H'(A)
in which the vertical maps are homomorphisms of degree 0. In view of the
dimension axiom, a homomorphism h induces a homomorphism h0: Z —> Z
202
HOMOLOGY CHAP. 4
that characterizes h on one-point spaces. The main result proved by Eilenberg
and Steenrod is that corresponding to any homomorphism h0: Z —> Z there
exists a unique homomorphism h from H and Э to H' and д', on the category
of compact polyhedral pairs, which induces h0. We shall not prove this, but
shall content ourselves with proving that a homomorphism h which is an iso-
morphism for one-point spaces is an isomorphism for any compact polyhedral
pair. This will illustrate how the axioms-can be used and will suffice for our
later applications.
The following is an easy consequence of the exactness axiom and the
five lemma (or of theorem 5 and axiom 2).
6 lemma Let A' CACX, Then H(A') H(A) if and only if H(X,A') ~
H(X,A) (both maps induced by inclusion). “
We now prove a stronger excision property. A map f: (X,A) —» (Y,B) js
called a relative homeomorphism if f maps X — A homeomorphically onto
Y — B. Following are some examples.
7 An excision map (X — U, A — U) C (X,A), where 17 C A, is a relative
homeomorphism.
8 If X is obtained from A by adjoining an n-cell e and f: (En,Sn x) -a (e,e)
is a characteristic map for e, then f is a relative homeomorphism.
9 theorem Let X be a compact Hausdorff space and let A be a closed
subset of X which is a strong deformation retract of one of its closed neigh-
borhoods in X. Let f: (X,A) —> (Y,B) be a relative homeomorphism, where Y
is a Hausdorff space and В is closed in Y. Then, for any homology theory
H(f): H(X,A) ~ H(Y,B).
proof Let N be a closed neighborhood of A in X such that A is a strong
deformation retract of N and let U be an open subset of X such that
ACl7Cl7CX(l7 exists because X is a normal space). Let F: N X I N
be a strong deformation retraction of N to A.	x
Define A? = f(N) U B, U' = f(U) U B, and F: N' X I -+ N' by
F'(y,t) — У	у £ B,t £l
F(y,t) = fF(f~\y),t)	y£f(N),tEl
Then F is well-defined and continuous on each of the closed sets В X I and j
f(N) X I- Therefore F' is continuous and is easily verified to be a strong '
deformation retraction of N' to B. Because X — 17 is open in X — A, i
Y ~ №) U B) is open in Y — B, and because В is closed, it is open in Y. I
Therefore/((7) U В is closed in Y, and U' C f(U) U В C N'. Because X — 17 1
is a closed, and hence compact, subset of X, f(X — U) = Y — U' is a compact
subset of Y. Because Y is a Hausdorff space, Y — 17' is closed in Y, and V is
open in Y. We have В C U' C U' C N' and a commutative diagram
SJjC. 8 AXIOMATIC CHARACTERIZATION OF HOMOLOGY
203
H(X,A) H(X,N) H(X - U,N- U)
НП 1	1-
H(Y,B)	H(Y - U', N' - 17')
where the vertical maps are induced by f and the horizontal maps are induced
by inclusion maps. Because A and В are deformation retracts of N and N',
respectively, 11(A) ~ H(N) and 11(B) ~ H(N'). It follows from lemma 6 that
the left-hand horizontal maps are isomorphisms. The right-hand horizontal
maps are isomorphisms by the excision axiom. The right-hand vertical map is
an isomorphism because it is induced by a homeomorphism. From the com-
mutativity of the diagram, it follows that 11(f ) is an isomorphism. “
1© theorem Let h be a homomorphism from H and S to H' and S' which
is an isomorphism for one-point spaces. Then, for any compact polyhedral pair
(X,A), /i(X,A): H(X,A) = H'(X,A).
proof By the five lemma, it suffices to prove h(X): H(X) H'(X) for any
compact polyhedron X. Hence, let К be a finite simplicial complex. We need
only prove that /i(|K|): H(|K|) ~ H'(|K|). We prove this by induction on the
number of simplexes of K. If К has just one simplex, |K| is a one-point space,
and 7i(|K|) is an isomorphism by hypothesis.
Assume that К has m simplexes, where m > 0, and that h is an isomor-
phism for the space of any simplicial complex with fewer than m simplexes.
Assume dim К = n and let s be an n-simplex of K. Let L be the subcomplex
consisting of all simplexes of К different from s. By the five lemma and the
exactness axiom, /i(|K|) is an isomorphism if and only if /i(|K|,|L|) is an
isomorphism. If /: (|s|,|s|) C (|K|,|L|), it follows from theorem 9 that Ilf) and
H'f) are isomorphisms. Hence we need only prove that /i(|.sj,|.sj) is an
isomorphism.
If n = 0, (|s|,|s|) is a one-point space, and /i(|s|,|s|) is an isomorphism by
hypothesis. If n > 0, because |.sj has the same homotopy type as a one-point
space, h(|s|) is an isomorphism. By the five lemma and the exactness axiom,
?i(|s|,|s|) is an isomorphism if and only if h(|s|) is an isomorphism. Because s is a
proper subcomplex of K, /i(|s|) is an isomorphism by the inductive hypothesis. “
To extend this result to arbitrary polyhedral pairs (not merely compact
ones), we add an additional axiom. A pair (X,A) with X compact and A closed
in X is called a compact pair.
и AXIOM OF compact supports Given any pair (X,A) and given z £ Hq(X,A),
there is a compact pair (X’,A') C (X,A) such that z is in the image of
11(Х',Л')	H(X,A).
A homology theory H and Э satisfying axiom 11 is called a homology
theory with compact supports (Eilenberg and Steenrod use the term “homology
theory with compact carriers”). It is clear that singular homology theory is a
204	HOMOLOGY chap. 4
homology theory with compact supports. We shall see that any homology
theory with compact supports satisfies the analogue of theorem 4.4.6. The
following lemma is the main point in proving this.	j
12	lemma Let H be a homology theory with compact supports and let j
(X',A') be a compact pair in (X,A). Given z € Hq(X',Af in the kernel of >
Hg(X',A') —» He(X,A), there is a compact pair (X",A"), with (X',A') c
(X",A") C (X,A), such that z is in the kernel of H(X',A') —> H(X",A").
<
proof In the proof all unlabeled maps are induced by inclusion, z is in the
kernel of the composite
He(X',A') He(X,A') -» Hg(X,A)
By theorem 5, Hq(i)z is in the image of He(A,A') —> Hg(X,A'). By axiom Ц,
there is a compact space A" such that A' С A" C A and such that Hq(f}zis
in the image of the composite He(A",A') —> He(A,A') —> He(X,A'). By theorem 5,
the composite He(A",A') —> He(X,A') —> l/,/X,A") is trivial. Therefore z is in
the kernel of Hq(X',A') —> Hq(X,A"} for some compact A" containing A'.
Because z is in the kernel of the composite
Hq(X’,Af Hg(X' U A", A") -> Hg(X,A")
it follows from theorem 5, that He(/)z is in the image of
dq+1: Hq+1(X, X' U A") Hq(Xr U A", A")
By axiom 11, there is a compact X" containing X' U A" such that Hq(f)z is in i
the image of the composite
Hg+i(X", X' U A") -> He+1(X, X' U A") Hg(X' U A", A")
This composite is also equal to the map dq+1: Hq+1(X", X' U A") —>
Hg(X' U A", A"). By theorem 5, the composite
Hg+i(X", X' U A") Hg(X' U A", A") -> Hg(X",A")
is trivial. Therefore, z is in the kernel of Hg(X',A') —» Hg(X",A"). 	|
For any pair (X,A) the family of compact pairs (X',A') contained in (X,A) 
is directed by inclusion. For any homology theory H and Й the groups |
(H(X',Ar) | (X',A') compact C (X,A)} constitute a direct system, and the maps |
H(X',A') —> H(X,A) define a homomorphism i: lim, {H(X',A')} —> H(X,A).
13	theorem A homology theory H and d has compact supports if and only
if for any pair (X,A), i: lim{HIX',A')} ~ H(X,A), where (X',A') varies over ,
the family of compact pairs contained in (X,A).	!
proof It is clear that axiom 11 is equivalent to the condition that i be an j
epimorphism. Hence, if i is an isomorphism, H and 3 has compact supports.
Conversely, if H has compact supports, i is an epimorphism, and lemma 12
implies that i is also a monomorphism. 
exercises
205
14	theorem Let hbe a homomorphism from H and 3 to H' and 3' that is
(in isomorphism for one-point spaces. If H and 3 and H' and 3’ have compact
supports, h is an isomorphism for any polyhedral pair.
proof This follows from theorems 10 and 13 and from the fact that for any
polyhedral pair (X,A) the compact polyhedral pairs (X',AZ) contained in it are
cofinal in the family of all compact pairs in (X,A). и
exeibcises
A CHAIN HOMOTOPY CLASSES
1	For chain complexes C and C show that [C;C'] is an abelian group (with group
operation [ri] +- [r2] = pTi + t2]) and that there is a homomorphism
<p: [C;C'J -> Hom (H(C),H(CZ))
such that <p|r] = r,,:.
2 If C is a free chain complex, prove that the homomorphism <p is an epimorphism,
3 If C is a free chain complex and H(C) is also free, prove that is an isomorphism.
В EULER CHARACTERISTICS
I Let (X,A) be a pair and assume that two of the three graded groups H(A), H(X), and
H(A’A) are finitely generated. Prove that the third is also finitely generated and that
x(X) = X(A) + x(X,A).
2 Let {Xj,X2} be an excisive couple of subsets of X such that H(Xf) and /I(X2) are
finitely generated. Prove that /I(Xj U X2) is finitely generated if and only if H(Xi П X2)
is finitely generated, in which case
X(X1) + x(X2) = х(Х1 и x2) + x(Xr n X2)
3	Let у be an integer-valued function defined on the class of compact polyhedra with
base points such that
(a)	If (X,x0) is homeomorphic to (Y,yo), then у(Х,х0) = y(Y,//0).
(b)	If (X,A) is a compact polyhedral pair and x0 E A, then y(X,x0) = у(А,х0) +
y(X/A,xb), where X/A denotes the space obtained by collapsing A to a single point
Prove that for any X
y(X,x-0) = y(So,po)x(X,xo)
(ffint:1 Prove first that if z0 is a base point of En in S'1”1, then y(/?',Zo) = 0. Show that
the result is true for X = S", and then use induction on the number of simplexes in a
triangulation of X.]
4	If X and Y are compact polyhedra, prove that
x(X X Y) = x(X)x(Y)
1See С. E. Watts, On the Euler characteristic of polyhedra, Proceedings of the American
Mathematical Society, vol. 13, pp. 304-306, 1962.
206
HOMOLOGY СИДР. 4
C EXAMPLES
1	Let s be an n-simplex and let (s)™ be its m-dimensional skeleton. Compute H((s)m),
2	Compute the homology group of an arbitrary surface.	,
3	Compute the homology group of the lens space L(p,q).	
4	Let A be a subspace of S" which is homeomorphic to the one-point union S» v S«, I
Compute HIS" — A).
5	Let X be the space obtained from a closed triangle with vertices v0, v±, and t>2 by
identifying the edges t j t:2, and v2Uo linearly with the edges tj c2, c2to, and roti,
respectively. Compute HIX').
6	Given an integer n > 0 and an integer m > 1, prove that there exists a compact
polyhedron X such that
ад 4"
lZm q = n
7	Let H be a finitely generated nonnegative graded abelian group such that Ho is a
free abelian group. Prove that there exists a compact polyhedron X such that HIX) ~ H,
D JOINS AND PRODUCTS	]
1	Prove that for any space X there are isomorphisms
Hg(X) Hg+1(X * SO)
(Hint: If Y is contractible, so is X * Y.)
2	Prove that for any space X there are isomorphisms
Hg(X X s« X x po) ~ He_„(X)	'
[Hint: Use induction on n and the fact that if Y is contractible, H(X X Y, X X yo) = 0.]
3	Compute the homology group of the n-dimensional torus (S1)H.
4	If a space is homeomorphic to a finite product of spheres, prove that the set of
spheres which are the factors is unique.
E ORIENTATION
1	Let К be an n-dimensional pseudomanifold. Prove that it is possible to enumerate
the n-simplexes of К in a (finite or infinite) sequence «о, «1,  .,«<?, -   and to find a i
sequence si, s2, . . . , s'Q, . . . of (n — l)-simplexes of К such that for q > 1, s, is a face I
of sq and also a face of s, for some i q.
2	IfK is a finite n-dimensional pseudomanifold, prove that exactly one of the following '
holds:
(n) //„(/<,/<) ~ Z and H„_i(K,jK) has no torsion.
(b) Hn(K,K) — 0 and Нп_!(КЖ) has torsion subgroup isomorphic to Z2.
3	Let К be a finite simplicial complex which is homogeneously n-dimensional and such I
that every (n — l)-simplex of К is the face of at most two n-simplexes of K. Let К be the ’
subcomplex of К generated by the (n — l)-simplexes of К which are faces of exactly one ,
n-simplex of K. Prove that if (K,K) satisfies either (a) or (b) of exercise 2 above, then К
is an n-dimensional pseudomanifold.
A finite n-dimensional pseudomanifold is said to be orientable (or nonorientable) if it
exercises
207
satisfies (a) (or (/>)) of exercise 2. An mentation of an orientable n-dimensional pseudo-
manifold К is a generator of //(/<,/<), and an oriented n-dimensional pseudomanifold is
an n-dimensional pseudomanifold together with an orientation of it.
4	Let z £ Hn(K,K) be an orientation of a finite n-dimensional pseudomanifold. If s is
any n-simplex of K, prove that there is a unique orientation of s, denoted by z | s £ Hn(s,s)
and called the induced orientation of s, characterized by the property that z and z | s
correspond under the homomorphisms
H„(K,K) - Hn(K, K-s)^ Hn(s,s)
A collection of orientations {a(s) £ Hn(s,s)} for each n-simplex s of an n-dimensional
pseudomanifold is called compatible if for any (n — l)-simplex s' of К — К which is a
face of the two n-simplexes sj and s2 of K, a(si) and — a(s2) correspond under the
homomorphisms
Hn(si,si) fti-iCri) ->	si - s')
~	H„i(s',.£)
Hn(s2;s2) Л H„_i(s2) Hn_-ds2, s2 — s')
5	If z is an orientation of a finite n-dimensional pseudomanifold, prove that the col-
lection {z | s} is compatible. Conversely, given a compatible collection {a(s)} of orienta-
tions of the n-simplexes s of a finite n-dimensional pseudomanifold K, prove that there
is a unique orientation z of К such that z | s = cr(s) for each n-simplex s of K. Use this to
define orientability for arbitrary (nonfinite) n-dimensional pseudomanifolds. [Hint: Iden-
tify //„(KjK"-1) with indexed collections (a(s) £ Hn(s,s)}, where s varies over the
n-simplexes of K, and show that the image of the homomorphism Hn(K,K) —> Hn(K,K”-1)
consists of the compatible collections.]
; F DEGREES OF MAPS
J	Let Ki and K2 be finite n-dimensional pseudomanifolds with orientations z: and z2,
f	respectively. Given a continuous map/: (||,|/<i|) —> (|K2|,|K2|), its degree, denoted by
\	deg /, is the unique integer such that f*(zf) = (deg /)z2 [where we regard
[ zi £ HndKiMKil)) and z2 £ H„(|K2|,|K2|)].
।	1 Let rp: (K^Kf) —> (K2,K2) be a simplicial approximation to /, let s2 be a fixed
'	n-simplex of K2, and let m+(<p) (or ?n. (<p)) be the number of n-simplexes si of Ki such
that <p maps the induced orientation Zi | «i into the induced orientation z2 | s2 (or into
।	— Z2 | s2). Prove that deg / = m+(<f) —
p 2 In case К is a finite orientable n-dimensional pseudomanifold and /: (|K|,|K|) —>
(|K|,|K|), there is a unique integer deg/such that/,. (z) = (deg/)z for any z £ H„(|K|,|K|).
Prove that if/, g: (|K|,|K|) -> (|K|,|K|), then deg (g ° /) = (deg g) (deg/).
3 Let /: Sn —> Sn be a map such that flE”) C f(EL) С E" and let /': Sn-1 —> S'"1
be the map defined by /. Prove that deg f = deg /'.
(.	4 Show that for any n > 1 and any integer in there is a map /: Sn —> S” such that
deg/= m.
G TOPOLOGICAL INVARIANCE OF PSEUDOMANIFOLDS
J I Let К be a simplicial complex and let r £ <s}, where s is a simplex of K. Prove that
208
HOMOLOGY CHAP. 4
there is an isomorphism
H(\K\, |K| — st s) ~ H(|K|, |K|-x)
2	Let К be a simplicial complex and let x € (л), where s is a principal n-simplex of К
(that is, s is not a proper face of any simplex of A'). Prove that
И,(|К|,Н -«)=!»
3	Prove that a locally compact polyhedron X has dimension n if and only if n is the
largest integer such that there exist points x € X, with Hn(X, X — x) 0.
4	Let X be a locally compact polyhedron and for each n let Xn be the closure of the
set of all x E X having a neighborhood U such that Hn(X, X — y) ~ Z for all у E U. If
К is any simplicial complex triangulating X and Kn is the subcomplex of К generated by
the principal n-simplexes of K, prove that Kn triangulates Xn.
5	Prove that the property of being homogeneously n-dimensional is a topologically
invariant property of simplicial complexes (and so we can speak of a homogeneously
n-dimensional polyhedron).
6	Let К be an arbitrary simplicial complex triangulating a homogeneously n-dimensional
polyhedron X. Prove that every (n — l)-simplex of К is the face of at most two n-simplexes
of К if and only if HJA, A — x) = 0 for all x E A and all q > n — 1, where A is the
closure in X of the set {x £ X | I/,,(X, X — x) is noncydic}.
T Let X be a homogeneously n-dimensional polyhedron satisfying exercise 6 and let
X = Bn_i, where В is the closure in X of the set {x E X | Hn(X, X — x) = 0} and where
Bn_i is defined in terms of B, as in exercise 4. If К is any simplicial complex triangulating X,
prove that the subcomplex of К generated by the (n — l)-simplexes of К which are faces
of exactly one n-simplex of К triangulates X.
8 Prove that the property of being a finite n-dimensional pseudomanifold is a topo-
logically invariant property of simplicial complexes.
И EDGE-PATH GROUPS
1 Let К be a connected simplicial complex with a base vertex c0 E K- Given an edge
e = (co,ti), of K, let [e] be the oriented 1-simplex If f = eje2  •  is a closed edge
path of К at v0, let ^(f) = [ег] + [e2] + • • • + |ej E Cj(A'). Prove that ^/(f) is a cycle
and that Я f and are equivalent edge paths, then V'© and ^©) are homologous. 4
2 Prove that there is a natural transformation tp: E(K,vg) —> Hi(K) (on the category of
connected simplicial complexes with a base vertex) defined by = (V'(f)}-
3 Prove that the homomorphism tp is an epimorphism and has kernel equal to the com-
mutator subgroup of E(K,o0).
I AXIOMATIC HOMOLOGY THEORY
In this group of exercises H will denote an arbitrary homology theory.
1 Let Xi and X2 be subspaces of a space X. Prove that the following are equivalent:
(a) The excision map (Xi, Xi П X2) C (Xi U X2, X2) induces an isomorphism of
homology.
(b)	The excision map (X2, Xi П X2) C (Xi U X2, Xi) induces an isomorphism of
homology.
(c)	The inclusion maps
eXi;hCISES
209
ii:	(Xi, Xi П Xz) C (Xi U X2, Xi П X2)
and
i2:	(X2, Xi П X2) C (Xi U X2, Xi П Xz)
induce monomorphisms on homology and
H(X1 U X2, Xi П X2) ii*H(Xi, Xi n Xz) © i2,H(X2, Xx П X2)
(d)	The inclusion maps
ii-	. (Xi U X2, Xi П X2) C (Xi U Xz, Xx)
and
/2: (Xi U X2, Xi П X2) C (Xi U X2, X2)
induce epimorphisms on homology and /х» and /2» induce an isomorphism
H(Xi U X2, Xx П X2) H(Xi U X2, Xi) © H(Xi U X2, X2)
(e)	For any A C Xx A X2 there is an exact Mayer-Vietoris sequence
---> Hq(Xi A X2, A) -+ Hq(Xi,A) © Hq(X2,A)
-> Hq(Xi U X2, A) -> He_x(Xx П X2, A) ->   
(f)	For any У Э Xi U X2 there is an exact Mayer-Vietoris sequence
---> Hq(Y, Xi A X2) Hq(Y,Xi) © Hq(Y,X2)
Hq(Y, Xi U Xz) -h> Hq~i(Y, Xi A Xz) ->   •
2	Let Xi, . . . , X„, and A be closed subspaces of a space X such that
(a)	X = U Xf.
(b)	Xi П X; = A if i ф j.
(с)	Х, — A is disjoint from X, — A if i j.
Prove that the homomorphisms H(Xt,A) —> H(X,A) are monomorphisms and H(X,A) is
isomorphic to the direct sum of the images.
3	Let {XyJ^j (with / possibly infinite) be a collection of closed subsets of a space X
and let A be a subspace of X such that (a), (b), and (c) of exercise 2 above are satisfied.
^Assume also that every compact subset of X is contained in a finite union of {X,} and
'that H is a homology theory with compact supports. Prove that H(X,A) ~ H(Xj,A).
4	Let (X,A) be a topological pair and let {Xs} be a family of subspaces of X indexed
by the integers such that
(a)	A = X_x.
(b)	Xs C Xs+x for all s.
(с)	X = U Xs and every compact subset of X is contained in Xs for some s.
(d)	Hq(Xs,Xs_i) = 0 if q s and s > 0.
"Int C — {CQ,0Q} be the nonnegative chain complex with Cq = HQ(XQ,XQ_x) for q >0 and
<i(/ the boundary operator of the triple (Xe,XQ_x,XQ_2) for q > I. If H has compact sup-
ports, prove that H(X,A) ~ II(C'). [Hint: Prove that there are exact sequences
Hq+i(Xq+1,Xq) Д Hq(Xq) Hq(X) 0
;and
0 Hq(Xq) Hq(Xq,Xq_i) HQ_x(XQ_x)]
5	Let H be a homology theory defined on the category of compact pairs. Prove that
there is an extension of H to a homology theory H with compact supports such that
Й(Х,Л) = lim , [H(X',A') j (X',A') a compact pair in (X,A)}.
CHAPTER FIVE
PRODUCTS 4
WE ABE NOW READY TO EXTEND THE DEFINITION OF HOMOLOGY TO MORE GENERAL
coefficients. In this framework the homology considered in the last chapter
appears as the special case of integral coefficients. The extension is done in a
purely algebraic way. Given a chain complex C and an abelian group G, their
tensor product is the chain complex C ® G = {Cf; ® G, Se ® 1}, and the
homology of C ® G is defined to be the homology of C, with coefficients G.
We shall also introduce the concepts of cochain complex and cohomology.
These are dual to the concepts of chain complex and homology and arise on
replacing the tensor-product functor by the functor Hom.
We shall establish universal-coefficient formulas expressing the homology
and cohomology of a space with arbitrary coefficients as functors of the
integral homology of the space. Although these new functors do not distinguish
between spaces not already distinguished by the integral homology functor,
it is nonetheless important to consider them, as it frequently happens that the
most natural functor to apply in a given geometrical problem is determined
by the problem itself and need not be the integral homology functor. For
example, in the obstruction theory developed in Chapter Eight we shall be
211
212
PRODUCTS CHAP. 5
led to the cohomology of a space with coefficients in the homotopy groups of
another space.
A further consideration is that the cohomology of a space has a multipli-
cative structure in addition to its additive structure, which makes cohomology
a more powerful tool than homology. We shall present some applications of
this added multiplication structure, the most important of which is the study
of the homology properties of fiber bundles, where we establish the exactness
of the Thom-Gysin sequence of a sphere bundle.
At the end of the chapter is a brief discussion of cohomology operations.
These are natural transformations between two cohomology functors and
strengthen even further the applicability of cohomology as a tool. We shall
define the particular set of cohomology operations known as the Steenrod
squares and establish their basic properties.
Sections 5.1 and 5.2 are devoted to homology with general coefficients
and to the universal-coefficient formula for homology. Section 5.3 deals with
the tensor product of two chain complexes and contains a proof of the
Kiinneth formula expressing the homology of the tensor product as a functor
of the homology of the factor complexes. This is applied geometrically to ex-
press the homology of a product space in terms of the homology of its factors.
Sections 5.4 and 5.5 contain the dual concepts of cochain complex and
cohomology and the appropriate universal-coefficient formulas for them. In
Sec. 5.6 the cup and cap products are defined, the cup product being the
multiplicative structure in cohomology mentioned previously, and the cap
product being a dual involving cohomology and homology together. These
products are used in Sec. 5.7 to study the homology and cohomology of fiber
bundles. We establish the Leray-Hirsch theorem, which asserts that certain
fiber bundles have homology and cohomology which are additively isomorphic
to the homology and cohomology of the corresponding product of the base
and the fiber.
Section 5.8 is devoted to a study of the cohomology algebra. The exact-
ness of the Thom-Gysin sequence is used to compute the cohomology algebra
of projective spaces, and this, in turn, is used to prove the Borsuk-Ulam
theorem. There is also a discussion of the structure of Hopf algebras, which
arise in considering the cohomology of an H space. In Sec. 5.9 the Steenrod
squares are defined and their elementary properties are proved. They will be
applied later.
1 HOMOLOGY WITH COEFFICIENTS
In this section we shall extend the concepts dealing with chain complexes to
the case where the chain groups are modules over a ring. The tensor product
of such a chain complex with a fixed module is another chain complex, and
its graded homology module is a functor of the original chain complex and
SEC. 1 HOMOLOGY WITH COEFFICIENTS
213
the fixed module. These homology modules have properties analogous to those
established in the last chapter for complexes of abelian groups. The section
closes with the definition of a homology theory with an arbitrary coefficient
module. This is analogous to the concept of homology theory (which has
integral coefficients) introduced in the last chapter.
Throughout this section R will denote a commutative ring with a unit.
We consider R modules and homomorphisms between them. A chain complex
over R, C = {CQ,3Q} consists of a sequence of R modules CQ and homomor-
phisms Cg —» such that й,;Й,;+1 = 0 for all q. There is then a graded
homology module
H(C) = {Hq(C) = ker 0g/im
The concepts of chain maps and chain homotopies can be defined for chain
complexes over R, and the results about chain complexes of abelian groups
generalize in a straightforward fashion to chain complexes over R. In particu-
lar, on the category of short exact sequences of chain complexes over R,
0 C -> C -> C" -> 0
there is a functorial connecting homomorphism
9^:	Hq_^C')
and a functorial exact sequence
. . . H Hg(C') -» Hq(C) Hq(C") \ Hq.^C’)	 • •
If C is a chain complex over R and G' is an R module, an augmentation
of C over G is an epimorphism e: Co —» G such that e ° 3i = 0. An
augmented chain complex over G consists of a nonnegative chain complex C
and an augmentation of C over G.
If C = {Ce,0a} is a chain complex over R and G is an R module, then
C ® G = {Cq ® G, дд ® 1} is also a chain complex over R, and if C is
augmented over G, then C ® G is augmented over G ® G. The graded
homology module H(C ® G) is called the homology module of C with coeffi-
cients G and is denoted by H(C;G). If т: C —e> C' is a chain map,
т ® 1: C ® G —» C' ® G is also a chain map, and r* : H(C;G) —-> H(G;G) de-
notes the homomorphism induced by т ® 1. Given a homomorphism
<p: G —> G, there is a chain map 1 ® q>: C ® G —> C ® G inducing a
homomorphism
<p^:H(C;G)-»H(C;G')
These remarks are summarized in the following statement.
1 theorem There is a covariant functor of two arguments from the
category of chain complexes over R and the category of R modules to the cat-
egory of graded R modules which assigns to a chain complex C and module G
the homology module of C with coefficients G. 
214	phoducts сидр. 5
Note that if c £ Cg is a cycle of C and g € G, then c ® g £ C5 ® G is a
cycle of C ® G, and if c is a boundary, so is c ® g. Therefore there is
a bilinear map
H5(C) X G Hq(C;G)
which assigns to ({c},g) the homology class (c ® g). This corresponds to a
homomorphism
p: H(C) ® G H(C-G)
such that /-<({/'} ® g) = (c ® g} for c € Z(C). The homomorphism g is easily
verified to be a natural transformation on the product of the category of chain
complexes with the category of modules.
If C is a chain complex over Z and G is an R module, then C ® G is a
chain complex over R. It follows from theorem 4.5 in the Introduction that
the homology module over Z of C with coefficients G is isomorphic, as a
graded R module, to the homology module over R of C ® R with coefficients G.
2	example Let C(K) denote the oriented chain complex of the simplicial
complex K. Given an abelian group G and a simplicial pair (K,L), the oriented
homology group of '-,(K,L) with coefficients G, denoted by H(K,L; G), is
defined to be the graded homology group of [C(K)/C(L)] ® G (which is
augmented over Z ® G ~ G). Then H(K,L; G) is a covariant functor of two
arguments from the category of simplicial pairs and the category of abelian
groups to the category of graded abelian groups. If G is also an R module,
H(K,L; G) is a graded R module. Similar remarks apply to the ordered chain
complex Д(К)/Д(Ь).
3	example If (X,A) is a topological pair and G is an abelian group, the
singular homology group of (X,A) with coefficients G, denoted by H(X,A; G),
is defined to be the graded homology group of [Д(Х)/Д(А)] ® G (which
is augmented over G). It is a covariant functor of two arguments from the
category of topological pairs and the category of abelian groups to the cate-
gory of graded abelian groups. If G is an R module, H(X,A; G) is a graded
R module.
Because the ring R is commutative, there is a canonical isomorphism
G ® G' ~ G’ ® G for R modules G and G'. Therefore, if C is a chain com-
plex over R, G ® C is canonically isomorphic to C ® G. Hence no new
homology modules are obtained from G ® C.
We recall some general properties of tensor products which will be
important in the next section.
4	lemma The tensor product of two epimorphisms is an epimorphism.
pboof Let a: A —> A" and /3-. R —> B" be epimorphisms. A" ® B" is gener-
ated by elements of the form a" ® b", where a" A" and b" c R". Since
a and ft are epimorphisms, A" ® B" is generated by elements of the form
SEC. 1 HOMOLOGY WITH COEFFICIENTS
215
(t(o) ® ДЬ), where fl C A and b £ B. Since (a ® ft)(a ® b) = ft'(a) ® ft(b),
£' ® B" is generated by (a ® /3)(А ® B), showing that a ® /3 is an
epimorphism, и
In general, it is not true that the tensor product of two monomorphisms
js a monomorphism (see example 7 below). The following lemma shows that
something can be said about the kernel of a ® ft when a and ft are epimorphisms.
5	lemma If a and ft are epimorphisms, the kernel of a ® ft is generated
by elements of the form a ® b, where a € ker a or b E ker ft.
pboof Let a: A A" and ft: В B" be epimorphisms and let D be the
submodule of A ® В generated by elements of the form a ® b, where
a £ ker a or b E ker ft. Let p: A ® В —> (A ® B)/D be the projection. There
is a well-defined bilinear map
A" X B” -» (A ® B)/D
sending (a",b") to p(a ® b), where a C A and h E В are chosen so that
a(a) = a" and ft(b) — b". This bilinear map corresponds to a homomorphism
f: A” ® B" -> (A ® B)/D
such that f(a" ® b") = p(o ® b), where n(«) = a" and ft(b) = b". It is then
obvious that p equals the composite
A ® В A" ® В" Л (A ® B)/D
This shows that ker (a ® ft) C D. The reverse inclusion is evident, showing
that ker (« ® ft) = D. B
6	corollary Given an exact sequence
A' -> A -> A" -> 0
and given a module B, there is an exact sequence
A'®B-+A®B—>A"®B—>0
proof It follows from lemma 4 that A ® В —> A" ® В is an epimorphism,
so the sequence is exact at A" ® B. If A C A is the image of A' -> A, then,
by lemma 4, A' ® В —> A ® В is an epimorphism. Because A is also the
kernel of A —» A", it follows from lemma 5 that the kernel of A ® В —> A" ® В
is the image of A ® В —> A ® B. Therefore the sequence is exact at
A® B. 
If the original sequence is assumed to be a short exact sequence, it need
not be true that the tensor-product sequence is a short exact sequence. We
present an example to illustrate this.
7	example Over Z, consider the short exact sequence
z Д z 4 z2 о
216
PRODUCTS СИДР. 5
where a(l) = 2 and /3(1) is a generator 1 of Z2. The tensor product of this se-
quence with Z2 is not a short exact sequence because a ® 1: Z ® Z2 —» Z ® Z2
is not a monomorphism [Z ® Z2 Z2 A 0, but (a ® 1)(1 0 1) = 2 ® I 5
10 2-1 = 0].
8	theorem The tensor-product functor commutes with direct sums.
proof Assume A = © Aj and consider the bilinear map A X В —» © (A; 0 B)
sending (S dj, b) to S (aj 0 b) and the homomorphisms Aj 0 В —> A 0 В
for all By the characteristic properties of tensor product and direct sum,
there are commutative triangles
Ax В	Aj® В
J- 4	Z I
A® B-^@(Aj® В) A® В <^@(Aj® B)
Clearly, the maps <p and f are inverses, showing that A ® В ~ © (Aj ® B).
If, also, В = © В/,, then similarly,
A®B-@Aj®Bk 
9	theorem The tensor-product functor commutes with direct limits.
proof Let A = lim, {A“} and consider the bilinear map A X В ~^> lim,
(A“ ® B} sending (fa},b) to {a ® b} for a £ A“ and the homomorphisms
A“ 0 В —> A ® В for all a. By the characteristic properties of tensor product
and direct limit, there are commutative triangles
A x В	Aa ® В
A® B^> lim, {A“ ® В) A ® В Д iinu ® в}
Clearly, <p and f are inverses, showing that A 0 В ~ lim , (A° 0 B}. If, also,
В = lim, {BP}, then similarly, A 0 В ~ lim, {A“ ® BP}. 
We now consider a special class of short exact sequences. These sequences
have the property that their tensor product with any module is again exact.
A short exact sequence
0 -» А' А А Л A" -> 0
is said to be split if /? has a right inverse (that is, if there exists a homomor-
phism /3': A” A such that /3 ° /3' = Li")- We also say that the sequence
splits.
I® example Any short exact sequence 0-> AA A 4 A" —> 0 with A"
free is split. To see this, let {a}'} be a basis for A" and for each / choose ц € A
so that /3(oj) = a’j. Let f3’-. A" A be the homomorphism such that f3'(a'j') = Щ
for all /'. Then /?' is a right inverse of [3.
gEC. 1 HOMOLOGY WITH COEFFICIENTS
217
1 ] lemma Given a short exact sequence
0 -> А' А А Л A" -» 0
define A' -G A' © A" A> A” by i(a') = (a',0) and p(a’,a") = a". Then
the following are equivalent:
(a)	The sequence is split.
(b)	There is a commutative diagram
A e
A' 7'T A"
A' © A"
(c)	There is a commutative diagram
Jf
A' 4 A"
A'® A"
(d)	a has a left inverse.
proof If /?': A” A is a right inverse of /3, let y': A' © A" —> A be defined
by -/(g'.g") = a(a') + /?'(«") Then y' has the desired properties. Conversely,
given y’, define ft'-. A" —» A by f>'(a"') = y'(G,a"). Then fl' is a right inverse
of ft, so the sequence is split. Therefore (a) is equivalent to (b). A similar argu-
ment shows that (c) is equivalent to (d). It follows from the five lemma that
in the diagram of (b) [or (c)], y' [or y] is necessarily an isomorphism. There-
fore (b) is equivalent to (c) with y' equal to y-1. 
12	corollary Given a split short exact sequence
(M A' 4 A A" -A)
and given a module B, the sequence
O-*A'®B	A®B-»A"®B^O
is a split short exact sequence.
proof By corollary 6 and lemma 11 we need only show that a ® 1 has a
left inverse. By lemma 11, a has a left inverse a'. Then я' ® 1 is a left
inverse of a ® I. 
In case 0 —e> C' —» C —» C" —> 0 is a split short exact sequence of chain
complexes, it follows from corollary 12 that for any module G the sequence
0 C' ® G -> C © G -> C" ® G -> 0
is a short exact sequence of chain complexes. This short exact sequence gives
rise to an exact homology sequence, and we obtain the next result.
218
PRODUCTS CHAP. 5
13	theorem Given a split short exact sequence of chain complexes
()- > C'	C—> C" -> 0
and given a module G, there is a functorial exact homology sequence
-----» H5(C';G) HS(C;G) Hq(C"-G) H^C'-G)	• a
This implies the exactness of the singular homology sequence (and
reduced homology sequence) of a pair with arbitrary coefficients. Similarly,
there is an exact sequence of a triple with arbitrary coefficients. All these
sequences (except the reduced sequence of a pair) are consequences of the
exactness of the relative Mayer-Vietoris sequence, which we now establish. If
{(Xj.Ai), (Хг.Лг)} is an excisive couple of pairs in a topological space, the
short exact sequence of singular chain complexes
0 -> Д(Х2 n Х2)/Д(А! n a2)
Д(Х1)/Д(А1) © Д(Х2)/Д(А2) -» Д(Х1 и Х2)/Д(Ат и А2)^ о
is split [by example 10, because Д(Х1 U Х2)/Д(А1 U A2) is a free abelian
group]. Therefore we obtain the following result.
14	corollary If {(Xi,Ai), (X2,A2)} is an excisive couple of pairs in a space
and G is an R module, there is an exact relative Mayer-Vietoris sequence of
{(Xi,Ai), (X2,A2)} with coefficients G. в
If G is fixed, the singular homology of (X,A) with coefficients G satisfies
all the axioms of homology theory except the dimension axiom (all of them
are easily seen to hold except exactness, which follows from corollary 14). If
P is a one-point space, there is a functorial isomorphism Ho(P; G) G. This
leads to the following definition.
Let G be an R module. A homology theory with coefficients G consists
of a covariant functor H from the category of topological pairs to graded
R modules and a natural transformation 3: H(X,A) —> H(A) of degree — 1 sat-
isfying the homotopy, exactness, and excision axioms, and satisfying the
following form of the dimension axiom: On the category of one-point spaces
there is a natural equivalence of H with the constant functor which assigns
to every one-point space the graded module which is trivial for degrees other
than 0 and equal to G in degree 0. A homology theory with coefficients Z is
called an integral homology theory. An integral homology theory is the same
as a homology theory as defined in Sec. 4.8.
Singular homology with coefficients G is an example of a homology
theory with coefficients G. The uniqueness theorem 4.8.10 is valid for
homology theories with coefficients.
In the next section we shall show how the singular homology modules
with coefficients are determined by the integral singular homology groups.
SEC. 2 THE UNIVERSAL-COEFFICIENT THEOREM FOR HOMOLOGY
219
2 ТИЕ UNSVEBSAE-COEFFIFIENT THEOREM
FOR fflOMOIAMSY
In order to express H(C;G) in terms of H(C) and G, it is necessary to intro-
duce certain functors of modules that are associated to the tensor-product
functor. This section contains a definition of these functors, and a study of
them in the special case of a principal ideal domain. This leads to the
universal-coefficient theorem. In the next section these new functors will
enter in a description of the homology of a product space.
Let A be an В module. A resolution of A (over B) is an exact sequence
--------------------» C, A-------> C A Co A A 0
If, in addition, each Ce is a free В module, the resolution is said to be free.
Thus a resolution of A consists of a chain complex C = {Ce,3e} over В which
is augmented over A and is such that C is acyclic. The resolution is free
if and only if the chain complex C is free.
Any В module A has free resolutions. In fact, given an В module B, let
F(B) be the free В module generated by the elements of В and let F(B) —» В
be the canonical map. The canonical free resolution of A is the following re-
solution (defined inductively):
-----> F(ker dg) A F(ker S^i) A • • F(ker e) F(A) A A -> 0
The method of acyclic models applies to chain complexes over В and,
when applied to a category consisting of a single object and single morphism,
implies the following result.
1 theorem Let C be a free nonnegative chain complex augmented over
A and let C be a resolution of A'. Any homomorphism <p: A A' extends to
a chain map
• • • —> CQ+1 —Cq—>    Cq A Л 0
1 I ll*
----> C' + 1-> Cq->---->C'o^ A'—> 0
preserving augmentations, and two such chain maps are chain homotopic. 
Specializing to the case <p = 14: A C A, we obtain the next result.
2 corollary If C and C are free resolutions of A, then C and C are
canonically chain-equivalent chain complexes. 
For modules A and В and a free resolution C of A, it follows from corol-
lary 2 that the graded module H(C;B) depends only on A and B. Let C be the
canonical free resolution of A. For q > 0 we define the qth torsion product
TorQ (A,B) — Hq(C;B). It is a covariant functor of A and of B. From the short
220	products CHAP, 5
exact sequence
0 -> Э1С1 -> Co A -> 0
it follows from corollary 5.1.6 that there is an exact sequence
01CX ® В -> Co ® В ^4 A ® В -> 0
By definition, Tor0 (A,B) is the zeroth homology module of the chain complex
------------------> c2 ® В Cl ® В Ai-4 Co ® в 0
Hence Tor0 (A,B) = (Co ® B)/im (9i ® 1). By the above exact sequence,
im (0i ® 1) = im (0jCi ® В Co ® B) = ker (e ® 1)
Therefore
Toro (A,B) = (Co ® B)/ker (e ® 1) A ® В
and so Tor0 (A,B) is naturally equivalent to A ® B.
All the previous remarks are valid for any commutative ring with a unit.
For the remainder of this section we specialize to the case where R is a prin-
cipal ideal domain. Over a principal ideal domain any submodule of a free
module is free. Therefore any module A has a short free resolution of the form
0 —> Ci —> Co —> A —> 0
(simply let Co = F(A) and Ci = ker [F(A) —> A]). Such a short free resolu-
tion of A is the same as a free presentation of A. Because there exist short
free resolutions, Torf; (A,B) = 0 if q > 1. We define the torsion product
A * В to equal Тогг (A,В). It is characterized by the property that, given any
free presentation of A,
0—»Ci~»C0—»A—>0
there is an exact sequence
O-»A*B-»Ci®B^Co®B-»A®B-»O
In fact, A * В ~ Hi(C ® B) = ker (Ci ® В -> Co ® B), since C2 ® В = 0.
The torsion product is a covariant functor of each of its arguments.
Because the tensor product commutes with direct sums and direct limits (by
theorems 5.1.8 and 5.1.9) and the direct limit of exact sequences is exact (by
theorem 4.5.7), the torsion product also commutes with direct sums and
direct limits. Its name derives from the fact that it depends only on the
torsion submodules of A and В (see corollary 11 below).
3 example If A is free, it has the free presentation
0	0 —>A^>A —> 0
from which we see that A * В = 0 for any B.
SEC- THE UNIVEKSAL'COEFFICIENT THEOREM FOR HOMOLOGY	221
I 4 example If A is the cyclic R module whose annihilating ideal is generated
1 by 311 element v € R, then A R/vR and there is a free presentation of A
I	0 > h ' ->/{ > Д -> ()
i in which a(v') — vv' for v' € R. For any module В there is an isomorphism
Л ® В ~ В sending 1 ® b to b. Under this isomorphism, the map
й ® 1: R ® В R ® В corresponds to а': В —» В, where a'(b) = vb for
 h € B- Therefore ker a' is the submodule of В annihilated by v, and so
£	(R/uR) * В ~ {b £ В | vb = 0}
f The above examples suffice to compute A * В for a finitely generated
j imodule A (because of the structure theorem 4.14 in the Introduction). This
1 theoretically determines A * В for arbitrary A, because any A is the direct
f limit of its finitely generated submodules (see theorem 4.2 in the Introduction)
V and the torsion product commutes with direct limits.
5 lemma If A or В is torsion free, then A* В — 0.
J proof Because the torsion product commutes with direct limits, it suffices
I to consider the case where A and В are finitely generated, in which case being
torsion free is equivalent to being free. If A is free, the result follows from
example 3. If В is free and finitely generated, it is isomorphic to a direct sum
of n copies of R. If
0 Cy -> Co -> A -> 0
is a free presentation of A, then Ci®B—»C0®B—»A®B—»0is isomorphic
to a direct sum of n copies of the sequence Ci®R—»C0®R—»A ® R —> 0.
у Since Ci ® R —> Co ® R is a monomorphism, so is Ci ® В —> Co ® B, and
’ A » В = 0. 
» It follows that if R is a field, then A * В = 0 for all modules A and B.
The following result is proved similarly by proving it first for finitely generated
modules (where being torsion free is equivalent to being free) and taking
direct limits to obtain the result for arbitrary modules.
। в lemma Given a short exact sequence of modules
,	о A' A A" 0
and given a module B, if A!' or В is torsion free, there is a short exact
sequence
]	0 -» A' ®B-> A®B-> A" ® /S 0
j proof As remarked above, it suffices to prove the result if A" or В is free and
finitely generated. If A" is free, the original sequence splits, by example 5.1.10,
and the result follows from corollary 5.1.12. If В is free and finitely generated,
the map A'®B—»A®B is a finite direct sum of copies of A' ® R —> A ® R,
222
PRODUCTS CHAP. 5
and hence a monomorphism. The result follows from this and corollary 5.1.6. <
We use this result to obtain an exact sequence of homology correspond-
ing to a short exact sequence of coefficient modules.
7	theorem On the product category of torsion-free chain complexes C
and short exact sequences of modules
0 ^ G' A C A G'A 0
there is a natural connecting homomorphism
ffi H(C;G")	H(C-G')
of degree — 1 and a functorial exact sequence
----> HfGG) Hq(C;G) HB(C;G") A Hg-^GG) -+ •  .
proof By lemma 6, there is a short exact sequence of chain complexes
0 C ® G -^A 0 ® G -A^A C ® G" —» 0
Since this is functorial in C and in the exact coefficient sequence, the result
follows from theorem 4.5.4. 
The connecting homomorphism /3 occurring in theorem 7 is called the
Bockstein homology homomorphism corresponding to the coefficient sequence
(A С' A C A G —> 0. Theorem 7 remains valid over an arbitrary commu-
tative ring R with a unit if C is assumed to be a free chain complex over B.
Let C be a chain complex over R and let G be an R module. Recall the
homomorphism p: H(C) ® G —> H(C;G) defined in the last section. This
homomorphism enters in the following universal-coefficient theorem for
homology.
8	theorem Let C be a free chain complex and let Gbe a module. There
is a functorial short exact sequence
0 -> H/G A C A Hg(C ® G) —> Hg-^G * C A)
and this sequence is split.
proof Let Z be the subcomplex of C defined by Z,; = Z,;(C) with trivial
boundary operator and let В be the complex defined by Bq = Bf;1(C) with
trivial boundary operator. Both В and Z are free chain complexes and there
is a short exact sequence
(azacAba)
where aq(z) = z for z € Zf; and ffc) = Й,/; for c £ Cq. Since В is a free com-
plex, this short exact sequence is split. By theorem 5.1.13, there is an exact
sequence
----> He(Z;G) A He(C;G) A HJffi,G) A Hg-1(Z;G)	•
SEC. 2 THE UNIVERSAL-COEFFICIENT THEOREM FOR HOMOLOGY
223
where д* {b} = {nf7AVVh} = {agJi(b)} for b £ ВГ1_Ъ Since Z and В have
trivial boundary operators, so do Z ® G and В ® G. Therefore Hg(Z;G) =
Zg® G and Hq(B;G) — Bq ® G = Bf;1(Cj ® G, and the above exact se-
quence becomes
. •  Bq{C} ® G ZQ(C) ® G Hg(C;G) ->
B^1(C) ® G S Z^i(C) ® G • • •
where уд: Bg(C) C Zg(C). From the exactness of this sequence we obtain a
short exact sequence
0 —> coker (y8 ® 1) —> H,/C;G) —> ker (yg_i ® 1) —> 0
and it only remains to interpret the modules on either side of He(C;G).
Since Ze(C) is free, the short exact sequence
0 BQ(C) ZQ(C) -» Hg(C) -» 0
is a free presentation of He(C). By the characteristic property of the torsion
product, there is an exact sequence
0Hg(C) * C Bq(C) ® G Zg(C) ®G-> Hg(C) ® G-» 0
Therefore coker (yg ® 1) ~ Hq(C) ® G and ker (yq ® 1) ~ Hr.(C) * G. Sub-
stituting these into the short exact sequence above yields the short exact
sequence
0 -> He(C) ® G —> He(C;G) -> Hg^C) * G -» 0
It is easily verified by checking the definitions that the homomorphism
Hg(G) ® G —» Hg(C;G) is equal to p.
If т: C —» C' is a chain map, т defines a commutative diagram
o-»zac4>b-»o
t 'I'7
о z' А с' Д в' о
from which we obtain the commutative diagram
0 Hg(C) ® G A Hg(C;G) -» Hg.^C) *G 0
т* ® 4 jT* |t* *1
0 -> He(C') ® G A Hg(C';G) -» He_i(C') * G 0
Therefore the short exact sequence for He(C;G) is functorial.
We now prove that the short exact sequence is split (but is not functorially
split). Because BQ_1(C) is free and dqCq = Bg_i(C), there exist homomorphisms
hq\ Be_1(C) —> Cq such that = 1. Then
hq ® 1: Be_i(C) ® G -> Cq ® G
224	PRODUCTS CHAP. 5
maps the kernel of yQ_i ® 1 into cycles of CQ ® G and induces a homomor-
phism HQ_i(C) * G —> Hq(C;G) which is a right inverse of the homomorphism
Hg(C;G) —> Hq-i(C) * G of the short exact sequence in the theorem. B
We can use this result to establish some properties of the torsion product,
beginning with the following six-term exact sequence connecting the tensor
and torsion products.
9	corollary Let 0 —> В' В В" 0 be a short exact sequence of
modules and let A be a module. There is an exact sequence
0 A * B' A * В A * B" ->
A® B' A ® В A ®B"-^0
proof Let 0 —> Ci —> Co —> A —> 0 be a free presentation of A and let C be
the corresponding free chain complex obtained by adding trivial groups on
both sides. Since C is free, it follows from lemma 6 that there is a short exact
sequence of chain complexes
0 C ® B' ^-^4 С® В С® В"О
Because HQ(C) = 0 if q =A 0 and H0(C) = A, the homology sequence of the
above short exact sequence of chain complexes (interpreted by means of
theorem 8) gives the desired exact sequence, и
This yields the commutativity of the torsion product.
10	corollary There is a functorial isomorphism
proof Let 0	Ci —> Co —> В —> 0 be a free presentation of B. By corol-
lary 9, there is an exact sequence
Q-^A*C1-^A*Co—>A*B-^A®C1—>A®Co—>A®B-^0
Since Co is free, it follows from lemma 5 that A * Co = 0, and there is an
exact sequence
0 A * В A ® Ct —> A ® Co —» A® В-^0
By the characteristic property of В * A, there is an exact sequence
Ci ® ACo ® A-^B® A-^Q
The functorial isomorphism A * В ~ В * A then results by chasing in the
commutative diagram
A*B-^ A® A® Co ~^A®B-^Q
0 —> В * A —> Ci ® A —> Co ® A —> В ® A —> 0
in which the vertical maps are the functorial isomorphisms expressing the
SEC. 2 THE UNIVERSAL-COEFFICIENT THEOREM FOR HOMOLOGY
225
commutativity of the tensor product, и
We can now show that the torsion product of A and В depends only on
the torsion submodules of A and B.
11	corollary Let A and В be modules and let i: Tor АСА and
j: Tor В С B. Then i * /: Tor A * Tor В A * B.
proof There is a short exact sequence
0 Tor В Л В В/Tor ВО
where В/Tor В is without torsion. By lemma 5, A * (В/Tor B) = 0, and, by
corollary 9,	Tor В sz A * B. By a similar argument, there is an iso-
morphism i * 1: Tor A * Tor В sz A * Tor B, and the composite of these gives
the result. «
We use these results to extend the universal-coefficient theorem. Given a
chain complex C over B, a free approximation of Cis a chain map т; С —> C
such that
(a)	C is a free chain complex over B.
(b)	т is an epimorphism.
(с)	т induces an isomorphism : H(C) H(C).
12	lemma Any chain complex C has a free approximation, uniquely deter-
mined up to homotopy equivalence.
proof For each q > 0 choose a homomorphism aq: Fq —> ZQ(C) such that Fq
is a free В module and aq is an epimorphism. Let Fq = aq1(BfC)') and
choose a homomorphism /3q: Fq —> CQ+i such that 8q+i/?q = aq | Fq [such a
homomorphism exists because Fq is free and	CQ+i —> BQ(C) is an epimor-
phism]. Define Cq — Fq © Fq_j and define homomorphisms
Cq	Cq-r	by	6Q(«,b) - (b,0)
tq: Cq	Cq	by	TQ(a,b) = aq(a) + /h/i(b)
Then C = {Cq,dq} is a free chain complex and т = {rQ} is a chain map from
C to C. т is epimorphic because tq(Cq) D ker 8Q and 8qtq(Cq) Э im dq. Since
ZfC) = Fq, Bq(C) = Fq, and tq(Z9(C)) — aq(Fq), it follows that
* t* : ZQ(C)/BQ(C) Zq(C)/Bq(C)
Therefore т: C —» C is a free approximation of C. The uniqueness will follow
from lemma 13 below. Ч1
If т: С —> C is a free approximation of C, there is a subcomplex
C = {Cq = ker xq: Cq —> CQ} of C and a short exact sequence of chain
complexes
O^CACA C-^ 0
Because : H(C) H(C), it follows from the exactness of the homology
226
PRODUCTS CHAP. 5
sequence of the above short exact sequence that C is acyclic (see corollary
4.5.5a). Since C is a free chain complex (because it is a subcomplex of a free
chain complex), it follows from theorem 4.2.5 that C is contractible. We use
this in the following lemma.
13	lemma Given a free approximation г. С -> C of C and given a free
chain complex C and a chain map /: C' C, there exist chain maps
f:C'—>C such that r ° f = т, and any two are chain homotopic.
proof As above, there is a short exact sequence of chain complexes
0-h> CA C-^ C-^0
where C is chain contractible. Let D = {Dq: Cq —» CQ+1} be a contraction
of C. Because Cq is free and rq. Cq —> Cq is an epimorphism, there is a homo-
morphism <pq: C'q Cq such that rq<pq = rq. Then
hq — dqtyq <Pq—ldq. Cq > Cq—1
and
Tq-lhq — rfjdftyq rq_iyq_df — ^qTqCpq rqaf
= dqr'q —	= 0
Therefore hq is a homomorphism of Cq into i(CQ_r). It follows immediately
that f = {tq =	— lDq\i~Ahq} is a chain map т: C —> C such that rr = /.
If f, f': (7 —> C are chain maps such that тт = tt', then t — r' = if for
some chain map f: C —> C. It follows immediately that
D = {Dq = iDqfq: Cq —> Cq+i}
is a chain homotopy from f to f'. B
If C is a chain complex over R and G is an R module, let C * G be the
chain complex C * G = {Cq * G, cq* 1}. We use this in the general universal-
coefficient theorem.
14	theorem On the sub category of the produ ct category of chain complexes
C and modules G such that C * G is acyclic there is a functorial short exact
sequence
0 HQ(C) ® G Д Hq(C;G) -> Hq^C) * G —> 0
and this sequence is split.
proof Let т: C —> C be a free approximation to C (which exists, by lemma 12),
and consider the short exact sequence
0-^ С Л C^> C—> 0
in which C is acyclic. By the characteristic property of the torsion product,
there is an exact sequence of chain complexes
()^G*G^C@G —C ® G — -> C® G-^ 0
SEC. 3 THE RUNNETH FORMULA	227
from which we get two short exact sequences
0-> C* G-> 0 ® G->im (i ® l)-> 0
0 -> im (i ® 1) С C ® G —C ® G -> 0
In the first of these C * G is acyclic by hypothesis, and C ® G is also acyclic
(by theorem 8, because C is free and acyclic). From corollary 4.5.5c it follows
that im (i ® 1) is also acyclic. In the second exact homology sequence this
implies that
(t ® 1)*.- H(C ® G) H(C ® G)
The desired short exact sequence is now defined, so that the following diagram
is commutative
0 Hq(C) ® G A Hg(C ® G) -> HQ_r(C) * G -н> 0
h**1
0 -> Hg(C) ® G A HQ(C ® G) —> HQ-i(C) * G-> 0
where the upper row is the short exact sequence of theorem 8 (it is possible
to define the unlabeled homomorphism in the bottom sequence to make the
diagram commutative because all the vertical homomorphisms are isomor-
phisms). Then the bottom sequence splits because the top one does.
The functorial property of the resulting short exact sequence (and the
fact that it is independent of the particular free approximation of C) follows
from lemma 13. и
.—‘
It should be emphasized again that the sequence of theorem 14 does not
split functorially.
15	corollary Let т: С C' be a chain map between torsion-free chain
complexes such that t* : H(C) ~ H(C'). For any R module G, r induces an
isomorphism
7* : H(C;G) ~ H(C';G)
proof This follows from the functorial exact sequence of theorem 14 and
the five lemma. B
In corollary 15, if C and C' are free, then r is a chain equivalence (by
theorem 4.6.10), and so is т ® 1: C ® G -> C' ® G. Therefore t* : H(C;G) ~
H(C;G). Corollary 15 shows that the latter fact remains true (even though т
need not be a chain equivalence) for chain complexes without torsion.
3
THE KiiNNETM FORMULA
In this section we extend the universal-coefficient theorem to obtain the
Kiinneth formula expressing the homology of the tensor product of two chain
228
PRODUCTS CHAP. 5
complexes in terms of the homology of the factors. This is given geometric
content by the Eilenberg-Zflber theorem asserting that the singular complex
of a product space is chain equivalent to the tensor product of the singular
complexes of the factor spaces.
If C and C' are graded R modules, their tensor product С ® C is the
graded module {(C ® where (C ® C')f/ = C, ® Cj. Similarly,
their torsion product C * C' is the graded module {(C* C')Q = Ci * CJ}.
If C and C are chain complexes, their tensor product [and torsion product]
are chain complexes {(C ® C')e, 8'9} [and {(C * C')Q, 8Q}], where if c G Q
and c' £ Cj with i + / = q, then
39'(c ® c') = 8jC ® c’ + (— 1) A ® d'jc'
[and dq | С; * Cj = di * 1 + ( —1)*1 * 8J. It is easy to verify that С ® C' [and
C * C'] really are chain complexes. We shall see later that the tensor product
arises naturally in studying product spaces.
If C is a chain complex such that C'g — 0 for q A 0, then С ® C is the
same as the tensor product of C with the module Cq. Therefore the tensor
product of two chain complexes is a natural generalization of the tensor
product of a chain complex with a module. It is reasonable to expect that
there is a generalization of the universal-coefficient theorem to express the
homology of С ® C' in terms of the homology of C and of C'.
We define a functorial homomorphism of degree 0
p- H(C) ® H(C) H(C ® C')
If с C Zj(C) and c' £ Z/C'), then с ® с' £ Zi+j(C ® C'), and if c or c' is
a boundary, so is c ® c'. Therefore there is a well-defined homomorphism p
such that
M({c} ® {c'}) = (c ® c'}
This homomorphism enters in the following Kiinneth formula.
I iemma Let C and C' be chain complexes, with C' free. Then there is a
functorial short exact sequence
0	[H(C) ® H(C')]9 A HQ(C ® C')	[H(C) *	-+ 0
If C is also free, this short exact sequence is split.
proof As in the proof of theorem 5.2.8, let Z' and B' be the complexes
(with trivial boundary operators) defined by Zg = ZQ(C') and B'g = Bg_i(CT).
There is a short exact sequence of chain complexes
0 -> Z' -> C' -> B' 0
Since C' is free, so is B', and there is a short exact sequence
0-^C®Z'^C®C' -^C®B'-^0
from which we obtain an exact homology sequence
SEC. 3 THE KtiNNETH fobmula
229
-----> НДС ® Z') H^C ® C') -+ Hq(C ® B') Hq^(C ® Z')	.
Note that C ® Z' = © Cd, where (Ci)q = CQ„3 ® Zj(C') and С ® B' = © C’,
where (C7)<; = CQ_3- ® B3_i(C'). Since Z3(C') and Bj(C') are free, it follows from
theorem 5.2.14 that
Hq(C ® Z') = ©НДС’) = © Hi(C) ® Z;(C')
Hq(C ® B') = © НДС7) = © НДС) ® ВДС')
j	i+i=q—1
The map corresponds under these isomorphisms to the homomorphism
(— 1)* ® Yj, where y3- is the inclusion map у/. ВДС') C Z3(C'). Therefore there
is a short exact sequence
0—> © [coker (-l)i ® y3] -> НДС ® C')-^ © [ker (— 1)* ® y3] —» 0
г+i—q	i+3=Q-l
To compute the two sides of this sequence, consider the short exact
sequence
0 Bj(C') Zj(C') H^C') -> 0
Because Z/C") is free, it follows from corollary 5.2.9 that there is an exact
sequence
0 НДС) * НДС') НДС) ® ВДС') Ы>‘®ь-> H.(C) 0 Z.(C)
-> НДС) ® НДС') -> 0
Hence
© [coker (-1)’ ® уД = © НДС) ® НДС')
i+i-q	i+i=q
and
© [ker ( —1)* ® уД = © НДС) * НДС')
l+j=q-l	ilj=q-l
Substituting these into the short exact sequence above gives a short exact
sequence
0	[H(C) ® H(C')]e А НДС ® C')	[H(C) * Н(С')]^!	0
We now verify that v is the map ft. Given {c} £ H(C) and {c'} £ H(C'),
then (c} ® d £ H(C) ® Z(C') and {c} ® d - {c® c'}c®z(r). Therefore
Д{с} ® (c'}) = {c ® c'}ce(7 = ft({c} ® (c'J). Thus we have the desired
short exact sequence, and it is clearly functorial.
Assuming that C is also free, we can show that the sequence splits. By
lemma 5.1.11, it suffices to find a left inverse for ft. Because C and C' are free,
so are B(C) and B(C'), and there are homomorphisms p: C Z(C) and
p': C' Z(C') such that p(c) = c for c £ Z(C) and p'(c') = d for d £ Z(C').
Then
p®p'-.C®C-+ Z(C) ® Z(C')
230	PRODUCTS CHAP. 5
maps B(C ® C')(which is contained in the union of im [B(C) ® С' —> С ® C']
and im [С ® B(C') С ® C']) into the union of im [B(C) ® Z(C') —>
Z(Q ® Z(C')] and im [Z(C) ® B(C') —> Z(Q ® Z(C')]. Therefore the composite
Z(C ® С) С С ® C Z(C) ® Z(C') -> H(C) ® H(C')
maps B(C ® C') into 0 and induces a homomorphism
H(C ® C) H(C) ® H(C')
which is a left inverse of /с в
A similar functorial short exact sequence can be defined if C (instead of C')
is assumed free. The two short exact sequences are identical when C and C'
are both free.1
2 corollary If C' is a free chain complex and either C or C is acyclic,
then С® C is acyclic. a
We now extend lemma 1 to obtain the following general Kiinneth
formula.
3 theorem On the subcategory of the product category of chain complexes
C and C such that C * C is acyclic there is a functorial short exact sequence
0	[H(C) ®	II,fC ® C)	[H(Q * Я(С')]9-!	0
and this sequence is split.
proof Let т: С C and т': C' —> C' be free approximations. Then there is
a short exact sequence
о c 4 c' 4 С' o
where C' is acyclic. Since C' is free, the six-term exact sequence becomes the
exact sequence
0-^C*C'-^C®C'-^C®C' -iAA C ® C -> 0
Since C * G' is acyclic by hypothesis and С ® C' is acyclic by corollary 2, it
follows (as in the proof of theorem 5.2.14) that there is an isomorphism
(1 ® 7%: H(C ® C') ~ H(C ® O')
There is also a short exact sequence
()-> C A C A C-^ 0
where C is acyclic. Since C' is free, there is a short exact sequence
o-^c®c' -^c®cr aa с ® с -> о
By corollary 2, С ® G is acyclic, and we have an isomorphism
(r ® 1)*: H(C ® O') H(C ® O')
1 This is proved in G. M. Kelley, Observations on the Kiinneth theorem, Proceedings of the
Cambridge Philosophical Society, vol. 59, pp. 575-587, 1963.
SEC. 3 THE RUNNETH FORMULA	231
Hence the composite (r ® Т')# = (1 ® 7% (7 ® A is an isomorphism of
jfC ® C’) onto H(C ® O'). The desired short exact sequence is now defined
so that the following diagram is commutative
0 -> H(C) ® H(G) Л ЩС ® G) H(C) * H(G) -> 0
’
’	0 -> H(C) ® H(C') A H(C ® O') -+ H(C) * H(G) 0
| where the top row is the short exact sequence of lemma 1 (it is possible to
f define the homomorphisms in the bottom row to make the diagram commu-
i tative because the vertical homomorphisms are isomorphisms). The bottom
! sequence splits because the top one does.
' The functorial property of the sequence (and the fact that it is independ-
; ent of the free approximations C and O') follow from the functorial property
| of the sequence in lemma 1 and from lemma 5.2.13. в
i
i If C and G are chain complexes over R and G and G' are R modules,
I the composite
’ H(C ® G) ® H(C’ ® G') A H[(C ® G) ® (G ® G')]
1	H[(C ®G)®(G® G’)J
[where the right-hand homomorphism is induced by the canonical isomorphism
I (C ® G) ® (C' ® G') ~ (C GA C') ® (G ® G')] is a functorial homomorphism
’	M'-- H(C;G) ® fr(C';G') ЩС ® C'; G ® G')
called the cross product. If z £ H(C;G) and zf E H(C';G'), then
z X z! e H(C ® C; G ® G')
denotes the image of z ® z' under this homomorphism [that is, z X * =
1 4 corollary Given torsion-free chain complexes C and G and modules
G and G' such that G * G' = 0, there is a functorial short exact sequence
0 -> [H(C;G) ® Н(С';С')]е 4 HQ(C ® C'; G ® G')
' and this sequence is split.
proof This follows from theorem 3 once we verify that (C ® G) * (Cz ® G')
| is trivial. To show that (C ® G) * (C' ® G') = 0, let 0 —F' F —G be a
free presentation of G. Because G * G' = 0, there is an exact sequence
।	0^F'®G'-^F®G'-^G®G'-^0
and since C and C' are without torsion, there is an exact sequence
a-» (C ® F) ® (C' ® G') (C ® F) ® (C ® G)
	(C ® G) ® (C' ® G')	0
232
PRODUCTS CHAP. 5
Because there is also a short exact sequence
0-^ C®F'-^ C®F~> C® G-^ 0
where C ® F is without torsion, it follows that (C ® G) * (C' ® G') is isomor-
phic to the kernel of the homomorphism
(C ® F) ® (O' ® G') -> (C ® F) ® (C' ® G')
and hence is 0. B
The cross product has the following commutativity with connecting
homomorphisms.
5 theorem Let 0-*C-^C—>C—>0bea split short exact sequence of
chain complexes and let z £ H(C;G) and z' £ H(C';G'). Then
Э* (z X z6 7) = X 2'
S^(z' X z) — (-l)deB^z' X a* z
proof We have a commutative diagram of chain maps
0 —> C®G	-> C®G	-> C®G	-> 0
T 4-	'tт	'tт
0 (C ® G) ® (C' ® G') (C ® G) ® (O' ® G')	(C®G)® (C' ® G') -> 0
with exact rows, with the vertical maps defined by forming the tensor product
on the right with d £ Z(C ® G'), where d — (d) [that is, т(с) = c ® d for
c £ C ® G]. Because d is a cycle, each vertical map is a chain map. Because
the connecting homomorphism is functorial, we obtain a commutative diagram
H(C ®G)^ H((C ® G) ® (O' ® G')) Я((С ® C') ® (G ® Gy.
a*|	a*j.	j,a*
H(C ®G) h H((C ® G) ® (C ® G')) H{{C ® C) ® (G ® Gy
in which each vertical map is a suitable connecting homomorphism. The top
row sends z into z X z', and the bottom row sends d^z into d^z X d. This
gives half the result. The second half follows by a similar argument, the' only
difference being that the tensor product formed on the left with d is not a
chain map but either commutes or anticommutes with the boundary operator,
depending on the degree of d. This accounts for the presence of the factor
(— l)de8 x’ in the second equation. B
The following Eilenberg-Zilber theorem1 is the link between the algebra
of tensor products and the geometry of product spaces.
6 theorem On the category of ordered pairs of topological spaces X and
Y there is a natural chain equivalence of the functor Д(Х X b) with the
functor Д(Х) ® Д(Х).
1 The theorem appears in S. Eilenberg and J. A. Zilber, On products of complexes, American
Journal of Mathematics, vol. 75, pp. 200-204, 1953.
SEC. 3 THE KVNNETH FOBMULA
233
proof We show that both functors are free and acyclic with models
{Др,Д®}м>о- Let dn 6 Д„(Д« X Д") be the singular simplex which is the
diagonal map Д” —> Дп X Д”. If а: Д” —> X X Y is any singular n-simplex,
then о is the composite
Д” Д« x Д» °' x °"> X X Y
where o' = pi ° о and a" = p2 ° o, and and p2 are the projections of
XX Y to X and Y, respectively. Conversely, given o'-. Xn X and о": Д” —> У,
there is a corresponding о — (o' X o")dn: X" X X Y. Therefore the
singleton {dn} is a basis for A„(X X Y), so Д(Х X Y) is free with models
{Д’1,Д’1}, and hence also free with models {Дг'.Дв}. Since Д*> and Дв are each
contractible, so is Д'' X Д®. Therefore Д(Дг> x A®) is acyclic, and we have
proved that Д(Х X Y) is a free acyclic functor with models {Др,Д«}.
Since Д/,(Х) is free with a basis ip £ Др(Др) and Д,/Т) is free with basis
E Дд(Дв), it follows that ДР(Х) ® Aq(Y) is free with the basis
lP ® Ь e ® лв(М
Then [Д(Х) ® Д( Y)]?! is free with the basis {fp® iq}P+g-n- Hence Д(Х) ® Д( Y)
is free with models (Д»,Дв}. Since e: Д(Д*>) —> Z and e: Д(Д«) —> Z are chain
equivalences, it follows that
e ® e: Д(Д») ® Д(Д«) -н» Z ® Z = Z
is also a chain equivalence. Hence, by lemma 4.3.2, the reduced complex of
Д(Дг') ® Д(Дв) is acyclic, and we have shown that Д(Х) ® A(Y) is also free
and acyclic with models {Д₽,Д«}. The theorem now follows by the method of
acyclic models, и
The same technique based on the method of acyclic models can be used
to prove the following results.
7	theorem Given X, Y, and Z, there is a chain homotopy commutative
diagram
Д((Х X Y) x Z) A(X x (Y X Z))
[Д(Х) ® Д(У)] ® A(Z) Д(Х) ® [Д(У) ® A(Z)]
where the vertical maps are the natural chain equivalences of theorem 6. B
8	theorem For any X and Y there is a chain homotopy commutative
diagram
Д(Х X Y) ~ Д(У X X)
Д(Х) ® Д(У) Д(У) ® Д(Х)
234
PRODUCTS CHAP. 5
where the bottom map sends x ® у to (— l)deg«deg у у (X x an(] the vertical
maps are the natural chain equivalences of theorem 6. 
The sign in theorem 8 is necessary to make the map a chain map (that
is, to make it commute with the boundary operators).
Given topological pairs (X,A) and (Y,B), we define their product
(X,A) X (Y,®) to be the pair (X X Y, X X В U A X Y). Then we have the
following relative form of the Eilenberg-Zilber theorem.
9	theorem On the category of ordered pairs of topological pairs (X,A)
and (Y,B) such that {X X В, A X Y } is an excisive couple in X X Y there is
a natural chain equivalence of Д(Х x Y)/A(X X В U A X Y) with
[Д(Х)/Д(А)] ® [A(Y)/A(B)].
proof Because {X X В, A X Y } is an excisive couple, the natural map
Д(Х X Y)/[A(X X в) + Д(А x Y)] Д(Х X Y)/A(X x В U A X Y)
is a chain equivalence. By theorem 6 there is a functorial equivalence of
Д(Х) ® A(Y) with Д(Х X Y) taking Д(Х) ® Д(В) and Д(А) ® A(Y) into 
Д(Х X В) and Д(А X Y), respectively. Hence there is a functorial chain
equivalence of the quotient
Д(Х) ® A(Y)/[A(X) ® Д(В) + Д(А) ® A(Y)]	(Д(Х)/Д(А)] ® [A(Y)/A(B)]
with the quotient
Д(Х X Y)/[A(X X A) + Д(В X Y)]
Combining these two chain equivalences gives the result, и
For any two pairs (X,A) and (Y,B) we define the homology cross product
g': HP(X,A; G) ® HQ(Y,B; G')	HJ)+Q((X,A) X (Y,B); G ® G')
to be equal to the cross product
НР([Д(Х)/Д(А)] ® G) ® Hq([A(Y)/A(B)] ® G')
Нр+в(([Д(Х)/Д(А)] ® [A(Y)/A(B)]) ® (G ® G'))
followed by the functorial homomorphism of the bottom module to
НР+9(Д(Х X Y)/A(X x В U A X Y); G ® G')
If z G Hp(X,A; G) and Z' C Hq(Y,B; G'), then we write
z X / = g'(z ® z') G HP+Q((X,A) X (Y,B); G ® G')
Because Д(Х)/Д(А) and A(Y)/Д(В) are free, we can combine theorem 9
with corollary 4 to obtain the following Kiinneth formula for singular
homology.
sgC. 3 THE RUNNETH FORMULA	235
jО theorem If {X x В, A X Y } is an excisive couple in X X^_X and G
arid G' are modules over a principal ideal domain such that G * G' = 0,
there is a functorial short exact sequence
О [H(X,A; G) ® H(Y,B; G')]Q 4 HQ((X,A) x (Y,B); G ® G')
[H(X,A; G) * H(Y,B; G')]q_i	0
and this sequence is split. B
In particular, if the right-hand term vanishes (which always happens if R
is a field), then the cross product is an isomorphism
ju': H(X,A; G) ® H(Y,B; G) zz H((X,A) X (Y,B); G ® G')
The following formulas are consequences of the naturality of p and of
theorems 5, 7, and 8.
Ц Let f: (X,A) —> (X',A') and g: (Y,B) —> (Y',B') be maps and let
g £ HP(X,A; G) and z' £ HQ(Y,B; G'). Then, in the module
Hp+q((X',A') X (Y',B'); G ® G')
we have
(f X g)* (z X z‘) = Д z X g*z' s
12	Let p: (X,A) X Y —> (X,A) be the projection to the first factor and let
e: H(Y;G') —> G' be the augmentation map. For z £ Hq(X,A; G) and
z’ £ Hr(Y;G'), in Hq+r(X,A-, G ® G),
P* (z x z') = p(z ® e(z')) и
13	For z £ HP(X,A- G), z1 £ Hq(Y,B; G), and z" £ HfZ,C- G"), in
Hp+q+r((X,A) X (Y,B) X (Z,C); G ® G ® G"),
we have
z X (~ X z") = {z X z') x z" s
14	Let T: (X,A) X (Y,B) (Y,B) X (X,A) and <p: G ® G G ® G inter-
change the factors. For z £ HP(X,A; G) and z’ £ Hq(Y,B; G'), in
Hp^((Y,B) x (X,A); G ® G), we have
(z X z') = (- l)w<p^ (z'x z) и
15	Let {(Х1Д1), (Х2,Аг)} be an excisive couple of pairs in X and let
z £ Hp(Xi U X2, Ai U A2; G) and z' £ Hq(Y,B; G). For the connecting homo-
| morphisms of appropriate Mayer-Vietoris sequences we have
;	9*(z X Z') = a*z X z'
। Ф HJ)+Q^1((X1 П X2, At П A2) x (Y,B); G ® G') and
X z) = ( — l)«z' X 8*z
\ in Hp+q—itfY,B) x (Xi Ci X2, Ai П A2); G' ® G) B
236
PRODUCTS CHAP. 5
СОМОМОЬвбТ
A chain complex has a differential of degree — 1. Related to this is the con-
cept of a cochain complex, which has a differential of degree +1. Cochain
complexes have many of the properties of chain complexes, and this section
is devoted to a discussion of these properties. The functor Hom associates to
every chain complex a cochain complex, and vice versa. The cohomology
module of a topological pair with coefficients G is the homology module of
the cochain complex associated in this way to the singular' complex of the pair.
The last part of the section is devoted to a brief discussion of axiomatic
cohomology theory.
Throughout this section R will denote a commutative ring with a unit.
A cochain complex (over R), denoted by C* — {C«,8e}, is a graded R module
together with a homogeneous differential 8 of degree +1 called the cobound-
ary operator (thus 6f/: C« —> Ce+1 and l = 0 for all q). The kernel of 8
is the module of cocycles Z(C*), and the image of 8 is the module of
coboundaries	Then B(C*) C Z(C*), and the cohomology module
H(C*) is defined to be the quotient Z(C*)/B(C*).
If C* is a cochain complex, there is a chain complex C defined by
Cq = C~i and CQ CQ_! equal to 8“«: C f/ —> C~«+1. Then HQ(C) =
and the cohomology module of C* corresponds to the homology
module of C. In this way results about chain complexes give results about co-
chain complexes. Thus we have the concepts of cochain map and cochain
homotopy, and there is a category of cochain complexes and cochain homotopy
classes of cochain maps. The cohomology module is a covariant functor from
this category to the category of graded modules. Furthermore, given a short
exact sequence of cochain complexes
( )^(A ДС*4с*^0
there is a functorial connecting homomorphism
8*: HfC*) H(C*)
of degree +1 and a functorial exact cohomology sequence
-----> н«(с*) A h«+1(C*) 4 h«+i(c*) A №+i(c*)4 ...
Passing from a cochain complex to a chain complex by changing the.
sign of the degree gives us the following analogues of theorems 5.2.14 and
5.3.3.
1	theorem Given a cochain complex C* and a module G such that
C* * G is acyclic, there is a functorial shod exact sequence
0	№(C*) ® G -4 №(C* ® G) №+1(C*) * G -н> 0
and this sequence is split, и
SEC. 4 COHOMOLOGY
2.37
2	theorem Given cochain complexes C* and C* such that C* * C*
is an acyclic cochain complex, there is a functorial short exact sequence
[H*(C*) ® H*(C'*)]i Л №(C* ® C'*)
(H*(C*) * H* (O' *)]«+!	0
and this sequence is split, и
There is also an analogue of corollary 5.3.4 for cochain complexes which
we shall not state as a separate theorem. If C* is a cochain complex over R
and G is an R module, an augmentation of C* over G is a monomorphism
q: G—»C° such that 8° ° q = 0. An augmented cochain complex over G con-
sists of a nonnegative cochain complex C* (that is, Ci = 0 for q < 0) and an
augmentation of C* over G. Such an augmentation can be regarded as
a monomorphic chain map of the cochain complex (also denoted by G) whose
only nontrivial cochain module is G in degree 0 to C*. For this trivial
cochain complex G it is clear that Hf/(G) = 0 for q 7^ 0 and H°(G) = G.
Therefore q induces a monomorphism 17*: G —> H°(C*). The reduced
cochain complex C* of an augmented cochain complex C * is defined to be
the quotient cochain complex Ci = Ci for q 7^ 0, & — coker q, and 81 is
suitably induced by 81. We define H(C*) = H(C*). Because there is a short
exact sequence of cochain complexes
0-ч> G Д С* -л C* 0
we see that Hi(C*) Hi(C*) for q 7^ 0, and there is a short exact sequence
0 -» G H°(C*)	H°(C*)	0
Our interest in cochain complexes is in their relation to chain complexes.
If C is a chain complex over R and G is an R module, there is a cochain
complex Hom (C,G) = {Hom (CQ,G), 8e}, where, if /€ Hom (Ce,G), then
8qf 6 Hom (Cq+i,G) is defined by
(WMW cece+1
We also write </,c> instead of f(c) and set </,c> = 0 if deg /7^ deg c.
In this notation (fife) = </,8Q+ic>.
If C is augmented by e: Co —» G', then Hom (C,G) is augmented by
q: Hom (G',G) Hom (C0,G), where '<?(/)(<-;) = /(e(c)) for c £ Co and
f 6 Hom (G',G). It is easy to verify the following result.
3	theorem There is a functor of two arguments contravariant in chain
complexes C and covariant in modules G which assigns to a chain complex C
and module G the cochain complex Hom (C,G). и
For a chain complex C and module G we define the cohomology module
H*(C-.G) = {H«(C;G)} of C with coefficients Gby
238
PRODUCTS CHAP. 5
№(C-G) = H«(Hom (C,G))
It follows from theorem 3 that H* (C;G) is a contravariant functor of C and a
covariant functor of G to the category of graded modules. For a chain map
т: С C we use t*: H*(C';G) —> H*(C;G) to denote the homomorphism
induced by the cochain map Hom (t,1): Hom (C',G) —> Hom (C,G), and for a
homomorphism <p: G —> G' we use <p*: H*(C;G) —» H*(C,G') to denote the
homomorphism induced by the cochain map Hom Hom (C,G)
Hom (C,G'). To distinguish the homology of C from its cohomology, we shall
sometimes denote H(C;G) by H*(C;G).
4	example Given an abelian group G and a simplicial pair (K,L), the
oriented cohomology group of (K,L) with coefficients G, denoted by
H*(K,L; G), is defined to be the graded cohomology group of the cochain
complex Hom (C(K)/C(L), G) [which is augmented over Hom (Z,G) ~ Q,
Then	G) is a contravariant functor of (K,L) and a covariant functor
of G to the category of graded abelian groups. If G is also an R module,
H*(K,L; G) is a graded R module.
5	example If (X,A) is a topological pair and G is an abelian group, the
singular cohomology group of (X,A) with coefficients G, denoted by
H*(X,A; G), is defined to be the graded cohomology group of the cochaiu
complex Hom (Д(Х)/Д(А), G) (which is augmented over G). It is contravariant
in (X,A) and covariant in G, and if G is an R module, H* (X,A; G) is a graded
R module. If (X',A') C (X,A) and и £ H* (X,A; G), we use и | (X',A') to denote
the element of H*(X',A'; G) equal to i*u, where i: (X',A') C (X,A). We also
use 1 C H°(X;B) to denote the image of 1 £ R under the augmentation
i]: R H°(X;R).
We recall some properties of the functor Hom. The following analogue
of corollary 5.1.6 is easily established.
6	lemma Given an exact sequence of R modules
A' A A" 0
z
and an R module B, there is an exact sequence
0 —» Hom (A",B) —> Hom (A,B) —> Hom (A',B) и
If A' —> A is a monomorphism [that is, if 0 —» A' —> A is also exact], it ,
need not be true that Hom (A,B) Hom (A',B) is an epimorphism, [that is,
that Hom (A,B) —> Hom (A',B) —> 0 is exact]. However, there is the following
analogue of corollary 5.1.12 (which follows easily by using lemma 5.1.11).
7	lemma Given a split short exact sequence of R modules
0 -> A' -> A A" 0
and an R module B, the sequence
SEC. 4 COHOMOLOGY
239
0 —» Hom (A",B) —> Hom (A,R) —» Hom (A',B) —> 0
is a split short exact sequence. 
In case 0 —> C' —* С —> C" 0 is a split short exact sequence of chain
complexes, it follows from lemma 7 that for any module G there is a short
exact sequence of cochain complexes
0 -> Hom (C",G) Hom (C,G) -» Hom (C',G) -> 0
This gives the following result.
8	theorem Given a split short exact sequence of chain complexes
0—>C'~>C—»C"—>0
and a module G, there is a functorial exact cohomology sequence
----> H«(C";G)	№(C;G)	№(C';G)	He+1(C";G)	«
This leads to an exact Mayer-Vietoris cohomology sequence analogous to
the exact sequence of corollary 5.1.14.
9	corollary If {(Xi,Ai), (Х2Д2)} is an excisive couple of pairs in a
topological space and Gisan R module, there is a functorial exact cohomology
sequence
... ^ H«(Xx U X2, Al U A2; G) Д H«(Xb Ax; G) © №(X2,A2; G)
№(Xt П X2, Ai П A2; G)^ • • 
where j*(u) = (j i (w)>/2 (H)) and i*(wi + u2) = i* ui — i*ll2 and R, i2, /Т,
and /2 are suitable inclusion maps, н
If {X;} is the set of path components of X, then Д(Х) = A(Xy).
Therefore Hom (A(X);G) = X; Hom (A(X7-),G), and by theorem 4.1.6, we ob-
tain the following result.
IO theorem The singular cohomology module of a space is the direct
product of the singular cohomology modules of its path components, и
Because the homology functor does not commute with inverse limits, it is
not true that the singular cohomology of a space is isomorphic to the inverse
limit of the singular cohomology of its compact subsets (that is, there is no
general cohomology analogue of theorem 4.4.6).
There is an exact cohomology sequence corresponding to a short exact
sequence of coefficient modules (analogous to theorem 5.2.7).
11	theorem On the category of free chain complexes C over R and short
exact sequences of R modules
0-> G' g4 G" -> 0
240	PRODUCTS CHAP. 5 s
there is a functorial connecting homomorphism
/3*:
of degree 1 and a functorial exact sequence
_____> №(C;G')	№(C;G)	№(C-,G'f	№+\C;Gf -> • • 
pboof Because C is free, there is a short exact sequence of cochain complexes
0 -> Hom (C,G') Л Hom (C,G) Л Hom (C,G") -> 0
where <p and f are induced by <p and f. The result follows from this, н
The connecting homomorphism (3* in theorem 11 is called the Bockstein
cohomology homomorphism corresponding to the coefficient sequence
0 G' G 4 G" 0.
Let G be an R module. A cohomology theory with coefficients G consists
of a contravariant functor H* — {Hq} from the category of topological pahs
to graded R modules and a natural transformation 8*: H*(A) —> ff*(X,A) of
degree 1 such that the following axioms hold.
12	homotopy axiom If fo, fi- (X,A) —> f¥,B) are homotopic, then
H*(f0) = H*(fi): H*(Y,B) -» H*(X,A)
13	exactness axiom For any pair (X,A) with inclusion maps i: А С X and
j: X C (X,A), there is an exact sequence
 • • Д №(X,A)	№(X)	№(A)	№+1(X,A)	• • •
14	excision axiom For any pair (X,A) if U is an open subset of X such
that U C int A, then the excision map j: (X — U, A — U) C (X,A) induces
an isomorphism
H*(X,A) ^H*(X - U, A - U)
15	dimension axiom On the category of one-point spaces there is a natural
equivalence of the constant functor G with the functor H*.
Singular cohomology theory with coefficients G is an example of a coho-
mology theory with coefficients G (the exactness axiom following from the
application of corollary 9 to a suitable couple). The uniqueness theorem is
valid for cohomology theories (that is, a homomorphism from one cohomology
theory to another which is an isomorphism for one-point spaces is an isomor--
phism for compact polyhedral pairs).
The reduced cohomology modules H* of a cohomology theory are
defined as follows. If X is a nonempty space, let с: X —> P be the unique map
from X to some one-point space P. The reduced module H* (X) is defined to
be the cokemel of the homomorphism
H* (с): H* (P) —» H* (X)
Because c has a right inverse, H*(c) has a left inverse. Therefore
gjjc. 5 THE UNIVERSAL-COEFFICIENT THEOREM FOR COHOMOLOGY
241
and the reduced modules have properties similar to those of the reduced
singular cohomology modules.
5 THE ENIVERSAE-COEFFICIENT THEOREM
FOR COHOMOLO6Y
This section is devoted to relations between cohomology and homology of
chain complexes. In order to express the cohomology of a chain complex in
terms of its homology it is necessary to introduce certain functors of modules
which are associated to the module of homomorphisms of one module to
another in much the same way that the torsion products are associated to the
tensor product. Over a principal ideal domain there is one associated functor,
the module of extensions. We use this to formulate the universal-coefficient
theorems and Kiinneth formulas established in the section.
Let C be a free resolution of the module A and let В be a module. There
is a cochain complex Hom (C,B) = {[Hom (C,B)]« = Hom (C9,B), S«} whose
cohomology module depends only on A and B, up to canonical isomorphism
(and not on the choice of C). Let C be the canonical free resolution of A and
define Ext0 (A,B) = H«(Hom (C,B)). Then Ext« (A,B) is a functor of two
arguments contravariant in A and covariant in B, and Ext0 (A,B) is naturally
equivalent to Hom (A,B).
Over a principal ideal domain, Ext« (A,B) = 0 for q > 1, and the module
of extensions Ext (A,B) is defined to equal Ext1 (A,B). It is characterized by
the property that given any free presentation of A
0 -> Ci Co -> A 0
there is an exact sequence
0 -> Hom (A,B) Hom (C0,B) -om(abl)> Hom (CbB) -> Ext (A,B) -> 0
In fact, because Hom (C2,B) = 0,
Ext (A,B) = Hi(C;B) = Hom (Ci,B)/im [Hom (31,1)]
= coker [Hom (3i,l)]
Clearly, Ext (A,B) is contravariant in A and covariant in B. Its name derives
। from its connection with the extensions of A by В which we describe briefly
after the following examples.
I
I If A is free, it has the free presentation
0-^0^A-^A^0
from which it follows that Ext (A,B) = 0 for any B.
PRODUCTS CHAP. 5
242
2 For v £ В there is a short exact sequence
0 -> R A R R/vR -> 0
where n(o') = vv' for v’ £ R, which is a free presentation of R/vR. For any
R module B, Hom (R,B) ~ В and the homomorphism Hom (a,l): Hom (B,B) —>
Hom (R,B) corresponds to a*: В B, where a* (b) = vb. Hence there is an
isomorphism coker Hom (a,l) ~ B/vB, and we have proved
Ext (B/oB,B) B/vB (R/vR) ® В -
Since Hom commutes with finite direct sums, it follows that for any finitely
generated torsion module A there is an isomorphism (nonfunctorial)
Ext (A,B) A ® В
because such a module A is a finite direct sum of cyclic modules (by theorem
4.14 in the Introduction).
An extension of A by В is a short exact sequence
0 В E -» A 0
With a suitable definition of equivalence of extensions (by a commutative
diagram), of the sum of two extensions, and of the product of an extension
by an element of R, there is obtained a module whose elements are equiva-
lence classes of extensions of A by B. This module is isomorphic to Ext (A,B). In
fact, given an extension 0—»B—>E—>A—>0 and a free presentation of A,
0 —> Ст —> Co —» A —> 0, there is, by theorem 5.2.1, a commutative diagram
0 —» Cx —> Co
о
0 В E
uniquely determined up to chain homotopy. Then yx 6 Hom (CX,B) is unique
up to im [Hom (C’o,B) —» Hom (Ci,B)], and so determines an element of
Ext (A,B). This function from extensions of A by В to Ext (A,B) induces an
isomorphism of the module of equivalence classes of extensions with Ext (A,B).
Given a graded module C = {CQ}, there is a graded module Ext (C,B) =
{[Ext (C,B)]4 = Ext (CQ,B)}. If C is a chain complex, Ext (C,B) is a cochain
complex with
8ч = Ext (3Q+i,l): Ext (CQ,B) Ext (Cq+1,B)
A homomorphism
h: HfC-G) -> Hom (НДС;С'), G ® G')
natural in C and G is defined by
(h{/}){2 Ci ® gQ = 2/(Ci) ® gi
for {/} £ H«(C;G) and {2 Cj <8> g[} £ HQ(C;G') [after verification that
SJ.;C. 5 THE UNIVERSAL-COEFFICIENT THEOREM FOR COHOMOLOGY	243
% fib) ® g'l is independent of the choice of f in its cohomology class
;ш<1 2 Ci ® g[ in its homology class]. For и £ H₽(C;G) and z £ НДС;С') we
.-define (u,z) £ G ® G' to be 0 if p q and to be hffff if p = q. .In this
notation
)
<{/}, {2 Ci ® g<}> = 2 <f,Cl> ® gi
The homomorphism h enters in the following universal-coefficient
theorem for cohomology.
;j theorem Given a chain complex C and module G such that Ext (C,G)
is an acyclic cochain complex, there is a functorial short exact sequence
0 -> Ext (Hq_t(C),G) -> №(C,G) Hom (He(C),G)	0
I and this sequence is split.
proof We first consider the case in which C is a free chain complex. There
j. js then a short exact sequence of chain complexes
(j , Z-> C-+ В—> 0
?' where ZQ = Z^C) and Bq — Bq_i(C). This sequence is split because В is free,
and by theorem 5.4.8, there is an exact cohomology sequence
; ,..	№-!(2;С)	H«(B;G)	H«(C;G)	№(Z-,G)	№+4B-,G)	
Since Z and В have trivial boundary operators, НД Z;G) = Hom (ZQ(C),G)
| and №(B-,G) = Hom (Bq_dC),G). Furthermore, the homomorphism
6 *: H«(Z;G)	H«+1(B;G)
equals Hom (yQ,l): Hom (Zc(C),G) —> Hom (BQ(C),G), where yQ: ВДС) C ZQ(C).
Hence there is a functorial short exact sequence
0 —* coker [Hom (ув_т,1)] —> H'/(C;G)	ker [Hom (yQ,l)] —> 0
To interpret the modules in the above sequence we have the short exact
I
• sequence
Hom (у0,1)
0	ВДС) —> zQ(C) НДС) 0
kwhich is a free presentation of НДС). By the characteristic property of Ext,
[there is an exact sequence
II -> Hom (He(C),Gj -> Hom (ZQ(C),G)
Hom (ВДС),С) -> Ext (НДС),С) 0
| Therefore, ker [Hom (yQ,l)] ~ Hom (НДС),С) and coker [Hom (yQ,l)] ~
i Ext (НДС),С). Substituting these into the short exact sequence containing
1 HQ(C;G) yields the desired short exact sequence
{	0 Ext (HQ_t(C),G)	№(C;G) Hom (HQ(C),G) -> 0
I'vith the homomorphism H«(C;G) —> Hom (НДС),С) easily verified to equal h.
244	PRODUCTS CHAP. 5
This sequence is functorial and is split (because the sequence of chain
complexes	\
() -- -> Z - -> C-a B - > 0
is split).	(
For arbitrary C such that Ext (C,G) is acyclic, the result follows by using j
a free approximation to C (as in the proof of theorem 5.2.14) to reduce it to (
the case of a free complex. ®	;
I
It follows from theorem 3 that if X is a path-connected topological'
space, then H°(X;R) is a cyclic R module generated by 1 [or, in other words, ;
the augmentation map is an isomorphism rj: В ~ /f°(X;R)]. From theorems 3 |
and 5.4.10, it follows that for any X, H°(X;G) is isomorphic to the direct product '
of as many copies of G as path components of X.
4 corollary If (X,A) is a topological pair such that Hq(X^.;R) is finitely ‘
generated for all q, then the free submodules of IlfiX,A; R) and Hq(X,A;R) •.
are isomorphic and the torsion submodules of №(X,A; R) and II,. fiX.A-, R) •
are isomorphic.
proof Let Hq(X,A; R) = Fq® Tq, where Fq is free and Tq is the torsion
module of Hq. Then
Hom (He(X,A; В), B) Hom (Fe,B) © Hom (TQ,B) ~ Fq
and by example 2,
Ext (Hq(X,A; В), B) Ext (Fq,B) © Ext (TQ,B) ~ Tq
SEC. 5 THE UNIVERSAL-COEFFICIENT THEOREM FOR COHOMOLOGY	245
where G (hence also G) is a finitely generated free module. There is a com-
mutative diagram
Hom (A,G) ® G Hom (A,G) © G -a Hom (A,G) © G' -> 0
Ц	H	H
Hom (A, G © G) -a Hom (A, G © G) -> Horn (A, G © G') -a 0
with exact rows (exactness follows from corollary 5.1.6 and, for the bottom
row, from the fact that A is free). Because Д and Д are isomorphisms, it
follows from the five lemma that p is also an isomorphism. “
There is also a functorial homomorphism
p: Hom (A,G) © Hom (B,G') -a Hom (A ® B, G ® G')
defined by p(f © J')(a © b) = fid) ©/'(&) for f £ Hom (A,G),f' £ Hom (B,G'),
a £ A, and b £ B. In case В = R, Horn (B,G') ~ G', and p corresponds to
the homomorphism in lemma 5.
0 lemma If В is a finitely generated free module, for arbitrary modules
A and G, p is an isomorphism
p-. Hom (A,G) © Hom (B,B) ~ Hom (A © B, G)
proof The result is trivially true for В = В and follows for a finite sum of
copies of В because both sides commute with finite direct sums. ®
7 corollary If A and В are free modules and either A and В or В and G'
are finitely generated, p is an isomorphism
p: Hom (A,G) © Hom (B,GZ) ~ Hom (A © B, G © G')
The result follows from theorem 3. »	J
I
For many purposes it would be more useful to have a formula expressing 4
H* (C;G) in terms of H*(C;R). Such a formula can be proved in the case of I
C or G finitely generated. We begin by establishing some properties of ।
finitely generated modules.
Let p: Hom (A,G) © G' -a Hom (A, G ® G') be the functorial homo- ;
morphism defined by p(f © g')(c) = fid) © g' for/ £ Hom (A,G), g' £ G', j
and a £ A.
5 lemma If A is a free module and G' is finitely generated, then for anif I
module G, p is an isomorphism.	j
proof The result is trivially true if G' = B. Because the tensor product and
Hom functors both commute with finite direct sums, it is also true if G' is a I
finitely generated free module. G' is assumed to be finitely generated, so there )
is a short exact sequence	j
0-aG-aG-aG'-a0	I
proof Since A and В are free, so is A © B. If A and В are finitely gener-
ated, so is A © B, and there is a commutative diagram
, [Hom (R,G) ® Hom (A,R)J ® [Hom (R,G') ® Hom (B.R)] Hom (R, G ® G') ® Hom (A ® B, R)
i‘ ® '' 1	G
Hom (A,G) ® Hom (B,G')	-*a Hom (A ® B, G ® G')
in which p,((fr © /г) © (fi © fi)) = p(fi ® /3) © p(fi © /4). By lemma 6, p
ijs an isomorphism and so are both vertical maps. Therefore the bottom map
is also an isomorphism.
If В and G' are finitely generated, there is a commutative diagram
Hom (A,G) © Hom (B,B) © G' Hom (A,G) © Hom (B,G')
f1®1!	4,n
Hom (A © B, G) © G' Hom (A © B, G © G')
By lemma 5, both horizontal maps are isomorphisms, and by lemma 6, the
left-hand vertical map is an isomorphism. Therefore the right-hand map
is also an isomorphism. »
246
PRODUCTS CHAP. 5
It follows from lemma 5 that if A is free and finitely generated, p is an
isomorphism
p-. Hom (A,B) ® A ~ Hom (A,A)
The following lemma is a partial converse of this result.
В lemma If A is a module such that
p: Hom (A,R) 0A~> Hom (A,A)
is an epimorphism, then A is finitely generated.
proof By hypothesis, there exist fi £ Hom (A,R) and </;£ A for 1 < i < n
such that p££fi ® a;) = 1a- Then, for any a £ A
a = fi ® afi(a) = S fi(a)at
showing that A is generated by {«;}. “
A graded module {Ce} is said to be of finite type if Cq is finitely gener-
ated for every q. Thus a graded module C of finite type is finitely generated
(as a graded module) if and only if Cq = 0, except for a finite set of integers q.
The following lemma asserts that a chain complex whose homology is of
finite type can be approximated by a chain complex of finite type.
9 lemma Let C be a free chain complex such that H(C) is of finite type.
Then there is a free chain complex C of finite type chain equivalent to C.
proof For each q let Fq be a finitely generated submodule of Zq(C) such
that Fq maps onto Hq(C) under the epimorphism Ze(C) - -> Hq(C). Let Fgbe
the kernel of the epimorphism Fq —» JIq(C\. Define a chain complex
C = {Cq,d'q} by Cq = Fq © Fg_i and Эд(с,с') = (c',0) for c £ Fq and c' £ F'q~_v
Then C' is a free chain complex of finite type and HQ(C') = Fq/Fq ~ Hq(C).
To define a chain equivalence т: C —> C, choose for each q a homomorphism
<pQ: Fq Cq+i such that dq+1cpq(c') = d for c' £ F'q. Then define т by t(c,c') =
c + <pe_i(c') for c £ Fq and c' £ Fqi. т is a chain map and induces an isomor-
phism t* : IKS'') ~ ЩС). Because C' and C are both free, it follows from
theorem 4.6.10 that т is a chain equivalence. »
We are now ready for the universal-coefficient theorems toward which
we have been heading.
IO theorem Let C be a free chain complex and G be a module such that
either H(C) is of finite type or G is finitely generated. Then there is a func-
torial short exact sequence
0 -> №(C;R) ® G №(C;G)	№+i(C;R) * G -a 0
and this sequence is split.
proof If G is finitely generated, it follows from lemma 5 that
p: Hom (C,R) ® G Hom (C,G)
SEC. 5 THE UNIVERSAL-COEFFICIENT THEOREM FOR COHOMOLOGY
247
Because Hom (C,R) is without torsion, Hom (C,R) * G = 0, and the result
follows from theorem 5.4.1.
If //(C) is of finite type, we use lemma 9 to replace C by a free chain
complex C of finite type. By corollary 7, p: Hom (C',R) ® G ~ Hom (C',G),
and the result again follows for C' (and hence for C) from theorem 5.4.1. и
In a similar way we obtain the following Kiinneth formula for cohomology.
11	theorem Let C and C be free chain complexes and G and G' be
modules over a principal ideal domain such that G * G' = 0 and either H(C)
and H(C') are of finite type or H(C') is of finite type and G' is finitely
generated. Then there is a functorial short exact sequence
0 -> [H* (C;G) ® H* (C';G')]e -> №(C ® C'; G® G')
[H* (C-,G)*H* (C';G')]e+1 -> 0
and this sequence is split.
proof If H(C) and H(C') are of finite type, by lemma 9, we can replace C
and C' by free chain complexes of finite type. Hence we are reduced to
proving the result for the case where C and C' have finite type or where C'
has finite type and G' is finitely generated. By corollary 7, there is an isomor-
phism p: Hom (C,G) ® Hom (C',G') ~ Hom (С ® C', G ® G'). The result
will now follow from theorem 5.4.2 as soon as we have verified that
Hom (C,G) * Hom (C',G') is acyclic.
We show that Hom (C,G) * Hom (C',G') = 0. In case C and C' are both
of finite type, Hom (CP,G) is isomorphic to a finite direct sum of copies of G
and Hom (Cg,G') is isomorphic to a finite direct sum of copies of G'. Because
G * G' = 0 by hypothesis, Hom (Cp,G) * Hom (C'q,G') = 0, and so
Hom (C,G) * Hom (C',G') = 0 in this case.
In case C' is of finite type, Hom (C'q,G') is isomorphic to a finite direct
sum of copies of G'. Hence it suffices to show that Hom (C,G) * G' = 0 if G'
is finitely generated. Let
() _> G	G —> C' -> 0
be a free resolution of G' with G finitely generated. Because G * G' = 0,
there is a short exact sequence
0 - > G ® G - > G ® G - > G ® G' - > 0
and a short exact sequence of cochain complexes (because C is free)
0 Hom (C, G ® G) -> Hom (C, G ® G) -> Hom (C, G ® G') -+ 0
Using lemma 5, this implies the exactness of the sequence
0 -> Hom (C,G) ® G -> Hom (C,G) ® G - » Hom (C,G) ® G' - » 0
Hence Hom (C,G) * G' = 0, and so Hom (C,G) * Hom (C',G') = 0 in either
case, и
248
PHODUCTS CHAP. 5
If A is a free finitely generated module, then
A Hom (Hom (A,R), R)
Since Hom (A,R) is also free and finitely generated, it follows from corollary 7
that
A ® G ~ Hom (Hom (A,R), R) ® Hom (R,G) ~ Hom (Hom (A,R), G)
We use this to express homology in terms of cohomology.
12	theorem Let C be a free chain complex such that H(C) is of finite
type. For any module G there is a functorial short exact sequence
0 Ext (№+1(C;H), G) He(C;G) Hom (№(C;B), G) -> 0
and this sequence is split.
proof By lemma 9, we are reduced to the case where C is of finite type.
Then C ® G Hom (Hom (C,B), G), and the result follows, by theorem 3,
on changing Hom (C,R) to a chain complex by changing the sign of the
degree. “
The following result is a version of lemma 8 valid for homology that is a
partial converse to theorem 10.
13	theorem Let C be a chain complex such that for every module G the
map p: Hom (C,R) ® C - -> Hom (C,G) induces isomorphisms of all cohomol-
ogy modules. Then IL.. (C) is of finite type.
proof Because p-. №(Hom (C,R) ® He(C)) =2 №(Hom (C,He(C))), it follows
from theorem 3 that there exist fi £ Hom (Cq,R) and Zf £ IIq(C) such that
® Zi} = I77,,(о Then, for any z £ HQ(C) we have
Z = (p{Hfi ® Zi}, z) — S {fi,Z)Zi
showing that He(C) is generated by Zi. «
Note that if the short exact sequence of theorem 10 is valid for a given C
for all G, then the hypothesis of theorem 13 is satisfied, and so II(C'j is
of finite type.
6 CUP AM> CAP PROBtCTS
There is a cross product of cohomology classes from the tensor product of the
cohomology of two spaces to the cohomology of their product space. By
using the diagonal map of a space into its square, the cross product gives rise
to a product in the cohomology module of a space. This multiplicative struc-
ture provides cohomology with more structure than just the essentially additive
module structure. In this section we shall define these products and establish
some of their elementary properties.
SEC. 6 COT AND CAP PRODUCTS
249
If (X X В, A X Y} is an excisive couple of X X Y, there is a cohomology
cross product
y': №(X,A; G) ® №(Y,B; G) IH‘4(X,A) X (Y,B); G ® G')
induced by the functorial homomorphism
Hom (A(X)/A(A),G) ® Hom (A(Y)/A(B),G')
4
Hom ([A(X)/A(A)] ® [A(Y)/A(B)], G ® G')
followed by an Eilenberg-Zilber cochain equivalence of the bottom module
with Hom (Д(Х X Y)/A(X X В U A x Y), G ® G'). If и E №(X,A; G) and
v E №(Y,B; G), we define
и xv = p'(u ® v) E Н₽+в((Х,А) x (Y,B); G ® G')
From theorem 5.5.11 we obtain the following Kiinneth formula for
singular cohomology.
1	theorem Let {X X В, A X Y } be an excisive couple in X X Y and
let G and G be modules such that G * G' = 0. If If, (X,A; B) and Ff. (Y,B; B)
are of finite type or if If,, (Y,B; B) is of finite type and G is finitely generated,
there is a functorial short exact sequence
0	[H*(X,A; G) ® H*(Y,B; G')]« 4 №((X,A) X (Y,B); G ® G') —>
[H*(X,A; G) * H*(Y,B; G')]«+i -> 0
ond this sequence is split. »
The cohomology cross product satisfies the following analogues of state-
ments 5.3.11 to 5.3.15.
2	Let fi (X,A)	(X',A') and g: (Y,B) —> (Y',B') be maps and let
•«' E №(X',A'; G) andv' £ №(Y',B'-, G). Then, in №+<ifiX,A) x (Y,B); G ® G),
we have
(J X g)*(u' X v') = f*u’ X g*v' я
3	Let p: IX,A) X Y —> (X,A) be the projection to the first factor and let
y: G--> H*(Y;G') be the augmentation map. For и E №(X,A; G), in
№(fX,A) XY-,G® G), we have
P* ® g')) = ч X 4(g') “
4	For и E №(X,A; G), v E №(Y,B- G), and w E Hr(Z,C- G"), in
HP+4+fi(X,A) X (Y,B) X (Z,C); G ® G' ® G"), we have
и X {v X w) = (t/ X v) X w a
5	Lei T-. (X,A) X (Y,B) (Y.B) X (X,A) and <?. G ® G G ® G
interchange the factors. For и E №(X,A; G) and v E №(Y,B; G), in
Hp+o((X,A) X (Y,B); G ® G), we have
250
PRODUCTS CHAP, 5
T*(v X и) = ( — Г)р^{и X v) и
6	Let {(Xi,Ai), (Х2Д2)} be an excisive couple of pairs in X and let
и (£ Hp{Xi П X2, Ai П A2; G) and v 6	G'). For the connecting
homomorphisms of appropriate Mayer-Vietoris sequences we have
8*(u x v) — 8*и X v
in №+e+1((Xi U X2, Ai U A2) X (Y,B); G ® G') and
8*(v X u) = ( —l)9v X 8*u
in HP+<1+1(fY,B) X U X2, Ai U A2); G' ® G). «
Consider the two functors A(X) and Д(Х) ® Д(Х) on the category of
topological spaces. Because Д(Х) is free with models {Де}е>о and Д(Х) ® A(X)
is acyclic with models {A«}e>o [that is, the reduced complex of Д(Д«) ® Д(Д«)
is acyclic for all q], it follows from the acyclic-model theorem 4.3.3 that
there exist functorial chain maps t* : Д(Х) -—> Д(Х) ® Д(Х) preserving aug-
mentation, and any two are chain homotopic. Such a functorial chain map is
called a diagonal approximation. The name stems from the fact that if
т'х: A(X X X) A(X) ® A(X) is a functorial chain equivalence given by the
Eilenberg-Zilber theorem and d: X —> X X X is the diagonal map, then the
composite
Д(Х) Д(Х X X) A(X) ® A(X)
is a diagonal approximation.
We construct a particular diagonal approximation called the Alexander-
Whitney diagonal approximation. If a: Д« X is a singular q-simplex, the
front i-face is defined for 0 < i < q to equal the composite о ° X, where
X: Д’ —> Де is the simplicial map defined by X(p;) = p; for 0 < / < i. Similarly,
the back i-face cq is defined for 0 < i < q to equal the composite a ° X',
where X': Д’ —> A 4 is the simplicial map defined by X'(p,-) = py+Q_j for
0 < j < i. It is easy to verify that	'
т(а) = S iO ® Oj
defines a functorial chain map т: Д(Х) —> Д(Х) ® A(X), and this chain map is
the Alexander-Whitney diagonal approximation.
Let G and G' be R modules. A pairing of G and G' to an R module G"
is a homomorphism cp: G ® G' —» G". For example, G and G' are always
paired to C ® G'. Given such a pairing and given a diagonal approximation
t, there is a functorial cochain map
тх-. Hom (A(X),G) ® Hom (Д(Х),С') Hom (Д(Х),С")
defined to equal the composite
Hom (Д(Х),С) ® Hom (Д(Х),С')
Hom (Д(Х) ® Д(Х), G ® G')	Hom (Д(Х),С")
SEC. 6 CUP AND CAP PRODUCTS	251
If А С X, then for f G Hom (A(X),G) and/' £ Hom (A(X),G')> we have
® /') | A(A) = fA(J | A(A) ® /' | A(A))
If Ai, A2 С X and / vanishes on Ai, /' vanishes on A2, it follows that
fx(/ ® /0 vanishes on Д(А1) + Д(А2). If {Ai,A2} is an excisive couple in X,
it follows that rx induces a homomorphism
Цр(Х,Аг, G) ® HfiX,A2- G') Hp+^X, Ax U A2; G")
which is called the cup-product homomorphism. If и £ Hp(X,Ay, G) and
v £ №{X,A2; G'), their cup product is denoted by
wore №+«(XA! U A2; G")
This product is a bilinear function of и and v and depends on the pairing <p
but not on the particular diagonal approximation. The Alexander-Whitney
diagonal approximation yields a particular map f which defines a cup product
of cochains / о /' for / £ Hom (Др(Х),С) and /' £ Hom (Д,;(Х),С') by
(/o/>) = <р(/М ®/'(oQ))
Then {/} о {/'} = {/^/'} in Hp+^X, Ax U A2; G").
As pointed out above, there exist diagonal approximations which are
factored through Д(с/). This implies the following relation expressing the cup
product in terms of the cross product.
7	theorem If {X x A2, Ai X X} is an excisive couple in X X X and
(p: G ® G' G" is a pairing, then for и £ №(X,/\ G) and v £ №(X,A2; G'),
in HP+qfX, Ai U A2; G"), we have
и о v — <p*(d* (и x t:)) “
The cup product has the following properties analogous to the corre-
sponding properties of the cross product.
В Let f: X —> Y map Ai into IL and A2 into B2 and let и £ /fo(Y,Bj; G)
and v E №(Y,B2; G'). Let fr. IX,Af) -> (Y,Bi), /2: (X,A2)	(Y,B2), and
f: (X, Ai U A2) - -> (Y, Bi U B2) be maps defined by fi In №+cdX, Ai U A2; G"),
we have
f* (u о v) = /^ и о /| v и
О For any и E Hq(X,A-, G) with the pairings R0GzGzG®R we have
1 о и = и = u\~> 1 »
10 Given a commutative diagram, where cp, cp', fi, and fi' are pairings,
Gi ® (G2 ® G3) ~ (Gi ® G2) ® G3 ^4 Gi2 ® G3
1	® sp'I	jA
Gi ® Сгз	Gi23
252	PRODUCTS CHAP. 5
and given щ £ №(X,Ai; Cj), u2 6 №(X,A2; G2), and u3 £ Hr(X,A3; G3),
then, in Hp+i+\X, Ai U A2 U A3; C'|23), we have
«1	(«2 O U3) - («1	U2) О t<3 “
11	Given a commutative diagram of pairings
G® G'^G'® G
\ /
G"
and given и £ Hp(X,Ai; G) and v G №(X,A2; G'), in IlP'fX, Ai U A2; G"),
we have
и v = (— l)i"'v a и
12	Let {(Xi,Ai), (X2,A2)} be an excisive couple of pairs in X, let A C -Xx U X2,
and let i: (Xj П X2, А П X, Cl X2) C (Xi U X2, A). For elements
и G Hp(Xi П X2, Ai Cl A2; G) and v £ //''(Xj U X2, A; G') and with the
connecting homomorphisms of the appropriate Mayer-Vietoris sequences, in
Hp+q+i(X1 U X2, Ai U A2 U A; G"), we have
S*(uv i*v) = v
8*(i*v о и) = ( — l)"c о 8*и и
Let т': Д(Х X Y) Д(Х) ® Д(У) be a functorial chain equivalence
given by the Eilenberg-Zilber theorem and let
Г: [Д(Х) ® Д(У)] ® [Д(Х) ® Д(У)]	[Д(Х) ® Д(Х)] ® [Д(У) ® Д(У)]
be the chain map defined by
T((c ®d)®(c'® d')) = (-IJdegrfdegeYc ® c') ® (d ® d')
If т is any diagonal approximation, it follows by the method of acyclic models
that the diagram
Д(Х X У)	Д(Х X У) ® A(X X Y)
Д(Х) ® Д(У) [Д(Х) ® Д(Х)] ® [Д(У) ® Д(У)]
is chain homotopy commutative. This implies the following additional relation
between cup products and cross products.
13	theorem Let q>: Gj ® G2	G and G'i ® G2 —> G' he pairings and let
Gi ® GI and G2 ® G2 be paired to G ® G' by the homomorphism
(Gi ® Gi) ® (G2 ® Gi) (Gi ® G2) ® (Gi ® G'2) G® G'
Given iii E №(X,Ar, Gf), 112 6 №(X,A2; G2), Vi £ Hr(Y,Bi: Gi), and
v2 6 Hs(Y,B2; G'2) then with suitable excisiveness assumptions, we have, in
Hp+q+^x, A± U A2) X (У BiU B2); G ® G'),
(ui X fi) о (иг X v2) = ( —l)er(ui о u2) X («1	«2) “
SEC. 6 CUP AND CAP PRODUCTS
253
Combining theorem 13 with statements 3 and 9, we obtain the following
result expressing the cross product in terms of the cup products.
14	corollary Let (X x В, A X Y ) be an excisive couple in X x Y and let
(X,A) X Y —> (X,A) and pz: X X (Y,B) - -> (Y,B) be the projections. Given
и £ H₽(X,A; G) and v E №(Y,B; G'), then, in Нр+чЦХ,А) X (Y,B); G © G'),
we have
и X v — p*(u) о p*(v) »
With the last result we can give the following example of two polyhedra
having isomorphic homology and cohomology modules but not isomorphic
cup-product structures.
15	example Let p and q be integers > 1 and let X be the space which is
the union of Sp, S<i, and S^+«, all identified at one point. If i: S₽ С X, j-. S'7 С X,
and k: Sp'o С X, then i^.H(SP) © j^.ftjSo') © k*H(Sp+o') H(X). Computing
H(Sp X S«) by the Kiinneth formula, we see that H(X) H(Sp X S«). By the
universal-coefficient theorem, X and Sp X So have isomorphic homology and
cohomology groups for any coefficient group. Since
/<*: №+e(X;Z) ~ №+e(SP+e;Z)
and к * commutes with the cup product, it follows that the cup product of
integral cohomology classes of degrees p and q, respectively, in X is zero.
However, it follows from corollary 14 that there are integral cohomology
classes of Sp X S'1 of degrees p and q, respectively, whose cup product is non-
zero. Therefore H*(X;Z) and H*(Sp x So; Z) are not isomorphic by an iso-
morphism of graded modules preserving the cup product. Hence X and
Sp X So are not homeomorphic, nor even of the same homotopy type.
There is another product closely related to the cup product that multiplies
homology and cohomology classes together. We begin with the observation
that if C and O' are chain complexes and G and G' are paired to G" by cp,
there is a functorial homomorphism
h-. Hom (C',G) © (С © C' © G') -> C © G"
such that h(f ® (c ® c' ® g')) = c © ф«У,с') © g')- Д straightforward calcu-
lation shows that for f E Hom (C'e,G) and c £ (C © C')n © G'
dh(f ®c) = (~l)p-oh(8f © c) + h(J © Эс)
If X is a space and t: Д(Х) Д(Х) ® Д(Х) is a diagonal approximation,
a functorial map
t: Hom (A(X),G) © (Д(Х) © G') Д(Х) © G"
is defined by т( / © c) = h(f © t(c)). The boundary formula yields
3f(f © c) = ( — l)degc~deg/-^gy- (g)	_|_ f(J © Sc)
Note that if A is a subset of X and f E Hom (Д(Х),С) vanishes on A, then for
any с E Д(А) © G', r(f © c) = 0. It follows that if A1; A2 С X,
254
PRODUCTS CHAP. 5
f £ Hom (Д(Х)/Д(А1),С) is a cocycle, and c E Д(Х) ® G is a chain such that
Sc £ [Д(А1) + Д(А2)] ® G, then T(/® c) is a chain of Д(Х) ® G" whose
boundary is in Д(Л2) ® G" [because Sf(/ ® c) = т(/® Sc)]. Furthermore, if
/ is the coboundary of a cochain which vanishes on Д(Аг) or if c equals
a boundary modulo [Д(А1) + Д(А2)] ® G, then r(/ ® c) is a boundary
modulo Д(А2) ® G". Hence f defines a homomorphism [sending {/} ® {c}
to {?(/® c)}]
№(X,Ax; G) ® Нп(Д(Х)/[Д(Аг) + Д(А2)], G')	H„_e(X,A2; G")
If {Ai,A2} is an excisive couple in X, this yields a homomorphism
№(X,Ai; G) ® Hn(X, Ar U A2; G') Hn_e(X,A2; G")
called the cap product. If и £ №(X,Ai; G) and z E Hn(X, Ai U A2; G'), their
cap product is denoted by a z £ H^_Q(X,A2; G"). It depends on the pairing
<p but not on the particular diagonal approximation used to define r. The
Alexander-Whitney diagonal approximation yields a map ? which defines a
cap product on cochains and chains, denoted by f r\ c, by the formula
f c = f (S a ® g^) = S a ®	® ga)
for/E Hom (AQ(X),G) and c = Sao®g^E Д(Х) ® G. Then {/} r> {c} =
The cap product has the following properties analogous to those of the
cup product.
1® Let f. X —> Y map Ai to Bi and A2 to B2 and let и E №(Y,Bu G) and
z E H„(X, Ai U A2; G). Let fo. (X,Ai) (Y,Bi), f2: (X,A2)	(T,B2), and
f: (X, Ai U A2) —> (Y, Bi U B2) be maps defined by f. Then, in Hn^YdB2; G"),
we have
fz*(fiu '~'z) = и ^f*z “
1T For any z E НИ(Х,А; G) with the pairing В ® G ~ G
1 z = z и
18 Given a commutative diagram, where tp, q>', гр, and гр' are pairings,
Gi ® (G2 ® G3) ~ (Gi ® G2) ® G3 Ct2 ® G3
1®<?У
Gi ® G23	A	Gi23
for и E Up(X,Ai; Gf), v E №(X,A2; G2), and z E Hn(X, Ai U A2 U A3; G3),
then, in Нп^р^д(Х,Аз; Gi23), we have
и r-x (v r-x z) = (и о v) ry z 
IO Let и E №(X,A; G) and z E HQ(X,A; G) and let e: JTo(X; G ® G) —>
G ® G be the augmentation. Then, in G ® G',
255
7 HOMOLOGY OF FIBER BUNDLES
e(w z) = (u,z)
20 Let {(Xi,Ai), (X2,A2)} be an excisive couple in X arid let A C Xi U X2 and
i- (Xi П X2, А П Xi П X2) C (Xi U X2, A). For и <= №(Xi U X2, A; G) and
- £ Hn(Xi U X2, Ai U A2 U A; G'), with the connecting homomorphisms of
the appropriate Mayer-Vietoris sequences, in II,tll_fXj П X2, А1 П A2; G"),
u>e have
|	d^.(u z) = i*u 3^.z »
| 21 Let ui € №(X,Ai; Gf), u2 6 He(Y,Bi; G2), zi F FfL Ai U A2; Gi), and
J Й € Hn(X, Bi U B2; G2), and let Gi and G'i be paired to G", G2 and G2 be
I paired to G2, and (Cj ® G2) and (Gi ® G2) be compatibly paired to Gf ® G2.
j. Then, in H„l+7,_p_e((X,A2) X (X,B2); G^ ® G2), we have
f	(ui X и2) (Zi X zf) = (- l)P(»-e)(M1 zi) X («2 zf) «
t’
;	7 HOMOLOGY OF FIBER BUNDLES
i
i Cup and cap products are used in this section to study the homology of fiber
• bundles. We shall show that in case the cohomology of the total space maps
| epimorphically onto the cohomology of each fiber, the homology (or cohomol-
ogy) of the total space is isomorphic to the homology (or cohomology) of the
|' product space of the base and the fiber. For orientable sphere bundles this
j leads to a proof of the exactness of the Thom-Gysin sequences, which will be
applied in the next section to compute the cohomology rings of projective
spaces.
We begin with some algebraic considerations. Let M = {Mf) be a free
finitely generated graded R module and let M* = {M« = Hom (Me,R)}. Let
/ (X,A) be a topological pair and f: X —> У be a continuous map. Given
i a homomorphism (of degree 0) 0-. M* H*(X,A; R), there are homomor-
| phisms (of degree 0) for any R module G
(	Ф: H(X,A; G) -a H(Y;G) ® M
1	Ф* : H*(Y;G) ® M* —» H*(X,A; G)
| defined by Ф(,т) = S{ f* (0(mf) z) ® mj, where {nij} is a basis of M and
j {m? } is the dual basis of M* (Ф is uniquely defined by this formula), and
j ф»(ы g) m*) = f* и о в(т*).
I 1 lemma With the notation above, if Ф is an isomorphism for G = R,
j then Ф and Ф * are isomorphisms for all R modules G.
Proof For each i let cf be a cocycle of Hom (A(X)/A(A);R) representing
the class hlrrif) and assume that mi (and hence also nr? and c?) have degree qi.
Let т: Д(Х)/Д(А) —> Д( У) ® М be the homomorphism (of degree 0) defined by
t(c) = S A(/)(cf c) ® mi
i
256
PRODUCTS СНДр 5
An easy computation shows that т is a chain map and that the induced i
homomorphisms	j
r^.: ^(ХД; G)H^(A(Y) ® Af; G) H^(Y;G) ® M	!
t*-. H*(Y;G)®M* ^H*(Hom(A(Y) ® M, G)) -* H*(X,A; G) i
equal Ф and Ф*, respectively. Since Ф is assumed to be an isomorphism for
G = R, the chain map r induces an isomorphism of homology. The universal-
coefficient theorems for homology and cohomology then imply that Ф and Ф*
are isomorphisms for all G. »	:
A fiber-bundle pair with base space В consists of a total pair (EJEfi a
fiber pair (F,F), and a projection p: E В such that there exists an open
covering {V } of В and for each V 6 { V } a homeomorphism <pv: V X (F,F) —>
(p-1(V), p-1(V) П £) such that the composite	\
V X F p-i(V) A V
is the projection to the first factor. If A С B, we let EA — p~x(A) and
ЁА = p-1(A) Г| Ё, and if b £ B, then (Efff) is the fiber pair over b.
Following are some examples.	,
2	For a space В and pair (F,F) the product-bundle pair consists of the total •
pair В X (F,F) with projection to the first factor.
3	Given a bundle projection p: Ё В with fiber F, let E be the mapping
cylinder of p and p: E —> В the canonical retraction. Then (E,E) is the total
pair of a fiber-bundle pair over В with fiber (F,F), where F is the cone over F,
and projection p.	{
4	If £ is a q-sphere bundle over B, then (Ef,E^ is the total pair of a fiber-
bundle pair over В with fiber (£«+1,S«) and projection pp. E^ —> B.
Given a fiber-bundle pair with total pair (Е,Ё) and fiber pair (F,F), I
a cohomology extension of the fiber is a homomorphism в: IT*(F,F; R) —>
H* (Е,Ё-, R) of graded modules (of degree 0) such that for each b £ В the |
composite	i
H*(F,F; В) А Н*(Е,Ё; R) Н*(ЕьЯь; R)
is an isomorphism. The following statements are easily verified.	I
5	Let p: В X (F,F) —> (F,F) be the projection to the second factor. Then
6	= p*-.H*(F,F-,R)-> H*(B x(F,FfB>	j
is a cohomology extension of the fiber of the product-bundle pair, и
6	Let 6: H*(F,F; R) —> Н*(Е,Ё; R) be a cohomology extension of the fiber ‘
of a fiber-bundle pair over В and let f:B'—>Bbea map. There is an induced,
bundle pair over B', with total pair (Е',Ё'} and fiber (F,F), and there is a map
SEC. 7 HOMOLOGY OF FIBER BUNDLES	257
у: (E',E'} - -> (Е,Ё) commuting with projections. Then the composite
H*(F,F; R) Л H*(E,E; R) A H* (E',E'- R)
is a cohomology extension of the fiber in the induced bundle. “
7	Given a fiber-bundle pair over В with total pair (Е,Ё), let the path com-
ponents of В be {B?} and let (Ej,Ej) be the induced total pair over Bj.
A cohomology extension 0 of the fiber of the bundle pair over В corresponds
to a family of cohomology extensions {Oj} of the induced bundle pairs
over Bj. B
We now establish the local form of the theorem toward which we are
heading. It shows that any cohomology extension of the fiber in a product-
bundle pair has homology properties as nice as the one given in statement 5
above.
8	lemma Let (F,F) be a pair such that H *(F,F; R) is free and finitely
generated over R and let в: I{*(F,F; R) H*(B X (F,F); R) be a cohomology
extension of the fiber of the product-bundle pair. Then the homomorphisms
Ф: H#(B x (F,F); G) H*(B;G) ® H#(F,F; R)
Ф*: H*(B;G) ® H*(F,F; R) -> H*(B X (F,F); G)
are isomorphisms for all R modules G.
proof By lemma 1, it suffices to prove that Ф is an isomorphism for G = B.
If {Bj} is the set of path components of B, then
H^(B X (F,F); R) ^©jH*(Bj X (F,F); R)
and
Щ (B;R) ® H* (F,F; B) ~ ©,• IL., (B};R) ® H* (F,F; R)
Therefore it suffices to prove the result for a path-connected space B. For
such a B, R ~ HO(B;R).
By the Kiinneth formula, I/:,:(B X (F,F); R) Si If,(B;R') ® H*(F,F; R).
We define graded submodules Ns of Jf,(B-,R.) ® H^.(F,F; R) by
(Ns)q =	@ Hi(B;R) ® Hj(F,F; R)
i+j=Q, j>s
Then
H*(B-,R) ® H*(F,R, R) = No3N1J  DNsD Ns+1
and Ay = 0 for large enough s. If и E №(F,F; R), then в(и) — 1 x A(it) + й,
where й E ®i+J-=s,j<s H\B;R) ® FP(F,F; R) and 0(n) | [b x (F,F)] = 1 X X(u)-
Because в is a cohomology extension of the fiber, X is an automorphism of
H*(F,F; R). Let z' E HS(F,F; R) and consider z X H E fVg. Then
Ф(г X z') = 2 p* (ffimf)	(z x z')) ® пн
г
and if deg m, < s, then о (z X z') E Air and p#(2A) = 0. Therefore
258
PRODUCTS сидр. 5
Ф(г X z') £ As, and so Ф maps Ns into itself for all s. Because of the short
exact sequences	4
0 Ns+1  > A,  > Ns/Ns+1 0
and the five lemma, it follows by downward induction on s that Ф is an iso-
morphism if and only if it induces an isomorphism of Ns/Ns+i onto itself for
all s. For z' £ HS(F,F; R), computing Ф(г X z') in Ns/Ns+i, we obtain
Ф(г X Z') = S p* [(1 X ) + mf) о (z X z')] ® mf
deg mt>s
= S p* [1 X X(rnf)	(z X z')] ® m;
deg »n~s -i
because mf rs (z X z') £ and p.,. (Nf) = 0. Now, by properties 5.6.21,
5.6.19, and 5.6.17,
S p* [1 X \mf) r>(z x z')J <8> пи
deg mi~s
= S z ®	= z (8> X.,.(X)
deg mi —s
where X,,.: H*(F,F; R) —> H*(F,F; R) is the automorphism dual to X. Hence
Ф(г x z?) = z X X^(z') in Ns/Ns+i, showing that Ф induces an isomorphism
of Ns/Ns+i for all s. . я
The following Leray-Hirsch theorem shows that fiber-bundle pairs with
cohomology extensions of the fiber have homology and cohomology modules
isomorphic to those of the product of the fiber pair and the base.
® theorem Let (E,E) be the total pair of a fiber-bundle pair with base В
and fiber pair(F,F'). Assume that IfifiF,F-, R) is free and finitely generated
over R and that в is a cohomology extension of the fiber. Then the
homomorphisms
Ф: ЕГ*(Е,Ё; G) H,(B;G) ® H^(F,F; R) Ф(г) = S	z) ® пц
Ф»: H*(B;G) ® H*(F,F; G) -> H*(E,E; G) Ф*(к ® v) = p*(u)	0(n)
are isomorphisms (of graded modules) for all R modules G.
proof By lemma 1, it suffices to prove the result for the map Ф in the case
G = R. For any subset А С В let hA be the composite
H*(F,F; R) Л H*(Ej>, R) H*(Ea,Ea; R)
Then 6A is a cohomology extension of the fiber in the induced bundle over A.
It follows from lemma 8 that if the induced bundle over A is homeomorphic
to the product-bundle pair A X (F,F), then
Фа: Н*(Еа,Ёа-, В) ~ H*(A;R) ® Н*(Е,Е; R)
Hence Фр is an isomorphism for all sufficiently small open sets V.
If Vand V are open sets in B, then {(Ер,Ёр), (Ер,Ёр')} is an excisive couple
of pairs in E, and it follows from property 5.6.20 that Фр, Фр', Фрлр,
and Фрир' map the exact Mayer-Vietoris sequence of (Eyfifi) and (EV,EV) into
1 sf.t: 4 HOMOLOGY OF FIBER BUNDLES	259
' the tensor product of the exact Mayer-Vietoris sequence of V and V' by
[E,E; R). Since H* (F,F; R) is free over R, its tensor product with any exact
sequence is exact. Therefore, if Фу, Фу, and Фуг,у are isomorphisms, it follows
' from the five lemma that Фуиу is also an isomorphism. By induction, Фу is an
Isomorphism for any U which is a finite union of sufficiently small open sets. Let
$ be the collection of these sets. Since any compact subset of R lies in some
element of ^l, H^.(R;R') linu (H*(U;R)}Also, any compact subset of E
jjeS in Ep for some U £ <?L, so H.2.(E,E; R) ~ linn, (Et/,E(/; R)}. Because
 the tensor product commutes with direct limits and Ф corresponds to
 fim- {Фр}^^ under these isomorphisms, Ф is also an isomorphism, в
A The above argument proves directly that Ф is an isomorphism for any
coefficient module G. A similar argument does not appear possible for Ф *,
because it is not true that H* (B;R) is isomorphic to the inverse limit
liin» {H* (U;R)}pe<jl. It should be noted that in theorem 9 we have said
nothing about commutativity of Ф * with cup products, because it is not true,
' m general, that Ф * preserves cup products.
We now specialize to the case of sphere bundles. Because
; №(№.»;«) = {“ ;/’ + [
! if £ is a q-sphcrc bundle, a cohomology extension of the fiber in £ is an ele-
ment U £ He+1(Ej,Ej; R) such that for any b £ B, the restriction of U to
(p^(h), p-1(&) П Ё) is a generator of №+1(p~1(b), р-1(Ь) Г1 E; R). Such a
cohomology class is called an orientation class (over R) of the bundle. If
orientations of the bundle exist, the bundle is called orientable. An oriented
sphere bundle is a pair (£,!/) consisting of a sphere bundle £ and an orientation
class of of £.
I If U is an orientation class of £ over Z and if 1 is the unit element of R,
1 then p(U ® 1) is an orientation class of £ over R. Therefore a sphere bundle
orientable over Z is orientable over any R.
If (£, is an oriented sphere bundle over В and f: R' —> R, then
(/*£,/* Ц) is an oriented sphere bundle over B' [where/: (Е^,Ё^) —> (E^Ej)
is associated to / ].
From theorem 9 we get the following Thom isomorphism theorem.
10 theorem Let (£, Ly) be an oriented q-sphere bundle over B. There are
natural isomorphisms for any R module G
ф£: Hn(Es,Es; G) Hn^B;G) Ф£(г) = pt(U(r> z)
Ф5*: Hr(B;G) №+ч+1(ЕьЁр, G)	Ф£* (v) =p*v^Ui
proof Let m and m* be dual generators of He+i(Ee+1,S«; R) and
H«+1(E<J+1,S«; R), respectively, and define a cohomology extension 0 by
6(mi;) = U^. Then Ф? is the composite
Rn(Es,Ef; G)	® He+1(Ee+i,S«;R)	H^^B-G)
where the second map sends z ® m to z. By theorem 9, Ф is an isomorphism,
260	PRODUCTS СНАР.5
and so is an isomorphism. A similar argument shows that Ф?* is an isomor-
phism. These isomorphisms are natural for induced bundles because of
naturality properties of the cup and cap products. ®
This result implies the exactness of the following Thom-Gysin sequences
of a sphere bundle.
I I theorem Let (£, Ut) be an oriented q-sphere bundle with base В and
projection p = p \ Ё: Ё - -> B. Far any R module G there are natural
exact sequences
-----> Н„(Ё£;С) А» H„(B;G) Hn^B-G} A Hn^(fie,G) -+ .
-----> Hr(B;G) TLf №(Ee,G) Л №~<i(B-G) X H^(B-,G) -> • • • •
in which Ф£ and 'Iq* have properties
Ф£(« ГУ z) = ( - l)(e+1) deg v (v) ГУ Z
Ф£*(«1 О Г2) = «1 О Ф£*(«г)
proof There is a commutative diagram (with any coefficient module^
• • •	Н„(Ё) Hn(E) M Нп(Е,Ё) Д НП_!(Ё)	 • •
SEC. 7 HOMOLOGY OF FIBER BUNDLES	261
•j 2 lemma Two orientation classes U and U' of a sphere bundle over
a^path-connected base space В are equal if and only if for some bo £ В
U | (р~ЦЬ0), р-ЦЬо) П Ё) = U' | (p-i(bo), p^(bo) А Ё) 
If В is not path connected, let {BJ be the set of path components of В
: and let (ЁУ,Ё;) be the part of (Е,Ё) over Bj. Then
Н*(Е,Ё; R) ~ 0 H*(Ej,Er, R)
and we also obtain the following result.
13 lemma Two orientation classes U and U' of a sphere bundle with base
space В are equal if and only if for all b £ В
i	U | (p-1(b), p-1(h) Г1 Ё) = U' | (p-1(b), p-1(h) П Ё) «
[ In case В = Z2, then №+1(p“1(b), p i(b) А Ё; Z2) ~ Z2 for all b G B.
• Therefore this module has a unique nonzero element, and we obtain the fol-
j lowing consequence of lemma 13.
! 14 COROLLARY
t equal. и
Any two orien tation classes over Z2 of a sphere bundle are
H„(B) Hn_e_i(B)
the top row of which is exact. Since p is a deformation retraction of E onto B.
p.,. is an isomorphism. By theorem 10, Ф£ is an isomorphism. The desired se-
quence is obtained by defining 'Iq = Фу-^р.^' and p = ЙФ^1. Similarly, the
cohomology sequence is defined by 'Iq* = p* -1j* Ф£* and p* = Ф£*~ЧЬ
We verify the formula for 'Iq.
^(vryz) = Ф^р*"1^ гу z) = Ф£/:!:(р*(г;) o/q-Jz))
= Ф£(р*(г) гу fap*-1^)) = p*(U гу \p*(v) гу qp* ^z)])
= P* (/ * [G O p* (v)] ГУ p* -!(z))
= ( - l)(e+1> deg « p* [j * Ф£* (v) ГУ p* -l(z)]
= ( _ 1) (e+1) deg v	(u) z “
Note that the isomorphisms Ф and Ф * of the Thom isomorphism theorem
depend on the choice of the orientation class U of the bundle. Therefore the
homomorphisms p and Ф and p* and 'I'* of the Thom-Gysin sequences also
depend on the orientation class. In case В is path connected and U and U'
are orientation classes of a sphere bundle over B, it follows from theorem 10
that there is an element r G R such that
U' = p*(r X 1) U = r[p*(l) о G]
If b0 G B, then
U’ | (p-1(bo)> р-^Ьо) П Ё) = r[G| (p-i(bo), p~4bo) П Ё)]
Thus, for R = Z2 the homomorphisms Ф, p, and Ф and Ф *, p *, and Ф *
< are all unique.	___
The characteristic class of an oriented q-sphere bundle ({,1-У 's
defined to be the elerhent
= Ф£*(1) G №+1(B;B)
This is functorial (that is, = /*£q). From the multiplicative properties of
'Iq and 'Iq* in theorem 11 we obtain the following equations.
; 15 For z G H„(B;G)
^(z) = гу z
and for v G Hr(B;G)
Ф(* (v) = v о и
We now investigate the existence of orientation classes for a sphere
bundle. Let (X,X') be a pair and let {AJjej be an indexed collection of sub-
sets Aj С X. An indexed collection
{iq E Hn(Aj, Aj А X; G)};ej
is said to be compatible if for all /,G J
iq | (Aj П Ar, Aj A Ar A X') = ц. | (Aj A Ar, Aj A Aj. A X')
The compatible collections {u}} constitute an В module Hn((Aj},X'; G).
Clearly, the restriction maps
Therefore we have the next result.
Hn(X,X’- G) H”(Aj, Aj A xq G)
262	PRODUCTS CHAP. 5
define a natural homomorphism H'n(X,X'; G) —-> Hn(fiAj],X'; G).
16 lemma Let (Е,Ё) be a fiber-bundle pair with base B, projection p-.E~>fi
and fiber pair Assume that for some n > 0, Hi(F,F; R) = 0 for i n
Then
(a) For all А С В and all R modules G
Hi(p-\A), р-ЦА) П E- G) = 0 = Hi(p~1(A), p-\A) П Ё; G) i < n
(h) If {V} is any open covering of B, then in degree n the natural homa
morphism is an isomorphism
Нп(Е,Ё-, G)	G)
proof By the universal-coefficient formula, it suffices to prove (g) for G = ]{
If А С В is such that (p-1(A), p '(A) Cl Ё) is homeomorphic to A X (Fffi
then by the Kiinneth formula,
Hi(P-1(A), p-^A) aE;R)^ HfiA x (F,F); B) = 0 i < n
From this it follows (as in the proof of theorem 9) by induction on the
number of coordinate neighborhoods of the bundle needed to cover A (using
the Mayer-Vietoris sequence and the five lemma) that (a) holds for all com-
pact A С B. By taking direct limits, (a) holds for any A.
For (b), let {W} be the collection of finite unions of elements of {V},
By (a) and the universal-coefficient formula for cohomology, there is a com-
mutative diagram
№(E,E; G)	~ Hom (H„(E^;B), G)
I	1=
lim.	p~\W) П Ё; G)} ~ lim_{Hom (H„(p-i(W), p~\Wj П Ё; B),G)j
Hence we need only prove that a compatible collection {uy}ve(y} extends to a
unique compatible collection {w}ii\ (wj- This follows by using Mayer-Vietoris
sequences again and from the fact that //!'(р-1( W), p-1( W) П Ё; G) = 0 for
i <; n. “
For sphere bundles we have the following immediate consequence.
17 corollary A sphere bundle Ij with base В is orientable if and only if
there is a covering {V} of В and a compatible family {uy}, where uyisan
orientation class of £ | V for each V £ { V }. “
Since a trivial sphere bundle is orientable, corollaries 17 and 14 imply
the following result.
IВ corollary Any sphere bundle has a unique orientation class over Z2. I
By theorem 2.8.12, there is a contravariant functor from the fundamental
groupoid of the base space В of a sphere bundle £ to the homotopy category
which assigns to b £ В the fiber pair (Еь,Ёъ) over b and to a path class [w] in
В a homotopy class h[w] £ [E^JE^oy, £a(i),lL(i)]. For fixed R there is then a
SBC. 8 THE COHOM^.OGY ALGEBRA	263
covariant functor from the fundamental groupoid of В to the category of
Я modules which assigns to b £ В the module Нв+1(ЕЬ,ЁЬ; R) and to a path
class [w] the homomorphism
h[w] *: Не+1(ЕШ(1),ЁИ(1); R) —> №+1(Ё^О)^^оу, R)
j О theorem A sphere bundle £ is orientable over R if and only if for
every closed path w in B, /i[w] * = 1.
proof If £ is orientable with orientation class U £ №+1(EJE; R), for any
small path w in В (and hence for any path)
/i[w] *(C71	= U | (Еа(О)Ж(о))
Since U | (Еъ,Ёъ) is a generator of №+1(Еь,Ёь; R), this implies that h[w] * = 1
for any closed path w.
Conversely, if h[w] * = 1 for every closed path w in B, there exist generators
Ub G №+1(Еъ,Ёъ; R) such that for any path class [w] in B, h[w] * (U^if) = Uu(0).
If Vis any subset of В such that £ | V is trivial, it is easy to see that there is an
orientation class Uy of £ | V such that Uy | (Еъ,Ёъ) = U, for all b £ V. If {V}
is an open covering of В by sets such that £ | V is trivial for all V, then {C7y}
is a compatible family of orientations, and by corollary 17, £ is orientable, н
20 corollary A sphere bundle with a simply connected base is orientable
over any R. «
8 THE COHOMOLOGY ALGEBRA
The cup product in cohomology makes the cohomology (over R) of a topologi-
cal pair a graded R algebra. In the first part of this section we define the
relevant algebraic concepts and compute this algebra over Z2 for a real pro-
jective space and over any R for complex and quatemionic projective space.
This is applied to prove the Borsuk-Ulam theorem.
For the case of an H space, there is even more algebraic structure that
can be introduced in the cohomology algebra. The cohomology of such a
space is a Hopf algebra, and the second part of the section is devoted to its
definition and some results about its structure. The section concludes with
a proof of the Hopf theorem about the cohomology algebra of a compact con-
nected H space.
A graded R algebra consists of a graded R module A = and a
homomorphism of degree 0
p: A ® A --> A
called the product of the algebra (/r then maps Ap ® A'1 into A'i'+,i for all
p and q). For a, a' £ A we write aa' — ifa ® a'). The product is associative
if (gg')g" = <z(<z'u") for all a, a', a" £ A and is commutative if
aa’ = (-l)degadega'o'o for аЦ a' £ Д
264
PRODUCTS CHAP. 5
W = {
I example If (X,A) is a topological pair, then H*(X,A; R) is a graded
R algebra whose product is the cup product (with respect to the multiplica-
tion pairing of R with itself to R). It follows from property 5.6.10 that this
product is associative and from property 5.6.11 that it is commutative. If
A = 0, it follows from property 5.6.9 that 1 is a unit element of the algebra
H*(X;R). H*(X,A; R) is called the cohomology algebra of (X,A) over R.
2	example The polynomial algebra over R generated by x of degree
n > 0, denoted by Sn(x), is defined by
0	<7^0 (n) or q < 0
free R module generated by xp q = pn, p >0
with the product (aXp)(f>Xq) = (a/3)xp+g for a, /3 £ R. It is then clear that
%o is a unit element and that xp = (xi)p. If we denote Xi by x, then Xp = XP.
Thus, disregarding the graded structure, S„(x) is simply the polynomial alge-
bra over R in one indeterminate x. The truncated polynomial algebra over R
generated by x of degree n and height h, denoted by Тп,л(х), is defined to be
the quotient of S„(x) by the graded ideal generated by Xй. If h = 2, this
is called the exterior algebra generated by x of degree n and is denoted
by En(x).
If A and R are graded R algebras, their tensor product A ® R is also a
graded R algebra with product
(a ® h)(n' ® b') = ( —l)degbdega'ao' (g) jjJj'
If A and R have associative or commutative products, so does A ® R.
3	example If R is a field and (X,A) and (Y,B) are topological pairs such
that either I/.,.(X,A; R) or H*(Y,R; R) is of finite type, it follows from theorem
5.5.11 that
H*(X,A; R) ® H*(Y,B; R) H*((X,A) X (Y,B); R)
We compute the graded Z2 algebra H*(P”;Z2) for real projective space
Pn. Note that the double covering p: S“ —> Ptl is a 0-sphere bundle. We let
wn £ H1(Pn;Z2) be the characteristic class (over Z2) of this bundle.
4	theorem For n > 1, H*(Pn;Z2) is a truncated polynomial algebra over
Z2 generated by wn of degree 1 and height n + 1.
proof All coefficients in the proof will be Z2 and will be omitted. By
corollary 5.7.18 and theorem 5.7.11, there is an exact Thom-Gysin sequence
_> Ha(Sn) №(]R)	№+1(P"}	№+1(Sn)
starting on the left with 0	H°(P")	and terminating on the right
with Hn(Sn) H«(R) —> 0 [note that Hr'(P>!) = 0 for q > n, because P" is a
polyhedron of dimension n]. Because №(Sn) = 0 for 0 < q < n, it follows
that
'?*: №IPp) Но+ЦР»)
SEC. 8 THE COHOMOLOGY ALGEBRA
265
is an epimorphism for 0 < q < n — 1 and is a monomorphism for
0 C q < n — 1. Because Pn and S'" are connected for n > 1, p* tFiP") = H°(S”),
which implies that 'I'*: H°(P") НЦР”) is also a monomorphism. Therefore
7^ 0 for 0 < q < n, and because p* = Hn(Pn) and Hn(Sn) Zz. Z2,
it follows that p* is a monomorphism and that Ф*: I/"’1 (P") Hn(Pn) is also
an epimorphism.
We have shown that for 0 < q < n — 1
'?*; №(P”) ~ H"H(P")
Then w„ = Ф*(1) is the nonzero element of H1(Pn'), and by equation 5.7.15,
'['* (te,/7) = wnq+1. Therefore, for 1 < q < n, wni is the nonzero element of
Hq(Pn')- 
By corollary 3.8.9, Pn(C) and P„(Q) are simply connected. It follows
from corollary 5.7.20 that the Hopf bundles S2fi+1 Pn(C) with fiber S1 and
S4n+3 Pn(Q) with fiber S3 are orientable over any R. Let xn £ H2(Pm(C);B)
and yn € H4(-P«(Q);II) be the characteristic classses of these Hopf bundles
(based on some orientation class of each bundle). An argument analogous to
that of theorem 4, using the Thom-Gysin sequences of the Hopf bundles,
establishes the following result.
5	theorem For n > 1, H* (Pn(C);R) is a truncated polynomial algebra
over R generated by xn of degree 2 and height n + 1, and H*(Pn(Q);R) is a
truncated polynomial algebra over R generated by yn of degree 4 and height
n + 1. 
6	corollary Let n > m > 1 and let i: P™ C Pn be a linear imbedding.
Then for q <m
i*-. HdPn-Zf) ~ №{Ptn,Z2)
proof The hypothesis that i is a linear imbedding implies that the 0-sphere
bundle over Pm induced by i from the double covering S” —> Pn is the double
covering S™ Pm. By the naturality of the characteristic class, i* wn = wm.
The result now follows from theorem 4 and the fact that i * (wnq) = (i * wn)q. 
7	corollary Let 11 )> m > 1 and let f: Pn Pm be a map. There exists
a map f': Pn S™ such that p ° f = f, where p: Sm P"‘- is the double
covering.
proof By the lifting theorem 2.4.5, it suffices to prove fftfnfP'f) — 0.
If in = 1, this follows from the fact that тг(Р") = Z2 and tz/P1) — Z. Assume
that m > 1 and observe that because H'(P") has just the two elements 0 and
to,i, either /*(wm) = 0 or f* (wm) = wn. Because f* is an algebra homomor-
phism, the latter is impossible [since 0 7^ wnm+1 and f *	= 0]. There-
fore f*(wm) = 0.
We know that ir(Pn') = Z2, and a generator for this group is the homo-
topy class of the linear inclusion map i: P1 С P". Because /* (wm) = 0, it fol-
lows that i*/*(wm) — 0. If j: P1 C Pm is the linear inclusion map, by
266
PRODUCTS CHAP. 5
corollary 6, j*(wm) 0. Since (f ° i) *(wm)	i is not homotopic
to Since чт(Рт) = Z2, f ° i is null homotopic. Hence /#[г] = [/0 i] — 0, and
so /#(w(Pn)) = 0 in this case also, и
8	corollary For n > m > 1 there is no continuous map g: S” gm
such that g( — x) = — g(x) for all x £ Sn.
proof If there were such a map, it would define a map f. Pn p™ such
that the following square (where p and p' are the double coverings) is
commutative
S’! A Sm
pn -L> pm
By corollary 7, f can be lifted to а шар f'\ Pn S7n. Then
pf'p' = fp' = pg
Therefore f'p' and g are liftings of the same map. For any x F Sn either-
g(x) = f'p'(x) or g( —x) = f'p'(x) = f'p'( —x). In any event, f'p' and g must
agree at some point df Sn. By the unique-lifting property 2.2.2, f'p' = g. This
is a contradiction, because for any x £ Sn, p' maps x and — x into the same
point, while g maps them into separate points. 
This last result is equivalent to the Borsuk-Ulam theorem, which is next.
9	theorem Given a continuous map f: Sn Rn for n > 1, there exists
x E S” such that fix) = f( — x).
proof Assume there is no such x and let g: S” —> S’,rl be the map defined by
g(x) =	/(*) -/(-*)
Then g( — x) = — g(x), which would contradict corollary 8. 
Dual to the concept of graded R algebra is that of graded R coalgebra,
which is defined by dualizing the concept of product. A graded R coalgebra
consists of a graded R module A = {A®} and a homomorphism of degree 0
d: A A ® A
called the coproduct of the coalgebra (so d maps Ac‘ into @i+;=e A1 ®Ai for
all q). The coproduct is said to be associative if
(d ® l)d = (1 ® d)d: A A ® A ® A
and is said to be commutative if Td = d, where T: A ® A A ® A is the
homomorphism T(a ® a') = (_ l)degodego'o' ® a. A counit for the coalgebra
is a homomorphism e: A R (where R is regarded as a graded R module
SEC. 8 THE COHOMOLOGY ALGEBRA	267
consisting of R in degree 0) such that each of the composites
e	R ® A
А Л A ® A
i	A ®
is the identity map.
A Hopf algebra over R is a graded R algebra В which is also a coalgebra
whose coproduct
d: В В ® В
is a homomorphism of graded R algebras. A Hopf algebra В is said to be con-
nected if B° is the free R module generated by a unit element 1 for the
algebra and the homomorphism e: R —> R defined by e(nl) = a for a £ B is
a counit for the coalgebra.
IO example If X is a connected H space whose homology over a field R is
of finite type, then the multiplication map p: X X X X defines a coproduct
d = p*: H*(X;R)	H*(X;R) ® H*(X;R)
H*(X;R) with this coproduct is a connected Hopf algebra of finite type whose
product is associative and commutative (the fact that X has a homotopy unit
x0 implies that the map H*(X;R)	H*(x0;R) ~ R is a counit).
We shall study connected Hopf algebras having an associative and com-
mutative product and describe the algebra structure of those which are of
finite type over a field of characteristic 0. The following is the inductive step
of the structure theorem toward which we are heading.
II lemma Let R be a connected Hopf algebra with an associative and
commutative product over a field R of characteristic 0. Let R' be a connected
sub Hopf algebra of R such that R is generated as an algebra by B' and some
element x — B'. If x has odd degree n, then as a graded algebra
В z L ® En(x) and if x has even degree n, then as a graded algebra
BzB'® Sn(x).
proof Because B' is a sub Hopf algebra of B, the unit element of В belongs
to B'. Since x £ В — В1, x has positive degree n. Let A be the ideal in В gen-
erated by the elements of positive degree in B', and if rj: В B/A is the
projection, let
d' = (1 ® f)d: B^ В ® B^ В® (B/A)
Then d' is an algebra homomorphism, d'(j8) = f> ® 1 for f £ B', and d'(x) =
x ® 1 + 1 ® rj(x). Note that x A, because A consists of finite sums Х;>(| ffx',
where fa £ B' is of positive degree, so fax"' is of degree larger than n unless
i = 0. Therefore rj(x) 0 in B/A.
Assume that x is of odd degree. Because В has a commutative product
and R has characteristic different from 2, x2 = 0. We show that there is no
268	PRODUCTS CHAP. 5
relation of the form ft0 + ft±x = 0 with /hi, fa € B' and fa 7^ 0. If there were
such a relation, then
0 = d'(ft0 + fax) = fa® 1 +(fa® l)[x ® 1 + 1 ® y(x)]
= fa® ч(х)
Since fix) 0, this implies fa — 0, which is a contradiction. Therefore the
homomorphism В' ® Ея(х) В sending ft ® 1 to ft and ft ® x to fa is an
isomorphism of graded algebras.
Assume that x is of even degree. We shall show that there is no relation
of the form 20<j<r ftfa = 0 with fa £ B', r > 1, and fa fi= 0. If there were
such a relation, consider one of minimal degree in x. Then
0 = d'(2 fax') = '2,(ftl® l)[x ® 1 + 1 ® т](х)р
= (2 ifax4^1) ® fix) +    + fa® (fix))r
The only term on the right in В ® (B/A)n is the term (2гДх4-1) ® fix).
It must be 0, and because rj(x) 0, 2 ifax^1 = 0. If r > 1, this is a relation
of smaller degree in x (note that rfa fi= 0 because R has characteristic 0), and
this is a contradiction. If r — 1, we get fa = 0, which is also a contradiction.
Therefore there is no relation, and the homomorphism B' ® Sn(x) В
sending ft ® xi to ftx'< for ft £ B' and q > 0 is an isomorphism of graded
algebras. 
We use this result to establish the following Leray structure theorem for
Hopf algebras over a field of characteristic 01.
12	theorem Let В be a connected Hopf algebra with an associative and
commutative product and of finite type over a field R of characteristic 0. As
a graded R algebra either B^R or В is the tensor product of a countable
number of exterior algebras with generators of odd degree and a countable
number of polynomial algebras with generators of even degree.
proof Because В is of finite type, there is a countable sequence
1 = Xo, %i, %2, . . . of elements of В such that i<fa implies that deg x3 < deg x3
and such that as an algebra В is generated by the set {x3}3><i- For n > 0 let
Bn be the subalgebra of В generated by Xo, Xi, . . . , х,г. We can also assume
that Xn+i does not belong to Bn. Because of the condition that deg x3 is a non-
decreasing function of each Bn is a connected sub Hopf algebra of В (that is,
d maps Bn into Bn ® B,,)- Since Bn+± is generated as an algebra by Bn and
Хи+1, lemma 11 applies. Since Bo ~ R, Bt ~ R ® E(xft) or Bi ~ R ® S(xr).
Therefore В = Bo ~ R or Bx is either an exterior algebra on an odd-degree
generator or a polynomial algebra on an even-degree generator. By induction
on n, using lemma 11, each Bn+i is a tensor product of the desired form.
Since В has finite type, В ~ lim , B„, and В has the desired form. 
XA structure theorem valid over a perfect field of arbitrary characteristic can be found in
A. Borel, Sur les cohomologie des espaces fibres principaux et des espaces homogenes de
groupes de Lie compacts, Annals of Mathematics, vol. 57, pp. 115-207, 1953.
SEC. 9 THE STEENROD SQUARING OPERATIONS
269
For a connected H space whose homology is finitely generated over a
field F no polynomial algebra factors can occur in the above structure theorem,
and we obtain the following Hopf theorem on H spaces.
13	corollary Let X be a connected H space whose homology over a field
В of characteristic 0 is finitely generated. Then the cohomology algebra of X
over R is isomorphic to the cohomology algebra over R of a product of a
finite number of odd-dimensional spheres. 0
In particular, we obtain the following result about spheres that can be
H spaces.
14	corollary No even-dimensional sphere of positive dimension is an
H space. 
THE STEENROD SQUABINC OPERATIONS
In the last section the cup product in cohomology was used to prove the
Borsuk-Ulam theorem, a geometric result. Any other algebraic structure which
can be introduced into cohomology (or homology) and which is functorial can
be similarly applied. A particular example of such an additional algebraic
structure is a natural transformation from one cohomology functor to another.
These natural transformations are called cohomology operations. In this sec-
tion we introduce the concept of cohomology operation and define the par-
ticular set of cohomology operations called the Steenrod squares.
Let p and q be fixed integers and G and G' fixed R modules. A cohomology
operation 0 of type (p,q; G,G') is a natural transformation from the functor
Hp( ;G) to the functor №( ;G') (both functors being contra variant singular
cohomology functors defined on the category of topological pairs). Thus 0
assigns to a pair (X,A) a function (which is not assumed to be a homomorphism)
Hp(X,A; G) №(X,A; G')
such that if f: (X,A)	(Y,B) is a map, there is a commutative square
№(Y,B; G) -tea» №(Y,B; G')
J/*
№(X,A; G) -tea» №(X,A; G')
A homology operation is defined similarly, but we shall not discuss homology
operations.
Following are some examples.
I If (p; G G' is a homomorphism, cp.,. is a cohomology operation of type
(q,q: G,G') for every q, where
<P*. №(XA; G) №(X,A; G')
270
PRODUCTS CHAP. 5
is defined as in Sec. 5.4. (p* is called the operation induced by the coefficient
homomorphism (p.
2 Given a short exact sequence of R modules 0 G' G G" 0,
the Bockstein cohomology operation fi* of type (q, q + 1; G",G') for every q
is defined to equal the Bockstein homomorphism
fi*-. №(X,A; G")	№+ЦХ,А; G)
corresponding to the coefficient sequence 0 G G G" 0 as defined
in theorem 5.4.11.
3 For any p and q there is an operation 0P of type (q,pq; R,R). called the
pth-power operation, defined by
= up и e №(Х,Л; R)
An operation 0 is said to be additive if 0(х,л) is a homomorphism for
every (X,A). The operations in examples 1 and 2 are additive; however, the
operation 0P of example 3 is not additive, in general.
Any cohomology operation provides a necessary condition for a homo-
morphism between the cohomology modules of two pairs to be the induced
homomorphism of some continuous map between the pairs. For example, if 0
is of type (p,q; G,G), a necessary condition that a homomorphism
$. H*(Y,B: G) H*(X,A; G)
be induced by some map/: (X,A)	(Y,B) is that
/0(y,B) = Hp(Y,B-, G) №(X,A; G)
In these terms the algebraic idea underlying corollaries 5.8.7 and 5.8.8 is that
for n /> m > 1 there is no homomorphism
H*(B»;Z2) H*(PffiL2)
such that / sends the nonzero element of H1(Pm;Z2) to the nonzero element
of H1(Pn;Z2) and commutes with the (m + l)st-power operation 0m+± of type
(1, m + 1; Z2,Z2).
We shall now define a sequence of operations Sql called the Steenrod
squares, each Sq’ being a cohomology operation of type (q, q + i; Z2,Z2) for
every q. These operations include the squaring operation 02 and are related
to it by “reducing” the value of 02(u) in a certain way. For this reason, the
operations Sq1 are also called the reduced squares.
For the remainder of this section we make the assumption that all
modules are over Z2 and all homology and cohomology modules have coeffi-
cients Z2. The Steenrod squares, or reduced squares, {Sq’Jjj.o are additive
cohomology operations
Sqh №(X,A)	№+t(X,A)
SEC. 9 THE STEENROD SQUARING OPERATIONS	271
defined for all q such that
(g) Sqo = 1.
(b)	If deg и = q, then Sqr'u = и и.
(c)	If q > deg и, then Sq'/u = 0.
(d)	If и £	and v £ H*(Y,B) and {X x В, A X is an exci-
sive couple in X X Y, the following Cartan formula is valid:
Sqk(u X o) = 2 Sqlu X Sqh)
i+j=k
The above properties characterize the cohomology operations Sq1. We
shall not prove the uniqueness1, but shall content ourselves with their con-
struction. First we establish a formula equivalent to the Cartan formula.
4	lemma If u, v EH* (X,A), then
Sqk(u о o) = S Sqlu о Sqh)
i+j=k
proof Since и v = d* (и X v), where d: (X,A)	(X,A) X (X,A) is the
diagonal map, this follows from the Cartan formula and functorial properties
of Sq1. 0
For any chain complex Clet T: С ® C -+ С ® C be the chain map
interchanging the factors [T(ci ® c2) = c2 ® Ci is a chain map over Z2].
5	lemma There exists a sequence (Dj}j^0 of functorial homomorphisms
Dp. Д(Х) Д(Х) ® Д(Х) of degree / such that
(a) Dq is a chain map commuting with augmentation.
(If) For j > 0, ar>j + DjS +	= 0.
If {/!;} and {jD;} are two such sequences, there exists a sequence {£j}j>o of
functorial homomorphisms Ер Д(Х) Д(Х) ® Д(Х) of degree j such that
(c) Eo = 0.
(d) For j > 0,	+ E3+i8 + Ej + TEj + Dj + Dj = 0.
proof We use the method of acyclic models. Let R be the group ring of Z2
over the field Z2. We regar d R as the quotient ring of the polynomial ring Z2(t)
modulo the ideal generated by the polynomial t2 + 1 = 0. Thus the elements
of R have the form a + ht, where a and b £ Z2.
Let Z2 be regarded as a trivial R module (that is, every element of R in-
duces the identity map of Z2) and let C be the free resolution of Z2 over R in
which Ce is free with one generator d,, for all q > 0 and which has boundary
operator c(df;) = (1 + f)dQ for q > 1 and augmentation e(c?o) = 1. The
functor which assigns to a space X the chain complex Д(Х) ® C is augmented
Z2
and free over R with models	and basis ® d)}. We regard
1 For a proof see N. Steenrod and D. Epstein, Cohomology operations, Annals of Mathematics
Studies No. 50, Princeton University Press, Princeton, N.J., 1962.
272	PRODUCTS CHAP. 5
Д(Х) ® Д(Х) as a chain complex over R, with t acting on Д(Х) ® Д(Х) in the
same way T does. Then Д(Х) ® Д(Х) is augmented and acyclic, with models
{ До} f/ >о. It follows from theorem 4.3.3 (which is valid for chain complexes
over R) that there exist natural chain maps т: Д(Х) ® С Д(Х) ® Д(Х) pre;
serving augmentation, and any two are naturally chain homotopic.
A map т: Д(Х) ® С Д(Х) ® Д(Х) of degree 0 corresponds bijectively
to a sequence of maps
Df. Д(Х) Д(Х) ® Д(Х) j > 0
of degree j such that D3(c) = t(c ® dj). Then т is a chain map preserving
augmentation if and only if {Dj} satisfies (a) and (b). Thus there exist families
{Dj} satisfying (a) and (b), and any such family corresponds to some t.
Similarly, a map El: Д(Х) ® С Д(Х) ® Д(Х) of degree 1 corresponds
bijectively to a sequence of maps
Ef. Д(Х) Д(Х) ® Д(Х) / > 0
of degree j such that Eo = 0 and Ej(c) = H(c ® d;i) for j > 1. Then El is a
chain homotopy from т to t' if and only if {Ej} satisfies (c) and (d) for the
sequences {Dj} and {Dj} corresponding to т and r', respectively. Thus,
if {Dj} and {Dj} are two sequences satisfying (a) and (b), there is a sequence
{Ej} satisfying (c) and (d). 
Given a sequence {Dj}j>0 as in lemma 5, we define homomorphisms
Dj*: Hom (Д(Х) X Д(Х), Z2) Hom (Д(Х), Z2)
of degree — j by (Dj*/)(a) = /(Dja) for 0 E Дд(Х) and f E Hom (Д(Х) ®
Д(Х), Z2). If с* E Hom (Де(Х), Z2) is a q-cochain of Д(Х), then
с* ® с* E Hom (Д(Х) ® Д(Х), Z2),
and we define a (q + i )-cochain Sq!c* £ Hom (Д(Х), Z2) by
„ . f 0	i > q
Sq C‘ ~ lD*_i(c* ® c*) i<q
Let us now establish some properties of these cochain maps. It will be
convenient to understand Dj = 0 for j < 0. Then lemma 5b holds for all /.
6 If c* is zero on Д(А) for some А С X, then Sqlc* is zero on Д(А).
proof This follows from the naturality of {Dj}, and hence of {Sq’}. 
7 If 8c* = 0, then 8{Sqlc*) = 0.
proof This is trivial if i }> q. If i < q, we have
8(Sqlc*)(o) = D^i{c* ® c*)(8u) = (c* ® e*)(Dcj iB0)
= {c* ® c*)(aDe_ja) + (c* ® c*){Dq^0 + TDg^o)
= (c* ® c*)(8Dg_i0)
•sEC* 9 THE STEENROD squaring operations	273
the last equality because (c* ® c*)(Tc) = (c* ® c*)c for any c £ A(X) ® Д(-Х).
Then we have
(с* ® с*)(ЭСе_;р) = S(c* ® c*)(De„jo) = 0
because 8c* =0. 
jj If c* = 8c*, then Sqlc* = 8[D^__fc* ® c*) + E>*_,i(c* ® c*)].
proof If i > q, both sides are zero. If i < q, we have
(Sq'lc* )(a) = (Sc* ® 8c* )(a) — 8(c* ® Sc* )(Д; ;(о))
= (c* ® 8c*)(Dg_i3o + De_{_ia + TDe_i„ia)
= D*_^c* ® с*)(до) + S(c* ® c*)(De_t_1a)
the last equality because
(c* ® Sc*)(De_i_1a + ZDe_.;_ia) = (c* ® Sc* + Sc* ® c*)(De_i_ia)
We also have
S(c* ® с*)(Гд_{_1о) = (c* ® c*)(Pfr_j^ic<T + De_{_20 + TDg^zp)
= Dj_j-i(c* ® c*)(8a)
The result follows by substituting this into the right-hand side of the other
equation. 
9 Ifc* and c% are cocycles, then
Sq\c^ + cf) = Sq,lc* + Sq'h* + SD*_i+i(ct ® c*)
proof If i f> q, both sides are zero. If i < q, we have
Sq^c? + cf)(o) = [(of + c*) ® (c? + c*)](De_jo)
= (c? ® c* + c* ® c*)(n,; io) + (c* ® c*)(n,7 io + TDg^)
= (Sqtcf + 8</с*)(<т) + (ej ® c^)(De_i+1aa + aDe_i+ia)
= [Sq’c* + Stfc* + SD*_i+1(ct ® c*)](a)
the last equality because S(c^ ® c^) — 0. 
It follows that there is a well-defined functorial homomorphism
Sqh №(X,A)	№+\X,A)
defined by Sqi{c* } = {Sqlc* }. If {Dj} is another system satisfying lemma 5a
and 5b, and Sq'j is defined using this system, let {Ej} satisfy 5c and 5d.
If c* is a (/-cocycle of Д(А)/Д(А), then
(С* ® С*)(Ре_4О + Dq_i(7 + Eq+1^ido) = 0
Therefore
Sqic* + Sq'lc* + 8Е^+1_^с* ® c*) = 0
showing that Sql{c* } = Sq'l{c* }. Hence Sql is uniquely defined independent
274
PRODUCTS CHAP. 5
of the particular choice of {Pj}. We shall now verify that these cohomology
operations {Sq1} satisfy the axioms characterizing the Steenrod squares.
I© theorem The additive cohomology operations {Sq1} defined above
satisfy conditions (a) to {d}, inclusive, on page 271.
proof Let С(Дв) denote the oriented chain complex of the simplex Д(Де).
Over Z2 there is a unique orientation for each simplex, and С(Д'') is isomorphic
to the subcomplex of Д(Де) generated by the singular simplexes which are the
faces of Д®. We regard С(Д«) as imbedded in Д(Дв) in this way. С(Д«) is
acyclic, and if Л: Др —> Д« is a p-face of Де, then Д(Л)(С(Др)) С С(Де). It
follows that a sequence {Pj} can be found satisfying lemma 5a and 5b such
that Pj(£e) £ С(Д®) ® С(Д«) for all q and /. For such a sequence, Dffif) = 0 if
j > q (because [С(Де) ® С(Де)]8 = 0 if s > 2q), whence Pj(a) = 0 for any
о £ Дв(Х) with q < /.
We now shall prove — ^q® ^q for all q by induction on q. If q = 0,
then Po(£o) must have nonzero augmentation, by lemma 5a. The only element
of С(Д°) ® С(Д°) with nonzero augmentation is £0 ® £o- Therefore Po(£o) =
& ® £0. Assume that q > 0 and Pe~i(£e_i) = £e_i ® £e_i. Either Dq(£q) —
f. ® %q or = 0. In the latter case, by lemma 5b, we have [because
Pe(3^) = 0]
De_i(£Q) + TO5_i(^) = 0
From this it follows that Pf;-i(^f;) = 2 a.{fq ® £q(i> +	® £e), where «; = 0
or fl; = 1. This is a contradiction, because
+ TDq_2(gq) =	+ Pf;i(c^,;)
and ® has a coefficient of 2м- + 1 = 1 on the right and a coefficient
of 0 on the left.
Therefore, with this choice of {Dj} we have Dfo) = a ® a if a has
degree q. Then
(Sq°c* )(a) = (c* ® c* )(PQ(a)) = [c* (a)]2
Because a2 — a for a £ Z2, we see that Sq°c* = c*, and so Sq° = 1, showing
that condition (я) is satisfied.
By definition, Dq is a chain approximation to the diagonal. Therefore
(P§(c* ® с*)} = {с* }	{c* } for any cocycle c*, and so Sq^u = и о и if
deg и = q. Hence condition (b) is satisfied. From the definition of Sq1 condi-
tion (c) is trivially satisfied.
It merely remains to verify the Cartan formula. Let {Pj} be a system
satisfying lemma 5a and 5b and let {Df} be the collection of homomorphisms
for Д(Х). On the category of pairs of topological spaces X and Y the system
{P^xy} and the system [T Tl:Dfi ® Pjy}, where
T: [Д(А) ® Д(А)] ® [Д(¥) ® Д(У)]	[Д(А) (x) Д(У)] (x) [A(JQ (x) Д(У)]
interchanges the second and third factors, both satisfy lemma 5a and 5b.
SEC. 9 THE STEENROD SQUARING OPERATIONS
275
Then a system {E/AXY} satisfying 5c and 5d with respect to them can be
defined by the method of acyclic carriers. Therefore the system
{T 2 T^D^^DjY}
i+j—k
can be used to define Sqk(u X v) for и £ H*(X,A) and v £ H*(Y,B). Let c|
I be a p-cochain of X, c2 a (/-cochain of Y, <ii a singular //-simplex of X with
i p < P 2p, and 02 a singular (/'-simplex of Y with q < q’ < 2q, where
j p' 4- q' — p + q + k. Then
Sqk(c* ® cf)(oi ® o2)
= [(c? ® c*) ® (c? ® c*)](IXXY ® a2))
= [(4 ® 0*) ® (c* ® €*)]( + _S k Tp+rkDiX0i ® D/o2)
|	= [(Cl ® СТ )(-^2p-p'Ul)][(c* ® C* )(E>2,;._f;-<T2)|
= (Sqp'-pcf ® Sqi'~ic^)(0i ® a2)
Letting <T| and o2 vary, we see that Sqk(cf ® cf) = 2i+j-k Sq'c* ® Sqic%.
j Passing to cohomology and using the natural homomorphism
| H*(X,A) ® H*(Y,B)	Н*([Д(Х)/Д(А)] ® [Д(Т)/Д(В)]) ~H*((X,A) x (Y,B))
i, sending the tensor product to the cross product, we obtain
Sqk(u X v) = 2 Sq1 X Sq’h:
i+3=k
’ showing that condition (d) is satisfied. 
|	11 example Observe that, by condition (b) on page 271 and theorem 5.8.5,
j	Sg2: H2(P2(O)	H4(P2(C))
is nontrivial. If и £ №(P2(C)) is such that Sq2u =/= 0 and v £ H^lj) is the
к nontrivial element, it follows from condition (d) that
»	Sq2(u x v) = Sq2u X v
and Sq2: H3(P2(C) X (1,1))	H5(P2(C) X (U)) is nontrivial., Let X be the
| unreduced suspension of P2(C) obtained from P2(C) X 1 by identifying
Pz(C) X 0 to one point Xq and B2(C) X 1 to another point %i. There is then a
1 continuous map
f: P2(C) X (I,i)	(X, x0 U Xi)
inducing an isomorphism
f*: №(X, xo U Xi) ~ №(P2(C) x (ij))
for all q. Therefore Sq2: H:t(X) H5(X) is nontrivial. Let Y be the one-point
union of S3 and S5. An easy computation shows that X and Y have isomorphic
homology and cohomology for any coefficient group, and even isomorphic
cup and cap products. However, because Sq2: H:j,(X)	Hr,(X~) is nontrivial
276
PRODUCTS CHAP. 5
and Sq2-. №(Y)	H5(Y) is trivial, X and Y are not of the same homotopy type.
Further applications of the Steenrod squares will be given in the next
chapter and in Chap. 8.
It is obvious that cohomology operations of the same type can be added
and that the sum is again a cohomology operation of the same type. Given
cohomology operations 0 of type (p,q; G,G') and 0’ of type (q,r; G',G"), then-
composite 0'0 (of natural transformations) is a cohomology operation of type
(p,r; G,G"). In this way the Steenrod squares can be added and multiplied,
and they generate an algebra of cohomology operations called the modulo 2
Steenrod algebra.
In this algebra the following Adem relations1 hold:
W = 0 2/21 (^Sq^Sq* 0 < i < 2/
where [i/2] denotes as usual the largest integer <i/2 and the binomial coeffi-
cient is reduced modulo 2. Using these relations, it is easily shown that
the algebra of cohomology operations generated by Sq1, where i is a power of 2,
contains all the Steenrod squares. This implies that the only spheres that can
be H spaces have dimension 2й — 1 for some n. By using deeper properties
of the algebra of cohomology operations Adams2 has shown that the only
spheres that can be H spaces are the spheres S°, S1, S3, and S7. Each of these
is, in fact, an H space, with multiplication defined to be the multiplication of
the reals, complex numbers, quaternions, or Cayley numbers, respectively,
of norm 1.
EXERCISES
A DISSECTIONS
Let C be a graded module over II. A filtration (increasing) of C is a sequence {FSC} of
graded submodules of C such that FSC C FS+1C for all s. It is said to be bounded below
if for any t there is s(t) such that FS(()C( = 0, and it is convergent above if U FSC = C.
1 If {FSC} is a filtration of a chain complex C by subcomplexes, there is an increasing
filtration of H* (C) defined by Ffifi. (C) = im [H* (FSC) —> If, (C)]. If the original filtration
on C is bounded below or convergent above, prove that the same is true of the induced
filtration on Ifi (C).
An increasing filtration {FSC} of a chain complex C by subcomplexes is called a dissec-
tion if it is bounded below, convergent above, and if
Hq(Fs+1C,FsC) =0 q Ф s + 1
1	See J. Adem, The iteration of the Steenrod squares in algebraic topology, Proceedings of the
National Academy of Sciences, USA, vol. 38, pp. 720-726, 1952, or H. Cartan, Snr I’iteration
des operations de Steenrod, Commentarii Mathematici Helvetici, vol. 29, pp. 40-58, 1955.
2	See J. F. Adams, On the non-existence of elements of Hopf invariant one, Annals of Mathe-
matics, vol. 72, pp. 20-104, 1960.
exercises	277
Given a dissection {FSC} of a chain complex C, the sequence
----->hq+i(fq+1c,fqc)Aндал-iC)Ah^f^cj^zC)
is a chain complex C, called the chain complex associated to the dissection.
2	If C is the chain complex associated to a dissection of C, prove that 77,. (C) ~ /7,. (Cj.
1	3 Let {FSC} be a dissection of a free chain complex C by free subcomplexes such that
Fs+jC/FsC is free for all s. If C is the chain complex associated to the dissection, prove
that C and C have isomorphic homology and cohomology for all coefficient modules.
, [Hint: The freeness hypotheses ensure that the universal-coefficient theorems hold for
both homology and cohomology. Then {FSC ® G} is a dissection of C ® G whose asso-
ciated chain complex is isomorphic to C ® G. Dual considerations apply to {Hom (FSC,G)}
and Hom (C,G)J
Д block dissection of a chain complex C is a collection of subcomplexes {Ej«}, called
blocks, where q varies over the set of integers and for each q, j varies over a set Jtl, such
1 that if FSC is the subcomplex of C generated by {L/'},;.-s and if Ef = Ef П Fs_iC, then
Ef (T Ef C Ff/_jC / к
Ef — 0	q sufficiently small
U FSC = C
4 If {Ej®} is a block dissection of a chain complex C, prove that the corresponding
collection {FSC} is a dissection of C whose associated chain complex C is free with
generators for CQ in one-to-one correspondence with the set
A block dissection of a simplicial complex К is a collection of subcomplexes {Kf),
where q varies over the set of integers and for each q, j varies over some indexing set JQ,
such that if FSK = UJSS Kf and Kf = Fs_iK (T Kf, then
,	KfFKf^Fq^K j^k
Kf = 0	q sufficiently small
U FSK = К
5 If {Kj«} is a block dissection of K, prove that {C(I</')} is a block dissection of the
chain complex C(K) by free subcomplexes. If C is the chain complex associated to the
dissection, prove that C and C(K) have isomorphic homology and cohomology with any
coefficient group.
IB HOMOLOGY MANIFOLDS
A homology n-manifold is a locally compact Hausdorff space X such that for all x £ X,
H^X, X — x) — 0 for q 7^= n and either Hn(X, X — x) — 0 or Hn(X, X — x) ~ Z. Further-
more, if the boundary X of X is defined to be the subset
X = {x £ X | H„(X, X — x) = 0}
then we also assume that X — X is a nonempty connected set. If X = 0, X is said to be
without boundary.
278
PRODUCTS CHAP. 5
1	If X is a homology n-manifold and Y is a homology m-manifold, prove that X x Yis
a homology (fl + m)-manifold whose boundary equals X X Y U X X
2	Prove that if a polyhedron is a homology n-manifold, its boundary is a subpolyhedron.
3	If К is a simplicial complex triangulating a homology n-manifold X, prove that К is
an n-dimensional pseudomanifold and К triangulates X. (A polyhedral homology n-mani-
fold is said to be orientable or nonorientable, according to whether any triangulation of
it is orientable or nonorientable as a pseudomanifold.)
4 Let (K,K) be a simplicial pah' triangulating a polyhedral homology n-manifold (X,X)
and let L be the subcomplex of the barycentric subdivision K' consisting of all simplexes
disjoint from K'. If is a q-simplex of К — К, let E"^i(si) be the subcomplex of L gen-
erated by the star of the barycenter b(s«). Prove that (En~i(si)}ifi ek-k is a block dissec-
tion of L and that if C is the chain complex associated to this block dissection, then C
has homology and cohomology isomorphic to that of X — X. (Hint: let st — si * B(si\
where B(si) is a subcomplex of K. Then £"_®(s«) = b(si) * (B(s«)]' and E"~i(si) = (B(s«)]',
Also note that |L| is a strong deformation retract of |K| — |lt|.)
5 Lefschetz duality theorem, Let (K,K) be a simplicial pair triangulating a compact
homology n-manifold (X,X) and assume that z £ Hn(K,K) is an orientation of K. For each
q-simplex si of К — £ let z(s«) £ H„(K, К — st s«) be the image of z, and assume an
orientation ai of si chosen once and for all. Then z(si) = ai * z(ai), where z(ai) £
Wn_5_i(B(s«)). Define я'(о«) £ Hn~q(En~i(si),En~i(si')) to correspond to z(ai) under the
isomorphisms
Hn_5„i(B(s«)) ~ Hn^1(&-i(si)') ~ Hn^(E»~i(si),Ei~i(si))
Let <p: Hom (С,,(К,К ), G) —> Cn-q ® G be the homomorphism defined by
<p(u) = 2 z!(ai) ® u(ai) и £ Hom (Ce(K,K), G)
cr'f
Prove that <p is an isomorphism and that it commutes up to sign with the respective co-
boundary and boundary operators. Deduce isomorphisms
H"(X,X; G) Hn~Q(X - X; G) and Hq(X,X; G) ~ H"~«(X — X; G)
C PROPERTIES OF THE TORSION PRODUCT AND EXT
In this group of exercises all modules will be over a principal ideal domain R.
I Prove that the torsion product is associative.
2	If A, B, and C are modules, prove that
A®(B*C}®A*(B®C)
is symmetric in A, B, and C.
3	Given a module A and a short exact sequence of modules
0 B’ В В" 0
prove there is an exact sequence
0 Hom (A,B') -» Hom (A,B) Hom (A,B") ->
Ext (A,B') -» Ext (A,B) —> Ext (A,B") -a 0
4	Given a short exact sequence of modules
0 A’ -> A A" 0
and given a module B, prove there is an exact sequence
exercises	279
О —> Hom (А”,В)	Hom (A,B) —> Hom (A',B) —>
Ext (A",B) -» Ext (A,B) Ext (A',B) -> 0
jf C = {Cj} and C* = {€>} are graded modules, there is a graded module
Hom (C,C*) = {Horn® (C,C*)J, where Hom’ (C,C*) = X i+j=g Hom (Ci,C') [thus an
element of Hom" (C,C*) is an indexed family {<p,-: C, —> C"'},-]. Similarly, there is
a graded module Ext (C,C*) = {Ext” (C,C*)}, where Ext® (C,C*) = Xi ,)-,, Ext
5	If C is a chain complex and C* is a cochain complex, prove that Hom (C,C*) is a
cochain complex, with
(8<p)ij =	° Si + (- 1)«8Н1 о	v =	€ Home(C,C*)
and that Ext (C,C*) is a cochain complex with
(ОДЛ = Ext (Si,!)№_!,j) + (-!)< Ext	$ = №,j} € Exte (C,C*)
6	If C is a chain complex and C* is a cochain complex such that Ext (C,C*) is
acyclic, prove that there is a split short exact sequence
0 Ext"-1 (H#(C',H* (C*)) H"(Hom (C,C*)) -> Hom" (H* (C*))	0
7	If C and C are chain complexes and C* is a cochain complex, prove that the expo-
nential correspondence is an isomorphism
Hom (C, Hom (C',C*)) ~ Hom (С ® O', C*)
В Let (X’,A) and (Y,B) be topological pairs such that {X x В, A X Y) is an excisive
couple in X X Y. For any module G prove that there is a split short exact sequence
0 -a Ext®-1	№((X,A) X (Y,B); G) Hom’	-> 0
where H* = H* (X,A; B) and H* = H* (Y,B; G).
D CATEGORY
A topological space X is said to have category < n, denoted as cat X < n, if X is the
union of n closed sets, each deformable to a point in X.
1 If X is a connected polyhedron of dimension n, prove that cat X < n + 1.
2 If X is any space, prove that cat (SX) < 2.
3 If cat X < n, prove that all n-fold cup products of positive-dimensional cohomology
classes of X vanish.
4 Prove that cat Р» = n + 1 and cat (P®> X    X P"*) — ni +    +	+ 1.
E HOMOLOGY OF FIBER BUNDLES
1 Let p: E —> В be a fiber-bundle pan, with total pah- (Е,Ё) and fiber pair (F,F), such
that Hs (Р,Ё) = 0. Prove that H* (E,E) = 0.
2 If p: E —> В is a fiber-bundle pair over a path-connected base space B, prove that a
homomorphism 0-.	R) —> Н*(Е,Ё; R) is a cohomology extension of the fiber if
and only if for some b £ В the composite
Н*(ВД В) 4 Н*(Е,Ё; В) -> Н*(ЕЬ,ЁЬ; В)
is an isomorphism.
3 Let р: Е В be a fiber-bundle pah- over a path-connected base space. If for some
b G В the pair (Eb,Eb) is a weak retract of (E,E), prove there exists a cohomology exten-
sion of the fiber.
280
PRODUCTS CHAP. 5
4 Prove that a q-sphere bundle £ with base space В is orientable over R if and only if
for every map a: S1 —» В the bundle a* (f) is orientable over R.
5 Prove that a q-sphere bundle £ is orientable over Z if and only if there is an element
U C №+1(Е(,Ёр, Z4) whose image in №+1(££,Ё£; Z2) is the unique orientation class of J
over Z2. (Hint: Show that there is such an element U if and only if for every closed path
и in the base space, 7r[w] * is the identity map of Нг^ЕмьЁиаУ, Z4), and this, in turn, is
equivalent to the condition that 7i[w] * is the identity map of №+1(Ea<1)Ж(1>; Z).)
6 Let £ be a q-sphere bundle with base space В and with orientation class
Uj E	R) and let g №+1(B;R) be the corresponding characteristic class.
Prove that Ф f (flj) — U( о U(.
7 Prove that the characteristic class fl£ of an even-dimensional sphere bundle £ oriented
over Z has order 2.
S Let £ be a sphere bundle oriented over R, with base space B. If £ has a section in ££j
(that is, if the map pp Ё£ —> В has a left inverse), prove that its characteristic class
flj = 0. [Hint: Any two sections В —> Ee are homotopic in Ee. Since Ё£ is the mapping
cylinder of pp Ёч —> B, there is an inclusion map I:: В CE, which is a section. There is
a section in Ё£ if and only if к is homotopic to a map В —> Ё£, in which case the composite
№+1(£{,£{; R) Д №+i(£{;R)	№+i(B-R)
is trivial, because p* -1 — k* .J
F HOPF ALGEBRAS
1	Prove that the tensor product of connected Hopf algebras is a connected Hopf algebra.
2	If В is a connected Hopf algebra of finite type over a field R, prove that
B* = Hom (B;R) is a connected Hopf algebra over R whose product and coproduct are
dual, respectively, to the coproduct and product of B.
3	Let В be a connected Hopf algebra over a field of characteristic p and assume
that В has an associative and commutative product and is generated as an: algebra by a
single element x of positive degree. Prove that if deg x is odd and p =£ 2, then В = £(%),
and if deg x is even or p = 2, then either В = Sdeg X(x) or В = Tdeg X^(x), where h = pk
for some к > 1.
4	Let В be a connected Hopf algebra of finite type over a field of finite characteristic
p 0 and assume that В has an associative and commutative product. If the pth power
of every element of positive degree of В is 0, prove that В is the tensor product of exte-
rior algebras (with generators of odd degree if p 7^ 2) and truncated polynomial algebras
of height p (with generators of even degree if p 7^= 2).
G THE BOCKSTEIN HOMOMORPHISM
1 Show that the Bockstein homomorphism in homology (or cohomology) anticommutes
with the boundary homomorphism (or coboundary homomorphism) of a pair.
For any prime p let be the Bockstein homomorphism in either homology or
cohomology for the short exact sequence of abelian groups
0 —> Z, —> Zp2 —> Zp —> 0
Let j8p be the Bockstein homomorphism for the short exact sequence
0^Z^Z^Z(,->0
exercises	281
where \(n) = pn and pp is reduction modulo p.
2 Prove that pp = (рД, ° (}p.
3 prove that ftp ° flp = 0.
4 ( Prove that J3p(u v) = Pp(u) о v + ( — 1)d,L«" u о /3p(v).
5 Prove that Sq2il1 = /?2 ° Sq2i for i > 0. [Hint: Show that there exist functorial
homomorphisms (Dj};>o, with Dj of degree / from the integral singular chain complex A(A’)
to Д(Х) ® Д(А'), such that Do is a chain map commuting with augmentation and
SD2j_i + D2;18 = D2j - TD2j	j > 0
8D2y — D2;0 = D2; j + TD2j-i	j > 0
where T(oi ® a2) = (-l)aeg’i ® ai.]
6 Let £ be a q-sphere bundle and let U( E №+1(Е£,Ё£; Z2) be its unique orientation
over Z2. Prove that £ is orientable over Z if and only if /hill:) = 0.
И STIEFEL-WHITNEY CHARACTERISTIC CLASSES
Let £ be a q-sphere bundle, with base space B, and let U( E №+1{Е(,Ё(-, Z2) be its orien-
tation class over Z2. The ith Stiefel-Whitney characteristic class Wj(lf) € H>(B;Z2) for
i > 0 is defined by
Ф ?©«(£)) = Sqi(D{)
I Let/: В' В be continuous. Prove that(w,(£)) =
2 If£ is a product bundle, prove that w,(£) = 0 for I > 0.
3 Prove the following:
(fl) w0(£) is the unit class of //°(B;Z2).
(b)	Дг(т2;(£)) = w2i+i(£) + Wi(£) о w2i© for i > 0.
(c)	If £ is a q-sphere bundle, then w,© = 0 for i > q 4- 1, and we+i© is the
characteristic class of £ over Z2.
(d)	£ is orientable over Z if and only if wi© = 0.
If £ is a q-sphere bundle over В and £' is a q'-sphere bundle over B', their cross product
£ X £' is a (q + q' + l)-sphere bundle with Efxf' = Ef X Ef-, Ё£х£- = Ef X Ef- U Ef x Ee-
and p{x{- = p£ X pe.
4 If U( C H«+1(Ej,E{; Z2) and Ц- E Н«'+1(Е£-,Ё£-; Z2) are respective orientation classes,
prove that
u( X U(. e	z2)
is the orientation class of £ x £'•
5 Prove that wA.(£ x £') = 2;+;=* w;© X w;(£').
If £ and £' are sphere bundles with the same base space B, their Whitney sum
£ ф £' is the sphere bundle over В induced from £ X £' by the diagonal map В В X В.
6 Whitney duality theorem. Prove that
wa-(£ ©£') = .?, ^t©	my(£')
I HOMOLOGY WITH LOCAL COEFFICIENTS
If а:	X is a singular q-simplex of X, with q > 1, let be the path in X obtained
by composing the linear path in Д« from Vo to tq with a. Given a local system Г of
282	PRODUCTS CHAP. 5
R modules on X, define Д,/Х;Г) to be the R module of finitely nonzero formal sums S a?o'
in which a varies over the set of singular q-simplexes of X and a„ g Г(о(и0)) is aero
except for a finite set of a. For q > 0 define a homomorphism 0: Д,/Х; Г) Д,;__х(Х;1) by
Э(па) = о S (—+ Г(а>о)(а)о(0)
1 Prove that Д(Х;Г) = (Д,;(Х;Г), 3) is a chain complex which is free (or torsion free)
if Г is a local system of free (or torsion free) R modules, and if А С X, show that
Д(А; Г | A) is a subcomplex of Д(Х;Г).
The homology of (X,A) with local coefficients Г, denoted by	Г), is defined
to be the graded homology module of Д(Х,А; Г) = Д(Х;Г)/Д(А; Г | А).
2 For a fixed ring R let 6 be the category whose objects are topological pairs (X,A),
together with local systems Г of R modules on X, and whose morphisms from (X,A) and
Г to (Y,B) and P are continuous maps f: (X,A) —> (Y,B), together with indexed families
of homomorphisms {fx: Г(.т) —> F’(f(x))}xex. Prove that //.(А’, А; Г) is a covariant functor
from <3 to the category of graded R modules.
3	Exactness. Given А С В С X and a local system Г of R modules on X, prove that
there is an exact sequence
------> Ha(B,A; Г | B) —» Hq(X,A; Г)	Hg(X,B; Г) ->	Г | B)
4	Excision. Let Xj and X2 be subsets of a space X such that Xx U X2 = int Xx U int X2.
For any local system* Г of R modules on X prove that the excision map /1 from
(X1; Xi Fl X2) and Г | Xi to (A’i U X2, X2) and Г | (Xx U X2) induces an isomorphism
n*  Н*(ХЪ Xi n X2; Г I Xi) H*(X1 U X2, X2; Г | (Xx U X2))
5	Two morphisms f and g in <3 from (X,A) and Г to (Y,B) and P are said to be
homotopic in Gif there is a homotopy F: (X,A) X I	from f to g and an indexed
family of homomorphisms {F(a.>(): Г(х) —> Г'(Е(.т,/)}(..,. ()гЛх, such that F{xfi} — fx and
F^j) = gx. Prove that homotopy is an equivalence relation in the set of morphisms from
(X,A) and Г to (Y,B) and P and that the composites of homotopic morphisms are
homotopic (so that the homotopy category of G can be defined).
6	Homotopy. If f and g are morphisms from (X,A) and Г to (Y,B) and P and J is
homotopic to g in G, prove that/* = g* : H*(X,A; Г) —> H*(Y,B; F').
7	If Г and P are local systems of R modules on X, there is a local system Г ® P on
X with (Г ® Г')(х) = Г(х) ® Г'(х) and (Г ® Г')(ы) — Г(«) ® Г'(со). In case P is the
constant local system equal to G, then prove that
Д(Х,А; Г ® G) ~ Д(Х,А; Г) ® G
Deduce a universal-coefficient formula for homology with local coefficients.
8	If Г and P are local systems of R modules on X and Y, respectively, let Г X P =
p* (Г) ® p'* (P) be the local system on X X Y, where p* (Г) and p' * (P) are induced
from Г and P, respectively, by the projections p: X x Y —> X and p': X x Y —> Y.
Prove that there is a natural chain equivalence of Д(Х;Г) ® A(Y;P) with Д(Х X Y; Г X P)-
Deduce a Kunneth formula for homology with local coefficients.
J COHOMOLOGY WITH LOCAL COEFFICIENTS
If Г is a local system of R modules on X, define Д«(Х;Г) to be the module of functions <p
assigning to every singular q-simplex 0 of X an element <p(o) £ F(a(oo))- Define a homo-
morphism 8: Д''(Х; Г) Д«+1(Х;Г) by
EXERCISES
283
,	(S<p)(a) = o ,2+1(-1)*<р(о<;)) + Г(«о-1)(<р(о<0)))
| Prove that Д*(Х;Г) = (Д«(Х;Г), 8} is a cochain complex and that if А С X, the
restriction map Д* (А’; Г) —> Д* (А; Г | A) is an epimorphism.
The cohomology of (X,A) with local coefficients Г, denoted by H*(X,A-> Г), is
i defined to be the graded cohomology module of
Д* (X,A; Г) = ker [Д* (Х;Г) -н> Д* (А; Г | A)]
2 For a fixed ring R let (S' be the category whose objects are topological pairs (X,A),
together with local systems Г of К modules on X, and whose morphisms from (X,A) and
1 Г to (Y,B) and Г are continuous maps f: (A’, A) —> (Y,B), together with indexed families
of homomorphisms (f!r:	—>	Prove that H*(X,A; Г) is a contravariant
functor from (? to the category of graded R modules.
3 Prove that the cohomology with local coefficients has exactness, excision, and homot-
opy properties analogous to those of the homology with local coefficients.
j 4 If Г is a local system of R modules on X and G is an R module, there is a local sys-
: tern Hom (P,G) of R modules on X which assigns to x £ X the module Hom (F(x),G).
Prove that
Д*(Х,А; hom (F,G)) ~ Hom (Д(Х,А; Г), G)
। Deduce a universal-coefficient forttiula for cohomology with local coefficients.
I Let £ be a (/-sphere bundle with base space В and let F£ be the local system on В
such that I» = Hq+i(Eb,Eb). Let pf (l\) be the local system on E( induced from Г£ by
j pp Ee B. A Thom class of £ is an element U£ £ №+1(E£,E£; pf (F{)) such that for every
j b С В the element
|	U( I (Еь,Ёь) e №+1(ЕьЁъ; pt (Г() I Ef) = №^(ЕЬ,ЕЬ; HQ+1(Eb^b))
I corresponds to the identity map of Н9+1(Еь,Ёь) under the universal-coefficient isomorphism
j	ЫАЧЕьЁы Hq+1(Eb,Eb)) Hom (Н9+1(ЁЬД), Н9+1(ЕЬД))
•j 5 Prove that every (/-sphere bundle has a unique Thom class. (Hint: Prove the result
i first for a product bundle, and then use Mayer-Vietoris sequences to extend the result to
j arbitrary bundles.)
1 6 Let £ be a (/-sphere bundle with a base space В and let ty be its Thom class. If Г is
1 any local system of abelian groups on X, prove that the homomorphism
j	Ф£: Ни(Е£,Ё£; p* (Г)) Hn^(B- Г£ ® Г)
Isuch that Ф£(й) = рй. (Ц z), where Ut z is an element of Н?1_,;1(Е; p* (Г£ ® Г)), is
an isomorphism. If В is compact, prove that the homomorphism
Ф f; №(В;Г)	Нг+о+1(Е(,Ё(; p*(F ® Г£))
j
such that Ф f (v) = p* (v) о 17£ is an isomorphism.
CHAPTER SIX
GENERAL COHOMOLOGY
THEORY AND DUALITY
I
\
\
IN THIS CHAPTER WE CONTINUE THE STUDY OF HOMOLOGY AND COHOMOLOGY
functors, with particular emphasis on the homological properties of topological
manifolds. For this important class of spaces we shall establish the duality
theorem equating the cohomology of a compact pair in an orientable manifold
with the homology, in complementary dimensions, of the complementary pair.
The cohomology which enters in the duality theorem is the direct limit
of the singular cohomology of neighborhoods of the pair, with the family of
neighborhoods directed downward by inclusion. For the case of a closed pair
in a manifold, the resulting direct limit depends only on the pair itself. In fact,
it is isomorphic to the Alexander cohomology of the pair, Alexander cohomol-
ogy being another cohomology theory distinct from the singular cohomology.
Thus we are led to consider Alexander cohomology. We define it and
prove that it is a cohomology theory in the sense that it satisfies the axioms
of cohomology theory. We also establish the special properties of tautness and
continuity possessed by this theory and not generally valid for singular coho-
mology. For deeper properties of the Alexander theory we introduce the
cohomology of a space with coefficients in a presheaf. The definition of this
285
286
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP.,6
cohomology involves a Cech construction, using nerves of open coverings.
We use general properties of this cohomology to prove that for paracompact
spaces the Alexander and Cech cohomologies are isomorphic, and with this
result establish universal-coefficient formulas for the Alexander cohomology
of compact pairs and for the Alexander cohomology with compact supports
of locally compact pairs.
The cohomology of presheaves is also applied to compare the singular
and Alexander cohomology theories, and we prove that they are isomorphic
for manifolds. Another application of the cohomology of presheaves is in the
proof of the Vietoris-Begle mapping theorem. The final topic is a discussion of
homological properties of one manifold imbedded in another.
In Sec. 6.1 we define the slant product as a pairing from the cohomology
of a product space and the homology of one of its factors to the cohomology
of the other factor. This furnishes the map that is the isomorphism in the
duality theorem for manifolds, and the duality theorem itself is proved in
Sec. 6.2. In Sec. 6.3 we consider various formulations of orientability for
manifolds.
The Alexander cohomology theory is defined in Secs. 6.4 and 6.5, and
the axioms of cohomology theory are verified for it. Section 6.6 contains
a proof of the tautness property for Alexander cohomology, that the Alexander
cohomology of a dosed pair in a paracompact space is isomorphic to the direct
limit of the Alexander cohomology of its neighborhoods. We deduce the con-
tinuity property of Alexander cohomology and show that the continuity
property characterizes Alexander cohomology on compact pairs. We also
define the Alexander cohomology with compact supports.
Sections 6.7, 6.8, and 6.9 develop the theory of the cohomology of spaces
with coefficients in a presheaf and illustrate its application to the Alexander
theory. In this way we equate the Alexander and singular cohomology for
paracompact spaces that are homologically locally connected in all directions.
Section 6.10 contains definitions of the characteristic classes of a manifold
and the normal characteristic classes of one manifold imbedded in another.
These are related in the Whitney duality theorem, which is a useful tool for
establishing non-imbeddability results.
1 THE SLANT PRODUCT
We are ready now to introduce a new product which pairs cohomology of a
product space and homology of one of the factors to the cohomology of the
other factor. This product will be used in the next section to prove the duality'
theorem for topological manifolds. In this section we shall establish some of
its properties. We shall also introduce new cohomology modules of a pair
(A,B) in a space X which appear to depend on the imbedding of (A,B) in X.
These will be used in the proof of the duality theorem in the next section.
Later in the chapter, we shall introduce the Alexander cohomology modules
SEC. 1 THE SLANT PRODUCT
287
and prove that these are isomorphic to the abovementioned ones in all
relevant cases.
Given chain complexes C and C over R and a cochain
c* £ Hom ((C ® C)n, G)
and chain dd C'Q ® G', their slant product с* /с’ £ Hom G ® G') is
the (n — q)-cochain such that if d = c'i ® g,! with G £ C'Q and g; £ G', then
<C*/c',C> = 2 <C*, C ® Cj> ® g- c £ Cn_Q
i
It is easily verified that
S(c*/c') = [(&*)/c'] — ( — 1)п~чс* /3d
Therefore the slant product of a cocycle and a cycle is a cocycle, and if the
cocycle is a coboundary or the cycle is a boundary, the slant product is
a coboundary. Hence there is a slant product of H"(C ® C'; G) and HQ(C';G')
to H«-e(C; G ® G') such that {c* }/{d} = {c* /d} for {c* } £ H«(C ® C'- G)
and {с'} C He(C,;G/).
For topological pairs (X,A) and (Y,B) let
т: [Д(Х)/Д(А)] ® [Д(¥)/Д(В)] [Д(Х X Y)]/[A(X X В U А х Y)]
be a functorial chain map given by the Eilenberg-Zilber theorem. For
и d H"(.(X,A) x (Y,B); G) and z d H^Y,B; G'), their slant product
u/z d Hn(J(X,A; G ® G')
is defined to equal the slant product (r*u)/z. The following properties of this
slant product are easy consequences of the definitions.
1 Given f: (X,A)	(X',A'), g: (Y,B) -+ (Y',B’), и d H”((X',A') X (Y',B'); G),
end z d Hg(Y,B; G'), then, in Hn~Q(X,A; G ® G'),
[(f X g)* u]/z =f* (u/g *z) 
2 Given и d №(X,A; G), v d №(Y,B; G'), and z d Hq(Y,B; G"), then, in
Hp(X,A-, G®G'® G"),
(u X vj/z = p(u ® <u,z>) 
3 Let {(Xi,Ai), (ХгДг)} and {(Yi,Bi), (Y2,B2)} be excisive cmtplesinX and
Y, respectively. Given
и d H”([(Xi П X2, Ar П A2) X (Yi U Y2, Br U B2)]
U [(Xi U X2, Ai U A2) X (Yi П Y2, Bi П B2)]; G)
and
z d Hq(Yi U Y2, Bi U B2; G')
then, in H^-^Xi U X2, Ai U A2; G ® G'),
[w | (Xi U X2, Ai U A2) X (Yi A Y2, Bi Г) В2)]/Э ^.z
= (-1)и-в-1 2 3й*([гг | (Xi П X2, Ai П A2) X (Yi U Y2, Bx U B2)]/z »
288
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. 6
The following formulas express relations between the slant product and
the cup and cap products. We sketch proofs in which the Alexander-Whitney
diagonal approximation о —> Si+j=deg a ,o ® Oj is used in Д(Х) and its tensor
product with itself
о ® o' 2 (— iy<P~l\iO ® jo') ® (op_i ® o'Q_j) deg о = p, deg o' = q
i,3
is used in Д(Х) ® A(Y).
4 Given v £ №(X,A; G), и £ НИ((Х,А') X (Y,B); G'), and z £ HQ(Y,B; G"),
then, in №+n~<i(X, A U A'; G ® G’ ® G"),
V о (u/z) — [(o X 1) О u]/z
proof Let c* be a p-cochain of Д(Х), cf an n-cochain of Д(Х) ®
and o' £ Де(Y). It suffices to prove that
c* ^(Cf/</) = [(c| ® 1) ^ <?*]/</
If a £ ДР+Н Г/(Х), then
<c| о (с|/а')> a> = <CT> p°> ® <c%/o',on_g)
= <c|, pO) ® <cf , On_q ® o')
= <c J ® 1, pO ® oa') ® (cf, on_g ® o')
= <(c | ® 1) о cf, о ® o') = <[(c* ® 1) о c*]/o', o) 
5 If и £ H«((X,A) X (Y,B); G), v £ Hp(Y,B; G'), andz £ Hq(Y, В U B'; G"),
then, in H»~(^p\X,A-, G ® G' ® G"),
u/(v /vz) = [u v (1 X v)]/z
proof Let c J be an n-cochain of Д(Х) ® A(Y), c| be a p-cochain of Д(У),
and o' £ AQ(Y). It suffices to prove that
cf/(c* o') = [cf (1 ® C*)]/o'
If о £ Ад—(q—p)^X), then
<C?/(C2	a')> °> = <cf,O® (c*	o'))
= <cj, (1 ® cf) (a ® o'))
= (of (1 ® cf), о ® o')
= <[c|	(1 ® cf )}/o', o) 
6 Given и £ H”((X,A) X (Y,B); G), w £ H^X,A-, G'), and z £ Hg(Y,B- G"),
let p: X X Y X be the projection to the first factor and let
T-. G ® G" ® G' G ® G' ® G"
interchange the last two factors. Then, in Hr^(n_q)(X; G ® G' ® G"),
р*(и (w X z)) = T^[(w/z) w]
proof Let c* be an n-cochain of Д(Х) ® Д(Х), о d ^fiX), and o’ d Aq(Y).
SEC. 1 THE SLANT PRODUCT
Then
289
(p ® o')) = Д(р)[ 2 (-l)^-i\r_io ® q4o’) ®х(с*7ЪГ'® a<>]
i+j—n
— r-(n-q)O ® <C*, On_g ® o')
= T-in-qyO ® (C* /o', On_Q)
= (c* /о') О 
For a topological space X let 8(X) be the diagonal of X defined by
8(X) = {(x,xz) e X X X I x = x'}. Given и £ H'"(X % X, X X X - S(X); R)
and a pair (A,B) in X, define
yu-. Hq(X — В, X — A; G)—> №^A,B- G)
by Y«(z) = [u | (A,B) X (X — В, X — A)]/z (with R ® G identified with G).
If i: (A,B) C (A',B') and j: (X — В', X — А') С (X — В, X — A), it follows
from property 1 that there is a commutative diagram (all coefficients G)
Hg(X -B',X- A')	№~<i(A',B')
Ц ]/*
Hg(X -B,X- A) №~Q(A,B)
Thus у и is a natural transformation from HQ(X — В, X — A) to Hn,‘\A,B) on
the category of pairs of subspaces and inclusion maps in X. It follows from
property 3 that yu commutes up to sign with the connecting homomorphisms
of relative Mayer-Vietoris sequences.
For a pair (A,B) in a topological space X we define a neighborhood
(V, V) of (A,B) to be a pair in X such that U is a neighborhood of A and V is
a neighborhood of B. The family of all neighborhoods of (A,B) in X is directed
downward by inclusion. Hence
{№(U,V; G) | (U,V) a neighborhood of (A,B)}
is a direct system, and we define
№(A,B; G) = lim_ {№(17, V; G)}
where (U,V) varies over neighborhoods of (A,B) [or over the cofinal family of
open neighborhoods of (A,B)]. The restriction maps №(U,V; G) —> №(A,B; G)
define a natural homomorphism
it №(A,B- G) №(A,B- G)
The pair (A,B) is said to be tautly imbedded in X, or to be a taut pair in X
(with respect to singular cohomology), if i is an isomorphism for all q and G.
The definition of tautness can be formulated for any cohomology theory (or
any contravariant functor). We shall see examples later of a subspace taut
with respect to one cohomology theory but not with respect to another.
Following are some examples.
7 If (A,B) is an open pair, or, more generally, if it has arbitrarily small
290	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP, (j
neighborhoods which are homotopy equivalent to (A,B), then (A,B) is a taut
pair in X.
8 Let A' = {(x,y) £ R2 | x > 0, у = sin 1/x}, let A" = {(x,;/) £ R2 | x = 0,
|j/| < 1}, and let A = A' U A" C R2. Then A' and A" are the path compo-
nents of A, and so H°(A;Z) ~ Z © Z. Since A is connected, in any open
neighborhood U of A in R2, A' and A" must be in the same path component
of U (the path components of U are the same as the components of U because
Uis locally path connected). It follows that H°(A;Z) = lira, {B°((7;Z)}, where
U varies over the connected open neighborhoods of A in R2. Therefore
H°(A;Z) ~ Z and i: H°(A;7,) —> H°(A;Z) is not an epimorphism. Thus A is not
a taut subspace of R2 with respect to singular cohomology.
9 lemma Let (A,B) be a pair in X. Then, if two of the three pairs (B,0f
(A, 0), and (A,B) are taut in X, so is the third.
proof This follows from the exactness of the cohomology sequence of a
triple, from the fact that a direct limit of exact sequences is exact, and from
the five lemma. 
Recall (exercise set l.C) that a normal space X is an absolute neighbor-
hood retract if it is normal and has the property that whenever it is imbedded
as a closed subset of a normal space, it is a retract of some neighborhood.
Also recall that a space X is binormal if X X f (hence also X) is normal.
IO theorem Any imbedding of an absolute neighborhood retract as a
closed subspace of a binormal absolute neighborhood retract is taut.
proof Assume А С X, where A and X are absolute neighborhood retracts
and A is closed in the binormal space X. There is a neighborhood 17 of A in
X such that A is a retract in 17. Then H * (17) —> H *(A) is an epimorphism,
and this implies that
i: H«(A)-> H* (A)
is an epimorphism.
To show that it is also a monomorphism, let <7 be an open neighborhood
of A in X. There is a closed neighborhood U' of A in U of which A is a retract.
Let r: U' A be a retraction and define a map
F: (Lz X 0) U (A X f) U (U' X 1) U
by F(x,0) = x and F(x,l) = r(x) for x £ 17' and F(x,t) = x for x £ A and t £ I.
Because A is closed in X, (17' X 0) U (A X 1) U (17' X 1) is closed in 17' X I,
the latter being a normal space because it is a closed subset of the normal
space X X I- Since U is an open subset of the absolute neighborhood retract X,
it follows (see exercise l.C.4) that U is an absolute neighborhood retract and
F can be extended to a map F'-. N U, where N is a neighborhood of
(17' X 0) U (A X I) U (17' X L) in U' X I- Ncontains a set of the form V X I,
where V is a neighborhood of A in 17', and F' | V X I is a homotopy from the
7' SEC. 1 THE SLANT PRODUCT	291
inclusion map j: V C U to /<?•', where r' = г | V: V A and /<: A C U.
Therefore there is a commutative triangle
!	П*((7) Д H*(A)
j*\	/г’*
H*(V)
which shows that ker к* C ker j *. Thus, if an element in H* (17) restricts to 0
in H*(A), it restricts to 0 in H*(V) for some smaller neighborhood V, hence
. it represents 0 in linu (H*(U)} = H*(A). Therefore i: H*(A)	H*(A) is
a monomorphism and A is taut in X. 
11 corollary If A, B, and X are compact polyhedra, any imbedding of
(A,B) in X is taut.
' proof This follows from the fact (exercise 3.A.1) that a compact polyhedron
j is an absolute neighborhood retract and from theorem 10 and lemma 9. 
One reason for introducing the modules №(A,B; G) is the following
result, which asserts that any pair (A,B) in X is taut with respect to the
functor №.
12 theorem As U varies over the neighborhoods of A, there is an
। isomorphism
lira . {№(U;G)} ~ HQ(A;G)
proof Restricting U to the cofinal family of open neighborhoods, we have
№(U;G) = №(U;G), and the limit on the left is, by definition, equal to the
module on the right. 
If (A,B) and (A',B') are pairs in X and (U,V) and (U',V'j are respective
' open neighborhoods, there is a relative Mayer-Vietoris sequence of
{(17,V), (17', V')}. As (17,V) and (V,V') vary over open neighborhoods of (A,B)
and (A',B'), respectively, (17 U U', V U V') varies over a cofinal family of neigh-
borhoods of (A U A', В U B'). If (A,B) and (A',B') are closed pairs in X, it is
: also true that (17 G 17', V G Vх) varies over a cofinal family of neighborhoods
of (А Г) А', В Г) В'). Because the direct limit of exact sequences is exact, we
obtain the following result, which is another reason for our interest in the
modules H* (A,B).
• 13 theorem If (A,B) and (A',B') are closed pairs in X, there is an exact
relative Mayer-Vietoris sequence (for any coefficient module G)
-----> №(A U A', В U B')	№(A,B) © H«(A',B')
№(A П А', В Ci В') —> • • • 
Given и £ №(X X X, X X X — S(X); B), as (U,V) varies over neighbor-
hoods of (A,B), the homomorphisms
yu: Hq(X — V,X—U;G)—> №^(U,V, G)
292
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP, в
define a homomorphism
lim , {IIfX - V, X - U-, G)} lim , {H«-«(G,V; G)}
Because singular homology has compact supports, it commutes with direct
limits, and the limit on the left is isomorphic to HQ(X — В, X — A; G),
Therefore we obtain a natural homomorphism
yu: HqlX -B,X- A; G)-> №(A,B; G)
such that if (17, V) is a neighborhood of (A,B), there is a commutative diagram
(all coefficients G)
HQ(X - V, X - U) -> HQ(X — B,X— A)
H	V"
H«-«(17,V)	№(A,B)	№(A,B)
If (A,B) and (A',B') are closed pairs in X, then y!( maps the exact Mayer-
Vietoris sequence of the couple of open pairs
{(X - В, X - A), (X -B',X- A')}
into the exact Mayer-Vietoris sequence of theorem 13 in such a way that each
square is commutative up to sign.
2 DUALITY IN TOPOLOGICAL MANIFOLDS
This section is devoted to a study of homology properties of topological
manifolds. Over a connected manifold as base space there is a fiber-bundle
pair called the homology tangent bundle. An orientation class of this bundle
gives rise to a duality in the manifold asserting that the cohomology of
a compact pair in the manifold is isomorphic to the homology of its comple-
ment. This duality theorem is proved by using the orientation class and the
slant product to define a natural homomorphism from homology to cohomology.
The resulting homomorphism is shown to be an isomorphism by proving it
first in euclidean space and then in an arbitrary manifold using the piecing-
together technique based on Mayer-Vietoris sequences.
A topological n-manifold (without boundary) is a paracompact Hausdorff
space in which each point has an open neighborhood homeomorphic to R”
(called a coordinate neighborhood in the manifold). Following are some
examples of n-manifolds.
1	R” and Sn are n-manifolds.
2	An open subset of an n-manifold is an n-manifold.
3	The product of an n-manifold and an m-manifold is an (n + m)-manifold.
SEC. 2 DUALITY IN TOPOLOGICAL MANIFOLDS
293
4	P11 is an n-manifold, Pn(C) a 2n-manifold, and P„(Q) a 4n-manifold for all
n. In fact, if X denotes one of these spaces and is coordinatized by homogene-
ous coordinates [toA, •  • ,tn]> then for each 0 < i < n the subset А; С X
of points having ith coordinate 0 is a projective space of dimension n — 1 and
X — Aj is homeomorphic to R, R2, or R4, respectively. Hence, X — Ai is a
coordinate neighborhood of X, and X is covered by these n + 1 coordinate
neighborhoods.
5	lemma In an n-manifold X each point x has an open neighborhood V
such that (V X X, V X X — S(V)) is homeomorphic to V X (X, X — x) by a
homeomorphism preserving first coordinates.
proof Let U be a coordinate neighborhood containing x. Without loss of
generality, we can suppose that there is a homeomorphism <p: U ~ Rn such
that <p(x) = 0. Let D' = (z £ R" | ||z|| < 2} and V = (z £ R'" | ||z|| < 1} and
define D = (р~Ц1У) and V = <//1( V7'). Then V is an open neighborhood of x
contained in the compact set D. If (x'.x") £ V X D — S(V), there is a unique
point z"' 6 R" such that ||z"'|| = 2 and <p(x") belongs to the closed segment
from <p(x') to z"'. If <p(x") = ftp(x') + (1 — t)z"', with t £ I, let h(x',x") £ D — x
be the point such that <ph(x',x") = (1 — t)z"', as illustrated
and define h(x',x') = x. A homeomorphism
(V X X, V X X - S(X)) = V X (X, X - x)
having the desired properties is defined by
Цх',х") =
(x',x")	x" D
(x7, 7i(x',x"))	x" £ D 
It follows from lemma 5 that if x' £ V then (X, X — x') is homeomorphic
to (X, X — x). Hence we obtain the following result.
в corollary In a connected n-manifold X the group of homeomorphisms
acts transitively; in particular, the topological type of (X, X — x) is independ-
ent of x. Furthermore, projection to the first factor p: X X X —> X is the pro-
jection of a fiber-bundle pair (X X X, X X X — S(X)) with fiber pair
(X, X — x). 
If V is a coordinate neighborhood of x in an n-manifold X, the couple
{V, X — x) is excisive, and so there is an excision isomorphism
294
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP, (j
H* (V, V - x- G) ~ H* (X, X - x- G)
Since II* (V, V — x; G) ~ H* (R'!, Rn — 0; G), it follows that
He(X, X-x;G)~[0
' (G q = n
and so the fiber pair (X, X — x) of the fiber-bundle pair of corollary 6 has the
same homology as (RB, RB — 0). For this reason the fiber-bundle pair of corol-
lary 6 will be called the homology tangent bundle of X (the tangent bundle
itself is an /г-plane bundle defined if X is a differentiable manifold and having
homology properties isomorphic to those of the homology tangent bundle).
A connected n-manifold X is said to be orientable (over R) if its
homology tangent bundle is orientable [that is, if there exists an element
U 6 H"(X X X, X X X — <5(X); R) such that for all x £ X, U | x X (X, X — x)
is a generator of Hn(x X (X, X — x); R)]. Such a cohomology class U is called
an orientation of X. An n-manifold X (which is not assumed to be connected)
is said to be orientable if each component is orientable, and an orientation of
X is defined to be a cohomology class U 6 Hn(X X X, X X X — 8(X); R)
whose restriction to each component is an orientation of that component.
7 example For R" the fiber-bundle pair (RB X Rn, R” X Rn — S(RB)) is
trivial, because the map
f(z,/) = (z, z' - z)
is a homeomorphism f: (Rn X R”, R” X R” — d(RB)) ~ RB X (RB, Rn — 0)
preserving first coordinates. Therefore R” is an orientable n-manifold.
The results of Sec. 5.7 dealing with the homology properties of sphere
bundles carry over to the homology tangent bundle. We list some of these
explicitly.
8	Two orientations U and U' of a connected manifold X are equal if and
only if for some xq £ X
U | x0 x (X, X - x0) = U' | x0 X (X, X - x0) 
9	Any manifold has a unique orientation over Zg. 
10	A simply connected manifold is orientable over any R. 
11	An n-manifold X is orientable if and only if there is an open covering
(V) of X and a compatible family (Uy € Hn(V X X, V X X — 8(V); R)},
where Uy corresponds to an orientation of V under the excision isomorphism
№(V X X, V X X - d(V); R) ~ №fV x V, V X V - d(V); R) 
The duality theorem asserts that if U £ H"(X X X, X X X — d(X); R) is
an orientation of X, then for any compact pair (A,R) in X, у у is an isomorphism
of HQ(X — R, X — A; G) onto №(A,R; G). We prove this first for RB by a
sequence of lemmas.
SEC. 2 DUALITY IN TOPOLOGICAL MANIFOLDS
295
12	lemma Let A C R” be homeomorphic to a simplex and let oq £ A.
}7ien HQ(Rn — ao, R” — A; G) = 0 for all q and G.
proof Regarding Rn as an open subset of S'1, there is an excision isomorphism
Hg(R’! — ao, R" — A; G) ~ Hq(Sn — ao, Sn — A; G). Because Sn — ao is
homeomorphic to R", fy/S'1 — ao; G) = 0. From lemma 4.7.13 and the
universal-coefficient formula, Hq(Sn — A; G) = 0. The lemma now follows from
exactness of the reduced homology sequence of the pair (S’1 — ao, S" — A). 
13	corollary If A C R" is homeomorphic to a simplex and U is an
orientation of R’! over R, then for all q and R modules G
yy. HQ(RB, R” - A; G) =: H«-«(A;G)
proof Let «о € A and consider the diagram (all coefficients G)
HS(R" - <70, R" — A) -» H„(R”, R" — A) —» Hq(R\ R" - a0) ->	- a0, R" - A) •
Td	Td	«d
№’^(A,a0)	->	H”-v(A)	-> №‘-i(a0)	->	№~«+i(A,ao)	-» • • •
The rows are exact, and each square either commutes or anticommutes.
Since A is contractible, H*(A,«o) = 0. Using lemma 12, we see that trivially
Yu: W^R” — ao, Rn — A) ~ Hn~i(A,ao'). By the five lemma, to complete the
proof we need only verify that yy. Hq(Rn, R” — a0) ~ Ни-е(ао). Because U is
an orientation, U | [«о X (Rn, Rn — «о)] = 1 X и, where и £ Hn(JRn, Rn — ao; R)
is a generator. By property 6.1.2,
Ы~) = <«,z>l
Since и is a generator of Hn(JRn, R’1 — ao; R) ~ Hom (Hn(Rn, R” — ao; R), R),
it follows that the map z —> <w,z) of H„(R", Rn — ao; R) to R is an isomor-
phism; and hence so is yy. Hn(Rn, R" — ao; R) ~ H°(ao;R). If q =/= n, it is
trivially true that yy. HfJRn, Rn — ao; R) ~ H’l-e(ao;R), since both modules
are trivial. 
14	theorem If U is an orientation of R” over R and (A,B) is a compact
polyhedral pair in R", then for all q and all R modules G there is an
isomorphism
yy. Hq(Rn - B, R« - A; G) H»~«(A,B; G)
proof Because of the naturality properties of yu, it suffices to prove this for
the case where В is empty. The theorem follows for A from corollary 13 by
induction on the number of simplexes in a triangulation of A, using Mayer-
Vietoris sequences and the five lemma. 
15	corollary If U is an orientation ofRn over R and (A,B) is a compact
pair in R’1, then for all q and R modules G there is an isomorphism
yy. Hq(Rn - B, R” - A; G) ~ H’'-«(A,B; G)
proof Since the family of compact polyhedral pairs is cofinal in the family
296	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. 6
of all neighborhoods of a compact pair (A,B) in Rn, the corollary follows from
theorem 14 by taking direct limits. 
Because of the commutativity of the triangle
He(R« - B, R« - A; G)
fy fy
№‘fA,B; G) -U Н«-ч(А,В; G)
it follows from theorem 14 and corollary 15 that any imbedding of a compact
polyhedral pair in R’1 is taut (which is also a consequence of corollary 6.1.11).
As an immediate result of corollary 15, we obtain the following Alexander
duality theorem.
16	theorem If A is a compact subset of R?l, then for all q and R modules G
Hg(Rn - A; G)~ H^~i(A;G)
proof Because H>;. (R'" ;G) = 0, there is an isomorphism
a*: He+i(Rn, R« - A; G) 7Z Hg(Rn - A; G)
The result is obtained by composing the inverse of this isomorphism with the
isomorphism of corollary 15. 
For general orientable manifolds there is the following duality theorem.
17	theorem Let U be an orientation over R of an n-manifold X and let
(A,B~) be a compact pair in X. Then for all q and R modules G there is an
isomorphism
yu-. Hq(X -B,X- A-,G)~ Нп~ч(А,В; G)
proof Because of the naturality properties of yu, it suffices to prove the
theorem for the case where В is empty. If A is contained in some coordinate
neighborhood V of X and U' = UfV % V, V % V - 8(V)) is the induced
orientation of V, there is a commutative triangle (all coefficients G)
Hg(V, V - A) Hg(X, X — A)
/ iv
H«-e(A)
By corollary 15, yu is an isomorphism, hence у и is also an isomorphism. The
result for arbitrary compact A follows by induction on the finite number of
coordinate neighborhoods needed to cover A, using naturality of yu, the usual
Mayer-Vietoris technique, and the five lemma. 
In case X is compact, by applying theorem 17 to the pair (X, 0) and
observing that i: H\X;Gj ~ №(X;G), we obtain the following Poincare
duality theorem.
SEC. 2 DUALITY IN TOPOLOGICAL MANIFOLDS	297
18	theorem If U is an orientation over R of a compact n-manifold X,
then for all q and R modules G there is an isomorphism
yw- Hg(X-G) ~ H”~^X-,G) 
A pair (X,A) is called a relative n-manifold if X is a Hausdorff space, A is
closed in X (A may be empty), and X — A is an n-manifold. For relative
manifolds there is the following Lefschetz duality theorem.
19	theorem Let (X,A) be a compact relative n-manifold such that X — A
is orientable over R. For all q and R modules G there is an isomorphism
Hq(X - A-,G)~ Нп~ч(Х,А-, G)
proof Let {Af} be the family of closed neighborhoods of A directed down-
ward by inclusion. There are isomorphisms
lim, (Rq(X - N; G)} ~ He(X - A; G)
linu {H«-e(X,A; G)}	H«^(X,A; G)
the first because singular homology has compact supports and the second as a
consequence of theorem 6.1.12. Let V be an open neighborhood of A contained
in the interior of N and let 17 be an orientation of X — A over R. By
theorem 17 and standard excision properties, there are isomorphisms (all
coefficients G)
He(X - N) Hq((X — A) — (N — V), (X - A) - (X - V))
~|Л
Н»-е(Х,Я) Нп~я(Х - V, N - V)
which yield the result on passing to the limit. 
An n-manifold X with boundary X is a paracompact Hausdorff space
such that (X,X) is a relative n-manifold and every point x £ X has a neighbor-
hood V such that (V, V Cl X) is homeomorphic to Rn-1 x (1,0). Since X may
be empty, the concept of manifold with boundary encompasses that of mani-
fold without boundary.
If X is an n-manifold with boundary X, then X has neighborhoods N such
that (N,X) is homeomorphic to X X (Д0).1 Such a neighborhood N is called a
collaring of X, and its interior is called an open collaring of X. (In case X is
compact, any neighborhood of X contains a collaring of X.) Because of the
existence of such collarings, X — X is a weak deformation retract of X, and
the pair ((X - X) X (X - X), (X - X) x (X - X) - S(X -X)) is a weak
deformation retract of (X X X, X X X — S(X)).
An n-manifold X with boundary X is said to be orientable over R if
X — X is orientable over R. An orientation over R of X is a class
1 See M. Brown, Locally flat imbeddings of topological manifolds, Annals of Mathematics,
vol. 75, pp. 331-341, 1962.
298
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. 6
U 6 Hn(X X X, X X X — й(Х); B) whose restriction to ((X — X) X (X — X),
(X — X) X (X — X) — S(X — X)) is an orientation of X — X over R. For
manifolds with boundary the Lefschetz duality theorem takes the following
form.
20 theorem Let X be a compact n-manifold with boundary X and orien-
tation U over R. For all q and R modules G there are isomorphisms (where
p.x-x ex)
HQ(X;G) HQ(X - X; G) Нп~ч(Х,Х; G)
Hq(X,X; G) H»~e(X — X; G) №>~<i(X;G)
proof Because j is a homotopy equivalence, у* and /* are isomorphisms.
Let N be a collaring of X with interior N. Let U' be the orientation of X — X
obtained by restricting U. In the following commutative diagram each hori-
zonal map is induced by inclusion and is an isomorphism because it is an
excision (labelled e) or a homotopy equivalence (labelled h) (all coefficients G):
Hg(X - X) 4 Hg(X - N) Hg((X - X) - (N - N), (X - X) - (X - ft))
H«-«(X,X) £ H«~<i(X,N)	Hn~o(X — N,N — N))
Because (X — N, N — N) has arbitrarily small neighborhoods of which it is a
deformation retract i: Hn~i(X — N, N — TV) ~ Hn(l(X — N, N — N), and it
follows from theorem 17 that the right-hand vertical map is an isomorphism
(because it corresponds to the isomorphism p.f Therefore the left-hand
vertical map is also an isomorphism proving the first part of the theorem.
Similarly, there is a commutative diagram
ВДХ)	Hg(X,N)	Hg(X - X, (x - X) - (X - X))
Hn~4(X - X)	H«-«(X - N) £ H^(X - ft)
Because X — N has arbitrarily small neighborhoods of which it is a deforma-
tion retract, it follows from theorem 17 that the right-hand vertical map is an
isomorphism. Therefore the left-hand vertical map is also an isomorphism,
proving the second part of the theorem. 
From the isomorphisms of theorem 20 and the universal-coefficient
theorem for homology, we obtain a short exact sequence
0	№(X-,R) ® G Д №(X;G)	№+i(X;R) * G -> 0
and a similar short exact sequence for №(X,X; G). Since this is so for every
R module G, from theorem 5.5.13 we have the following result.
21 corollary If X is a compact n-manifold with boundary X orientable
over R, then H* (X;R) and H* (X,X; R) are finitely generated. 
SEC. 3 THE FUNDAMENTAL CLASS OF A MANIFOLD
299
Later in the chapter (see theorem 6.9.11) we shall prove that corollary 21
is also valid for nonorientable manifolds.
3 THE FUNDAMENTAL CLASS OF A MANIFOLD
In view of the importance of the concept of orientability of manifolds, we
shall now investigate some equivalent formulations. We shall show that a
compact connected n-manifold is orientable if and only if its n-dimensional
homology module is nonzero. In fact, any orientation class of the manifold
will be shown to correspond to a generator of the n-dimensional homology
module. Moreover, if z is the element of Hn corresponding to the orientation,
then the cap product of z and a cohomology class defines a homomorphism
which equals, up to sign, the inverse of the duality isomorphism. The methods in
this section rely heavily on the technique of piecing together homology classes,1
analogous to the piecing together of cohomology classes in lemma 5.7.16.
Let X be a space, X' a subspace of X, and (f = (A} a collection of sub-
sets of X — X'. A compatible @ family is a family {za G Hq(X, X — A; G)}
(for some fixed q and G) indexed by (J such that if A, A' G <? and A' C A,
then Za maps to z&, under the homomorphism
HQ(X, X - A; G) -> He(X, X - A'; G)
The compatible C? families form a module with respect to componentwise
operations that will be denoted by Hf (X,X'; G). For the collection & of all
compact subsets of X — X' we use HQC(X,X'; G) to denote the corresponding
module.
We are interested in the module НИС(Х,Х; B) for an n-manifold X with
boundary X. The following lemma is important in this connection.
I lemma Let X be an n-manifold with boundary X and let A be a com-
pact subset of X — X. For all R modules G
Hq(X> X — A;G) = 0 q > n
proof Assume first that A is contained in some coordinate neighborhood V
in X — X. By excision, H(l( V, V — A) ~ He(X, X — A), and since V is homeo-
morphic to Rn, we can use corollary 6.2.15 to obtain
Hq(V, V — A) zz Hn~v(A) =0 q > n
For arbitrary compact A the result follows by induction on the number of co-
ordinate neighborhoods needed to cover A, using Mayer-Vietoris sequences. 
In an n-manifold X with boundary X a small cell in X — X is defined to
he a compact subset A having an open neighborhood V С X — X such that
1	This technique can be found in H. Cartan, Methodes modernes en topologic algebrique,
Commentarii Mathematic! Helvetici, vol. 18, pp. 1-15,1945.
300
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. 6
(V,A) is homeomorphic to (R",E’1). Every point of X — X has arbitrarily small
neighborhoods which are small cells. If A and V are as above, there is an
excision isomorphism
Hg(X, X - A; G) HS(V, V - A; G) ~
If Xq € A, then the inclusion map induces isomorphisms
Hg(X, X -A; G)~ Hg(X, X - xo; G)
We use HQSC(X,X; G) to denote the module of compatible C? families, where fi1
consists of the collection of small cells of X — X. Since the collection of small
cells is contained in the collection of compact subsets of X — X, there is
a natural homomorphism
HQC(X,X; G) HQ^X,X, G)
which assigns to a compatible family {za} indexed by all compact A the com-
patible subfamily of elements indexed by small cells.
2	lemma Let X be an n-manifold with boundary X. Then, for all G
Hnc(X,X-, G) ~ Hns<fX,X- G)
proof For each positive integer i let Cl’-; be the collection of compact subsets
of X — X contained in the union of i small cells. Then ft C (fi+i and U Cl’.; is
the collection of all compact subsets of X — X. There are homomorphisms
---->	-+-------Hnsc
and an isomorphism Hnc ~ lim^ {Я®}.
Since every element of ef, is contained in some small cell, it is obvious
that H®1 ~ Hnsc. By the usual Mayer-Vietoris technique and lemma 1, it
follows that for any i > 1	11 ~	. Combining these isomorphisms
yields the result. 
This gives the following important result.
3	theorem Let X be an n-manifold with boundary X and let
{zA} e нпс(х,х-, G)
(a)	{zA} = 0 if and only if zx = 0 for all x £ X — X.
(b)	If X is connected, = 0 if and only if zx = 0 /or some x £ X.
proof (a) follows from lemma 2 and the observation that if A is a small cell
and x 6 A, then
Hn(X, X- A-,G)~ Hn(X, X - x; G)
and so zA = 0 if and only if zx = 0.
To prove (b), assume zXa — 0 for some Xq 6 X — X. Because X is
connected, so is its weak deformation retract X — X. This implies that if
SEC. 3 THE FUNDAMENTAL CLASS OF A MANIFOLD	301
% С X — X, there is a finite sequence of small cells Ai, . . . , Am in X — X
such that x0 € A± and x £ Am, and A, meets Ai+i for 1 < i < m. Choose a
point Xj E Ai П Aj+j for 1 < i < m. There are isomorphisms
Hn(X, X — Xq) Ез Hn(X, X — Ai) 77» Hn(X, X — x'i) Es   
Hn(X, X - Am) Hn(X, X - x)
from which it follows that if zXll = 0, then zx = 0. Since this is so for all
x E X — X, the result follows from (a). 
If X is an n-manifold with boundary X, a fundamental family of X over
R is an element (za } E Hnc(X,X; R) such that for all x E X — X, zx is a gener-
ator of Hn(X, X — x; R). The relation between fundamental families and
orientations is made precise in the next result.
4	theorem Let X be an n-manifold with botmdary X. There is a one-to-
one correspondence between orientations U (over R) of X and fundamental
families {za} (over R) of X such that U and (za} correspond if and only if
yu(zA) = 1 E H°(A-,R) for all compact A in X — X.
proof If U is an orientation of X, let U' be the induced orientation of
X — X. For any compact А С X — X we have the commutative diagram (all
coefficients R)
Hn(X,X — A) Hn(X - X, (X - X) - A)
H°(A)	H°(A)
By theorem 6.2.17, the right-hand vertical map is an isomorphism, and since
1 E H°(A) is the image of 1 E H°(A), there is a unique Za E Hn(X, X — A)
such that Уп'/ф-1(2а) = 1 E H°(A). Because of the uniqueness of Za and the
naturality of yu and yV', the collection {za} is a compatible family. From the
commutativity of the above diagram, yu(zA) = 1 E H°(A) for all compact A
in X — X. Hence we need only verify that {za} is a fundamental family. In
case A = x, it follows from the commutativity of the above square and the
fact that i: H°(x) ~ H°(x) that yv- Hn(X, X — x) zz H°(x). Therefore
zx = Yu-1(l) is a generator of Hn(X, X — x). Hence {za} is a fundamental
family with the desired property, and the collection (zx}xex-x (and hence, by
theorem 3a, {za}) is uniquely characterized by the property yu(zx) = 1 E H°(x).
Conversely, given a fundamental family {za}, let Vbe any open subset
of X — X homeomorphic to Rn. If xo E V, then H*(V;R) ~ H*(x();R), which
implies that
H*(V X x, V X x - S(V); R) ~ H*(x0 X (X, X - Xo); R)
If и E Hn(V X X, V X X — S(V); R), it follows from the Kiinneth formula
for cohomology (theorem 5.6.1) that и | Xo X (X, X — x0) — 1 X u’ for a
Unique и' E Hn(X, X — Xo; R) ~ Hom (Hn(X, X — xo; R), R). By property 6.1.2,
302
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. 6
[и I Xo X (X, X - хо)]/хЖо = <-u'A0> 1
Since z;ro is a generator of Hn(X, X — xo; R), {и',хЖо> completely determines
u'. Therefore there is a unique element U £ Hn(V X X, V X X — 8(V); Д)
such that [171 x0 X (X, X — x0)]/z;f0 = 1 6 H°(x0;B).
We now show that for any x 6 V, [171 x X (X, X — x)]/zXo = 16 H°(x;B).
If x and x' belong to a small cell А С V, then zA maps to z,r and to z^.
Therefore [171 A X (X,X — A)]/za 6 H°(A;R) maps to [171 x X (X, X — x)]/Zs,
and to [17 | x' X (X, X — х')]/2ж- by naturality of yv. Since H°(A;R) ~ H°(x;fi)
and H°(A;R) ~ H°(x';R), it follows that both [171 x x (X, X — x)]/z,. =
1 £ H°(x;B) and [17 | x7 X (X, X — x')]/zX' = 16 H°(x';B) or neither equation
is true. Hence the set of x € V for which [171 x X (X, X — x) |/z;f = 16 H°(x;B)
is open and its complement in V is open. Since V is connected and
[17 | x0 X (X, X — хо)]/хЖо = 1, it follows that [171 x X (X, X — х)]/гж = 1
for all x 6 V.
This means that 17 is an orientation of V, and if 17' is a similarly defined
orientation for another coordinate neighborhood V' in X — X, then for any
x £ V Г) V', U | x X (X, X — x) = U'l x X (X, X — x). This implies that V
and 17' induce the same orientation of V Г) V. Hence the collection { Uy} for
coordinate neighborhoods V in X — X is compatible. Therefore there is an
orientation U of X such that 17 | (V X X, V X X — d(V)) = Uv. From the
construction of Uv we see that yi^zf) = 16 H°(x;R) for all x 6 X — X. By the
first half of the proof, there is a fundamental family {z^} such that
Yt/zl) = 16 H°(A;B). Then z'x = zx for all x £ X — X, and by theorem 3a,
z'a — %a for all compact А С X — X. Therefore Yl(Ai) = 16 H°(A;B) for all A,
proving that every fundamental family {za} corresponds to some orientation U.
The orientation U is uniquely characterized by the fundamental family
{za}, for if U and 17' are two orientations of X such that Ycfe) = yufzf) for all
x 6 X — X, then 171 x X (X, X — x) = U' | x X (X, X — x) for all x 6 X — X.
Therefore, by lemma 5.7.13, U = U'. 
This last result gives the following useful characterization of orientability
for connected manifolds.
5	theorem Let X be a connected n-manifold with boundary X. If
Hnc(X,X; R) 0, then Hnc(X,X; R) ~ R and any generator is a fundamental
family of X.
proof From theorem 3b it follows that, given Xq 6 X — X, the homomorphism
НИС(Х,Х; В) H„(X, X - x0; R)
sending {zA} to zXo is a monomorphism. Since Hn(X, X — Xo; R) ~ R, either
Hnc(X,X; R) = 0 or Hnc(X,X; R) ~ R. Assume Hnc(X,X; R) ~ R and let {zA}
be a generator of Hnc(X,X; R). Assume that for some x 6 X — X, zx is not a
generator of Hn(X, X — x; R). There is then a noninvertible element r 6 R
such that zx = rzx for some zx 6 Hn(X, X — x; R). It follows that for any small
cell A containing x, zA = rzA for some 6 IlfX, X — A; R). Because X
SEC. 3 THE FUNDAMENTAL class of a manifold	303
js connected, it follows, as in the proof of theorem 3b, that for any small cell
& in X — X, zA = rz'A for some z'A £ H„(X, X — A; R). If A' is a small cell in A,
then rzA maps to rdA’ in Hn(X, X — A'; R). Because Hn(X, X — A'; R) is
torsion free, by lemma 1, z'A maps to zA>. Therefore {^} £ H?lsc(X,X; B).
gy lemma 2, it follows that the original element {zA} £ Hnc(X,X; R) is divisible
py the element r £ R. Since r is not invertible, this contradicts the hypothesis
that {zA} is a generator of Hnc(X,X; R). 
0 corollary If X is a connected n-manifold with boundary X, then X is
orientable over R if and only if Hnc(X,X-, R) 0.
proof This is immediate from theorems 4 and 5. 
We now specialize to the case of a compact manifold.
7	lemma If X is a compact n-manifold with boundary X, there is an
isomorphism
Hn(X,X; G) ~ Hnc(X,X; G)
sending z £ H?!(X,X; G) to {zA = image of z in Hn(X, X — A; G)}.
proof Let V be an open collaring of X and let R = X — V. Then В is com-
pact and there is a homomorphism
Hnc(X,X; G) Hn(X, X — B , G)
sending (zA) to Zu- Since X — В = V and (X,X) C (X,V) is a homotopy
equivalence, the composite
Hn(X,X; G) Hnc(X,X-, G) -» Hn(X, X - B- G)
is an isomorphism. To complete the proof we need only show that the right-
hand map is a monomorphism. Assume that {zA} is a compatible family such
that Zb = 0 and let A be any compact set in X — X. There is then an open
collaring V' of X such that V' С V and V' is disjoint from A. Let В' = X — V'.
Then А, В С B', and we have homomorphisms (all coefficients G)
Я„(Х, X — A) H„(X, X — B')	H„(X, X - B)
the second map being an isomorphism because (X,V9 С (X, V) is a homotopy
equivalence. Since Zb = 0, zB, = 0 and zA = 0. Therefore {zA} = 0 in
H„C(X,X; G). 
8	corollary A compact n-manifold X with boundary X is orientable
over R if and only if Hn(X,X; R) 0.
Proof This is immediate from corollary 6 and lemma 7. 
If X is a compact n-manifold with boundary X, a fundamental class over
ffi of X is an element z £ Hn(X,X; R) whose image in Hnc(X,X; R) under the
isomorphism of lemma 7 is a fundamental family [that is, for every x £ X — X
the image of z in Hn(X, X — x; B) is a generator of the latter].
304	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP, (j
9	theorem If X is a compact n-manifold with boundary X, there is a
one-to-one correspondence between orientations U over R and fundamental
classes z over R such that U corresponds to z if and only ifytfz) = 1 £ H°(X-,Rf
proof This follows from theorem 4 and lemma 7 on observing that an
element v E H°(X;R) equals 1 if and only if v | я: — 1 £ H°(x;R) for all
x E X — X 
10	corollary If X is a compact n-manifold with boundary X, then if X
is orientable, so is X, and any fundamental class of X maps to a fundamental
class of X under the connecting homomorphism
d*: U„(X,X- B) -+ Hn-i(X;R)
proof Let N be a collaring of X with interior N. Then N is an n-manifold
with boundary X U fN — N), and there is a commutative diagram (all
coefficients B)
Hn(X,X)	-±>	Hn(X,X U (X — TV)
a*|	j«T~
Hn-ilX U (TV - TV), TV - TV) Л H„(TV, X U (N - TV))
It is clear from the definition of fundamental class that if z E Hn(X,X) is a
fundamental class of X, then j* -ii* z = z' is a fundamental class of TV. Because
TV is homeomorphic to X X I in such a way that X and N — TV correspond to
X X 0 and X X 1, respectively, the Kiinneth formula implies
Hn(TV, X U (TV - TV)) ~ Нп.л(Х) ® Hr(I,i)
Let w E Hi(U) be a generator and let {Xy} be the components of X.
Then z' corresponds to S z'j X w for some z'j E ILifXj), and	=
±S z'j. Hence <fz = ±S z'j, and since z is a fundamental class of X,
z'j X w corresponds to a fundamental class of Xy X I- Therefore z'j is non-
zero and is a generator of Hn~i(Xj). Then z'j is a fundamental class of Xy,
whence ±S Zy = d^z is a fundamental class of X. 
We are now heading toward a proof that cap product with a fundamental
class is an isomorphism which, up to sign, is inverse to the duality isomorphism
in a compact manifold. First we need a lemma.
11	lemma Let X be a compact orientable n-manifold with boundary X
and let pi, p2: X X X —» X be the projections. Given
и E №(X X X, X X X - S(X); B), z E Hm(X X X, X X X - S(X); G),
and v E Hr(,X;G), then
p^ (u r>z) — p2* (u	z) in	Hm_g(X;G)
uvpfv = и ^p*v in	№+r(X X x, X X x - S(X); G)
proof Let Г: (X X X, X X X - S(X)) (X X X, X X X - S(X)) be the
SEC. 3 ,, ИЕ FUNDAMENTAL class of a manifold	305
map interchanging the factors. If w £ Hn(X,X; R) is a fundamental class of X,
then w X w E H2n((X,X) X (X,X); R) is a fundamental class of X X X (whence
у x X *s orientable), and L, (w X w) = (— l)”w X w. By theorem 9, T maps
the orientation of X X X corresponding to w X w into (— 1)” times itself. Let
y: Hm(X x X, X X X - S(X); G) tP^’fi8(Xf,G)
be the duality map associated to this orientation. Then we have a commuta-
tive diagram (all coefficients G)
Hm(X X X, X X x - S(X); G) A Hm(X X x, X X x - 8(X); G)
H2«-™-(5(X);G)
Therefore L, (z) = (— l)nz for any z 6 И* {X X X, X X X — 8(X); G) (which
implies T*(w) = (-!)’•« for any и £ H* (X X X, X X X - S(X); G)). Then
P2* (M rs z) = Pl* T* (w r> z) = pi* (T* и r>T*z) = pi* (u z)
and и ^p*v = (— l)nT* (u о p*v) = u о T*pfv = и о р* и 
12	theorem Let zbe a fundamental class over R of a compact n-manifold
X with boundary X. For all q and R modules G the homomorphism
кг(и) = v z defines isomorphisms
kz-. №(X;G) ~ Hn^(X,X; G)
KZ: №(X,X; G) ~ IL C(X;G)
which are, up to sign, the inverse of the duality isomorphisms of theorem 6.2.20
defined by the orientation corresponding to z.
proof Let U be the orientation of X corresponding to z as in theorem 9,
and let /: X — X С X. We prove commutativity up to sign in the triangle (all
coefficients G)
He(X - X) H»-5(X,X)
Л, X	iSк,
Hq(X)
For w € He(X — X), by property 6.1.6,
W«) = {[[7| (X,X) x (X - X)]/w} rs z
= Pi*№| (X,X) X (X-X)]n(zX w)}
By lemma 11, this equals
Рз*{[С71 (X,X) X (X - X)] (z X w)}
= Р1*Т^ {[171 (X,X) X (X - X)] n(z X «)}
= ±/*pi*{[l7| (X - X) x (X,X)] r> (w X z)}
where pi: (X — X) X X —> X — X is projection to the first factor. Again by
property 6.1.6,
306	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP, g
{[17| (X - X) X (X,X)] (w x Z)} = Yt/z) w = w
Therefore
Kzyu(w) = ±i*(w)
Similarly, we prove commutativity up to sign in the triangle
H«(X) Hn-q(X,X)
/Tr
№(X - X)
For v E №(X), by property 6.1,5,
YuM«) = [G | (X - X) X (X,X)]/(u z)
= {[Uop*(n)]|(X-X) x(X,X)}/z
By lemma 11 and property 6.1.4, this equals
±{[pH*(O	17] | (X — X) X (X,X)}/Z = ±/*(г) о yv(z) = ±/*(n)
Therefore
7uKz(v) = ±j*v 
41 THE ALEXANDER COHOMOLOGY THEORY
We shall now describe a cohomology theory particularly suited for applications
in which a space is mapped into polyhedra (the singular theory is more suitable
for applications where polyhedra are mapped into a space). One approach to
the theory, called the Cech construction, is based on approximating a space
by nerves of open coverings; another approach, called the Alexander-Kolmo-
goroff construction, is based on complexes built of “small” simplexes consisting;
of finite sets of points. We shall begin with the Alexander construction, and
show later in the chapter (see corollary 6.9.9 and the following paragraph)
that if (A,B) is a closed pair in a manifold X, then №(A,B; G) as defined in
Sec. 6.1 is the Alexander cohomology of (A,B) with coefficients G.
Let G be an В module and let X be a topological space. For q > 0 let
Ce(X;G) be the module of all functions <p from Xe+1 to G with addition and
scalar multiplication defined pointwise. Thus, if Xo, Xi, . . . , xq £ X, then
<p(xo,xi, . . . ,x5) E G, and if <pi, <p2 € Ce(X;G) and r E B, then
npi(x0, . . . ,xQ) = r(<pi(xo, . . . ,x5))
(<Pi + <Рг)(х0, • •  ,xe) = <pi(x0, . . . ,xe) + <рг(х0, . . . ,x5)
We shall omit the symbol G from Ce(X;G) where its absence will not cause
confusion.
A coboundary homomorphism й: C®(X) —> Ce+1(X) is defined by the
formula
SEC. 4 THE ALEXANDER COHOMOLOGY THEORY	30/
(3<p)(Xg, . . .	— S (	1)Чр(т‘о, . . . ?X*2, . . . ,Хд+1)
0<j<q+1
Then 33 = 0 and C*(X) = {Се(Х),й} is a cochain complex over R. If X is
nonempty, it is augmented over G by y: G —> C°(X), where (r](g))(x) = g for
g € G and all x G X. So far the topology of X has played no role, and the fol-
lowing result shows that С * (X) has uninteresting cohomology.
I lemma If X is a nonempty space, y*: G ~ H*(C*(X;G)).
proof Let x be a fixed point of X and define a cochain homotopy
p: C*(X)^ C*(X) by
(P<p)(x0, . . . ,xQ) = <p(x,x0, . . . ,Xq) q>0
Then	8D<p + D8q> = Г45	dcg T > 0
1<P — 7}(<p(x)) deg <p = 0
Therefore, if t: C(X;G) —> G is the cochain map defined by
T(<p) = Г0	Ф > 0
'	(<p(x) deg <p = 0
then ту = 1g and D is a cochain homotopy from to yr. Therefore у is a
cochain equivalence, whence the result. 
We now use the topology of X to pass to a more interesting quotient com-
plex. An element <p £ C®(X) is said to be locally zero if there is a covering of
X by open sets such that <p vanishes on any (q + l)-tuple of X which lies in
some element of <21, Thus, if we define 9lf/+1 =	<A+1 c Xq+1, then <p
vanishes on 0р+1. The subset of Ce(X) consisting of locally zero functions is a
submodule, denoted by CoQ(X), and if <p vanishes on then 8<p vanishes
on W+2, whence C$(X) = {Сов(Х),б} is a cochain subcomplex of C*(X). We
define C*(X) to be the quotient cochain complex of С* (X) by (X). If X is
nonempty, the composite
СЛ C*(X)^ C*(X)
is an augmentation of С * (X), also denoted by y. The cohomology module of
C*(X) of degree q is denoted by №(X;G).
Given a function f: X —> Y (not necessarily continuous), there is an in-
duced cochain map
/#: C*(Y;G)^ C*(X;G)
defined by the formula
(f#<p)(x0, . . . ,Xq) = <p(f(xo), . . . ,f(xe)) <p E 0®(Y); Xo, . . . ,xq£X
If <p vanishes on “А1 «+1, where is an open covering of Y, and if there is
an open covering of X such that f maps each element of Qt into some ele-
ment of % then f#q> vanishes on 9L'711. In particular, if f is continuous, f~
is an open covering of X which can be taken as ^L, and therefore f#
308
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. 6
maps C^(Y) into G$(X). It follows that if/is continuous, there is an induced
cochain map
/#: C*(Y;G)—> C*(X;G)
Let A be a subspace of X and let i: А С X. Then!#: C* (X;G) —> C * (A;G)
is an epimorphism. Therefore the kernel of i# is a cochain subcomplex
of C*(X;G), denoted by C*(X,A; G). The relative module №(X,A; G) is de-
fined to be the cohomology module of C*(X,A; G) of degree q.
Since there is a short exact sequence of cochain complexes
0-> C*(X,A; G)	C*(X;G)	C*(A;G)~» 0
it follows that there is an exact sequence
2 -------> №(X,A; G) A №(X;G)	№(A;G)	№+1(X,A; G)->---
The graded module H*(X,A) = [№(X,A; G)} is the module function of
the cohomology theory we are constructing, and the homomorphism
5* : №(A;G) —> №+1(X,A; G) is the connecting homomorphism of the theory.
Given a continuous map/: (X,A) —> (Y,B), there is induced by /a commuta-
tive diagram of cochain maps
0	C*(Y,B; G) C*(Y;G)	C*(B-,G)	0
0 -> C*(X,A; G) -> C*(X;G)	C*(A-G)	0
The homomorphism /*: H*fY,B; G) —> H*fX,A-, G) is defined to be the
homomorphism induced by the cochain map /# in the above diagram. It is
then clear that for fixed G, H*(X,A; G) and/* constitute a contravariant
functor from the category of topological pairs to the category of graded
В modules. Furthermore, the connecting homomorphism 8* is a natural
transformation of degree 1 from H*(A;G) to H*(X,A; G). Therefore we have
the constituents of a cohomology theory, and we shall verify that the axioms
are satisfied. The resulting cohomology theory is called the Alexander (or
Alexander-Spanier1) cohomology theory, and №(X,A; G) is called the Alex-
ander cohomology module of (X,A) of degree q with coefficients G.
The exactness axiom is a consequence of the exactness of the sequence 2.
The dimension axiom will follow from the next result.
3 lemma If X is a one-point space, y*: G ~ H* (X;G).
proof Because X is a one-point space, a locally zero function on X is zero.
Therefore C*(X;G) = C*(X;G) and the result follows from lemma 1. 
Before proving the excision axiom it will be useful to introduce another
cochain complex for the relative theory. If А С X, let C * (X,A) be the sub-
1 See E. Spanier, Cohomology theory for general spaces, Annals of Mathematics, vol. 49,
pp. 407-427, 1948.
Sgc. 4 THE ALEXANDER COHOMOLOGY THEORY
309
complex of С* (X) of functions <p which are locally zero on A. Thus there is a
short exact sequence
0-> C*(X,A)	C*(A)	0
and Cg(X) С С*(ХЛ). It follows that C*(X,A) = C*(X,A)/Cg(X). The
excision axiom follows from the next result.
4 lemma Let U be a subset of А С X such that U has an open neighbor-
hood W with W C int A. Then the inclusion map p. (X — U, A — U) C (X,A)
induces an isomorphism
i#; C*(X,A) ^C*(X - U,A - U)
proof There is a commutative diagram with exact rows
0-»Cg(X)	~^C*(X,A)	—» C*(X,A)	0
>1	lk*	JZ
o Cg (X - U) С* (X - U, A - 17) Д. с* (X - U, A - 17)	0
It suffices to prove that X/c# is an epimorphism and that (fc#)-1(Cg (X — U)) =
Cg(X). If ф E Сч(Х — 17, A — U), let <p E CX(X) be defined by
_ 10	Xi E W for some 0 < i < q
<p(x0, . . . ,xq) -	x0, ... ,xq EX - W
If % is an open covering of A — U such that <p vanishes on W1, then
<?L = {VU W| V E CV'} is an open covering of A such that ф vanishes on
7lf/H. Therefore ф E CQ(X,A), and from the definition of ф, к#(р — <p vanishes
on 5 6№e+1 where "Ilf = {V Г) int A | V E “V) U {X — W), which is an open
covering of X — U. Therefore \k#<p = X<p, and because X is an epimorphism,
so is hk#.
Assume that <p E C^(X,A) is such that k#q> E Coe(X — U). Because <p is
locally zero on A, there is an open covering "?li of A such that <p vanishes on
Because k#<p E Coe(X — U), there is an open covering of X — 17
such that <p vanishes on 712e+1. Let
%. = {I7i П int A I I7i E 711} T2 = {U2 П (X - I7)| U2 E 712}
Then C'V = t'i|i U “Х'г is an open covering of X such that <p vanishes on
4®+1. Therefore <p E Coe(X) and so
(fc#)-i(Cg(X- l7)) = Cg(X) =
The homotopy axiom will be proved in the next section. We conclude
this section with a study of H°. A function <p from a topological space X to a
set is said to be locally constant if there is an open covering Ql of X such that
<p is constant on each element of 7l.
5 theorem If А С X, then H°(X,A; G) is isomorphic to the module of
locally constant functions from X to G which vanish on A.
310	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. 6
proof A locally zero function from X to G is zero. Therefore C0°(X) = 0,
and so
C°(X,A) = C0(X,A)/C0°(X) = C°(X,A)
Therefore Й°(Х,А; G) is the kernel of the composite
C°(X,A) A Сг(Х,А) C^X’A)
C°(X,A) is the module of functions from X to G which vanish on A. If
<p G C°(X,A), then <p is in the kernel of the above composite if and only
if there is some open covering of X such that S<p vanishes on qi2.
Since (8<p)(x,y) = <p(y) — <p(x), this is equivalent to the condition that there is
an open covering such that <p is constant on each element of Ql. Hence the
kernel of the above composite equals the module of functions vanishing on A
that are locally constant on X. 
6	corollary Let Xbea topological space in which every quasi-component
is open and let А С X. Then H°(X,A; G) is isomorphic to the module of func-
tions from the set of those quasi-components of X which do not intersect
A to M.
proof This follows from theorem 5 and the fact that a locally constant
function on X is constant on every quasi-component of X. 
7	corollary A nonempty space X is connected if and only if
q*-. G^H°(X-G)
proof This follows from theorem 5 and the trivial observation that every
locally constant function on X is constant if and only if X is connected. 
It follows that there exist spaces for which the singular cohomology and
Alexander cohomology differ. In fact, for any connected space which is not
path connected, corollary 7 and theorem 5.4.10 show that they differ in
degree 0.
We now present a version of theorem 5.4.10 valid for the Alexander
theory.
8	theorem Let {Ц} be an open covering of X by pairwise disjoint sets.
Then there is a canonical isomorphism
№(X-G) ~ X №(UpG)
proof Because { Uj} consists of pairwise disjoint sets, the map induced by
restriction
i#: C*(X)-> X C*(U,)
is an epimorphism. Because {Ц} is an open covering of X, it follows that
R-^X Cg(l7;)) = Cg(X)
Therefore i# induces an isomorphism С* (X) X C* (Uj). 
SEC< 5 THE HOMOTOPY AXIOM FOR THE ALEXANDER THEORY	,3/7
<; corollary Let {GJ be the collection of components of a locally con-
nected space X. Then there is a canonical isomorphism
>	№(X;G) x №(Q;G)
proof Because X is locally connected, its components are open, and the re-
sult follows from theorem 8. 
I 5 THE HOMOTOPY AXIOM FOB THE ALEXANDER THEOBY
!
| In this section we shall prove the homotopy axiom for the Alexander cohomology
' theory. The proof will be based on a description of the Alexander cochain
1 complex as the limit of cochain complexes of abstract simplicial complexes.
[ We shall also use this description to construct a homomorphism of the
j Alexander cohomology theory into the singular cohomology theory. Because
j the Alexander theory satisfies all the axioms, this homomorphism is an isomor-
phism from the Alexander theory to the singular cohomology theory on the
category of compact polyhedral pairs.
We shall be considering a fixed R module G as coefficient module for
1 cohomology and will usually not mention G explicitly. Let be a collection
t of subsets covering a set X. Let X(9() be the abstract simplicial complex
whose vertices are the points of X and whose simplexes are finite subsets F of
X such that there is some U £ Ql containing F. Let C(9l) be the ordered chain
complex of X(9l) over R. Given a subset А С X and a subcollection ЧТ C 4*1
which covers A, we let A(Ql') be the subcomplex of X(9l) whose vertices are
J the points of A and whose simplexes are finite subsets of A lying in some ele-
• ment of Ql'. Then С'(%') will denote the chain subcomplex of ClflC) cor-
responding to A(^L').
Let (%T') be another pair consisting of a covering *¥ of X and a subset
T' C which is a covering of A. Assume that (fifT') is a refinement of
>, ill the sense that every element of Wis contained in some element of and
; every element of is contained in some element of Qi'. Then the pair
	is mapped injectively into the pair (C^l),C'('?l')) by the identity
I map of (X,A) to itself.
I Let X be a topological space and A a subspace of X. Consider pairs
where is an open covering of X and Ql' is a subset of Ql which covers A.
Such a pair is called an open covering of (X,A). Let C * (QL,l71') be the
I cochain complex of the pair (С(Ч’1),С'(ч)1')) (with coefficients in G). An element
j и of Ci^[,<?!') is a function defined on (q + l)-tuples of X which lie in some
1 element of ^l, taking values in G, and vanishing on (q + l)-tuples of A which
| lie in some element of ^l'. If (%%') is a refinement of the restriction
1 map is a cochain map
C*	C* (%W')
312	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP, j
If (91,91') and (%У') are two open coverings of (X,A) as above,
9lf = {U n V| U E 91, V E У} and let 9lf' = {U П V' | U' E 9l', V E Vj; '
Then (9lf,9b'') is another open covering of (X,A) and (9lf,9lf') is a refinement of
(9l,9L') and of (%У'). Therefore the cochain complexes {C*(9l,9l')} from a
direct system, and we have a Emit cochain complex
lira , {C*(91,9l')}
We shall show that this limit cochain complex is canonically isomorphjt,
to C* (X,A). If <p E Ce(X,A), let 9l' be a collection of open subsets of X covers
ing A such that <p vanishes on (9l')«+1 Г) Ae+1 (such а 9Г exists because <p is
locally zero) and let 9l = 9t' U {X}. Then (91,9l') is an open covering of (X,/Q
and <p determines by restriction an element <p | (9l>91') E Ce(91,9l'). Passing to
the limit, we obtain a homomorphism (by restriction)
X: C*(X,A)^ linu {C*(9l,9l')}
which is a canonical cochain map. The following result explains our interest
in the cochain complexes C*(9l,9l').
I theorem The canonical cochain map
X: C * (X,A) linu { C * (9l,9l')}
is an epimorphism and has kernel equal to C$(X).
proof To prove that \ is an epimorphism, let и E Ce(9l,9l'). Define
€ C5(X) by
.	. f t/.(x0, . . . ,x0) if x’o, . . . , x0 E U, where U E 9l
<p„(x0, . . . ,x5) = J()	otherwise
Then <p„ vanishes on (9l')e+1 Г) A’11, and therefore <p„ E Ca(X,A). By defini-
tion, <p„ | (9l,9l') = u, and X is an epimorphism.
An element <p E С<ЦХ,А) is in the kernel of X if and only if there is some
(91,91') such that <p | (9l,9l') = 0. Thus X(<p) = 0 if and only if there is some
open covering 9l such that <p vanishes on 9l®+1. By the definition of C'fi(X),
X(<p) = 0 if and only if <p E Cft (X). 
From lemma 1 and the analogue of theorem 4.1.7 for cochain complexes,
we have the following corollary.
2	corollary For the Alexander cohomology theory there is a canonical
isomorphism
№(X,A; G) ~ linu {№(C* (9l,9l'; G))} 
We are now ready for the proof of the homotopy axiom for the Alexander
cohomology theory. In the presence of the other axioms, it suffices to prove it
for the case of the two mappings
ho, hi: (X,A) (X X I, A X I)
SEC. 5 THE HOMOTOPY AXIOM FOB THE ALEXANDER THEORY
313
(Y,n) = I
where h0(x) = (x,0), /i|(x) = (x,l). The proof consists in showing that if (Ql,^t')
is any open covering of (X X I, A X I), there is an open covering (%“¥') of
(X,A) such that ho and /i| induce chain-homotopic chain maps from
(Cf^1 ),С(Т')) to (C(Ql),C(9l')). This is a result about free chain complexes, and
the technique of acyclic models is available for obtaining the desired chain
homotopy.
Let Y be an arbitrary set and n a nonnegative integer. Let C(Y,n) be the
chain complex over R of the abstract simplicial complex (У X I)(Ql(Y,n)),
where L?l( Y,n) is the covering of Y X I defined by
л7 fm m + 111 „ .	,
Y X —,---------- 0 < m < 2?i
12»	2« JI —
3	lemma If Y is nonempty, the chain complex C(Y,ri) is acyclic.
proof For 0 < m < 2" let Km be the subcomplex of (Y X I)(^L(Y,n)) con-
sisting of all the finite subsets of Y X [m/2n, (m -|- l)/2»]. For 0 < m < 2” let
Lm be the subcomplex of (Y X I)(Ql(Y,n)) consisting of all the finite subsets of
Y X (m/2«). Then (Y x I)(^l(Y,n)) = U»> and К; П K} = 0 if |i - ;| > 1
and Ki Г) Ki+i = Li+i- Because Y is nonempty, each Km (and Lm) is non-
empty and is the join of Kn (or Lm) with any vertex in it. Therefore,
by theorem 4.3.6, C(Km) and C(Lm) are acyclic. Let Ng = UCT.<5Km. Then
Nq+i = Ng U Kg+1 and Ng Г) Kg+± = Lq+i- By induction on q, using the
exactness of the reduced Mayer-Vietoris sequence, it follows that C(Nq) is
acyclic for all q. Therefore C(Y,n) = C(V2«-i) is acyclic. 
From this we have our next result, which will provide the acyclic model
for the homotopy axiom.
4	lemma Let Yi, . . . , Ys be subsets of a nonempty set Y, where
Y = Yi, and fen' each i let rq be a nonnegative integer. Let К be the simplicial
complex defined by
к=щ%х
i
Then C(K) is acyclic.
proof We prove the lemma by induction on q. If q = 1, it follows from
lemma 3. Assume that q f> 1, and the result is valid for fewer than q sets Y;.
Let К = Uug-r (Yj X 1)(<?1(^,п4)). Then К U (Y5 X I)(3l(Y5,n5)) = K. If Y5
is empty, C(K) = C(K) is acyclic, by the inductive assumption. If Ye is non-
empty, С(Тд,Пд) is acyclic, by lemma 3, and C(K') is acyclic, by the inductive
assumption. To prove that C(K) is acyclic, from the exactness of the reduced
Mayer-Vietoris sequence it suffices to prove that CfK П (YQ X I)(Ql(Y5,n5)))
is acyclic. However, К Г) (Ye X I)(^l(Ye,ne)) = UL<i<e (Y( x I)(^l(Yj.>n'j)),
where Y( = Y$ Г) Ye are subsets of YQ (and Yj = YQ) and nj = max (п,,пе).
Therefore, by the inductive assumption, C(K Г) (Ye X I)(^l(Y5,n5))) is
acyclic. 
314
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. 6
We now come to the following main step in the proof of the homotopy
axiom.
5	lemma Let (71,71') be any open covering of (X у I, A X I)- There is an
open covering (CV',‘V') of (X,A) such that h0 and hi induce chain-homotopic
chain maps from (C^^C'^1')) to (C(7l),C'(7l')).
proof For each x £ X it follows from the compactness of x X I that there
is an open set V:c about x and an integer n > 0 such that for 0 < m < 2n the
set Vx X [m/2”, (m + l)/2”] is contained in some element of 7l. Furthermore,
if x £ A, we can choose and n so that V® X [m/2”, (m + l)/2n] is con-
tained in some element of 7l'. Let c¥be the collectionand T' the sub-
collection { V®To show that (fTfCf has the desired property, let Й be the
category consisting of the subcomplexes of X(“V) partially ordered by inclu-
sion. For each subcomplex К of X(T) let G(K) be the ordered chain complex
of K. For each simplex s of XlfT) [or AfV')] define n(s) to be the smallest non-
negative integer such that for 0 < m < n(.v) each set a X [m/2n, (m + l)/2n]
is contained in some element of 7l [or 7Г]. Such an integer exists because of
the way (fifil') was chosen. For a subcomplex К of X(fT) let К be the
subcomplex of (X X I)(71) defined by
К = U {(s X I)(fil(s,n(s)) | s £ K]
and let G'(K) be the ordered chain complex of K. Then G and G' are covari-
ant functors from (2 to the category of augmented chain complexes.
Let 2)11 be the set of subcomplexes {s C X(“V) | s C XfV)}. Then G is free
on C with models 2Ж, and by lemma 4, G' is acyclic on (3 with models OIL If
о = (xo,xi, . . . ,xq) is an ordered q-simplex of X(T), then
h0(o) = ((xo,O), . . . ,(xe,I)) and /ц(о) = ((x0,l), . . . ,(xe,l))
are natural chain maps preserving augmentation from G to G. It follows from
theorem 4.3.3 that there is a natural chain homotopy from ho to hi. 
If и € №(C* (71,71')), where (71,71') is an open covering of (X X I, A X I),
it follows from lemma 5 that there is an open covering (<V,‘V') of (X,A) such
that ho^V) C (7l,7l'), hi(%V') C (^l,Ql'), and h$u = h*u in №(C*(W)).
Passing to the limit and using corollary 2 gives us the final result.
6	theorem The Alexander cohomology theory satisfies the homotopy
axiom. 
We have now verified all the axioms of cohomology theory for the
Alexander cohomology theory. We construct a homomorphism p from the
Alexander cohomology theory to the singular cohomology theory. Let (7l,7l')
be an open covering of (X,A). There is a canonical chain transformation
Г) A)) -+ (C(7t),C'(7l'))
which assigns to a singular q-simplex о: A'; —> X the ordered simplex
((fiyofcfvi), . . . ,o(uff)) of C(7l). This induces a homomorphism
SEC. 6 TADTNESS AND CONTINUITY	315
C*(W; G) С*(Д(<?1), Д(ч>1' П A); G)
passing to the limit and using corollary 2, we obtain a canonical homomorphism
p': №(X,A; G) -+ linu	Д(<?Г П A); G)}
py theorem 4.4.14, there is a canonical isomorphism
p": №(Д(Х), Д(А); G) lirru	Д(ОГ П A); G)}
and the homomorphism
p: №(X,A-, M) №(Д(Х), Д(А); M)
is defined to equal the composite p"1//. It can be verified that this homomor-
phism has the commutativity properties necessary to be a natural transforma-
tion of cohomology theories.
We now introduce a cup product in the Alexander theory, which will
have the usual properties of a cup product (as in Sec. 5.6) and will be com-
patible with the singular cup product by the homomorphism p.
Let G and G' be R modules paired to an R module G". Given
<pi € Cp(X;G) and <p2 € 0®(X;G'), we define <pi о <p2 € Cr,+e(X;G") by
(<P1 O <P2)(x0, • • • ,Xp+q) = (pl(x0, . . . ,Xp)(p2(xp, . . . ,xp+5)
If <pi is locally zero on Ai, so is <pi о <p2, and if <p2 is locally zero on A2, so is
(pi о (p2- Therefore <pi о <p2 induces a cup product from Cp(X;G) and
&(X;G') to Cp+e(X;G"). An easy verification shows that
8(ф1 о (p2) = S<pL о <p2 + ( —IJPqDj	S(p2
Therefore the cup product induces a cup product on cohomology classes, and
this cup product is clearly mapped by p to the singular cup product.
In order to get a cup product from С₽(Х,А1; G) and C''(X,A2; G') to
С₽+е(Х, Ai U A2; G"), we need to ensure that an element of Cr+f/(X;G")
which is locally zero on Ai and locally zero on A2 will be locally zero on
Ai U A2. If Ai U A2 = int/^ ./jA! U int/, ил2А2, this is so. With this modi-
fication properties 5.6.8 to 5.6.12 are all valid for the resulting cohomology
product.
W TAUTNESS AND CONTINUITY
In this section we shall consider tautness for the Alexander theory and estab-
lish the strong result that any paracompact space imbedded as a closed sub-
space of a paracompact space is taudy imbedded. This implies a strong excision
property for paracompact pairs (X,A) with A closed in X. It also implies the
continuity property (that the Alexander cohomology theory commutes with
limits of compact Hausdorff spaces directed by inclusion). This continuity
property, together with the other axioms of cohomolgy theory, characterizes
316	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP, fj
the Alexander theory on the category of compact Hausdorff pairs (that
pairs with X compact Hausdorff and A closed in X). The section closes with ‘
a brief discussion of the Alexander cohomology with compact supports. C)ur
proof of the special tautness properties of the Alexander cohomology theory js
based on techniques of Wallace.1
Let be a collection of subsets of a set X. Let ЭД* = [U* where
U* = U {U'(= П 177^0}	j
A collection T is said to be a star refinement of if %* is a refinement of ^i. i
A topological space X is said to be fully normal if every open covering of X J
has an open star refinement. It is known that for Hausdorff spaces paracom-
pactness is equivalent to full normality.	1
I lemma Let Abe a subset of a topological space X and let “Ybe an open !
covering of X. There exist a neighborhood N of A and a function f.N^A ’
(not necessarily continuous) such that	1
(a)	fix) = x for x E A.	i
(b)	IfVE^, then fiV П N) С V*.	I
proof If A is empty, let N = A and /be the identity map. If A is nonempty, j
let N = U {V E A"| V П A 0} and define /: N —> A by fix) = x for x £ A, !
or if x £ A, choose fix) £ A so that there is V C АГ with x, fix) £ V. Such a
choice of fix) is always possible because of the way N was defined. Clearly, if •
x E V Г) N, there is V E АГ with x, fix) £ V. Therefore x E V П V and
V' С V*. Hence, /(V П IX) С V* and (a) and (b) are satisfied. 
This last result may be interpreted as asserting that A is a discontinuous
neighborhood retract of X with a retraction that is not too discontinuous, s
If A is a closed subset of a paracompact space, it is similar enough to an abso- |
lute neighborhood retract so that we have the following generalization of )
theorem 6.1.10 for the Alexander theory.
2	theorem A closed subspace of a paracompact Hausdorff space is a
taut subspace relative to the Alexander cohomology theory.
proof Let A be a closed subspace of a paracompact space X and let j
<p E C"(A) be a cochain such that S<p vanishes on ЭД ®+1, where ЭД is an open Г
covering of A. Let *?l = { W U (X — A) | W E ЭД} and observe that Ql is an '
open covering of X because A is closed in X. Let be an open star refinement i
of and let N be a neighborhood of A and f: N^An function (not neces- '
sarily continuous) satisfying lemma 1 relative to A? Then f#rp E CfiN), and we 1
show that 8f#<p = f#8<p vanishes on П №+1. By lemma lb, for any j
V E ATthere is U E ЭДsuch that/(V Г) N) C 17. Then fiV Г) N) C U fl A C W
for some W E ЭД- Therefore 8f #<p vanishes on (V Г) A')®+1. This means that
f#rp represents a cocycle of CrfN) and, by lemma la, (/#<p) | A = <p. Henci I
1 See A. D. Wallace, The map excision theorem, Duke Mathematical Journal, vol. 19, ।
pp. 177-182, 1952.
SKC. ® TAUTNESS AND CONTINUITY	3J7
Фе cohomology class (<p) E №(A) is the image under restriction of the coho-
mology class {/#<p} E HfiN), showing that lim, {№(N)} HfiA) is an
epimorphism.
To prove that it is a monomorphism, let N' be a paracompact neighbor-
hood of A and assume that <p E Ca(N') is such that Sep vanishes on ЭД’®+2 and
ф | A = 8ff on (ЭД')®+1, where ЭД is an open covering of N' and ЭД' is an open
covering of A. Let Ql = {W' U (N' — A) | W E ЭД'} and observe that is an
open covering of N' (because A is closed.) Let Abe an open star refinement
of both ЭД and Ql(A'is a covering of N') and let N be a neighborhood of A in
N' and f: N A a function (not necessarily continuous) defined with respect
to A‘ to satisfy lemma 1. If V E % then fi V П N) C W for some W E ЭД.
Therefore /#(<p | A) = 8f#tp' on V®+1 П №+1.
To show that ffiep | A) is cohomologous in CfiN) to <p | N, for / E Cp(N)
define Df E Ci’fiX) by
(Df)(x0, . . . ,xp_i) = S (-l)ty(x0, . . . ,Xj,fixj), . . . ,/(xp_i))
0<;<p—1
An easy computation establishes the formula
8Df + D8f = f#(f | A) - /
For every V E АГ (V П N) U /(V Г) N) C Wfor some WE Ilf (by lemma lb),
and because Sep vanishes on ЭД'®4-1, 8D(tp | N) = ffitp | A) — <p | N on
Ar®+1 П №+1. Therefore the cohomology class {<p} E №(N') maps to zero in
• H®(N). This suffices to show that liiru (№(N)} —> №(A) is a monomorphism,
nd so A is a taut subspace of X. 
3	corollary Let X D A D B, where X is a paracompact Hausdorff
space and A and В are closed subspaces of X. Then, relative to the Alexander
cohomology theory, (A,B) is a taut pair in X.
proof This is an immediate consequence of theorem 2 and lemma 6.1.9. 
4 example Let X be the subspace of R2 C S2 defined in example 2.4.8.
The space X obtained by retopologizing X by the topology generated by the
path components of open sets in X is a half-open interval. Since X has the
same singular homology as X, H\XfI) = 0. Since S2 — X has two components,
it follows from the Alexander duality theorem that lim , {Hi(l7;Z)} = Z as 17
Varies over neighborhoods of X. Therefore linu (W-fUfZ)} H\X-,T) is not
a monomorphism, and so X is not a taut subspace of R2 with respect to sing-
ular cohomology. Since X is closed in R2, it is taut with respect to Alexander
cohomology.
Note that the above example is one in which linu {HhJJfL)} H2(X;Z) is
'Ot a monomorphism, whereas in example 6.1.8 a subspace A C R2 was given
rich that lim_, {H°(17;Z)} —> HfiAfZ) was not an epimorphism.
The tautness property 3 implies that the Alexander cohomology theory
satisfies the following strong excision property.
318	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP, (j
5	theorem Let (X,A) and be pairs, with X and Y paracompact: '
Hausdorff and A and В closed. Let f: (X,A) —> (Y,B) be a closed continuous '
map such that f induces a one-to-one map of X — A onto Y — B. Then, fOr
all q and all G
f*: №(Y,B; G) ~ H«(X,A; G)
proof Because f is closed, continuous, and one-to-one from X — A onto
Y — B, it follows that f is a homeomorphism of X — A onto Y — B. Let { [7,J
be the family of open neighborhoods of В in Y and let Va = f^fUf). Then
Va is an open neighborhood of A in X, and because f is a closed map, the col-
lection { Va) is cofinal in the family of all neighborhoods of A in X. We have
a commutative diagram
№(Y,B) e- linu {№(Y,UO)} linu {№(Y - B, Un - B)}
№(X,A) linu {№(X,Va)} linu (№(X - A, Va - A)}
in which the vertical maps are induced by f and the horizontal maps are
induced by inclusions. By corollary 3 and lemma 6.4.4, the horizontal maps
are isomorphisms. Because f | X — A is a homeomorphism of X — A onto 1
X — B, it follows that for each a, f \ (X — A, Va — A) is a homeomorphism of |
(X — A, V,, — A) onto (Y — B, Ua — B). Therefore is an isomorphism,
and by commutativity of the diagram, f* is also an isomorphism. 
The following weak continuity property of the Alexander cohomology
theory is another consequence of its tautness properties.
6	theorem Let {(Xn,Aa)}„ be a family of compact Hausdorff pairs in
some space, directed downward by inclusion, and let (X,A) = (П Xa, П Aa).
Tire inclusion maps ia: (X,A) C (Xa,A„) Induce an isomorphism
{i* }: linu №(Xa,Aa- M) ~ №(X,A; M)
proof If F is a closed subset of Xp for some /?, the collection (Xtt. Г1 F}„
consists of compact sets directed downward by inclusion, and X P F =
П (XQ О F). It follows that if X P F = 0, there is some a such that
X„ b F = 0. Therefore, if U is any neighborhood of X in Xp, there exists a
such that X,. C U. Similarly, if (17,V) is any neighborhood of (X,A) in Xp,
there is a such that (XQ,AQ) C (G,V).
To show that (i* } is an epimorphism, let и E №(X,A) be arbitrary. For
any /3, (X,A) is a taut pair in Xfi, by corollary 3. Therefore there is a neighbor-
hood (U,V) of (X,A) in Xp and an element v £ №(U,V) such that v | (X,A) = u.
Let a be such that (X„,A„) C (U,V) and va = v | (Xa,Aa). Thenr,. £ №(Xa,Aa)
and i$ va = h, which proves that {i« } is an epimorphism.
To prove that {i* } is a monomorphism, let и € Нч(Хр,Ар) be such that
iffu = 0. By corollary 3, (X,A) is a taut pair in Xp. Therefore there is a
neighborhood (U,V) of (X,A) in Xp such that и | (U, V Г) Ap) = 0. Choose a
SEC. 6 TAUTNESS AND CONTINUITY
319
so that (X^Af) C (17, V Г) Af). Then u| (Xa,Aa) = 0, and {/*} is an
isomorphism. 
The continuity property involves an assertion analogous to that of
theorem 6 for an arbitrary inverse system {(X,„A(,)} of compact Hausdorff
pairs, where (X,A) = lim. {(XOTAa)}. It is not hard to prove that the continuity
property is equivalent to the weak continuity property.1 A cohomology theory
having the weak continuity property is called weakly continuous. Such
theories are characterized on the category of compact Hausdorff spaces in
view of the following result.
7	lemma Any compact Hausdorff pair can be imbedded in a space in
which it is the intersection of a family of pairs directed downward by inclu-
sion, each pair of the family being a compact Hausdorff space of the same
homotopy type as a compact polyhedral pair.
proof It is a standard fact that any compact Hausdorff space can be imbedded
in a cube IJ; hence we assume (X,A) imbedded in П. For each finite sub-
set a C J let pa: IJ Ia be the projection map and let (U,V) be a compact
polyhedral neighborhood of (pa(X),pQ(A)) in Iя. It can be verified that the col-
lection of pairs {(.pa^1(.U),pa~1(V')')} corresponding to all finite a CJ and com-
pact polyhedral neighborhoods of (pa(X),pa(A)) in Ia is directed downward by
inclusion and has (X,A) as intersection. Furthermore, (pa-1(t7),ptt"1(V)) is a
compact pah in IJ homeomorphic to (U,V) X IJ~a, and the projection map
Pa- (Pa-KUlPa-^V)) (U,V)
is a homotopy equivalence. Therefore the family {(pa^1(t7),pQ-1(V))} has the
desired properties. 
This yields the following extension of the uniqueness theorem for weakly
continuous cohomology theories.
8	theorem Given two weakly continuous cohomology theories, any
homomorphism between them which is an isomorphism for some one-point
space is an isomorphism for all compact Hausdorff pairs. 
We now describe the Alexander cohomology with compact supports.
This is a cohomology theory on a suitable category of topological pairs and
maps, and we shall discuss the category first.
A subset A of a topological space X is said to be bounded if A is compact.
A subset В С X is said to be cobounded if X — В is bounded. A function f
from a space X to a space Y is said to be proper if it is continuous and if for
every bounded set A of Y, f -1(A) is a bounded set of X (or, equivalently, for
every cobounded set В of Y, /“1(B) is a cobounded set of X). Clearly, the
composite of proper maps is proper, and there is a category of topological
spaces and proper maps. There is also a category of topological pairs and
1 See S. Eilenberg and N. E. Steenrod, “Foundations of Algebraic Topology,” Princeton Univer-
sity Press, Princeton, N.J,, 1952, or exercise 6.C.2 at the end of this chapter.
320
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. 6
proper maps, a proper map from (Х.Л) to (T,B) being a proper map from X
to Y which maps A to B. This is the category on which the Alexander coho-
mology theory with compact supports will be defined.
Given a topological pair (X,A), let Cce(X,A; G) be the submodule of
C«(X,A; G) consisting of all cp G C®(X,A; G) such that <p is locally zero on
some cobounded subset of X. If <p is locally zero on B, so is 8<p, and therefore
there is a cochain complex Cj?(X,A; G) = {Cfv(X,A; G), 5} which is a sub-
complex of C*(X,A; G). Clearly, C$(X;G) C Cj?(X,A; G), and we define
C*(X,A; G) = C*(X,A; G)/Cg(X;G)
The Alexander cohomology of (X,A) with compact supports, denoted by
17? (X,A; G), is the cohomology module of C?(X,A; G). If/: (X,A) —> (Y,B)
is a proper map, /# maps C?(Y,B; G) to Cj!:(X,A; G) and induces a
homomorphism
f*: H*(Y,B; G) -+ H*(X,A; G)
The Alexander cohomology with compact supports satisfies suitable modifica-
tions of all the axioms of cohomology theory.
The homotopy axiom holds for proper homotopies, a proper homotopy
being a proper map (X,A) X I (Y,B). In general, an inclusion map
(X',AZ) C (X,A) is not a proper map. It is a proper map, however, if X' is
closed in X. Because of this, the coboundary homomorphism
fi* : HC®(A;G) -> f/.v' VX.A; G)
is defined only when A is a closed subset of X. When A is a closed subset of
X, there are proper inclusion maps i: А С X and j: X C (X,A) and there is a
short exact sequence of cochain complexes (for any coefficient module G)
О С* (X,A) Д C* (X) Д C* (A) -a 0
The connecting homomorphism of this short exact sequence is a natural trans-
formation from H ? (A) to H * (ДА), of degree 1 on the category of pairs
(X,A), with A closed in X and proper maps between such pahs. The exact-
ness axiom then holds for pairs (X,A) with A closed in X.
The excision axiom holds for proper excisions, a proper excision map
being an inclusion map j: (X — U, A — U) C (X,A) such that U is an open
subset of X with U C int A, in which case it can be shown (analogous to the
proof of lemma 6.4.4) that
/#: С* (X,A) ~ С*(X - U, A - IT)
The dimension axiom is obviously satisfied.
We now consider relations between the Alexander cohomology with
compact supports and the Alexander cohomology theory previously defined.
The following is one case in which they agree.
9 lemma If A is a cobounded subset of X, then
H*(X,A-, G) = H*(X,A; G)
SEC. 6 TAUTNESS AND CONTINUITY	321
proof Because A is cobounded in X,
C*(X,A) = С*(ХД)
andso С*(ХД) = C*(X,A). в
IО lemma Let В be a closed subset of a Hausdorff space A. Then a subset
[J of A — В is cobounded in A — В if and only if U U В is a neighborhood
of В cobounded in A.
proof If U' is a neighborhood of В in A, then the closure of A — V in A
equals the closure of (A — B) — (U' — B) in A — B. Hence one is compact if
and only if the other is. Therefore the result will follow once we have verified
that if 17 is a cobounded subset of A — B, then 17 U В is a neighborhood of
В in A. However, if C is the compact set which equals the closure of
(A — B) — U in A — B, then C is closed in A (because A is Hausdorff).
Therefore A — C is an open subset of A containing B. Since (A — В) — С C U,
it follows that (A — С) C G U B, and 17 U В is a neighborhood of В in A. 
Let В be a closed subset of a normal space A. If 17 is a neighborhood of
В in A which is a cobounded subset of A, then C* (A, 17) С C* (A,B). There-
fore li m , (C*(A,17)} = U C*(A,U) is imbedded as a subcomplex of
C;!:(A,B). By the excision property 6.4.4,
U C*(A,U) U C*(A - B, U — B)
As 17 varies over cobounded neighborhoods of В in A, it follows from lemma
10 that U — В varies over cobounded subsets of A — B. Therefore
U C*(A - B, G - B) = C*(A - B)
and we have defined a functorial imbedding
/: C*(A — В) C C*(A,B)
such that /(Cj? (A — B)) = lim , {C* (A,I/)}, where U varies over cobounded
neighborhoods of В in A. Hence j induces an isomorphism of cohomology if
and only if
liiru {H*(A,G)} ~ H*(A,B)
We shall now consider cases in which j induces an isomorphism of cohomology.
11	lemma If A is a compact Hausdorff space and В is closed in A, for all
q and all G there is an isomorphism
Н<я(А ~ B; G) zz №(A,B; G)
proof By lemma 9 and the above remarks, it suffices to prove that as
V varies over neighborhoods of В in A (any such neighborhood being
cobounded because A is compact), there is an isomorphism
linu {№(A,G; G)}	H®(A,B; G)
322	GENERAL COHOMOLOGY THEORY AND DUALITY CHAR.,(j
Since A is paracompact, this is a consequence of the tautness property 3 of
Alexander cohomology. 
This result allows the following interpretation of the cohomology with
compact supports of a locally compact space.
12	corollary If X is a locally compact Hausdorff space and X+ is the
one-point compactification of X, there is an isomorphism
ILfiX-G) ~ RfiXfG)
proof By lemma 11, HC«(X;G) ~ №(X+, X+ — X; G) and because
H* (X+ — X; G) = 0, there is an isomorphism
№(X+, X+ - X; G) ^(X+;G) 
13	example It follows from corollary 12 that
HC«(R»;G)~[°
11	' (G q = n
because (R”)+ is homeomorphic to Sn. Hence, if n m, Rn and Rnl are not of |
the same proper homotopy type.
14	example Regarding R1 as a linear subspace of R2, then
HI.(RM1i;G) = (0GeG	i
15	theorem Let В be a closed subset of a locally compact Hausdorff space |
A. For all q and all G there is an isomorphism
lirru [№(A,U; G)} Hci(A,B; G)
where U varies over cobotmded neighborhoods of В in A.	|
proof If A is compact, this follows from lemmas 9 and 11. If A is not com-
pact, let A+ be the one-point compactification of A. Set p+ = A+ — A and I
B+ = В U p+ C A+. Then B+ is closed in the compact space A+. There is a |
commutative diagram of chain maps	i
C*(A-B)^ C*(A)	C*(B)	j
I I i
0 -> C*(A+,B+) -> C*(A+,p+) -> C*(p-,p+) -> 0
and, by corollary 12 and lemma 11, each vertical map induces an isomorphism j
on cohomology. Since the bottom row is exact and C? (A — В) C C?: (A), it
follows that Cg(A)/Gg(A — B)—> C?(B) induces isomorphisms of coho-
mology. Since there is a short exact sequence of cochain complexes	‘
0^ C*(A,B)/C*(A - B) C*(A)/C*(A - B) -+ C*(B)	0
it follows that C?(A,B)/CJ(A — B) has trivial cohomology. Therefore
SEC. 7 PRESHEAVES	323
ff$(A — В) ~ H'^(A,B), and this is equivalent to the statement of the
theorem. 
The last result is a form of tautness for Alexander cohomology with
compact supports. This and the five lemma easily imply the next result.
16	theorem Let (A,B) be a pair of closed subsets of a locally compact
Hausdorff space X. For all q and all G there is an isomorphism
lim , {№(U,V; G)} zz НУ(А,В; G)
where (U,V) caries over neighborhoods of (A,B) in X, both U and V being
cobounded subsets of X. 
In a similar fashion, we may consider the singular cohomology with
compact supports. A singular cochain с* E Hom (Де(Х)/Де(А),С) is said to
have compact support if there is some cobounded set U С X such that
c* (a) = 0 for any singular q-simplex a in U. The singular cochains with com-
pact support form a subcomplex of the singular cochain complex, whose
cohomology module is denoted by (X,A; G).
S PHES1IEAVES
In this section the Cech construction will be introduced. Because of the
ultimate applications, we define the Cech cohomology of a space not merely
for coefficients in a module, but, more generally, for coefficient modules
which may vary from one point of X to another. This leads to the concepts of
presheaf and sheaf. We shall introduce these and give the definition of
the Cech cohomology of a space with coefficients in a presheaf. Applications
will be given in the next two sections.
A presheaf Г of В modules on a topological space X is a contravariant
functor from the category of open subsets U of X and inclusion maps U С V
to the category of В modules such that Г(0) = 0. Thus Г assigns to every
open subset U С X an R module Г([7) and to every inclusion map 17 С V a
homomorphism Pvv: T(V) —> Г([7), called the restriction map, such that
Puu = lr<U)
Puw = Pvv ° PyT0 T(W) T(U) U GV GW
Given у E Г(У) and 17 С V, we use у | 17 to denote the image pudy) E Г(17).
In a similar manner, we define presheaves on X with values in any cate-
gory. We are interested primarily in the case of a presheaf of modules or of
cochain complexes. Following are some examples.
1	Given an R module G, the constant presheaf G on X assigns to every
nonempty open 17 С X, the module G (and to 0 the trivial module).
2	Given a subset А С X, the relative Alexander presheaf of (X,A) with
324
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. 6
coefficients G, denoted by C* (•, • П A; G), assigns to an open CJ С X the
cochain complex C*(U, U PlA; G).
3	The relative singular presheaf of (X,A) with coefficients G, denoted by
Д * (•, • ПА; G), assigns to an open U С X the cochain complex
Д*(Ц, и П A; G) = Hom ((Д* (и)/Д* (CJ П A)), G).
Given two presheaves Г and Г' on X taking values in the same category,
a homomorphism а: Г —» Г' is defined to be a natural transformation from
Г to Г'. It is then clear that there is a category of presheaves on X with values
in any fixed category and homomorphisms between them. In particular, there
is a category of presheaves of modules and a category of presheaves of
cochain complexes. If а: Г Г' is a homomorphism of presheaves of
modules (or cochain complexes), it is clear how to define ker a, im a, and
coker a so as to be presheaves of modules (or cochain complexes) on X.
Therefore it is meaningful to consider exact sequences of presheaves of
modules (or cochain complexes) on X.
If Г and Г are presheaves of modules (or cochain complexes) on X, their
tensor product Г ® Г' is the presheaf of modules (or cochain complexes) on X
such that for open U С X
(Г ® T')(CJ) = T(CJ) ® T'(CJ)
Consider two examples.
4	There is a homomorphism
t: C*(-, • П A; G)-> Д*(-, • П A; G)
such that if <p C Ce(CJ, U П A; G) and ст: Д« U, then т(<р)(ст) =
<р(ст(р0)> • • • ,CT(P«))> where p0, . . . , pq are the vertices of A'J
5 There is a homomorphism
t: C*( •, • C A; R) ® G —> C*( • , • П A; G)
such that if <p € C<fU, U П A; R) and g C G, then
т(<р ® g)(x0, . . . ,Xg) = <p(x0, . . . ,Xg)g	Xj с и
Similar to the concept of presheaf on X with values in a category is the
concept of sheaf on X with values in a category. We are interested only in
sheaves of modules, and for this case the following formulation will do.
Let Г be a presheaf of modules on X. If = { CJ } is a collection of open
subsets of X, a compatible Qlfamily of Г is an indexed family {yr £ ЦСС)}^
such that
yu I и П U' = I и П U' CJ, CJ' e
The presheaf Г is said to be a sheaf it both the following conditions hold:
SEC. 7 PBESHEAVES
325
(a)	Given a collection 01 of open subsets of X with V = Ureqi U and
given у E T(V) such that у | U = 0 for all U £ Ql, then у = 0.
(b)	Given a collection Ol of open subsets of X with V = U r£i?l U and
given a compatible 0L family {yu}uc;?i> there is an element у £ T(V)
such that у | U = yv for all U E Ol.
It follows from (a) that the element у in (b) is unique.
We now associate to every presheaf Г of modules another presheaf Г,
called its completion, whose elements are compatible families of Г. Given a
collection of open sets Ql = {G}, let Г(01) be the module of compatible
01 families of Г. If У is another collection of open sets which refines Ol, there
is a homomorphism Г(01) —» I'(CV) which assigns to a compatible family
{yu} the compatible Tfamily {yy} such that if V E *Vis contained in U E 01,
then yy = yv | V (yy is uniquely defined by this condition because of the com-
patibility of {yu})- As Ol varies over the family of open coverings of a fixed
open set W С X, the collection {T(0l)} is a direct system of modules, and we
define
f(W) = linu {T(Ot)}
If W' C fy and Ol is an open covering of W, then Ol' = {G Л W' | U E Ol}
is an open covering of W' which refines Ol. Hence there is a homomorphism
Г(01)	Г(01') which defines (by passage to the limit) a homomorphism
r(W) —> T(W'). A trivial verification shows that Г is a presheaf [if Ol = { 0 },
then trivially T(0l) = 0, and so Г(0) = 0]. There is a natural homomor-
phism а: Г Г such that a assigns to у E T(V) the element of T(V)
represented by the compatible LYfamily {y}, where Tconsists solely of V. The
presheaf Г is called the completion of Г. It depends only on the values T(G)
for small open sets U С X.
6	lemma A presheaf Г is a sheaf if and only if
а: Г f
proof In fact, condition (a) above is satisfied if and only if a is a monomor-
phism. If condition (b) is satisfied, a is an epimorphism. If a is an isomorphism,
then (b) is satisfied. 
7	example The constant presheaf G defined by a module G is not gen-
erally a sheaf [if U is a disconnected open set, G(U) G(G)].
8	example If C * is the relative Alexander presheaf of (X,A) (with some
coefficient module G), the kernel of a: С* —> C* is C$ (the locally
zero functions). To show that a satisfies condition (b) (and hence induces an
isomorphism С* ~ C*), let <p' E C9(V, V П A) and assume <p' represented by
a compatible Ol family {фи}иси, where Ol is an open covering of V. Then
<Pv: Uq+1 —> G for U E Ol is locally zero on U П A and
<pu | (G П G')9+i = tpv | (G П U')i+1 G, G' E %
326
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. 6
Therefore there is a function y:	G such that <p | CM1 = <pv for U £ %
and tp(x0, . . . ,xg) = 0 if x0, . . . , xg do not all lie in some element of ^1.
Then <p is locally zero on A, whence <p £ C9(V, V П A) and a(<p) = <p'.
This example shows that, in general, H*(C*) H* (C*), so it is not
generally true that a presheaf of cochain complexes and its completion have
isomorphic cohomology.
9 example If Д * is the relative singular presheaf of (X,A) (with some
coefficient module G), the kernel of а: Д* Д* is the subcomplex of
locally zero cochains [that is, с* E Hom (Д9(У),С) is in the kernel of a if and
only if there is some open covering Ql of V such that c* is zero on
Д,;(9[) С Дв(У)]. Also a satisfies condition (b) (as can be shown by an argu-
ment similar to that of example 8). If is an open covering of X, it is clear
that Д*(с11) = Hom (Д * (^1)/[Д * (Ql) Г| Д*(А)], G). As Ql varies over open
coverings of X, there is an inverse system of chain complexes
{Д^тДА^т n Д*(А)]}
and a direct system of cochain complexes
{Hom (Д*(^)/[Д*т n Д*(А)], G)}
Therefore there is an isomorphism
lim. {Hom (Д,«Д» W П Д *(А)], G)} = A*( •, • П A; G)(X)
It follows from theorem 4.4.14 that
Hom (Д * (Х)/Д * (A), G) Hom (Д * (^1)/[Д * (^l) П Д „ (A)], G)
induces isomorphisms of the cohomology modules. Therefore a induces an
isomorphism
Н*(Д*(-, • n A; G)(X)) ~H*(A*(-, • П A; G)(X))
10 example Let £ be an n-sphere bundle with base space В and let A be
fixed. A presheaf Г on В is defined by F(V) = Ml+I(j^_|(V), /^'(У) G Eg R)
for an open V С В. Г is called the orientation presheaf of £ over R. It can be
verified that if В is connected, £ is orientable over R if and only if Г(В) 0.
11 example Let X be an n-manifold with boundary X and let R be fixed.
Define a presheaf Г on X — X such that Г(У) = Hn(X, X — V; R) for open
V С X — X. Г is called the fundamental presheaf of X over R. It can
be verified (using lemma 6.3.2) that f (X) ~ //„'(X,X; R). By theorem 6.3.5, it
follows that if X is connected, it is orientable over R if and only if Г(Х) 7Д 0.
There are cohomology theories of X with coefficients in sheaves,1 and
cohomology theories with coefficients in presheaves. For paracompact spaces
1 See R. Godement, “Theorie des faisceaux,” Hermann et Cie, Paris, 1958.
SEC. presheaves	327
these theories are equivalent. We now define the Cech cohomology with
coefficients in a presheaf of modules.
Let Г be a presheaf of modules on a space X and let be an open cov-
ering of X. For q > 0 define С«(01;Г) to be the module of functions i// which
assign to an ordered (<; + l)-tuple C70, C1; . . . , Uq of elements of Ql an ele-
ment f(Uo,    ,Uq) E Г(С70 П • •  П Uq). A coboundary operator
8: Св(ч>1;Г)	C«+1(QL;F)
is defined by
(&//)(Uo, . . • ,CQ+i)
= .2 1 (-Wo, • • • ,Ui,   • ,ue+1) I (Co П ... П C9+1)
O<S<Q+1
Then 88 — 0 and С*(01;Г) = {C,'(Ql; I'),8} is a cochain complex. Its cohomol-
ogy module is denoted by H* (Ql;T).
12	example It is an immediate consequence of the definition that
= r(Lll) (the module of compatible 8*1 families).
Let % be a refinement of Ql and let A: T—> be a function such that
V C A(V) for all V E % There is a cochain map Л*: С*(^1;Г)	С*(^';Г)
defined by
(A^)(V0, . . . ,Vg) = ^(A(V0), . . . ,A(Ve)) I (Vo П ... n Ve)
If ft: A' sH is another function such that V C ft(V) for all V E % a cochain
homotopy D: С«(01;Г)	Co'-1(CV;F) from A* to ft* is defined by
(» • • • ,V9-i)
=	(-WA(Vo), . • • AWfiW . . . ^(V,-!)) I (Vo n ... П VQ_i)
It follows that there is a well defined homomorphism
А*: Н*(01;Г)^ Н*(Т;Г)
such that A* {<p} = {A*<p} that is independent of the particular choice of A.
As 9l varies over open coverings of X, the collection {Я*(91;Г)} is a direct
system, and the Cech cohomology of X with coefficients Г is defined by
Н*(Х;Г) = lim, {H*(Ql;F)}
13	example For any presheaf Г, Й°(Х;Г) = f (X).
14	example The (fecit cohomology of X with coefficients G, denoted by
H*(X;G), is-defined to be the cohomology of X with coefficients the constant
presheaf G.
We now establish some basic properties of the cohomology with coeffi-
cients in a presheaf.
32S
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. (J
15	theorem There is a covariant functor from the category of short exact
sequences of presheaves on X to the category of exact sequences which
assigns to a short exact sequence 0 Г' Г Г" 0 of presheaves on X
an exact sequence
----> №(Х;Г')	№(Х;Г) Нч(Х;Г") -» Но+1(Х;Г') -> • • •
proof For any open covering Qt there is a short exact sequence of cochain
complexes
0	С*(91;Г') -» С*(01;Г)	С*(<?1;Г") О
This yields an exact cohomology sequence, and the result follows from this on
passing to the direct limit. “
Given a short exact sequence of modules
0 G' G G" 0
the corresponding constant presheaves on X constitute a short exact sequence
of presheaves. The corresponding exact cohomology sequence of Cech coho-
mology modules given by theorem 15 is an analogue for Cech theory of the
exact sequence of theorem 5.4.11.
Given a presheaf Г on X and given a subspace А С X, define a presheaf
Гл on X by
r/m-R	UGA^0
Mb) - [()	[J n A = 0
Also define a presheaf Гл on X by
W)_(r(b)	UGA=0
(	~ Io	UOAf0
Then P1 is a sub-presheaf of Г, and there is a short exact sequence of presheaves
0 TX~A
The corresponding exact cohomology sequence given by theorem 15 is an
exact Cech cohomology sequence of the pair (X,A) with coefficients Г when
we define 1Р(Л;Г) = №(X,TA) and Й«(Х,А; Г) = Й9(Х;Гх-л). Thus the exact
sequence of theorem 15 gives rise to exact sequences corresponding to a
change of coefficients or to a change of space.
A presheaf Г of modules on X is said to be locally zero if, given у 6 Г( V),
there is an open covering Qt of V such that у | 17 = 0 for all U £ This is so
if and only if the completion Г of Г is the zero presheaf and is equivalent to
the condition that for all x E X, linn, {Г(С7)}=0 as 17 varies over open
neighborhoods of x.
16	theorem If X is a paracompact Hausdorff space and, Г is a locally zero
presheaf on X, then Й*(Х;Г) = 0.
proof Let 7l be a locally finite open covering of X and <p a q-cochain of
SEC. 8 FINE PRESHEAVES
329
(7*(^1;Г). For x £ X let 117,. be an open neighborhood of x intersecting only
a finite number of elements of Ql. Because Г is locally trivial, there is an open
neighborhood V} of x contained in such that for all Uo,    , Ug £ Qt,
<p(Uo, • •  ,Uq) | V® = 0 (this is only a finite number of nontrivial conditions,
because W® intersects only finitely many elements of ^L). Let ‘Vbe an open
covering of X which is a refinement of and of the covering {V®}®ex.
If A: CV'^- is a function such that V С Л( V) for all V £ Л7 then X*<p — 0 in
Therefore Нв(Х;Г) = 0 for all q. 
A homomorphism а: Г —> Г' between presheaves on X is called a local
isomorphism if ker a and coker a are both locally zero. This is equivalent to
the condition that for all x С X, a induces an isomorphism
But, {Г(С7)} ^linu {Г(С7)}
where 17 varies over open neighborhoods of x. There are short exact sequences
of presheaves
0	ker « —> Г -> im о 0
0 im « 7> Г' coker a 0
with a = a"a'. Combining theorems 15 and 16, we obtain the following result.
17	corollary If а: Г Г' is a local isomorphism of presheaves on a
paracompact Hausdorff space X, then
а*: Н(Х;Г) ~Н*(Х;Г') »
18	corollary If X is a paracompact Hausdorff space, the natural homomor-
phism а: Г Г induces isomorphisms
а*: #*(Х;Г) ~ H*(X;f)
proof It suffices to prove that «: Г —» I is a local isomorphism. Let
у E (ker a)(V). Then у £ T(V), and there is an open covering Ql of V such that
у | U = 0 for all 17 £ Ql. Hence ker a is locally zero.
If у' £ (coker a)(V), there is an open covering 7l of V and a compatible
u7l family {yr } which represents y'. For each 17 £ % y' | U is represented by
Yu С а(Г(17)). Therefore y' | U = 0, and coker a is locally zero, и
8 FINE PHESIIEAVES
In this section we shall introduce the concept of fine presheaf and show that
the positive dimensional cohomology of a paracompact space with coefficients
in a fine presheaf is zero. This leads to uniqueness theorems for cohomology
of cochain complexes of fine presheaves on a paracompact space, which we
apply to compare the Alexander and Cech cohomology. Further applications
will be given in the next section.
330
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. 6
A presheaf Г on X is said to be fine if, given any locally finite open
covering of X, there exists an indexed family {еи}ие^ of endomorphisms of
Г such that
(a)	For у £ T(V), еи(у) | (V - U) = 0.
(b)	If V meets only finitely many elements of {A}, then for у G Г(У),
у = SueQi ец(у).
Note that the sum in condition (b) is finite because, by (o), Cr/y) = Oif
u n V = 0.
1	example The relative Alexander presheaf of (X,A) of degree q with
coefficients G is fine. In fact, if Ql is a locally finite open covering of X, for
each i С X choose an element Ux G containing x and for <p G V, V A A; G)
define evp G Ci(V, V A A; G) by
. v	x f<p(xo, .   ,xfi U = UXo
(е^)(х0, • • • Az) - [0	u=£ u::^
If V С V, there is a commutative square
C«(V,V A A; G) % C«(V, V A A; G)
j	i
Ce(V', V A A;G)^ сщу, V A A; G)
showing that ejj is an endomorphism of Ci. If
xo, . . . , xq G (V1+1 - Ui+i) C (Ve+1 - U"+l),
then С7Жо U and (evp)(x0, . . . ,xq) = 0. Hence evp | V — U = 0, and con-
dition (o) is satisfied. To show that (b) is also satisfied, observe that, given
To, . . . , xq, there is a unique U, namely C40, such that (c(,<p)(x(), • •  A?) 7^ 0.
Then
(2 evp)(x0, . . . ,xq) = (eUxo<p)(xo, . . . ,xfi = <p(xo, • • • A?)
It should be noted that ey does not commute with the coboundary operator
in С* (V, V A A; G). Therefore ev is not an endomorphism of the Alexander
presheaf C*( •, • A A; G) of cochain complexes.
2	example The relative singular presheaf of (X,A) of degree q with coeffi-
cients G is also fine. If v?l is a locally finite open covering of X and С7Ж is
chosen so that r G G G then
ev-. Hom (Aq(V)/A9(V A A), G) Hom (AQ(V) AQ(V A A), G)
is defined by
— f6*^)	U = Ua{Po)
{euc^o)-^	U^UpM
Then the family {eu)ue ® satisfies conditions (a) and (b) of the definition of fine-
SEC. 8 fine presheaves
331
ness [but ev is not an endomorphism of Д* (•, • П A; G) so Д* (, • П A; G)
jS not a fine presheaf of cochain complexes].
Given a presheaf Г on X and a continuous map f: X —» Y, there is a pre-
sheaf/^ Г on Y defined by ( /. Г)(\7) = Г(/-1У) for an open V C Y. Clearly,
defines a covariant functor from the category of presheaves of any type on
X to the category of presheaves of the same type on Y. Some of the nice
properties of fine presheaves are made explicit in the following result.
;{ theorem Let Г be a fine presheaf of modules on X.
(a)	For any presheaf Г' of modules on X, Г ® Г' is fine.
(b)	If f: X —» Y is continuous, fi. Г is fine on Y.
(с)	Г is a fine presheaf on X.
proof For (a), observe that if Ql is a locally finite open covering of X and
(eujvevi are the corresponding endomorphisms of Г, then {ev ® l/vs is a
family of endomorphisms of Г ® Г', showing that Г ® Г' is fine.
For (b), observe that if Qt is a locally finite open covering of Y,
then f~1GlI = {fAU | U E Qi} is a locally finite open covering of X. If
is a family of endomorphisms of Г corresponding to the covering /-1Ql, they
induce endomorphisms of fi Г, showing that fi Г is fine.
(c)	follows easily on observing that any endomorphism of Г induces an
endomorphism of Г. 
Given an open covering Ql of a space X, a shrinking of Qi is an open cov-
ering Tof X in one-to-one correspondence with Ql such that if 17 E 01 corre-
sponds to 17 C % then Vv C U. Any locally finite open covering of a normal
Hausdorff space has shrinkings. Any shrinking of a locally finite open covering
is clearly locally finite.
The following theorem is the main result on fine presheaves.
4 theorem If Г is a fine presheaf on a paracompact Hausdorff space X,
then №(Х;Г) = 0 for q fi 0.
proof Let Ql = {U} be a locally finite open covering of X and let
Ql' = {17} be a shrinking of Ql. Let	be fineness endomorphisms of Г
corresponding to the covering 01' (but indexed by the covering Qi). Let
Y= {V} be an open refinement of Ql covering X such that each V E Tr
meets only a finite number of elements of Ql and for any U £ Ql either V C U
or V С X — U'. Let A: ‘V —» Ql be a function such that V C A(V) for
all V E %
Since each ev is an endomorphism of Г, e(- induces a cochain map,
denoted by ejj: С*(01;Г) —» С*(01;Г) such that for f E С9(01;Г) and
Uo, ... ,Uq£ Ql
(eu/XUo, . . . ,Ufi = efiffifi . . . ,C7e))
Then ер acts similarly as a cochain map on С*(7';1’) and commutes with the
cochain map A* : С* (01;Г) —» С* (Т;Г).
Let q > 0 and / E С«(01;Г) be a cocycle. Define fu E C’CVjF) by
iEC( 8 FINE PRESHEAVES	333
jj corollary Let Г * be a cochain complex of presheaves of modules on
0 paracompact Hausdorff space X. For any q there is a short exact sequence,
functorial ini'*,
0 im [Й°(Х;Ве) HfX-Z^)] №(T * (X))
ker [H°(X;№) HfX;Bfi] -» 0
If I '; l is fine, this becomes
0 H^XiZr1) №(T * (X)) ker [H°(X-,№) HfX-,Bfi] 0
proof From the short exact sequence of presheaves
0 Be Ze	0
it follows, by theorem 6.7.15, that there is an isomorphism
H0(X;Ze)/H0(X;Be) ker [Н°(Х;Не) -> HfX;Bi)]
From lemma 5, there is an isomorphism
H°(X;Ze)/ker [H0(X;Be) HfiX^Zr1)] №(Г*(Х))
It follows that №(Г * (X)) maps epimorphically to ker [H°(X;№) —» Hi(X;B,‘)]
with kernel isomorphic to
Я°(Х;Ве)/кег [H^X-Bfi H\X-,Zn)] ~ im [H°(X;Be) HfX-Zn)]
This gives the first short exact sequence. For the second, there is a short exact
sequence of presheaves
0 Z«-i -> Г'/i Bi -> 0
and if Ге-1 is fine, it follows from theorems 6.7.15 and 4 that
im [H°(X;Be) HfXiZ^1)] = HfX;Z4-i) о
7 theorem Let Г* be a nonnegative cochain complex of fine presheaves
of modules on a paracompact Hausdorff space X. Assume that for some
integers 0 < m < n, Н'/(Г *) is locally zero for q <fm and m <fq < n. Then
that ’ there are functorial isomorphisms
>	№-”»(Х;Н™(Г *)) ~ №(f * (X))	q < n
I1 and a functorial monomorphism
№~^Х;Нт(Г*))	№(f *(X))
j proof For each q there is a short exact sequence of presheaves
'	0 Za I'" Be+i -> 0
J Because Ге is fine, it follows from theorems 6.7.15 and 4 that
(а)	ЙР(Х;Ве+1) №+i(X;Ze) p > 1
332	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP, (j f
фг- = er-(f\*f). Then фу is a cocycle for each U £ 9l, and if Vo, . . . , VQ £ <у 1
then i/r^Vo, • • • ,Vq) = 0, except for a finite number of U £ 01. Therefore
S фи exists, and 5 фу = Л* ф.
Define ф'у € Се'(с\';Г) by
Ф'и(Уо, . . . ,Vg_l)
_ (еи(ф(и, X(V0), . . . A(Ve_i)) | (Vo П • • • П V9_r) Vo П • • • П c 0
~ IO	Vo П • • • П VQ_r С X - 0'
Then 8ф'и = фи for all U, and because S фу can be formed [for given;
Vo, . . . , Ve_i, i//h(Vo, . . . ,VQ_i) = 0, except for a finite number of U £ *?l]
we see that
X* ф = S фу = 8(S ф'ц)
Therefore Л* ф is a eoboundary, and №(X;T) =0. и
Our next results are technical lemmas about cochain complexes of
presheaves. If Г * is a cochain complex of presheaves of modules on X, we
use Z® and Bi+1 to denote the kernel and image, respectively, of 8: Ге —> [>/+!
and № to denote Za/Bq, all of these being presheaves of modules on X. (Note
that a fine presheaf of cochain complexes is a cochain complex of fine pre-
sheaves, but the converse is not generally true.)
5 lemma Let Г* be a cochain complex of presheaves of modules on X.
For every q there is an exact sequence, functorial in Г *,
0 ker (H0(X;B«) HfiX,Z«~i)) H°(X;Ze) №(T * (X))	0
proof By example 6.7.13, Ге(Х) = Н°(Х;Гв). From the short exact sequence
of presheaves
0 -» Z« -» Ге Be+1 0
there follows, by theorem 6.7.15, an exact sequence
0 -> H°(X;Z«) Я°(Х;Гв)	HfiX;Zfi
Because B«+1 С Г«+1, it follows from a similar exactness property
H°(X;Be+1) С Н°(Х;Г9+1). Combining these, we see that
F°(X;Ze) ~ ker [Н°(Х;Г«)	Й°(Х;Ве+1)]
~ ker [#<>(№)	Я°(Х;Гв+1)]
and also that
im [Я°(Х;Ге)	Н°(Х;Г'/+')] ker [H°(X;Be+i) HfX-,Z<i)\
Since
№(t *(X)) = ker [Н°(Х;Ге)	H°(X; Г'/ ' । )]/jm [Й°(Х;Ге-1)	Н°(Х;Гв)]
the result follows. 
334	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP, (j
For each q there is also a short exact sequence of presheaves
О Вч -» Z«	0
Because № is locally zero for q <Z m and m < q n, it follows fioirs
theorems 6.7.15 and 6.7.16 that	„
(b)	№(Х;В'1') ~ 1Jp(X;Z'i) q X m or m < q < n, all p
Since B° is the zero presheaf, it follows by induction on q from equations
(b) and (a) that for q < m
(c)	№(X;Ze) = 0 = №(X;B'/+I) p > 1
From this and corollary 6, it follows that Hl(T * (X)) = 0 for i < in. Hence
the theorem holds for q < m (both modules being trivial). For q = in we
have [by corollary 6 and equation (c)]
H«(f *(X)) ~ H°(X;H»')
and the theorem holds in this case too.
To obtain the result for m < q < n, note that, by equation (c).:
H'P(X;Bm') = 0, if p > 1. From the short exact sequence of presheaves
o -» Bm -4> Z"‘ Hm 0
it follows that
HP(X;Zm) = №(X-,Hm) p > 1
For in <^i < n it follows from corollary 6 that
//'(XjZ^1) ~ и(Г^(Х))
and for i = n there is a monomorphism
#i(X;Z»-i) HffT * (X))
Using equations (b) and (a), we see that for m <^i < n
nux-z1-') ~	~ н2(Х;г{-2) ~	~ №~™(x-,z™) ~ й-™(Х;Ят)
and this gives the result for m < q < n. 
This last result has as an immediate consequence the following isomor-
phism between the Cech and Alexander cohomologies with coefficients G.
8 corollary For any paracoinpact Hausdorff space and module G there
is a functorial isomorphism
H*(X;G) ~ H*(X;G)
of the Cech and Alexander cohomology modules.
proof Let C* be the Alexander presheaf of X with coefficients G. Since Cs
is fine for all q (by example 1), this is a nonnegative cochain complex of fine
sheaves. Furthermore, for any nonempty U, by lemma 6.4.1,
SEC. 8 FINE PRESHEAVES
335
№(C*(C7;G))^{°
Therefore №(6*) is locally zero for q > 0 and H°(C*) is isomorphic to the
constant presheaf G. The hypotheses of theorem 7 are satisfied with m = 0
and any n, and there is a functorial isomorphism
№(X;G)	№(C*)
for all q. As pointed out in example 6.7.8, there is a canonical isomorphism
С* C*, and so №(X;G) ~ He(C*). Combining these isomorphisms yields
the result. 
The last result is also true without the assumption of paracompactness
(see exercise 6.D.3). The next result is the main uniqueness theorem of the
cohomology of presheaves.
9 theorem Let X be a paracompact Hausdorff space and let т: Г* Г' ♦
be a cochain map between nonnegative cochain complexes of fine presheaves
of modules on X. Assume that for some n > 0,	: №(T *) —» №(T' *) is a
local isomorphism for q <Zn and a local monomorphism for q = n. Then the
induced map
г*;Я’(Г*(Х)Н№(Г*(Х))
is an isomorphism for q < n and a monomorphism for q = n.
proof Let Г * be the mapping cone of т (defined for cochain complexes analo-
gous to the definition in Sec. 4.2 for chain complexes). Then Гт« = Г'/+| © Г'в,
and for у £ Г9-1(17) and у' £ V'fiU), й(у,у') = ( — й(у), т(у) + 8(y')). Г* is
a nonnegative cochain complex of fine presheaves on X, and for any open
U С X there is an exact sequence
----> №(Г'*(С7)) —> №(Г*(иУ)^> H9+1(r*(CJ))Jb №+1(Г*(Г/))^ •••
Taking the direct limit as U varies over open neighborhoods of x £ X, we see
that : №(T *) -ч> HfiT' *) is a local isomorphism for q <f n and a local
monomorphism for q = n if and only if №(Г*) is locally zero for q < n. By
theorem 7, it follows that HfiT * (X)) = 0 for q n (if n = 0 this is trivially
true, and if n > 0 it follows from theorem 7 with m — 0).
It is obvious that I * is the mapping cone Г* of the induced map
f: Г *	Г * between the completions. Therefore
-----> №(f' * (X))	№(T * (X)) H?+i(f * (X)) К №+1(Г * (X)) -» • • •
Since Нв(Г * (X)) was shown to be zero for q < n in the first paragraph above,
the result follows from the exactness of this sequence. 
For compact spaces there is the following universal-coefficient formula
for Cech cohomology.
1® theorem Let X be a compact Hausdorff space. On the product category
336	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. 6
of presheaves Г on X consisting of torsion free R modules and the category of
R modules G there is a functorial short exact sequence
0	№(Х;Г) ® G №fX-, Г ® G) №+1(Х;Г) *G^ 0
pboof Let Ql be a finite open covering of X. The cochain map
r. C*(Ql;T) ® G C* (Ql; Г ® G)
defined by r(f ® g)(Uo, •   ,Ug) = f(U0, . . . ,Uq) ® g is an isomorphism
(this is a consequence of the finiteness of Ql analogous to lemma 5.5.6). From
the universal-coefficient formula for cochain complexes (theorem 5.4.1), there
is a functorial short exact sequence
0	№(Ql;F) ® G№(% Г ® G) —» №+1(<?1;Г) *	0
The result follows by taking direct limits over the cofinal family of finite open
coverings of X (because the tensor product and the torsion product both
commute with direct limits). 
From corollary 8, this gives a universal-coefficient formula for Alexander
cohomology of compact spaces. The following theorem generalizes this result
to compact pairs and includes the statement that the short exact sequence in
question is split.
11 theorem On the product category of pairs (X,A), where A is a closed
subset of a compact Hausdorff space X, and the category ofR modules G, there
is a functorial short exact sequence
0	№(X,A; R) ® G -» №(X,A; G) -» №+l(X,A; R) * G 0
and this sequence is split.
pboof Let т: C* (•,  П A; R) ® G -» C* (•, • П A; G) be the homo-
morphism of presheaves defined as in example 6.7.5 [that is, т(<р ® g)
(x0, . . . ,Xg) - <p(x0, . . . ,Xg)g]. Both C*(-, • n A; R) ® G and
C*( •, • П A; G) are nonnegative cochain complexes of fine presheaves.
First we prove that
r*:	• П A;R) ® G)^H*(C*(-, • П A; G))
is a local isomorphism. If U С X — A, C*(C7, U П A; R) — C*(U;R), and
C*(G, V Г) A; G) = C*(G;G), it follows from lemma 6.4.1 and theorem 5.4.1
that
U П A; R) ® G))	U П A; G))
Since A is closed in X, for any x 6 X — A, is an isomorphism of
lim , {H*(C*(17, и П A; R) ® G)} onto linu {H*(C*(17, U П A; G))},
both limits as U varies over open neighborhoods of x in X.
For any U intersecting A there is a commutative diagram with exact
rows
sec. 8 rINE pkesheaves	337
0 -» C*(C7, и n A; R) ® G^ C*(U;R) ® G^C*(U П A; R) ® G-> 0
q.-* C*(U, и Г) A; G) -+C*(U;G)	~^C*(U Г) A; G) -» 0
By lemma 6.4.1, the middle cochain complexes have trivial reduced modules.
Therefore there is a commutative square
№(C*(U П A; R)®G)	№+1(C*(U, U П A; R) ® G)
1	J,
№(C*(U n A; G))	№+1(C*(U, и П A; G))
To complete the proof that is a local isomorphism, therefore, we need only
prove that for x £ A
linu [№(C*fU GA-,R)®G)}~ linu {№(C*(U П A- G))}
as U varies over neighborhoods of x in X. This is equivalent to the condition
that
№(linu {C*(G П A; R)}) ®G~ #e(linu {C*(U П A; G)})
where U П A varies over neighborhoods of x in A. This is trivially true
because both sides are zero for all q (this follows from the tautness property
of x in the paracompact space A but can be proved without assuming the
paracompactness of A, because any one-point subspace is taut in any space
with respect to Alexander cohomology).
We have verified that т satisfies the hypotheses of theorem 9 for all n.
Therefore т induces an isomorphism
f*: H*([C*( •, • П A; R) ® G](X)) ~ H*(C*( •, - nA; G)(X))
By example 6.7.8, the right-hand side is isomorphic to H*(C*(X,A; G)). By
example 6.7.13, the left hand side is the qth cohomology module of
H°(X‘,C* (•,  П A; R) ® G). By theorem 10 and the fineness of
C*( •, • Г) A; R) ® G, this is isomorphic to
, • П A; R)) ® G (C*( •, • П A; R)(X)) ® G
~ C*(X,A; R) ® G
It follows that the map
7: C*(X,A; R) ® G^C*(X,A; G)
induced by т induces an isomorphism of cohomology. The result now follows
from the universal-coefficient formula for cochain complexes (theorem 5.4.1). 
This implies the following universal-coefficient formula for Alexander
cohomology with compact supports.
338	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP, (j
12 corollary On the product category of pairs (X,A), where A is a closed
subset of a locally compact Hausdorff space X, and the category of R modules '
G, there is a f unctorial short exact sequence
0 Hc<dX,A; R)® G Нсч(Х,А; G) Нсч+\Х,А; R) * G —> 0
and this sequence is split.
proof Let TV be a closed cobounded neighborhood of A in X. There is
a commutative square of cochain maps
C*(X,N; R) ® G	-»	C*(X,N; G)
J,	J,
С* (X - N, X — N Cl N; R) ® G -> С* (X - N, X-N Cl N; G)
in which, by theorem 6.6.5, each vertical map induces an isomorphism of
cohomology. By theorem 11, the top horizontal map induces an isomorphism
of cohomology. Therefore the bottom horizontal map also induces an isomor-
phism of cohomology.
There is also a commutative square (in which the limit is over closed
cobounded neighborhoods N of A in X)
linu{C* (X - N, X — N nN;K)]$G-> lim ,{C* (X - N, X — N П N; G))
C* (X,A; R) ® G	C* (X,A; G)
It follows from the first paragraph above that the top horizontal map in-
duces an isomorphism of cohomology. Since the closed cobounded neighbor-
hoods of A in X are cofinal in the family of all cobounded neighborhoods of A
in X, it follows from theorem 6.6.15 that each vertical map induces an isomor-
phism in cohomology. Therefore the bottom horizontal map induces an
isomorphism in cohomology. The result follows from this and theorem 5.4.1. 
9 APPLICATIONS OF THE COHOMOLOGY OF PRESHEAVES
This section is devoted to two main applications of the theory developed in
the last two sections. One is the study of the relation between Alexander and
singular cohomology. We shall prove that in a homologically locally connected
space (for example, a manifold) the two are isomorphic. The other application
is to a .study of the relation between the Alexander cohomology of two spaces
connected by a continuous map. We conclude with a proof of the Vietoris-
Begle mapping theorem.
Let (X,A) be a pair and let G be an R module. Recall the homomorphism
t:C*(-, • П A; G)-> &*(-, • П A; G)
i 0 applications of the cohomology OF PRESHEAVES	339
'' jgfined in example 6.7.4. This induces a homomorphism
f: C*(-, • П A; G)-> Д»(-, • П A; G)
1 such that the following square is commutative
j	C*(-,-nA;G)-b А*(-,-ПА;С)
C*(-, • Г1 A; G) Л Д*(-, • Г1 A; G)
( By examples 6.7.8 and 6.7.9, there are isomorphisms
C*(-, • Cl A; G) C*(-, • CA;G)
as:H*(A*(-, • C A; G))	Я* (A * (•, • C A- G))
[ In Sec. 6.5 a natural homomorphism
4	Jt: Я* (X,A; G) H* (X,A; G)
s was defined, and it is a simple matter to check that commutativity holds in
the diagram
H* (C * (X,A; G))	H* (Д * (X,A; G))
H*(C*(-, • C A; G)(X)) А» Н»(Д»( •, • C A; G)(X))
Therefore p is an isomorphism if and only if f is.
1 theorem Let X be a paracompact Hausdorff space and suppose there
( is n > 0 such that each x ( X is taut with respect to singular cohomology
i with coefficients G in degrees n. Then
p: №(X;G)	№(X;G)
is an isomorphism for q <fn and a monomorphism for q — n.
} proof Both C*(• ;G) and Д *(• ;G) are nonnegative cochain complexes of
; fine presheaves. The tautness assumption of the points of x with respect to
! singular cohomology implies that t* : He(C* (• ;G))	П''(Д * (• ;G)) is a local
/ isomorphism for q <f n and a local monomorphism for q — n (in fact, it
। is always a local monomorphism for all q). By theorem 6.8.9,
|	т * : №(C* (X;G)) ->	* (X;G))
I is an isomorphism for q <f n and a monomorphism for q = n. 
There is a partial converse of theorem 1 which asserts that if
fit №(U;G) —> №(U;G) is an isomorphism for q <fn and every open U С X,
then each point x £ X is taut with respect to singular cohomology in degrees
< n. This follows from commutativity of the following diagram (where
U varies over open neighborhoods of x £ X):
?	9 APPLICATIONS OF THE COHOMOLOGY OF PRESHEAVES	341
V SB1-'*
j g corollary Let A be a closed subset, homologically locally connected in
: dimension n, of a Hausdorff space X, honwlogically locally connected in
dimension n. If X has the property that every open subset is paracompact,
, Hy(X,A; G) Hcq(X,A-, G) is an isomorphism for q < n and a monomor-
i
In case X is a Hausdorff space in which every open subset is paracompacpj
(for example, X is metrizable), we see that each point x £ X is taut with respect
to singular cohomology in degrees < n if and only if ft: №(U;G) —> !/"(( ;(;)
is an isomorphism for all q < n and all open U С X.	j
340
GENERAL COHOMOLOGY THEORY AND DUALITY СНдр, (j Л
lirru {He([J;G)}	№(x; G)
linu {№(U;G}}	№(x;G)
pllismfor q = n + 1.
proof From the definitions, there is a commutative square (where U varies
over open cobounded subsets of X)
lintu {№(X,CJ; G)} Hci(X-,G)
A space X is said to be homologically locally connected in dimension ц
if for every x E X and neighborhood U of x there exists a neighborhood V of
x in U such that H9(V) —> is trivial for q < n. It is said to be
homologically locally connected if it is homologically locally connected in
dimension n for all n.
2 example Any locally contractible space, in particular any polyhedron
or any manifold, is homologically locally connected in dimension n for all n,
3 example Let Xq — Si for q > 1 and let xq be a base point of Xq. The
subspace of X Xq consisting of all points having at most one coordinate dif-
ferent from the corresponding base point is homologically locally connected,
in dimension n for all n but is not locally contractible.
I.	Ч	Iм
linu {H<fx,u-, G)} Hce(X;G)
Since an open subset of a space homologically locally connected in dimension
: n is again a space homologically locally connected in dimension n corollary 5
| applies to X and to every open U С X By the five lemma,
№(X,C7; G) №(X,U; G)
I is an isomorphism for q < n and a monomorphism for q = n + 1. Passing to
! the limit, p: Hci(X;G) —> Hci(X;G) is an isomorphism for q < n and a mono-
morphism for q = n + 1. Since A has the same properties as X,
|	ft: Hce(A;G) Hci(A-G)
4 lemma If X is homologically locally connected in dimension n, then
ll'ilf. * (. ;Gj) ?-s locally zero for q < n Und aii q
is an isomorphism for q < n and a monomorphism for q = n + 1. The result
now follows from the five lemma. 
proof Let c* E Hom (AQ(U),G) be a cocycle (0 < q < ri) and let x E U. If
q = O,Jet V be a neighborhood of x in U such that H0(V) —> H0(H) is trivial.
If с E A0(V), there is с' E Ai(<7) such that c = Sc'. Then c*(c) = c*(Sc') =
(Sc*)fc') = 0. Therefore с* | До( V) = 0, proving that H°(A*( • ;G)) is locally
trivial.
If 0 < q, let V and V' be neighborhoods of x in U, with V С V' and such
that Hg_i(V)	and HQ(V') Hq(U) are both trivial. If c is a re-
duced singular (q — l)-cycle of V, let c' be a q-chain of V' such that Sc' = c,
Then с* (с') E G is independent of the choice of c'; if c" is another q-chain in
V' such that Sc" — c, then c' — c" = Sd for some (</ + l)-chain d in U and
c*(c' — c") = c*(3d) = (Sc*)(d) = 0
Hence there is a homomorphism c*: Z,; i(\z) G such that c*(c) = c*(c') if
Sc' = c. Because A,;i( V)/Ze_i(V) is free (since it is isomorphic to a subgroup
of AQ_2(V)ff q > lor to Z if q = 1), there is a homomorphism d* : Ae_i(V)—> G
which is an extension of c*. Then c* | A9(V) = 8d*, proving that j№(A * (• ;G))
is locally trivial. 
5 corollary IfX is a paracompact Hausdorff space homologically locally
connected in dimension n, then p: №(X;G) —> №(X;G) is an isomorphism for
q < n and a monomorphism for q = n + 1. 
Since a manifold is homologically locally connected in dimension n for
all n, and every open subset is paracompact, this implies the next result.
i 7 corollary If X is a manifold, p: H*(X;G) ~ H*(X;G). If A is a
( closed homologically locally connected subset of X,
l’	p: H* (X,A; G) H* (X,A; G). »
C corollary If X is a homologically locally connected space imbedded as
I a closed subset of a manifold Y, then X is taut in Y with respect to singular
i cohomology.
| proof By corollary 5, H* (X;G) ~ H* (X;G), and for an open set U in Y, by
corollary 7, H*(U;G) ~H*(fJ-,G). Since X is taut in Y with respect to
. Alexander cohomology, these isomorphisms imply that it is also taut with
| respect to singular cohomology. 
j 9 corollary If A is any closed subset of a manifold X, then as U varies
i over neighborhoods of A in X,
linu {H*(tZ;G)} ^H*(A;G)
where the right-hand side is Alexander cohomology.
proof By corollary 7, lim , {H*(17;G)} linu {H* (U;G)), so the result
ECi 9 applications of the cohomology of presheaves
343
342	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. g j
follows from the tautness of A with respect to the Alexander cohomology
theory. 
This shows that the modules H*(A;G) and H*(A,B; G) introduced щ '
Sec. 6.1 are the Alexander cohomology modules if A [or (A,B)] is a closed subset I
[or pair] of a manifold. The next result generalizes the duality theorem 6.2.17 I*
to arbitrary closed pairs.
IO theorem Let X be an n-manifold orientable over R. For any closed
pair (A,B) in X and any R module G there is an isomorphism	|
HQ(X - В, X - A; G) Нсп~ч(А,В; G)	I
proof Let A be a closed cobounded neighborhood of В in A. By theorem
6.6.5, there is an isomorphism
H”-e(A,A; G) ~ №-e(A - N, A — N A N; G)
Since (A — N, A — X A X) is a compact pair in X, by theorem 6.2.17,
HQ(X - (A - N A N),X- (A - X); G) ~ H«~e(A - A, A - A A A; G)
Since X — (A — A) and X — A are open, there is an excision isomorphism
HQ(X - A, X*— Л; C) ~ H9(X - (A - A A A), X - (A - A); G)
Combining these gives an isomorphism
HQ(X - A, X - A; G) ~ H"-e(A,A; G)
As A varies over closed cobounded neighborhoods of В in A, the limit of the
modules on the left is HQ(X — В, X — A; G) and the limit of the modules oil
the right is Hcn~Q(A,B; G), whence the result. 
11	theorem If X is a compact Hausdorff space which is homologically
locally connected in dimension n, then H,fX) is finitely generated for q < n,
proof This follows from corollary 5, theorem 6.8.11, and theorem 5.5.13. •
The last result gives a generalization of corollary 6.2.21 to arbitrary com-
pact manifolds (orientable or not). We now work toward a proof of the
Vietoris-Begle mapping theorem.
12	iemma Let (X,A) be a pair and let Г be the presheaf on X defined by
F(V) = Ce(V, V A A; G) for open F С X (<y and G being fixed).
(a)	For any open covering Ql ofX the map Г(Х) F(Ql) sending у E I'(A')
to the compatible ^[family {y | U}ve* is a monomorphism.
(b)	If Ql is a locally finite open covering ofX and ‘X is a shrinking of Ql,
the image of F(Ql) —> TfV) equals the image of the composite
Г(Х) T(Ql) -> T(T)
proof For (o), assume that у E Ci(X,A- G) is in the kernel of Г(Х) —» F(Ql)
(that is, у | U = 0 for all U £ Ql). Let <p C Ce(X,A; G) be a representative of y.
Then у | U = 0 implies that <p | U is locally zero on U. Since this is so for all
0 E Ql> <P is locally zero on X and у = 0, proving (a).
To prove (b), let {yuJueQi be a compatible Ql family and suppose that
£ Cq(U, U A A; G) is a representative of у и for U £ Qi. Then, for IJ, U' E Ql,
| U А U' — <pv | U A U' is locally zero on U A IT. If x E X, some neigh-
borhood of x meets only finitely many elements of Ql, and there is a smaller
neighborhood W,. of x such that
(i) Wa, intersects Vv x £ Vv
(ii)	xjU^W^U
(iii) _ x E Vv C Vv
(iv) x E Vv A Vv Wc = <pv | W?
The first three conditions are clearly satisfied by taking Wx small enough
(because there are only a finite number of conditions to be satisfied) and (iv)
can also be satisfied, because for x E Vv A Vv, qv\ U (J U' — cpv |- U A U'
is locally zero.
For x E X choose U so that x E Vv and set <рж = <pr| We E
Wx A A; G). By (iv), this is independent of the choice of U. If x" E W® A WX',
then x" E Vv for some U £ Ql. Then Ws and WX' meet Vv, and by (i), x,
x' E Vv. Therefore <рж = <pr | We and <рж- = rpv | We-, whence <рж | We A W^ =
<рг-1 W® A Wr- Hence the collection {<рж E CfiWx, Ws A A; G)} is a com-
patible (We) family [of Cfi •, • A A; G)]. By example 6.7.8, there is an ele-
ment <p E C«(X,A; G) such that <p | Wj. = <рж for all x E X. To complete the proof
of (b) it suffices to prove that for each U E Ql, <p I — xpv | Vv is locally zero
on Vv- However, if x E Vv, then, by (iii), We C V, and <p | WT = <рж = rpv | We.
Hence {W,.};,.ci,-( is an open covering of Vv on which <p | Vv and cpv | Vv
agree. 
। 13 theorem Let f: X' -ч> X be a closed continuous map between para-
‘ compact Hausdorff spaces. Let A' be a closed subset of X' and suppose there
are integers 0 < m <f n such that H^f^x, f^x A A'; G) = 0 for all x E X
and for q <T) m or m <fq<fn. Let Г be the presheaf on X defined by
T(U) = lJm(tf~1(U),f~1(tU) A A'; G). Then there are isomoiphisms
Нв^™(Х;Г) ~ He(X',A'; G)
q < n
i and a monomorphism
Н”-’»(Х;Г)	H^X',A'; G)
proof Let Г * be the nonnegative cochain complex of presheaves on X de-
lined by Г*(С7) = C*(J~4U),f~l(U) A A’-G). Thus R is the image under
' fi of the fine presheaf on X' which assigns CfiU', U' A A'; G) to U' С X'.
By theorem 6.8.3c, the latter is a fine presheaf on X' [being the completion of
the fine presheaf C«( •, • A A'; G); see example 6.8.1], and by theorem 6.8.3b,
Г® is fine on X. As U varies over neighborhoods of x in X, (/-1( U), /-1( U) A A')
344
GENERAL COHOMOLOGY THEORY AND DUALITY CHAR, (j
varies over a cofinal family of neighborhoods of (/’ fy f~lx Г) A') in (X',Af
(because f is closed and continuous). From the standard tautness properties
and the hypothesis about H* (f~Ax, f~Fx Г) A'; G), it follows that №([’*) jjj
locally zero for q < m and m < q < n. By theorem 6.8.7, there are functorial
isomorphisms
Й«-’»(Х;Н™(Г	№(Т*(X)) q < n
and a monomorphism
H»-»i(X;H™(r*)) Hn(T*(Xf)
Since Г = Нт(Г *), it merely remains to verify that
№(? * (X))	Hp(X',A'; G) all p
As Ql varies over the cofinal family of locally finite open coverings of X it fol-
lows from lemma 12 that
Г*(Х) = linu {Г*(Qt)} = lim , {C*(•, • П A'; G)(f~41)} ~ C*(X',A'; G)
and this yields the result. 
If f is an m-sphere bundle over a paracompact Hausdorff base space B,
then Нч(р^(х), pfhxj П Ё) = 0 if q fy m + 1. Therefore the hypotheses of
theorem 13 are satisfied for all n. Since the presheaf Г that occurs in theorem
13 is the tensor product of the orientation presheaf of f and G, we obtain the
following generalization of the Thom isomorphism theorem to nonorientable
sphere bundles.
14 theorem Let £ be an m-sphere bundle over a paracompact Hausdorff
base space В and let Г be the orientation presheaf of £ over R. For all
R modules G and all q there is an isomorphism
№(B; T®G)~ №+m+\E^Ep, G) 
Another interesting consequence of theorem 13 is the following Vietorls-
Begle mapping theorem.
15 theorem Let f: X' —>-X be a closed continuous surjective map between
paracompact Hausdorff spaces. Assume that there is n > 0 such that
Hfff^xiG) = 0 for all fyX and for q < n. Then
f*: №(X;G)	№(X'-,G)
is an isomorphism for q <^n and a monomorphism for q = n.
proof Let Z be the mapping cylinder of f and regard X' as imbedded in Z.
Then Z is a paracompact Hausdorff space, X' is closed in Z, and the retrac-
tion r: Z —> X is a closed continuous map. For x £ X, r“1(x)js contractible
[since it is homeomorphic to the join of x with/-1(x)], and so Н*(г!(г)) = 0.
Because r-1(x) Г) X' = /1(x) is nonempty, we have
Г SEC. 9 APPLICATIONS OF THE COHOMOLOGY OF PRESHEAVES	345
"	№+1(r~i(x), П X'; G) ~ №(f^(x); G) = 0 q < n
It follows from theorem 13 that №(Z,X'; G) — 0 for q < n. Since there is a
commutative diagram with an exact row
: .	----> №(Z,X')	№(Z)	№(X')	№+1(Z,X')	 • •
I	у
i	r* I ~	/у*
[	№(X)
I the result follows. 
i There is a partial converse of theorem 15 asserting that if f: X' X is a
closed continuous surjective map between paracompact Hausdorff spaces and
there is n > 0 such that for every open U С X, /*: №(U;G)	H^f-^U^G)
: is an isomorphism for q < n, then №(f~1(x);G) — 0 for all x € X and
for q <C n. This follows from commutativity of the following diagram (where
! U varies over open neighborhoods of x £ X):
j'	lirnu {&(U;G)>	ffr(x;G)
linu {№(f-\U);G)}
| In particular, if X and X' are metrizable (or have the property that every
i open subset is paracompact), then for n > 0, f*: №((./; G)	№(f~1(U);G)
fc an isomorphism for all open U С X and all q < n if and only if
H9(/-1(x);G) = 0 for all x £ X and all q < n.
We present an example to show that the condition that /be a closed map
is necessary in theorem 15.
i
! 16 example Let X' = {(x,y) £ R2 | x2 + y2 = 1 or x2 + y2 <( 1, x > 0}
! and let X = [0,1]. Define J : X' X by
„ .	fO	x<0
l	x	x>0
j	A \J
| Then / is a continuous surjective map but not a closed map. Furthermore,
f~\t) =
closed semicircle
closed interval
single point
t = 0
0<t< 1
t = 1
Because the unit circle S1 is a strong deformation retract of X',
fli(X';G) ~ H^G) ~ G.
Since /7'(Х;С) = 0, the homomorphism/*: H^XjG) ^(XhG) is not an
j isomorphism.
17 example Let X C R2 be the space of example 2.4.8, illustrated below:
346
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP, (j ?(	M(. 10
CHARACTERISTIC CLASSES
347
(0Д)
(0,-2)	a2	(1,-2)
There is a closed continuous surjective map f of X onto the space Y consisting
of the four sides of the rectangle
(0,0) -- (1,0)	!
(0,-2)	(1,-2)	I
such that	|
_ (single point	у (0,0)	?
[closed interval	у = (0,0)	।
It follows from theorem 15 that/*: H*(Y;G)	H*(X;G) for any G, and (
therefore the map / is not null homotopic.	i
18 theorem Let f: X' X be a proper surjective map between locally |
compact Hausdorff spaces and assume that for some n > 0, №(/-1(x);G) = 0 (
for all x E X and all q < n. Then	I
/*: HC«(X;G) -> HC«(X';G)
is an isomorphism for q < n and a monomorphism for q = n.
proof If either X or X' is compact, the other one is also compact, and the
result follows from lemma 6.6.9 and theorem 15. If neither X nor X' is '
locally compact, let X+ and X'+ be their one-point compactifications and /
extend/to a map/+: X'+ -a X1 mapping the point at infinity of X'+ to the Д
point at infinity of X+. Then/+ satisfies the hypotheses of theorem 15, and
the result follows from corollary 6.6.12 and theorem 15. 
1 О characteristic classes
This section is a culmination of our general work on homology theory. We
use the cup product and Steenrod squares to define characteristic classes of a
manifold and of one manifold imbedded in another. These characteristic
classes are important invariants of the manifold and have interesting applica-
tions to nonimbedding problems.
Let X be an n-manifold with boundary X and U £ IT'(X x X, X X X — S(X))
be an orientation (over R) of X. Let j-. X - X С X be the inclusion map.
Then the maps
j X 1: (X - X) x (X,X) С X x (X,X)
1 X /: (X,X) X (X - X) С (X,X) X X
are both homotopy equivalences. Therefore there are elements
Ui e h«(x x (x,x)) u2 e я»((хд) x x)
such that
(/X l)*<7i= C7| (X — X) X(X,X) (1 Xj)*U2 = C7|(X,X) x (X - X)
If X is compact, let z £ Hn(X,X) be the fundamental class of X corresponding
to U, as in theorem 6.3.9. The Euler class of a compact oriented manifold X,
denoted by x € Hn(X,X), is defined by
X = (hr v U2)/z
The reason for the name is furnished by theorem 2 below.
Assume that В is a field and that X is a compact n-manifold with bound-
ary X. By theorem 6.9.11, H^(X) and 7/. (X,X) are finitely generated. If {«,}
is a basis of H* (X) and {//} is a basis ot H* (X,X), then by the Kiinneth for-
mula for cohomology, {щ X ц} is a basis of H*(X X (X,X)). Hence
Ui — S aijUi x Vj
for some scalars o{y. Let bjk = (ц- о t/fe, z), where z is the fundamental class
corresponding to U. Then we have matrices A = oi;- and В = bjk, and the
following expresses their relation to each other.
1	lemma With the above notation,
(AB)ik = (-l/ideg^g.^
proof The proof is essentially the same as that for theorem 6.3.12. Because
z is the fundamental class corresponding to U, it follows that
Gt/z = i e h°(x)
By property 6.1.4, for any к
= Uk о ! = tik o C7i/z = [(lift X 1) о Lfffz
From lemma 6.3.11 it follows readily that
(Wft X 1) о Ut = (1 x uk) о Hi = (-l)mdesufcGi о (1 X tik)
= 2 (-1)” ^ ^aijUi X (v3- о uk)
г,з
Hence by property 6.1.2
Uk = 2 (- 1)« deg Ukaijb^Ui
i,3
Since {tq} is a basis, this implies the result. 
348	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. ()
2	theorem If x is the Euler class of a compact n-manifold X oriented
over a field, then <x,z> is the Euler characteristic of X.
proof We first compute U2. Let T: X X X X X X be the map inter-
changing the factors. There is a commutative diagram, with all vertical maps
induced by maps defined by T and all horizontal maps induced by inclusions,
HfiX x X, X X X - S(X))	U-'fiX - X) x (X,X))	H«(X X (X,X))
-ttJ,	Jr?
HfiX X X, X X X - S(X))	№\(X,X) X (X - X))	H>fiX,X) X X)
In lemma 6.3.11 it was shown that TfU = ( — I)’1 U. Therefore 7'* Uj =
( — 1)’!U2, and so
U2 = (—l)7!Tf ( S Янин x vi)
k,l
= (-1)” S (-1)dl«	deg V1 awvi X t/fc
Therefore
Ui U2 = (-1)” S ( —l)deSrl deg +deguA degplC..CfcZ(H. Г;) X (r.	Mfc)
= (— I)11 S (— l)deg vi deg v> + deg deg vi + deg "1 deg vi Oijautvi о ufi
x (ny О uk)
where the summation is over all i, j, k, and I such that
deg Hi + deg vj = n = deg uk + deg 14
It follows that
Ui о U2 = S (~l)deg"gOyOH(o! о Hi) X (ц- о ufi)
Using lemma 1,
<XX> = < Li о U2, z X z>
~ 2 ( l)deg ^aijbjjcaicjbii
= S (-l)degM(:(AB).fc (AB)ki
i,к
= S (-l)deSufc
and the last sum is the Euler characteristic of X. 
Classically, the Euler class is usually taken to be the Euler class (in our
sense) over Z. For any pair (Y,B) whose homology is of finite type, it follows
from the universal-coefficient formula for cohomology (theorem 5.5.10) that
H«(Y,B; R) №(Y,B; Z) ® R
Therefore the monomorphism Z R induces a monomorphism
№(Y,B; Z) H«(Y,B; R)
p' sgc. 10 CHARACTERISTIC CLASSES	349
’ jn particular, the monomorphism H’!(X,X; Z) Hn(X,X; R) maps Euler class
; to Euler class, and therefore theorem 2 remains valid for the integral Euler
class of X.
j We now specialize to the case where the coefficient field is Z2, in which
i case U, hence also 14, and (if X is compact) z, are all unique. There is the
j Thom isomorphism
Ф*: №(X -X)zz H«+n((X - X) X (X - X), (X - X) x (X - X) -8(X - X))
defined by Ф* (o) = (o x 1) о U', where
U' = CT I ((X - X) x (X - X), (X - X) X (X - X) — 8(X - X))
j ф* can be extended to
j	Ф*: №(X)	№+n(X X X, X X X -8(X))
j by Ф* (f) = (о X 1) о U. There is a commutative diagram whose vertical
I maps are isomorphisms
• Hq(X) -21»	№+n(X x X, X X X - 8(X))
? =! =!
; HQ(X - X) №+»((X - X) X (X - X), (X - X) X (X - X) — 8(X - X))
j from which it follows that Ф* is also an isomorphism on H«(X). For i > 0 the
ith Stiefel-Whitney class of X, Wf С //’(Х;Х2), is defined by the formula
Ф*(ю») = Sq‘U
[that is, SqlU = (uq X 1) о С7]. Following are some examples.
3 By condition (a) on page 271, w0 = 1.
4 By condition (b) on page 271, if X is a compact n-manifold without
, boundary, wn is the Euler class of X over Z2.
5 By condition (c) on page 271, Wi = 0 for i > dim X.
i
I 6 A manifold X is orientable over Z if and only if 101 = 0 (see exercise
! 5.H.3J).
} If X is compact and z £ Hn(X,fC) is the fundamental class of X over Z2,
j then, by property 6.1.4,
I	Wi = [(w{ X 1) UlJ/z = Sq^Ui/z
j where U± £ Hn(X X (X,X)) corresponds to U. We use this to determine the
I Stiefel-Whitney classes of a compact X in terms of cohomology operations
! in X. For i > 0 the homomorphism Sq‘: Hn~'l(X,X) —> Hn(X,X) has a transpose
' homomorphism Sql: Hn(X,X) Hn-fXJC) such that
[	{Sqlu,z} = (u,Sqlz) и £ Д”~*(Х,Х)
where z is the fundamental class of X. By the isomorphism of theorem 6.3.12,
350
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP, fj
Ke. №fX) ~ H„_{(X,X)
and there is a unique Vi E H»(X) such that iq(Vj) = Sql(z). Then for
и £ №~i(X,X; Z2)
(Sqlu,z) = {u,Sqlz') = (u^lVi))
= <«, Vi z) = (и о Vi, z>
This equation holds trivially if deg и -=f= n — i. The following Wu formula
shows that the classes V; and the Stiefel-Whitney classes w, determine each
other.
f theorem In a compact n-manifold, for q < 0
w„ = S Sq^~lVt
0<l_q
proof We have (h = S ацЩ X Vj, where {нг} is a basis of H* (X,Z2) and
{vj} is a basis of H * (X,X; Z2). By the Cartan formula, condition (d) on page 271
Sq«U = S aijSq^Ui X SqlVj
l:tl~q
Let Vi — S cimum. Then we have
wq = (Sqrdj)/z = S aij(SqlVj,z)SqkUi
Jc+l—q
= S aij(vjO Vi, z)SqkUi
k+l—q
= S aijCim(vj о um, z)Sqktii
k+l—q
= S tti}bjmCimS(jkUi
k+l=q
Using lemma 1, we find that
w„ = S ctiSq^Ui = S SqkVi 
k+l—q	k+l—q
Let Pn be the real projective n-space and let w be a generator of HfP")
for any n > 1. We use lemma 5.9.4 to compute Sqfwi) in the following
examples.
8 For the real projective plane P2, Sq4(w) = w2; therefore Vi(P2) = w,
mi(P2) = w, and w2(P2) = w2.
О For P3, Sq2(w) = 0 and Sq4(w2) — 0, so Vj(P3) = 0 for i > 0 and
wfP3) = 0 for i > 0.
IO For P4, Sq2(w2) — w4 and Sq1(w3) = w4, so Vi(P4) = w, Vz^P4) = w2,
w^P4) = w, w2(P4) = 0, w3(P4) = 0, and w4(P4) = w4.
11 For P5, Sq2(w3) = w5 and Sq^w4) = 0, and V2(P5) = w2 is the only non-
zero W(P5), where i > 0. Hence tt>i(P5) = 0, w2(P5) = w2, m3(P5) = 0,
m4(P5) = w4, and w5(P5) = 0.
The Euler class and Stiefel-Whitney classes of a manifold X are topolog-
ical invariants associated to X. We shall now define characteristic classes for a
sgc. 10 CHARACTERISTIC CLASSES	351
manifold X imbedded in a manifold Y. These will be topological invariants of
the imbedding. First, however, we need an algebraic digression.
In our consideration of the slant product we limited ourselves to one of
the two possible slant products. We now introduce the other one. Given chain
complexes C and C', a cochain c* £ Hom ((C	C')B, G), and chain
c € C(; ® G', there is a slant product c\c* £ Hom (CB_e, G ® G') which is
the cochain such that if c = S c, ® g;, with a £ Cq and gi £ G', then
<c\c*,c'> = S <c*, Ct ® d) ® g- d € C'_e
Then	S(c\c*) = (— l)«(c\Sc* — Эс\с*)
from which it follows that there is an induced slant product of II“(C ® C'; G)
and Hq(C;G') to Нп~ч(С; G ® G'). This gives rise to a topological slant
product of Hn((X,A) x (Y,B); G) and Hq(X,A- G') to	G ® G') having
properties analogous to 6.1.1 to 6.1.6. We list without proof two of these, to
which we shall have occasion to refer.
12 Given и £ Hn((X,A) x (Y,B); G), z £ Hq(X,A; G"), and v £ Hp(Y,B; G’),
let T: G ® G" ® G' —> G ® G' ® G" interchange the last two factors.
In Hn~q+p(Y,B; G ® G' ® G") we have
T* ((z\u) v) = z\[u v (1 x o)] 
13 Given и e Hn((X,A) X (Y,B); G), v £ №(X,A- G'), and z £ Hq(X,A- G"),
then, in H»+p-4(Y,B- G ® G' ® G"),
(о о z)\u = z\[u о (о X 1)] 
Let Y be an m-manifold without boundary and
U £ Hm(Y X Y, Y x Y - S(Y); R)
an orientation of Y over R. Given a pair (A,B) in Y, we define
yv- Hq(A,B- G) 11>,,!(Y — B,Y — A;G)
ЬУ
Yb(Z) = Z\[G|(A,B) X(Y-Д y-А)] zEHe(A,B;G)
Then we have the following complement to the duality theorem.
14 lemma Let X be a compact homologically locally connected space in
an m-manifold Y with orientation class U. Then we have an isomorphism for
all q and all G
y'u- Hq(X-G) -z Hm~<i(Y, Y - X- G)
proof Since X is compact and homologically locally connected, it follows
from theorem 6.9.11 that Н(Д(Х)) is of finite type. By lemma 5.5.9, there is a
free chain complex C of finite type which is chain equivalent to Д(Х).
Let X: С Д(Х) be a chain equivalence. Let Д' and C' be the chain complexes
obtained by reindexing the cochain complexes Hom (Д(Х),В) and Hom (C,B),
respectively, so that Lq — Hom (Дт_е(Х),В) and C'q = Hom (Cm_Q,B). The
352	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP.(j
chain equivalence X defines a chain equivalence X': Д' C. Because C is
free and of finite type, so is С' [Д' will not be free, in general, because Д(Л')
need not be 6f finite type].
Let с* E Hom ([Д(Х) <8> (Д(¥)/Д(¥ — X))]m, /I) be an m-cocycle corre-
sponding to U | X X (¥, ¥ — X) under the Eilenberg-Zilber isomorphism and
define a map
т: Д(¥)/Д(¥ - X) Д'
by т(с) = с* /с for с E Д(Х). If deg c = q,
Э(т(с)) = S(c*/c) = ( —1)’п"«+1с*/Эс = ( —1)Я1“«+1т(Эс)
so т either commutes or anticommutes with 3, depending on degree. Hence r
induces homomorphisms t* on homology and t* on cohomology for any
coefficient module. Clearly,
r* = yv-. HQ(X, ¥ - X; K)
Because X is homologically locally connected, by corollary 6.9.8, X is taut in Y,
and by the duality theorem, yv, and hence , is an isomorphism. Therefore
the composite X' ° т induces an isomorphism X'* ° of He(Y, ¥ — X; R)
with He(C') = H№-e(C;B). Since Д(¥)/Д(¥ — X) and C are both free, it
follows from the universal-coefficient formula for cohomology (theorem 5.5.3)
that for any G
(X' ° t)* = t* ° X'*: H*(Hom (C',G)) ~ H*(Hom (Д(¥)/Д(¥ - X), G))
There is also a commutative diagram
Нв(Д(Х) ® G) He(C ® G)
I	!=
H’»-«(Hom (Д',С))	H™-e(Hom (C',G))
where the vertical maps are induced by the canonical map
A ® G Hom (Hom (A,K), G)
for any module A (the right-hand vertical map being an isomorphism because
C is of finite type). Hence there are isomorphisms
He(X;G) H»^«(Hom (Д',С))	¥ - X; G)
It only remains to verify that this composite is yfr. If a E Aq(X) and
g E G, the composite
Дв(Х) ® G Hom (Д'т_е,С) ^от(тД)> Hom	Am_q(Y - X),G)
maps a ® g to the homomorphism h such that if o' E Д»< <-;(¥),
h(a') = т(о')(а ® g) = (c* /n')(a ® g)
= <c*, a ® a'>g — [(a ® g)\c*](a')
SEC> 10 CHARACTERISTIC CLASSES	353
therefore Ji = (a ® g) \c*, and this gives the result on passing to homology. 
Let X be a closed subset of a space Y tautly imbedded with respect to
singular cohomology and let А С X. Assuming X — A taut in Y — A, we define
Hp(Y, Y - X; G) о №(X, X - A; G') Hp+<i(Y, Y — A; G")
where G and G' are paired to G". If V is any neighborhood of X in Y and V
is a neighborhood of X — A in V — A, there is a cup product
№(V, V - X; G) о №(V,V; G')	№+<i(V, (V - X) U V- G")
There are excision isomorphisms (for all coefficients)
№(Y, Y -X)zz Hp(V, V - X)
H₽+«(Y, Y - A) ~ Hp+<i(V, V — A)
Since V - A = (V - X) U V' we have a cup product
№(Y, Y - X; G) №(V,V; G') -» H₽+«(Y, Y - A; G")
As V varies over neighborhoods of X in Y and V varies over neighborhoods
of X — A in V — A, it follows from the tautness assumptions and the five
lemma that lim , (H* (V,A; G')} ~ H* (X,A; G'). The desired cup product is
thus obtained by passing to the direct limit with the above cup product.
Let X be a compact n-manifold without boundary imbedded in an
m-manifold Y without boundary. Assume that U and U' are orientations of
X and Y, respectively, over R. There is then an isomorphism (for any R
module G)
6»: H«(X;G)	Hm-«+e(Y, Y — X; G)
characterized by commutativity in the triangle of isomorphisms (note that X
is homologically locally connected, and so lemma 14 applies to X C Y)
H^e(X;G)
Тгт/
He(X;G) A h™-»+q(Y, Y — X; G)
This map в is similar to a Thom isomorphism and has the following
multiplicative property.
15 lemma The isomorphism 6: №(X,G) ~ H’»-»+?(Y> Y — X; G) has the
property that for v £ №(X;G)
6(v) = ±6(1) о v
where 0(f) £ Hm~,l(Y, Y — X; R)
proof Let z £ Hn(X;R) be the fundamental class of X corresponding to U
and suppose v = i*v' for v' £ №(V;G) and i: X С V, where Vis a neighbor-
hood of X in Y. By theorem 6.3.12, Yu-1(u) = z = ±i* v' z. Then,
using properties 12 a"'1 13 (with all equations holding up to sign),
354
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. 6
0(u) | (V, V - X) = ±(i* I/ z)\[C7' I X x (V, V7 - X)]
= ±i*(i*v' r- s)\[C7'| V x (К V - X)]
= ±(г^^)\[Г|Ух(УЛ-Х)]
= ±i* Д{[17' | VX (V, V-X)] v (o' X ly)}
= ±i*A{[C7' I V x (К V - X)] о (ly X «/)}
= ±Д{[17' | X x (V, V - X)] о (1Л X </)}
= ±[0(1)|(V, V-X)]^o'
= ±[0(1) r] | (V, V - X)
Since H*(Y, Y — X) ± H*(V, V — X), this gives the result, и
Our next result, a consequence of lemma 15, follows immediately from
the definition of the cup product, №(Y, Y — X) о №(X) Hp+i(Y, У — X),
16 corollary Let X be a compact oriented n-manifold imbedded in an
oriented m-manifold Y, both without boundary. For any element о £ №(Y;Q)
we have
6(v | X) = ±0(1) о v 
The normal Euler class of X in У, denoted by xx,y F Hm~”(X;R), is
defined by the equation
= 0(1)	0(1) 6 H2('«-«)(Y, Y - X; R)
Since 0(1) о 0(1) = 0(1) о [0(1) | Y], we obtain from corollary 16 the fol-
lowing characterization of the normal Euler class.
17 theorem If a compact n-manifold X is imbedded in an m-manifold Y,
both without boundary and oriented over R, the normal Euler class
Xr,r=0(l)|X. -
In particular, if H’n-’l(Y;JR) //"' "(X;/!) is trivial, it follows that the
normal Euler class is zero. Thus, if Y is Euclidean space, the normal Euler
class of any compact X imbedded in Y is zero.
For i > 0 the ith normal Stiefel-Whitney class ofX in Y, Wj E 1/!(Х;/2),
is defined by
0(шг) = S<7{0(1)
Here are some examples.
18 By condition (a) on page 271, m0 = 1-
IO By condition (b) on page 271, if X is a compact n-manifold, wn is the
normal Euler class of X in У over Z2.
20 By condition (c) on page 271, tq = 0 for i > dim Y — dim X.
There is an important relation between the Stiefel-Whitney classes of X
and Y and the normal Stiefel-Whitney classes of X in Y toward which we are
heading.
•sgc. 10 CHARACTERISTIC CLASSES	355
2 В lemma Let X be a compact n-manifold imbedded in an m-manifold Y,
both without boundary. Let U and U' be the orientation classes ofX and Y,
respectively, over Z2 and let 6(1) £ Hm~n(Y, Y — X; Z2). Then
U' \(XxY,XxY - 8(X)) = [1 X 6(1)] о U
proof If X' is a component of X, it suffices to prove that
| (X' X Y, X' X Y - 8(X')) = ([1 x 6(1)]	17) | (X' X Y, X X Y - 8(X'))
Hence we may assume X connected, in which case (X X Y, X X Y — 8(X)) is
a fiber-bundle pair over X with fiber pair (Y, Y — x0), where _т0 € X. Since
U' | (X X Y, X X Y — S(X)) is an orientation over Z2 of this bundle pair, and
there is a unique orientation over Z2, it suffices to prove that [1 X 6(1)] U
is also an orientation over Z2 of this bundle pair. That is, we need only show
that for x £ X, ([1 X 6(1)] о 17) | x x (Y,Y — x) is nonzero. This will be so
if its image in x X (Y, Y — X), which equals ([1 X 6(1)] U) | x X (Y, Y — X),
is nonzero. Because U £ Hn(X X X, X X X — S(X)) is an orientation,
U | x X (X X — x) = 1ж X и, where и £ Hn(X, X — x) is nonzero. Because
Hn(X, X — x) Hn(X) is a monomorphism [dual to the monomorphism
Ho(x) —> Hq(X)], и | X is nonzero. We have
([1 X 6(1)]	17) | x X (Y, Y - X) = [l.r X 6(1)] (l;r X и | X)
= 1л. X [6(1) и | X] = 1ж x 6(u | X)
Since 6 is an isomorphism, this implies that ([1 X 6(1)] U) | x X (Y, Y — X)
is nonzero. 
From this result we have the following Whitney duality theorem.
22 theorem Let X be a compact n-manifold imbedded in an m-manifold
Y, both without boundary. For к > 0
wifY) | X = S tbj о Wj(X)
i+i—k
where w^Y), w.j(X), and ш, denote the Stiefel-Whitney classes of Y,X, and
X in Y, respectively.
proof The result follows easily from lemma 21 and the Cartan formula
(rather, the equivalent form of lemma 5.9.4):
(H-(Y) | X] X ly) 17' | (X X Y, X x Y - S(X))
= ([wft(Y) x ly] 17') | (X X Y, X X Y - 8(X))
= Sq4J' | (X X Y, X X Y - 8(X)) = Sq4U' | (X X Y, X X Y - 8(X)))
= S</'"([lx X 6(1)] о 17) =^2 * [U X S<7’6(1)] SqiU
= ,S= k (lx	X [6(1) Wi]) [w/X) X lx] 17
= S	X lx)	[w,(X)	X lx] [lx X	6(1)]	о 17
= (([ S tbi о Wj(X)] x ly) о 17') | (X X Y, X X Y - 8(X))
356
GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. (J
gXERCISES
357
By the Thom isomorphism theorem, this implies the result. 
In case Y is Euclidean space, wk( Y) = 0 for к > 0, and theorem 22 showy
that tfo and Wj(X) determine each other recursively. In particular, the classes
tOj are independent of the imbedding of X in the Euclidean space. If X is a
compact n-manifold imbedded in Rn+d, it follows from example 19 and 20
and from the fact that the Euler class of X in R«+d is zero that w? = 0 for
i > d. This gives the following necessary condition for imbeddability of X fo
R»+d
23 corollary Let X be a compact n-manifold imbedded in Rn+d and let
Wj € H‘(X;Z2) be defined by
v - /va	Zc — 0
i+?=fc Wi	[0 к > 0
Then tfo = 0 for i > d. 
We present some examples.
24 For P2, ibi(P2) = w and w2(P2) = 0, so P2 cannot be imbedded in R3.
25 For P3, tfo(P3) = 0 for i > 0.
26 For P4, tbi(P4) = w, tb2(P4) = w2, w3(P4) = w3, and ib4(P4) = 0. There-
fore P4 cannot be imbedded in R7.
27 For P5, wfiP5) = 0, w2(P5) = w2, th3(P5) = 0, m4(P5) = 0, and w^P5) = 0.
Hence P5 cannot be imbedded in R7 (which is also a consequence of example 26).
The last examples show the importance of calculating Wj(P'i), which we
now do.
28 theorem Let (]‘)2 be the binomial coefficient (’/) = n!/i!(n — i)! reduced
modulo 2. Then
Wi(Pn) =
proof Since (”J1)2 n + 1 = X(Pn), the result is true for i = n. For
i < n, where n > 1, we suppose Pn 1 linearly imbedded in P" . Then Pn — Pnl
is an affine space, hence Й*(Р® — P"-1) = 0 and IlfiP", Pn — РпЛ) zz №(Pl).
Then the normal Thom class 6(1) £ ТР^Р», Pn — P11-1) maps to w in №(Pl),
so wi = w. By theorem 22, Wi(Pn) | PnA = т{(Р"-1) + w о wi_i(P'!-1).
Since №(Pl) zz Hr‘(Pn') for q < n, it follows by induction on n that
W;(P«) = [(Л1)2 + (?)2]W’ =	
EXERCISES
A MANIFOLDS
1 If X is an n-manifold with boundary X, prove that X is a homology n-manifold whose
boundary, as a homology manifold, equals X.
1 In the rest of the exercises of this group, X will be an n-manifold without boundary and
f{ will be a fixed principal ideal domain.
' 2 If Г is a local system of R modules on X, prove that for any А С X
j	Hq(A X X, A x X - 8(A); R x Г) = 0 q < n
i (Hint: Prove this first for A contained in a coordinate neighborhood of X. Prove it next
| for compact A by using the Mayer-Vietoris technique. Then prove it for arbitrary A by
taking direct limits over the family of compact subsets of A.)
3 Prove that there is a local system Tr of R modules on X such that
fofo) = Hn(X, X — x; R) for x £ X.
For x 6 X let Zs, 6 Hn(X, X — x; IT) be the generator corresponding under the
| isomorphism
’	H„(X, X - x; for) ~ Hom (№(X, X - X; R), H»(X, X - x; R))
| to the identity homomorphism of Hn(X, X — x; R). A Thom class of X is an element
j	U e H"(X X X, X X x - S(X); R X Hom (TX,R))
such that (U | [x x (X, X — x)])/^ = 1 G H°(x;R) for all x 6 X.
1 4 If V is an open subset of X and U is a Thom class of X, prove that
V | (V X V, V X V — 8(V)) is a Thom class of V.
I 5 Prove that Rn has a unique Thom class.
[ 6 Prove that X has a unique Thom class. [Hint: Use exercise 2 to show that
I H”(X X X X X x - 8(X); R X Hom (FA-,R)) ~
|'	lim_ [H"(VXX, Vx X - 8(V); R X Hom (FA,R))}
- where V varies over finite unions of coordinate neighborhoods. Then the result follows
from exercises 4 and 5 by Mayer-Vietoris techniques.]
If (A,B) is a pair in X and G is an R module, define
)	y: Hq(X - В, X - A; PA ® G) —> Н”~ч(А,В; G)
I by y(z) = [U | (A,B) x (X — В, X — A)]/z, where U is the Thom class of X. As (V,W)
. varies over neighborhoods of a closed pair (A,B) in X, there are isomorphisms
f	lim_> (HQ(X - W, X - V; Er ® G)} ~ Hq(X - В, X - А: ГА ® G)
! and	firn. (№-i(V,W; G)} ~ H"''/(A,B; G)
and a homomorphism
y: Hq(X — В, X — А; ГА ® G) -а №~ч(А,В; G)
I
is defined by passing to the limit with y.
' 7 Duality theorem. Prove that for a compact pah (A,B) in X, у is an isomorphism.
j В THE INDEX OF A MANIFOLD
[ 1 Let X be a compact n-manifold, with boundary X oriented over a field R, and let
J [X] 6 Hn(X,X; R) be the corresponding fundamental class. For u £ №(X,X; R) and
j l; € Hn~i(X;R) prove that (pfiu,v) — (u w v, [X]) C Bis a nonsingular bilinear form from
( №(X,X) x Hn^i(X) to R [that is, и = 0 if and only if <pfou,o) = 0 for all о].
A With the same hypotheses as above, let [X] = 3[X] £ H„_1(X;H) and let <pA be the
corresponding bilinear form from №-\X;R) X H”-i(X;R) to R. Let /: X С X, and
if w £ //'/ 1 (X;/{) and v £ Hn^'i(X;R), prove that
358	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP g I
<Pa{«>/*(u)) = <РЛ'№),У)	3
3 Prove that the Euler characteristic of any odd-dimensional compact manifold is ()
and the Euler characteristic of an even-dimensional compact manifold which is a boundary
is even. (Hint: If X is the boundary of a (2n + l)-manifold X, then, with Z2 coefficients i
dim im [f*: H”(X) -> H«(X)]) = dim im [S: H"(X)	H«+1(X,X)]
and their sum equals dim №(X).)
Let Y be a compact 4m-manifold, without boundary oriented over R, and define the
index of Y to be the index of the nonsingular bilinear form <pr from H2m(Y;R) x №”,(Y;R)1
to R (when <py is represented as a sum of к squares minus a sum of j squares, the index
of <[ y is к — /).
4 If Y is oppositely oriented, prove that its index changes sign. Show that the index of i
the product of oriented manifolds is the product of their indices.
£- gxERCISES	359
pf K(ty [°r °f ^/(^|/)] generated by all simplexes {< 0,    ,Ur] of K(9l) [or of
. X'(9l')] such that Ui contains s' for 0 < i < r. Then C(X(.s)) and C(p(s')) are acyclic, and
the method of acyclic models can be applied to prove the existence of chain maps
1	r: (C(K(Ql)),C(K'(9l')))	(C(X(9l)),C(A(91')))
r': (C(X(9l)),C(A(Ql)))	(C(K(9l)),C(K'(9l')))
j such that r(C(s)) C C(X(.s)) and r'(C(.s')) C C(/j,(s')). Similarly, the method of acyclic
* models shows that r and r' are chain homotopy inverses of each other.1)
2 Let fT/V') be a refinement of (91,9l'), let w: (К^Х^К'^Х')) —> (K(f),K'(f')) be a pro-
jection map, and let j: (X^Y ),A(V')) C (X(9l),A(91')). For any abelian group G prove that
there is a commutative diagram
Н«(ВДЛ'(^;Ц ~ H*(X(Ql),A(9l'); G)
-•J,	V*
5 If X is a compact (4m + l)-manifold, with boundary X oriented over R, prove that
the index of X is 0. [Hint: Prove that (H2m(X;R)) is a subspace of №ra(X;R) whose di-
mension equals one-half the dimension of HZm(X;R) and on which <px is identically zero. !
This implies the result.]	•<
C CONTINUITY
1 Let {(X;,Aj), be an inverse system of compact Hausdorff pairs and let
(X,A) = linn {(Xj,Aj)}. Prove that (X,A) can be imbedded in a space in which it is
a directed intersection of compact Hausdorff pairs {(X;,A])}j(j, where (XJ,AJ) has the same
homotopy type as (Xj,Aj). [Hint: For each j £ J imbed Xj in a contractible compact !
Hausdorff space Xj, f^r example, a cube, and let (X}-,Ay C X/ej Xj be defined as the pair /
of all points (t]j} with t/л.- in X^ or in A,;, respectively, such that if j < k, then i/j = ^‘(yf), I
and if j k, then i/j is arbitrary.]
2	Prove that a cohomology theory has the continuity property if and only if it has the
weak continuity property.
3	The p-adic solenoid is defined to be the inverse limit of the sequence	I
Si <L S1 <----<- S1 <£ S1 <---- {
where/(s) = zp. Compute the Alexander cohomology groups of the p-adic solenoid for |
coefficients Z, Zp, and R.	[
4	Generalize the solenoid of the preceding example to the case where there is a I
sequence of integers ni, «2,   • such that the with map of S1 to S1 sends z to znm. Com- I
pute the integral Alexander cohomology groups of the resulting space.	j
5	Find a compact Hausdorff space X such that №(X;Z) = 0 if q 7^ 1 and H^XjZ) s R. •
D CECH COHOMOLOGY THEORY	!
1 Let (9|,9Г) be an open covering of (X,A) (9l is an open covering of X and 9l' C Ql is I
a covering of A) and let К('-:)\) be the nerve of Qt and K'(9l') the subcomplex of •
K(9l) which is the nerve of 9l' Г) A = {If П A | If £ 91'}. Prove that the chain com- i
plexes (C(K(9l)),C(K'(9l'))) and (C(X(91)),C(A(91'))) are canonically chain equivalent. (Hint: f
If s = {Go, . . . ,UQ} is a simplex of K(9l) [or of K'(Ql')], let X(s) be the subcomplex of I
X(91) [or of A(9l')] generated by all simplexes of X(9l) [or of A(9l')] in Г) Ui, If J
s' = {x'o, . . . ,xQ} is a simplex of X(9l) [or of A'(9l')], let p(.s') be the subcomplex j
Н*(К(Т),К'(Ч’'); G) == Н*(Х(С\')Д(Т'); G)
where the horizontal maps are induced by the canonical chain equivalences of exercise 1
above.
3 The Cech cohomology group of (X,A) with coefficients G is defined by H* (X,A; G) =
lim (H* (K(9l),K'(91'); G)}. Prove that there is a natural isomorphism
H*(X,A; G) ~ H*(X,A; G).
4 If dim (X — A) < n, prove that №(X,A; G) = 0 for all q > n and all G.
E THE KUNNETH FORMULA FOR CECH COHOMOLOGY
If Kt and K2 are simplicial complexes, their simplicial product Kt A K2 is the simplicial
complex whose vertex set is the cartesian product of the vertex sets of Kt and of K2 and
whose simplexes are sets {(dq.Wo), • • • ,(Vq,wq)}, where Vo, . . . , vq are vertices of
some simplex of Kt and w0, . . . , wQ are vertices of some simplex of K2.
1	Prove that Kt A K2 is a simplicial complex, and if Lt C Kt and L2 C K2, then
Li A L2 C Kt A K2.
2	For simplicial pairs (Kt,Lt) and K2,Lz) define
(Kt,Lt) A (K2,L2) = (Kt A K2, Kt A L2 U Lt A K2)
Prove that C((Kt,Lf) A (K2,L2)) is canonically chain equivalent to C(Kt,Lt) ® C(K2,L2).
(Hint: Use the method of acyclic models.)
3	If (9l,9l') is an open covering of (X,A) and (c\r,c\r') is an open covering of (Y,B), let
(9l,9l') X (‘Xff) = (9lf,6lb'') be the open covering of (X,A) X (Y,B), where
« = {Fx v| U£9i,
and 9lf' = (UXV\C £ 9l', V £ T'}. Prove that
(K(9lf),K'(9B')) = (K(91),K'(9l')) А да,К'(Т'))
4	If (X,A) and (Y,B) are compact Hausdorff pairs, prove that the family of coverings of
(X,A) x (Y,B) of the form (91,9l') X (‘Xfi ') is cofinal in the family of all open coverings
of (X,A) X (Y,B).
1 For details see С. H. Dowker, Homology groups of relations, Annals of Mathematics, vol. 56,
pp. 84-95, 1956.
360	GENERAL COHOMOLOGY THEORY AND DUALITY CHAP. 6
5 If (X,A) and (Y,B) are compact Hausdorff pairs and G and G' are modules such that
G * G' = 0, prove that there is a short exact sequence
0-»(Й* ® H*)5^ >((X,A) x (Y,B); G ® G') ->(H* * Н*)ч+1 -> 0
where Й* = H*(X,A; G) and Й* =	G').
6 Let (X,A) and (Y,B) be locally compact Hausdorff pairs-with A and В closed in ,\'
and Y, respectively. If G and G' are modules such that G * G' = 0, prove that there is a
short exact sequence
0	(H* ,1 ® H* ,2)« -> HC«((X,A) X (Y,B); G * G') -> (H? д * Я? ,2)®+i -> 0
where H* д = H*(X,A; G) and Й* ,2 =	(Y,B; G').
F LOCAL SYSTEMS AND SHEAVES
Throughout this group of exercises we assume X to be a paracompact Hausdorff space.
1	If Г is a local system on X, let Г be the presheaf on X such that for an open
set VCX, T(V) is the set of all functions / assigning to each x E X an element
f(x) E I (r) with the property that for any path o in V, /(«(!)) = Г(ы)(/(ы(0))). Prove that
Г is a sheaf on X and the association of Г to Г is a natural transformation from local sys-
tems to sheaves.
2	A presheaf Г on X is said to be locally constant if there is an open covering Qi = {Г/}
of X such that if V E QL and x E U, then Г(Н) lim , {T(V)}, where V varies
over open neighborhoods of r. If U E 9i and V is a connected open subset of U, prove
that the composite
Г(С1)-> Г(С7)—> Г(Г)
is an isomorphism. Deduce that if Г is a locally constant sheaf and U' is a connected
open subset of U E % then Г(<7) ~ Г(Н').
3	If X is locally path connected and Г' is a locally constant sheaf on X, prove that
there is a local system Г on X such that Г ~ Г'.
4	If X is locally path connected and semilocally 1-connected, prove that there is a one-
to-one correspondence between equivalence classes of local systems on X and equiva-
lence classes of locally constant sheaves on X.
5	If Г is a local system of В modules on X, let Д';( • ;Г) be the presheaf on X such that
Д'/( • ;L)(V) = Д«(V;Г | V) for V open in X. Prove that Д«(  ;Г) is fine.
6	If Г is a local system of В modules on X, let Д* ( ;Г) be the cochain complex of pre-
sheaves Дв( • ;Г) on X and let A* ( ; Г) be the cochain complex of completions A';(  ;Г).
Prove that there is an isomorphism
И*(Д*( ;Г)(Х)) ~ И* (Д* ( ;Г)(Х))
7	Let Г be a local system of В modules on X and assume that Н«(Д* (• ;Г)) is locally
zero on X for all q > 0. Prove that there is an isomorphism
Н*(Х;Г) ~Н*(Х;Г)
(Hint: Note that Г = Н°(Д* ( ;Г)) and apply theorem 6.8.7.)
G SOME PROPERTIES OF EUCLIDEAN SPACE
1 Find a compact subset X of R2 that is «-connected for all n and such that PF(X;Z) sZ.
exercises	361
2» If X is a compact subset of R" and dim X < n — 1, prove that R" — X is connected.
Let Ai and Az be disjoint closed subsets of R” and let Zi £ Hp(Ar,R) and
-z € Hq(A2-,R), With p + q = n - 1. If £ £tp(Ai;R), let z{ £ Hp+i(R” R" - A2;R) be
the image of Zi under the composite
£p(Ai) -> HP(R« - A2) Hp+i(R\ R'! - A2)
•phe linking number Lk (zi,z2) £ R is defined by
Lk (zbz2) = <Yu(zi),22>
where U is an orientation class of R„ over R fixed once and for all.
3 Prove that Lk (zi,z2) = (U, i*(z2 X zi)), where
i: A2 x (R”, R" - Az) C (R’> x R«, R" X R” - fi(R»))
4 Assume that Lk (z2jZj) is also defined [that is, z2 £ l7,/A2)|. Prove that Lk (zi,z2) =
Lk (z2,zi).
5 Let Ai be a p-sphere and A2 a q-sphere imbedded as disjoint subsets of Rn, where
p + q = n + 1. Prove that //;,(Aj) —> Hp(R" — Az) is trivial if and only if
HQ(A2) -» Я/R" - Ai) is trivial.
Л IMBEDDINGS OF MANIFOLDS IN EUCLIDEAN SPACE
I Prove that a compact n-manifold which is nonorientable over Z cannot be imbedded
in R',+1.
2 Let X be a compact connected n-manifold imbedded in R',+1 and let U and V be
the components of R'l+1 — X. Let i: X C Rn+1 — U and j: X C R',+1 — V and prove
that over any R, i* (H* (R'"+1 — U)) and j*(H*(Rn+1 — V)) are subalgebras of H*(X)
and there is a direct-sum representation
{;*,/*}: >(R«+1 - U) ©H«(R«+1 - V)~>(X)	0<q<n
3 Prove that for n > 2 the real projective n-space P" cannot be imbedded in Rn+1.
CHAPTER SEVEN(
HOMOTOPY THEORY
I
I
WITH THIS CHAPTER WE RETURN TO THE CONSIDERATION OF GENERAL HOMOTOPY
theory. Now that we have homology theory available as a tool, we are able to
obtain deeper results about homotopy than we could without it. We shall con-
sider the higher homotopy groups in some detail and prove they satisfy
analogues of all the axioms of homology theory except the excision axiom. We
introduce the Hurewicz homomorphism as a natural transformation from the
homotopy groups to the integral singular homology groups. It leads us to the
Hurewicz isomorphism theorem, which states roughly that the lowest-dimen-
sional nontrivial homotopy group is isomorphic to the corresponding integral
homology group.
We discuss next the concept of CWcomplex. The class of CW complexes
is particularly suited for homotopy theory because it is the smallest class of
spaces containing the empty space and, up to homotopy type, is closed with
respect to the operation of attaching cells (even an infinite number).
The last main result is the Brown representability theorem. It character-
izes by means of simple properties those contravariant functors from the
homotopy category of path-connected pointed CW complexes to the category
363
364	HOMOTOPY THEORY CHAP. 7
of pointed sets that are naturally equivalent to the functor assigning to a
CW complex the set of homotopy classes of maps from it to some fixed
pointed space.
Section 7.1 contains a general exactness property for sets of homotopy
classes. Section 7.2 contains definitions of the absolute and relative homotopy
groups and proofs of the exactness of the homotopy sequences of a pair, a
triple, and a fibration. In Sec. 7.3 we consider the extent to which the homo-
topy groups depend on the choice of the base point used in their definition and
prove analogues for the higher homotopy groups of properties established in
Chapter One for the fundamental group.
The Hurewicz homomorphism is defined in Sec. 7.4 and the Hurewicz
isomorphism theorem is proved in Sec. 7.5. The proof establishes the absolute
and relative Hurewicz theorems, as well as a homotopy addition theorem, by
simultaneous induction. The Hurewicz theorem implies the Whitehead
theorem, which asserts that a continuous map between simply connected
spaces induces isomorphisms of all homotopy groups if and only if it induces
isomorphisms of all integral singular homology groups.
Section 7.6 introduces the concept of CWcomplex. Among the elementary
properties established is the cellular-approximation theorem, which is an
analogue for CW complexes of the simplicial-approximation theorem. Section
7.7 deals with contravariant functors on the homotopy category of path-
connected pointed spaces. We prove the representability theorem cited above,
and apply it in Sec. 7.8 to obtain CW approximations to a space or a pair and
to discuss the related concept of weak homotopy type. The representability
theorem will be used again in Chapter Eight.
I EXACT SEQUENCES OF SETS OF HOMOTOPY CUASSES
One of the most important properties of the homology functor is the exactness
property relating the homology of the pair and the homology of each of the
spaces in the pair. A similar exactness property is valid for functors defined
by homotopy classes. This section is devoted to preliminaries about homotopy
classes and a proof of this exactness property. Throughout the section we
shall work in the category of pointed spaces, and unless stated to the contrary,
(X,A) will be understood as a pair of pointed spaces (that is, A has the same
base point as X) in which the subspace A is closed in X. Homotopies in this
category are understood to preserve base points. If А С X, we use X/A to
denote the space obtained from X by collapsing A to a single point (this point
serving as the base point of X/A). If X' and A are closed subsets of X, then
A/(A П X') is a closed subset of X/X'. Hence, if (X,A) is a pair and X' is
closed in X, there is a pair (X/X', A/(A П X')), which will also be denoted by
(X,A)/X'.
SEC. 1 exact sequences of sets of homotopy classes
365
The unit interval I will be a pointed space with 0 as base point. The
reduced cone CX over X is defined to be the space obtained from X X I by
collapsing X X 0 U x0 X 1 to a point (so CX — X X I/(X xO Uio X I))-
\Ve shall use [x,t] to denote the point of CX corresponding to the point
(x,t) С X 1 under the collapsing map X X I —> CX. X is imbedded as a
closed subset of CX by the map x —> [x,l]. If (X,A) is a pair, then CA is a
subspace of CX and C(X,A) is defined to be the pair (CX,CA).
1	lemma A map f: (X,A) —> (Y,B) is null homotopic if and only if there
is a map F: C(X,A) —> (Y,B) such that F[x,l] = fix) for all x 6 X.
proof There is a one-to-one correspondence between null homotopies
H: (Х,Л) X I (Y,B) of / and maps F:C(X,A) —> (Y,B) such that F[x,l] = fix),
given by the formula
F[x,f] = H(x, 1 - t) 
The following relative homotopy extension property can also be deduced
from the relative form of theorem 1.4.12.
2	lemma Given f: C(X,A) —> (Y,B) and a homotopy G: (X,A) Xl-r (Y,B)
of f | (X,A), there is a homotopy F: C(X,A) x I (Y,B) of f such that
F\(X,A) xI=G.
proof An explicit formula for F is

t(l + f) < 1
1 < t(l + T) 
The homotopy class of the unique constant map (X,A) —> (Y,B) is denoted
by 0 C | A',A; Y,B] [it consists of the null-homotopic maps (X,A) —-> (Y,B)].
Because the composite, on either side, of a null-homotopic map and an arbi-
trary map is null homotopic, the element 0 is a distinguished element of
|X,A; Y,B], and we regard [X,A; Y,B] as a pointed set with this distinguished
element. Given a map f: (A',A') —> (X,A), the kernel of the induced map
f #: [X,A; Y,B] -> [X',A'; Y,B]
is defined to be the pointed set /#'1(0) and is denoted by ker f#.
We now show how to map another set of homotopy classes into
[X,A; Y,B] so that its image equals ker f#. This will be the basis for the exact-
ness property we seek. The mapping cone Cf of a map /: X' —> X is defined
to be the quotient space of CX' v X by the identifications [x',1] = fix') for all
x' e X’. Given a map f: (A',A') -> (X,A), let X' -> X and f": A' A be
maps defined by f. Then Cf is a closed subspace of Cf and there is a pair
(Cf,Cf). There is a functorial imbedding i of (X,A) as a closed subpair of
(Q,Cr).
A three-term sequence of pairs and maps
IX',A')	(X,A) A (X",A")
366	HOMOTOPY THEORY CHAP. 7
is said to be exact if for any pair (Y,B) (where В is not necessarily closed in Y)
the associated sequence of pointed sets
[Y,B; X',A'] M [Y,B; A,A]	[Y,B; X",A"]
is exact (that is, ker g# = im fy). Similarly, it is said to be coexact if the
sequence of pointed sets
[A",A"; Y,B] -Si» [X,A; Y,B] A [X',A'-, Y,B]
is exact (that is, ker f# = im g#). A sequence of pairs and maps (which may
terminate at either or both ends)
• • • * (Xn+i,An+i) (An,An) > (An_i,An_i) » • • 
is said to be an exact sequence (or a coexact sequence) if every three-term
sequence of consecutive pairs is exact (or coexact).
3 theorem For any map j: (X’,A') —» (A,A) the sequence
(а',а')4(а,а)Л(слсг)
is coexact.
proof Let (Y,B) be arbitrary (with В not necessarily closed in Y) and con-
sider the sequence
[Cf,Cr; Y,B] -A [A,A; Y,B]	[A',A'; Y,B]
We now show that imi# C ker f#. The composite i ° f: (A',A') —> (Q,Q-)
equals the composite
(А',А') С C(A',A') C C(A',A') v (А,А) Л (Cr,Cr)
where к is the canonical map to the quotient. However, the inclusion map
(A',A') C C(A',A') is null homotopic [by lemma 1, because this inclusion map
can be extended to the identity map of C(A',A')]. Therefore i ° f is null
homotopic, and so im (f# ° i#) = 0, proving that im i# C ker/#.
Assume that g: (A,A) —> (Y,B) is such that/#[g] = 0 (that is, g ° / is null
homotopic). By lemma 1, there is a map G: C(A',A') —> (Y,B) which extends
g °/. Then G and g define a map G': C(A',A')v (A,A) —> (Y,B) such that
G' | C(A',A') = G and G' | (A,A) = g. Since
G'[x',l] =(g ° /)[<!] = g(/(x')) = G'(/(x')) x' 6 A'
there is a map Zi: (Cf,Cp') -» (Y,B) such that G' = h ° k. Then h | (A,A) = g,
showing that h ° i = g or [g] = -i#[h]. Therefore ker f# dim i#. 
For a map /: (A',A')	(A,A) we have a sequence
4	(A',A') A (A,A) A (Cf,Cr) A (CisCr) A (Q,Cr)
and by theorem 3, it follows that this sequence is coexact.
Thus we have succeeded in imbedding the map
/#: [A,A; Y,B] -> [A/A'; Y,B]
SEC. 1 exact sequences of sets of homotopy classes
367
jn a coexact sequence. We shall show that the pairs (C,-',C,-") and	in
sequence 4 can be replaced by other pairs more explicitly expressed in terms
of (X',A'), (X,A), and f.
ft lemma Let (Y,B) be a pair and let Y' be a closed subset of Y. Assume
that there is a homotopy H: (У,В) x 1 —* (Y,B) such that
(a)	H(y,O) = y, for у 6 Y.
(b)	H(Y' X I) C Y'.
(C) H(Y X 1) = yo-
phen the collapsing map k: (Y,B) —> (Y,B)/Y' is a homotopy equivalence.
proof Define a map J: (Y,B)/Y' —-> (Y,B) by the equation
>(</)) = Н(у,1) У e Y
[this is well-defined, because H(Y' X 1) = yo]- We show that/is a homotopy
inverse of k. By definition off, we see that His a homotopy from 1<у,в) to / ° k.
On the other hand, because H(Y' X 1) C Y', there is a homotopy
H': (Y,B)/Y' -> (Y,B)/Y
such that H'(fc(y),t) = fc(H(y,t)) for у £ Y and t £ I. Then
ЧШ) = W) = H'(b(y),i) У e Y
Therefore H' is a homotopy from the identity map of (Y,B)/Y' to к ° f and f
is a homotopy inverse of k. 
О corollary Let f: (X',A') —> (ХД) be a map and let i: (X,A) C (Cf,Cf").
Then CX C Cr, (,Ci',Ci")/CX = (Cf,C/")/X, and the collapsing map
k-. (Cr,Ci") (Ci',Ci")/CX
is a homotopy equivalence.
proof Cr is the quotient space of CX’ v CX with the identifications [x',1] =
[f(x'),l] for all x' d X', hence CX C Cy. Since Cp is the union of the closed
subspaces CX and Cf, it follows that
Ci'/CX = Cr/(Cr n CX) = Cf/X
Similarly, Ci"/CA = Cf/A, and because С',-- Г) CX = CA,
(Ci',Ci")/CX = (Cf',Cf")/X
This proves the first two parts of the corollary.
Let F: C(X,A) X I —> C(X,A) be the contraction defined by F([x,t], t') =
[x, (1 — t')l] and let g: C(X',A')	(Ci',Ci") be the composite
C(X',A') C C(X',A') v C(X,A) (Ct',Ct")
where the second map is the canonical map. The composite
(X',A') X I (X,A) X I C C(X,A) x I-4 C(X,A) C (Q,Q.)
368	HOMOTOPY THEORY CHAP. 1
is a homotopy G: (X',A') X / (CisCi") such that G(x',O) = [j(x'),l] = g[41].
By lemma 2, there is a homotopy F: C(X',A') X I —* (Cj-C,") such that
F' | (X',A') X 1 — G and F'([x',t], 0) = g([x',t]). Then a homotopy
H: (C^Ci") X (Cr,Cr')
is defined by the equations
H([x',t], t') = F'([x',t], t') x' E X'; /, f' C J
H([x,t], f) = F([x,t], f) x £X;t,f E I
[this is well-defined because F'([x',l], f) = G(x',t') = F([j(x'),l], f)]. Then Ц
satisfies a, b, and c of lemma 5 with (Y,B) = (Cj',C{") and Y' = CX. Therefore
k; (Ci',Ci") -a (Ci',Ci")/CX is a homotopy equivalence. 
Recall from Sec. 1.6 that the suspension SX is defined as the space
X X I/(X X 0 U xo X к U X X 1) (therefore SX = CX/X). For a pair (X,A)
we define S(X,A) = (SX,SA). Then, for any map/: (X',A')	(X,A), we have
(Cf,Cf')/X = S(X',A'), and we let k: (Cf',Cf") —-> S(X',A') be the collapsing map.
7 lemma For any map f: (X',A')	(X,A) the sequence
(X',A') -U (X,A) 4 (Cr,Cr) S(X'A') S(X,A)
is coexact.
proof We shall use the coexact sequence 4,
(X',A') 4 (X,A) 4 (Cf,Cr)	(Ci',Ci") 4 (Q,Cr)
By corollary 6, there is a homotopy equivalence
(Ci',Ci") 4 (Cf,Cf")/X = S(X',A')
and the composite (Cf,Cf) (Ci',Ci") S(X',A') is seen to be the collapsing
map k-. (Cf,Cf) —> S(X',A'). Also by corollary 6, there is a homotopy equivalence
(cy,cr) (см/ссг = (Ci',Ci")/cf = s(x,A)
and the composite (Ci',Ci")	(Cj',Cj") S(X,A) is easily seen to be the
collapsing map к: (С{',С{") -> (Ci',Ci")/Cr = S(X,A). Let g: S(X',A') -a S(X,A)
be the map defined by g([x',t]) = [/(x'), 1 — t]. The triangle
(Cv,Ci")
V V
S(X',A') A S(X,A)
is homotopy commutative because a homotopy
H: (Ci',Ci")	S(X,A)
from к to g ° k' is defined by
H([x',t], t') = [f(x'), 1 - tt']	xJ E X';t,fEI
H([x,t], f) - [x, (1 - t')f]	x E X; t, f E I
src, J EXACT SEQUENCES OF SETS OF HOMOTOPY CLASSES	369
[this is well-defined because Н([ж',1], t') = [/(%'), 1 — f] = H([/(x'),l], t')].
'Therefore there is a homotopy-commutative diagram
(СЛСГ) 4 (C„Cr) (Cy,Cr)
4	4	/;"l
S(X',A') 4 S(X,A)
in which k' and k" are homotopy equivalences. From the coexactness of the
sequence 4, the coexactness of the sequence
(Х'Д') 4 (ХД) 4 (Cf,Cr) 4 S(X',A') 4 S(X,A)
follows. Let Л: S(X,A) —> S(X,A) be the homeomorphism defined by /r([x,t]) =
[x, 1 — t]. The coexactness of the above sequence implies thfe coexactness of
the sequence
(X',A') 4 (X,A) 4 (Cf,Cr) S(X',A') S(X,A)
Because h ° g = Sf, this is the desired result. 
8 lemma If the sequence
(X',A') -U (X,A) 4 (X", A")
is coexact, so is the suspended sequence
S(X',A') -A. S(X,A) S(X",A")
proof For any pair (Y,B) let S2(Y,B) = (BY,fiB). By theorem 2.8 in the
Introduction, there is a commutative diagram (in which the vertical maps are
equivalences of pointed sets)
[S(X",A"); Y,B] -AA [S(X,A); Y,B] [S(X',A'); Y,B]
Ф	t	$
[X",A"; B(Y,B)] -^4 [ХД; B(Y,B)] -4^ [Xх,A'; B(Y,B)]
Hence im (Sg)# = ker (Sf)# in the top sequence is equivalent to im g# = kerf#
in the bottom sequence. 
We define Sn(X,A) inductively for n > 0 so that
S°(X,A) = (X,A)
S«(X,A) = S(S»-i(X,A)) n > 1
9 theorem For any map f: (X',A') -a (X,A) the sequence
(X'.A') 4 (X,A) 4 ... -^4 S«(X,A) AU Sn(Cf,Cr) Src+I^A') ...
is coexact.
Proof From lemmas 7 and 8, for n > 0 there is a coexact sequence
S«(X,,A') Sn(X,A) Sn(Cr,Cf) SB+1(X,,A/) A+Y> 5и+1(ХД)
370	HOMOTOPY THEORY CHAP, %
Since every three-term subsequence of the sequence in the theorem is con-
tained in one of these five-term coexact sequences, the result follows. 
In the coexact sequence of theorem 9 all but the first three pairs are
H cogroup pairs, and all but the first three of these are abelian. Furthermore
all maps between H cogroup pairs are homomorphisms. Thus, for any (Y,B)
the coexact sequence of homotopy classes of maps of the sequence of
theorem 9 into the fixed pair (Y,B) (with В not necessarily closed in Y) con-
sist of groups and homomorphisms except for the last three pointed sets, and
all but three of the groups are abelian.
We now show how the last group in the sequence, namely [S(/V',A/); Y,B],
acts as a group of operators on the left on the next set in the sequence,
namely [Cf,Cf"; Y,B], in such a way that the orbits are mapped injectively by
i# into [X,A; Y,B], If a-. S(X',A')	(Y,B) and /S’: (Cf,Cr) (Y,B), we define
« т fi-. (Cf,Cr) -+ (Y,B)
,	/ -г ОЧГ ' >1 — Г Ф'>2#]	0 <t< У2, X' EX', t E I
by	(a T fd)[x ,t] - 2t _ ц i/2 < t < 1, x' 6 X’, t £ I
and	(a T /?)(%) = /3(х) x С X
It is then clear that (a T /?) | (X,A) = fi | (X,A), and the following statements
are easily verified.
I® a ~ a' and /3 ~ /3' (or fi /3' rel X) implies a T [3 ~ a' T fi' (or
a T fi ~ a' T f3' rel X). 
Il Ifa0 is the constant map, then a0 T fi fi rel X. 
12 («1 * a2) T /5	«1 T (a2 T /3) rel X. 
13 T (a2 ° k) ~ (ai * a2) ° к rel X. 
Given maps j6i,j62: (Cf,Cf) (Y,B) such that j8i | (X,A) = [i2 | (X,A), we
define d(/3i,f32): S(X',A')	(Y,B) by
7/Д /? u ' л -	0 < t < \2, x3 E X', t E I
d(fii,/32)[x ,t] -	2 _ 2tj y2<t<l, x' E X', t E 1
The following results are easily verified.
14 (3i ~ fi'i rel X and fi2 fi2 rel X implies d(/?i,/?2)	d((3i,(i2). 
15 d(f3i,f3s) d(/3i,/32) * с1(/?2,Лз)- 
16 d(a T /3,(3) a. 
17 fix — d(J3i,f32) T (32 rel X. 
From statements 17, 10, and 11, it follows that if d(f31,f32) is null homo-
topic, then fii ~ f32 rel X. Conversely, if /31 ~ f32 rel X, it follows from
statements 11, 14, and 16 that
d(/3i,(32) ~ d(a0 T /3i,/ЗЕ) a0
SEC. 2 ШОПЕН HOMOTOPY GROUPS	3/j
Therefore we have fit ~ p2 rel X if and only if tf(/3i,/32) is null homotopic.
By statements 10, 11, and 12, there is an action of [S(X',A'); Y,B] on the
left on [Cf',Cf; Y,B] defined by [a] T [/3] = [a T /3].
1# theorem Given [/3i], [/f] £ [Cf',Cf"; Y,B], then = i#[/32] if and
Only if there is [a] £ [S(X',A'); Y,B] such that [/У = [a] T [/32],
proof By the definition of a T /32 we see tha^ j
i#[a T = [(a T /32) I (*,A)] = [f 2 ] (X,A)] = i#[/32]
showing that i#([a] T [/32]) = i#[/32]. Conversely, if i#[/3i] = i#[/32], we can
choose representatives /3 x and p2 such that /3, | (X,A) — /32 | (X,A) [because
the map i: (Х,Л) C (Cf ,Q") is a cofibration]. Then, by statement 17,
[/3i] = [d(/31;/32) T /32] = [d(/31;/32)] T [/32] 
If) theorem Given [cti], [a2] £ [S(X',A'); Y,B], then Jc#[ai] = fc#[a2] if and
only if there is [у] C [S(X,A); Y,B] such that [a2] — [nJ + (Sf )#[y].
proof By statement 13, if /3q: (Cf,Cff —» (Y,B) is the constant map
H«1 * (Y ° Sf)] = [aj T (k#Sf #[y]) = [nJ T [До]
= [«i | т /<#[fto] = /<#[ft| * ao]
Therefore /c#([ai] + (Sf )#[y]) = fc#[ai]. Conversely, if AX[ai] = fc#[a2], then
by statements 10 and 13,
0 = i#[fti' * ax] = [а,1] T fc#[ai] = [ai-1] T k#[a2] = fc#[ai-1 * a2]
Therefore there is [y| C [S(X,A); Y,B] such that [ai-1 * a2] = (Sf )#[y], and so
[a2] = [ai] + [ai^1 * a2] = [aj + (Sf)#[y] 
2 HIGHKK HOMOTOPY GROUPS
The higher homotopy groups of a space or pair are covariant functors defined
to be the set of homotopy classes of maps of fixed spaces or pairs into the
given one. In the absolute case these are the functors already defined in
Sec. 1.6. The exactness property established in the last section implies an
important exactness property relating relative and absolute homotopy groups.
This section contains definitions of the homotopy groups, some of their
elementary properties, and a proof of the exactness of the homotopy sequence
of a fibration.
We shall use 0 as base point for I and for the subspace t С I. Let X be a
space with base point x0- For n > 1 the homotopy group ття(Х) [or tt„(X,xo),
when it is important to indicate the base point] is the group [Sn(Z);X] [this being
equivalent to the definition given in Sec. 1.6, because Sn is homeomorphic to
Sn(S°) ~ Sn(j)]. For n = 0 the homotopy set 77o(X) is defined to be the
pointed set [Z;X] (that is, the set of path components of X with the path com-
372	HOMOTOPY THEORY CHAP. 7 '
ponent of Xo as distinguished element). Then ття is a covariant functor from»
the category of pointed spaces to the category of abelian groups if n > 2, the
category of groups if n = 1, and the category of pointed sets if n = 0.
Let (X,A) be a pair with base point Xo € A. For n > 1 the nth relative
homotopy group (or homotopy set for n = 1), denoted by 77Я(Х,А) Or
тгя(Х,А,Хо), is defined to equal [S”“1(Z,Z); X,A], Then is a covariant functor
from the category of pairs of pointed spaces to the category of abelian groups
if n > 3, the category of groups if n = 2, and the category of pointed sets if
n = 1.
There is a homeomorphism of S(1) with I/t which sends [0,t] € S(I) to
the base point of 1/1 and [l,t] £ S(Z) to that point of I/t determined by the
point t E I. Therefore, for n > 1, Sn(l) and 8г|1(//1) = S''ll(/)/S,l-1(i) are
homeomorphic. This induces a natural one-to-one correspondence between
[S"1(//Z); X,{xo}] and [8И(1);Х]. By means of this correspondence we iden-
tify the relative homotopy group 77n(X,{x0}) for n > 1 with the absolute
homotopy group w„(X). Then the inclusion map j: (X,{x0}) C (X,A) induces a
homomorphism
/#: я-и(Х) -> 77Я(Х,А) n > 1
Because Sn(I) is path connected for n > 1, it follows that if X' is the path
component of X containing Xq, the inclusion map X' С X induces isomorphisms
ття(Х') ~ wn(X) for n > 1. Similarly, if A' is the path component of A contain-
ing x0, the inclusion map (X',A') C (X,A) induces isomorphisms ттп(Х',А/)
77-„(X,A) for n > 1.
We present an alternative description of the relative homotopy groups.
For n > 1 there is a homeomorphism of Sn~\I/i) with (I X Iя-1, 1 X
(I x U 0 x h1-1) sending [ • • •	to [t,ti, . . .
(7° is a single point and t° is empty). Therefore, for n > 1, 7t„(X,A,x0) is in
one-to-one correspondence with the set of homotopy classes of maps
(I», Z« I x fr-1 и 0 x fi,_1) (X,A,xo)
Since I X i”-1 U 0 X Iя-1 is contractible, if z0 — (0,0, . . . ,0), the inclusion
map
(in,in,zo) с (Iй, Iй, i x i71-1 и о x fi1-1)
is a homotopy equivalence. Hence, for n > 1, wn(X,A,xo) is in one-to-one
correspondence with the set of homotopy classes of maps
(In,tn,Zo) (X,A,xo)
Since (Zn,Z,!,Xo) is homeomorphic to (E",Sn-1,po) for n > 1, 77-я(Х,А,хо) is in
one-to-one correspondence with the set of homotopy classes of maps
(E»,S«-i,p0)	(X,A,x0)
The following condition for a map (En,Sn-1,po) —* (X,A,xo) to represent
the trivial element of 77n(X,A,x0) is a relative version of theorem 1.6.7.
1 theorem Given a map a: (En,Sn~1,p0) —» (X,A,Xo), then [a] = 0 in
у.,-. HIGHER HOMOTOPY GROUPS	3/3
^„(ХЛА'о) if and only if a is homotopic relative to Sn~^ to some map ofE” to A.
proof Assume [a] = 0 in •nn(X,A,Xo). Then there is a homotopy
H-. (En,Sn~1,po) X I -* (X,A,Xo)
;from et to the constant map En —* xq. A homotopy H' relative to S’1-1 from a
to some map En to A is defined by
Conversely, if a is homotopic relative to S’1-1 to some map a' such that
a'(En) C A, then [a] = [a'] in 77n(X,A,xo), and it suffices to show that [a'] = 0
in wn(X,A,Xo). A homotopy H: (En,S’I-1,po) X I —* (X,A,Xq) from a' to the
constant map En —-> x0 is defined by
H(z,t) — a'((l — t)z + tpo) 
A pair (X,A) is said to be n-connected for n > 0 if for 0 < к < n every
map a: (E^S*-1) —-> (X,A) is homotopic relative to S/l ) to some map of
EA‘ to A. For к = 0, (E’O^S1) is a pair consisting of a single point and the
empty set, and the condition that every map a: (E^S1) —-> (X,A) be homo-
topic to a map E° —-> A is equivalent to the condition that every point of X
be joined by a path to some point of A. From theorem 1 we obtain the fol-
lowing relation between «-connectedness of (X,A) and the vanishing of rela-
tive homotopy groups of (X,A).
2 corollary A pair (X,A) is n-connected for n > 0 if and only if every
path component of X intersects A and for every point a £ A and every
1 < к < n, ,nifX,A,a') = 0. 
For n > 1 there is a map (which is a homomorphism for n > 2)
Э: 77-„(X,A,Xo)	77„_i(A,Xo)
defined by restriction. That is, given a: S’!-1(Z,Z) —(X,A), then
3[a] = [a | S«-i(Z)]
It is trivial that if f. (X',A',xo) —» (X,A,x0) is a map, there is a commutative
square
77'и(Х/,А',х6) Л 77„_1(A',Xo)
77-n(X,A,Xo) ТГИ(А,Х0)
In other words, c is a natural transformation between covariant functors
ЯЙ(Х,А) and 77n_i(A) on the category of pairs (X,A) of pointed spaces. Thus
the homotopy-group functors and the natural transformation d are in analogy
374	HOMOTOPY THEORY CHAI', /j?
with the constituents of a homology theory. We shall show that they also
satisfy many of the axioms of homology theory. It is obvious that the homol
topy axiom and the dimension axiom are satisfied for the hoiiiotopy-g)0llp
functors.
We shall now investigate the exactness property. Given a pair (X,A) 0|
pointed spaces, let i: А С X and /: (X,{x0}) C (X,A). The homotopy sequence
of (X,A) [or of (X,A,%o)] is the sequence of pointed sets (all but the last three
being groups)
• • • -> 77-„+i(X,A) A 77„(A)	77„(X) 4 77-n(X,A) 4 • -  4 77O(X)
3	theorem The homotopy sequence of a pair is exact.
proof Let/: (Z,{0)) C (1,1) andlet/': 1 C 1 and/": {0} C L By theorem
7.1.9, there is a coexact sequence
(i,{0}) 4 (z,i) 4 (cf,cr) 4 s(z,{0}) 4 s(i,i) •
Let g: (Cf,Cf) —-> (1,1) be the homeomorphism defined by g([(),t]) = 0 and
g([l,t]) = t. Then the composite g ° i is the inclusion map i': (1,1) C (1,1), and
the composite к ° g1 equals the composite
(i,f)	(i/i,{0})	(s(i),{0})
where k' is the collapsing map and h is the homeomorphism used in identify-
ing 77„(X,{xo}) with irn(X). Therefore there is a coexact sequence
(i,{0)) 4 (i,i) (i,i)	s(i,{0}) 4 • • •
This yields an exact sequence
-----> rrn+1(X,A) rrn(A) -irn(X)	77„(X,A) -a-----> vrofyV)
The proof is completed by the trivial verification that
(Sni')# — d, (S’1/)# = i#, and (Sn-1(h ° k'))# =	
4	corollary For n > 0, (En+1,Sn) is n-connected.
proof For n > 0, E,1+1 is path connected and Sn is nonempty; therefore
every path component of En+1 meets S'1. If x £ Sn, then irie(En+1,x) = 0 for
0 < k, because E’l+1 is contractible. By theorem 3.4.11, 77>(S’l,x) = 0 if
0 < к < n. It follows from theorem 3 that rT^(En+1,Sn,x) = 0 for 1 < к < n.
The result follows from corollary 2. 
We shall see that the excision property fails to hold for the homotopy
group functors. There is, however, a different property possessed by the
homotopy group functors but not by homology functors. This property is the
existence of an isomorphism between the absolute homotopy groups of the
base space of a fibration and the corresponding relative homotopy groups of
the total space modulo the fiber. This is true for a more general class of maps
than fibrations, and we now present the relevant definition.
A map p: E В is called a weak fibration (or Serre fiber space in the
SEC- 2 HIGHER HOMOTOPY GROUPS
375
literature) if p has the homotopy lifting property with respect to the collection
<gf cubes	E is called the total space and В the base space of the weak
fibration. For b £ B, p^b) is called the fiber over b.
If s is a simplex, |s| is homeomorphic to some cube, and so a map
p. E —> В is a weak fibration if and only if it has the homotopy lifting prop-
erty with respect to the space of any simplex. We shall show that, in fact, a
weak fibration has the homotopy lifting property with respect to any
polyhedron.
It is clear that a fibration is a weak fibration. If p: E В is a weak
fibration and f: В' —-> В is a map, let E' be the fibered product of B' and E.
Then there is a weak fibration p': E' —-> B', called the weak fibration induced
from p b y f.
5	lemma Let p: E—> В be a weak fibration and let g:E' \0 U In X 1-^ E
and H-. In X I —> В be maps, with n > 0, such that H is an extension of
p о g. Then there is a map G: In X I —* E such that p ° G = H and G is an
extension of g.
proof The lemma asserts that the dotted arrow in the diagram
In X 0 U tn	X I	E
nj	I*
In x I	В
represents a map making the diagram commutative. This follows from the
homotopy lifting property of p since the pair (In x T In X 0 U tn X Г) is
homeomorphic to the pair (Iй X I, In X 0). 
6	theorem Let (X,A) be a polyhedral pair and let p: E —» В be a weak
fibration. Given maps g: X X 0 U A Xl~~> E and II: X x L-^ В such that
H is an extension of p ° g, there is a map G: X xj E such that p ° G — H
and G is an extension of g.
proof The method of obtaining G involves a standard stepwise-extension
procedure over the successive skeleta of a triangulation of X. Let (K,L) be a
triangulation of (X,A) and identify (X,A) with (|K|,|L|). For q > —1 set
Xq = |K| X о и (|K® U L\ X I), so that = X X 0 U A x fandX9_i C Xq
for q > 0. By induction on q, we shall define a sequence of maps Gq: Xq—*E
such that
(°) G_r = g
(b) Gq | Xq_L = Gq_i for q > 0
(c) po Gq = H | for q > -1
Once a sequence { G(/} with these properties is obtained, a map G: X X I —> E
with the desired properties is defined by the conditions G | Xq — Gq, for
q > —1. Thus the problem is reduced to the construction of such a
sequence {GQ}.
376	HOMOTOPY THEORY CHAP. 7
Condition (a) defines G_i. Assume Gq defined for q < n, where n > (),
To define Gn to satisfy conditions (b) and (c), for every n-simplex s С К _ £
let gs: |s| X 0 U |s| X f E and Hs: |s| X 1 —» В be the maps defined by
gs = G„_i | (|s| X 0 U |s| X I) and Hs = H\ (|s| X I)- Because (|s|,|s|) is
homeomorphic to (In,ln), it follows from lemma 5 that there is a map
Gs: |s| X I -> E such that Gs | (|s| X 0 U |s| X f) = gs and p ° Gs = ft*
Then a map G„: Xn —-> E satisfying conditions (b) and (c) is defined by the
conditions G„ | X„_i = G„_i and G„ | (|s| xl) = G for s an n-simplex of
K- L. 
Taking A to be empty in theorem 6, we see that a weak fibration has the
homotopy lifting property with respect to any polyhedron.
7 corollary Let (X',A') be a polyhedral pair such that A' is a strong
deformation retract of X' and let p: E В be a weak fibration. Given maps
g': A' —» E and H': X' —-> В such that H \A' = p ° g', there is a map
G': X' —> E such that p ° G' = ЕГ and G' | A' = g'.
proof Let D: X' X I —> A' be a strong deformation retraction of X' to A'.
Then D(X' X 1 U A' X 1) GA', and we define g: X' X 1 U A' X I E to
be the composite
X' X 1 U A' X I A' E'
Let EI: X' X I —* В be the composite
X' X I Л X' В
Then H is an extension of p ° g, and it follows from theorem 6 that there is a
map G: X' X I E such that p ° G = H and G is an extension of g. Let
G': X' —> E be defined by G'(x') = G(x,O). Then G' has the desired properties. 
The following theorem is the main result relating the homotopy groups
of the base and total space of a weak fibration.
8 theorem Let p: E —> В be a weak fibration and suppose b0 G В' С B.
Let E' = p“1(B') and let e0 G p-1(b0). Then p induces a bijection
p#: rrn(E,E’,eo) ~ rrn{B,B',bf) n > 1
proof To show that p# is surjective, let a: (In,T,Zoj -» (B,B',b0) represent
an element of rrn(B,B',bf). Because Zo is a strong deformation retract of T, we
can apply corollary 7 to the pair (Bl,{z0)) and to maps g7: {z()} —> E and
H': [и _> B, where g'(zo) = eo and H' = a | In. We then obtain a map
G': In —> E such that p ° G7 = H' and G'(z0) = e0. Then
G'(I«) С р~ЧЩТ)) C = E7
Therefore G' defines a map a': (Injn,Zo') -> (E,E',e0) such that p ° a' = a.
Then a' represents an element [a'] £ wn(E,E',eo) and p#[a7] = [a].
To show that p# is injective, let (In,tn,Zo) —» (E,E',eo) be such that
p ° a0 ~ p ° aj. Let X' = In X I and A' = (I» X 0) U (z0 X I) U (Iй X !)
f 2 HIGHER HOMOTOPY GROUPS	377
’iГ. SB*-7*
j Then (X',A7) is a polyhedral pair, and because X' and A' are both contractible,
j Д' is a strong deformation retract of X'. Let g': A7 —> E be defined by
j /(2,0) = «o(z), g7(z,l) = fti(z), and g'(zo,t) = eo and let ЕЕ: X' —> В be a map
I’ which is a homotopy from p ° a0 to p ° a, in (B,B',b0). By corollary 7, there
is a map G7: X' E such that p ° G7 = EE and G71 A7 = g7. Since
G'(Z” X I) C	X I)) C p-i(B') = E7
ff is a homotopy from ao to ap in (E,E',eo); hence [ao] = [ap] in wn(E,E',eo). 
«I corollary Let p: E —> В be a weak fibration, bo G B, and eo G E =
p^bo)- Then p induces a bijection
P#- гтп(Е,Е,ео) ~ 77-и(В,Ь0) n > 1
proof This follows from theorem 8 on taking B7 = {b0} and using the
canonical identification MB,{bo},bo) = rrn(B,bo). 
If p: E —> В is a weak fibration with F = p-1(b0) and e0 G F, we define
Э: 77-и(В,Ьо) -> 77и-1(Е,е0) n > 1
Ю be the composite
’	b0) ^(EJfieo) rrn^fiF,e0)
The homotopy sequence of the weak fibration is the sequence
. • • -> rr.n(F,eo) гтп^Е,ео) rrn(B,b0) 77„_1(F,e0) -> • •  rro(B,bo)
where i: (F,e0) C (E,e0).
10 theorem The homotopy sequence of a weak fibration is exact.
.proof Exactness at гто(Е,ео') is an easy consequence of the homotopy lifting
j property. Exactness at any set to the left of 77-o(E,eo) is a consequence of the
1 exactness of the homotopy sequence of the pair (E,F). 
I1 11 corollary Let p: E —> В be a weak fibration with unique path lifting.
Then p induces an isomorphism
p#: rrq(E,eo) 77g(B,p(eo)) q > 2
(Proof Because F = p^1(p(eo)) has no nonconstant paths (by theorem 2.2.5),
jME,eo) = 0 for q > 1. The result then follows from theorem 10. 
j 12 COROLLARY For q > 2, MS1) = 0.
 proof This follows from application of corollary 11 to the covering projec-
* tion p: R —» S1 and the fact that because R is contractible, 779(R) = 0 for all
j (j > 0. 
I 13 corollary Let p: S2n+1 —> Pn(C) be the Hopf fibration. Then p induces
an isomorphism
p#-. MS2k+1) ~ MEn(C)) q > 3
378	номото»т theory сидр, 7
proof Because F = S1 for the Hopf fibration, this follows from corollary Jg ;
and theorem 10. 
14 COROLLARY 77'3(S2)	0.	j
proof Because the identity map (S3,po) C (S3,po) induces a nontrivial /
homomorphism of HzfSfpo), it is not homotopic to the constant map. There- '
fore 77‘3(S3)	0, and the result follows from corollary 13, with 11 — 1 (since ;
Ei(C) ~ S2). 	j
This last result shows that, unlike the homology groups, the homotopy
groups of a polyhedron need not vanish in degrees larger than the dimension i
of the polyhedron.	j
If H is a closed hemisphere of S2 and a is the pole in H, then the pair ;
(S2 - a, H — a) has the same homotopy type as (E2,Si). Therefore	|
77-3(S2 — a, H - o) ~ tt3(E2,S1) ~ 77-2(8!) = 0	!
On the other hand, (S2,H) has the same homotopy type as (S2,{a}). Therefore :
77-3(S2,H)	77-3(S2,{o}) = 77-3(S2) Д 0	I
Hence we see that the excision map j: (S2 — a, H — a) C (S2,H) does not |
induce an isomorphism of 77‘3(S2 — a, H — a) with 77-3(S2,H). Therefore the |
excision property does not hold for homotopy groups.	।
Recall the path fibration p: E(X,x0) —> X with fiber p-1(x0) = ЯХ (see i
corollary 2.8.8). Since E(X,x0) is contractible (by lemma 2.4.3), 77-я(Р(Х,Хо) = 0 [
for n > 0, and by theorem 10, there is an isomorphism
Э: 77„(X) ~ 77-и_1(ВХ) n > 1
This result can also be deduced directly from the canonical one-to-one cone- j
spondence [Sn(Z);X]	[S?,1(E );ЯХ] given by the exponential law. We shall .
use the path space to prove the exactness of the homotopy sequence of a triple. *
Given a triple (X,A,B) with base point xo € B, let i: (A,B) C (X,B) and"
j: (X,B) C (X,A) and let f: (A,{x0)) C (A,B). Define	;
д': ття(Х,А,Хо) —> tt„_i(A,B,xo) n > 2
to equal the composite	-
77-и(Х,А,Хо) \ 77„_1(A,XO)	1ГП-1(А,В^Хо)	!
The homotopy sequence of the triple (X,A,B~) is defined to be the sequence ;
-----> тги+1(Х,А) rrn(A,B) К irn(X,B) 77„(X,A)	-----> 77‘1(X,A) j
15 theorem The homotopy sequence of a triple is exact.	।
proof Let p: P(X,x0) —» X be the path fibration and let X' = P(X,Xo)> 1
A' = p~T(,A), and B' — p-1(B). Then (X',A',B') is a triple, and it follows from J
theorem 8 that p# maps the homotopy sequence of (X',A',B') bijectively to the
homotopy sequence of (X,A,B). Therefore it suffices to prove that the homo? |
topy sequence of the triple (X',A',B') is exact.
SEC. 3 CHANGE OF BASE POINTS	3/9
Let i: (A’,Bf C (X',B'), p (X',Bf C (X',A'), i': В' C A', and/': A' C (A',B').
There is a commutative diagram
• • 	77„+i(X',A')	irn(A',B') rrn(X',B') irn(X',A')  • •
4	>	4 I3
 • •	TT„(A')	77„(A',B')	L» 77„_1(B')	S 77„_1(A')	-»   
in which each vertical map is a bijection (because X' is contractible). Therefore
the exactness of the homotopy sequence of the triple (X',A',B') follows from
the exactness of the homotopy sequence of the pair (A',B'). 
This result can also be derived from the exactness of the homotopy se-
quence of a pair and functorial properties of the homotopy groups (as was the
case with the corresponding exactness property for homology, theorem 4.8.5).
3 CHANCE OF BASE POINTS
The absolute and relative homotopy groups are defined for pointed spaces and
pairs. This section is devoted to a study of the extent to which these groups
depend on the choice of base point. By generalizing the methods of Sec. 1.8,
we shall see that these groups based at different base points in the same path
component are isomorphic, but the isomorphism between them is not usually
unique. Much of these considerations apply to more general homotopy sets,
and we begin with this.
Let (X,A) be a pair with base point x0 C A and let (Y,B) be a pair. Two
maps a0, аг: (X,A) —-> (Y,B) are said to be freely homotopic if they are homo-
topic as maps of (X,A) to (Y,B) (that is, no restriction is placed on the base
point during the homotopy). If co is a path in В from ao(xo) to ai(x0), an
a-homotopy from «о to ai is a homotopy
H: (X,A) x (Y,B)
such that 7/(x,0) — ao(x), 7/(x,l) = ni(x), and 7Z(xo,t) = co(t). If such a homo-
topy exists, we say that a0 is u-homotopic to ai. It is clear that a0 and ai are
freely homotopic if and only if there is some path co in В such that ao and a^
are co-homotopic. In particular, two maps a0, a±: (X,A,x0) —> (Y,B,y0) are
freely homotopic if and only if there is some closed path co in В at yo such
that ao is co-homotopic to
Although the relation of free homotopy is an equivalence relation in the
set of maps from (X,A) to (Y,B), for a fixed co the relation of co-homotopy is
not generally an equivalence relation. For example, if co is not a closed path,
it is impossible for any map a0 to be co-homotopic to itself.
1 lemma (o) Given a map f: (X',A',x6) -> (X,A,x0), maps a0, аг: (ХД) ->
(Y,B), and a path co in В such that no is u>-homotopic to aj, then ao ° f is co-
homotopic to ai ° f.
380	HOMOTOPY THEORY CHAP. f
(b) Given a map g: (Y,B) —> (Y',B'), maps a0, аг: (X,A)	(Y,B), and a '
path co in В such that ao is u-homotopic to ai, then g ° ao is (g ° cofhoino-
topic to g ° ai.
(c) Given maps ao, a'o: (SX,SA,Xo) —> (Y,B,co(O)) and a,, af. (SX,SA,xo) —>
(Y,B,co(l)) such that ao is co-homotopic to ai and a'o is co-homotopic to a]
then a0 * a'o is o-homotopic to ai * al.
proof If H: (X,A) X I —» (Y,B) is an co-homotopy from a0 to ai, then for (я)
the composite
(X',A') x I (X,A) X I (Y,B)
is an co-homotopy from a0 ° j to a, " f, and for (b) the composite
(X,A) xl^ (Y,B)	(Y',B')
is a (g ° co)-homotopy from g° ao to g ° ap
In (c), if H, H': (S/V,SA) X I —» (Y,B) are co-homotopies from ao andop
to ai and al, respectively, the map
H * H': (SX,SA) xl-+ (Y,B)
defined by
(H * Tt'\(\x ri t1} —	0	0 < t < Уг
 H)([x,tj, f) _ 2t _	y2 < t < 1
is an co-homotopy from ao * a'o to aj * al. 
The base point x0 for a pair (Y,A) is said to be a nondegenerate base point
if the inclusion map (.ТоД’о) C (X,A) is a cofibration [that is, if, given a map
ao: (X,A) —-> (Y,B) and a homotopy co: xo X I —» B, there is a homotopy
H: (X,A) X I (Y,B) such that H(x,O) = a0(x) and H(xo,t) = co(t)]. It follows
from lemma 3.8.1 and corollary 3.2.4 that any point of a polyhedral pair is a
nondegenerate base point.
2 lemma Let (X,A) be a pair with nondegenerate base point and let (Y,B)
be an arbitrary pair.
(a)	Given a path co in В and a map ai: (X,A,xo) —» (Y,B,co(l)), there is a
map a0: (X,A,Xo) —» (Y,B,co(O)) such that a0 is ы-homotopic to ai.
(b)	If a0, a'o: (X,A,x0)	(Y,B,co(O)) are both co-homotopic to a1} then
[a0] = [ao] in [X,A,x0; Y,B,co(O)].
(c)	If a0 is u-homotopic to a, and a0 ~ a'o as maps from (X,A,xo) io
(Y,B,co(O)), ai ~ al as maps from (X,A,xo) to (Y,B,co(l)), and co ~ cd as paths
in B, then a'o is co'-homotopic to af
proof (o) Given a! and co, it follows from the nondegeneracy of Xo that
there is a map Id': (X,A) X I —* (Y,B) such that H'(x,O) — ai(x) and
H'(xo,t) = co(l — t). Define ao: (X,A,xo) —» (Y,B,co(O)) by a0(x) = H'(x,l). Then
H: (X,A) X I (Y,B) defined by H(x,t) = H'(x, 1 — t) is an co-homotopy
from ao to ai.
(b) Because Xo is a nondegenerate base point, there is a retraction
Sj.c. 3 CHANGE OF BASE POINTS	381
r. (X,A) X I —* (%'o X I U X X 1, *o X I U A X 1) (by theorem 2.8.1), and
vve let rt: (X,A) —> (x0 X I U X X 1, x0 X I U A X 1) be defined by rt(x) = r(x,t).
het G: (x0 X I U X X 1, x0 X I U A X 1) X I —> (X,A) X I be the homotopy
relative to (xo,O) defined by G(x,t,t') = and define
Gf. (xo X I U X X 1, xo X I U A X 1)	(X,A) X I
by Gj(x,t) = G(x,t,t'). Then Go ° i'o ° r0 rel Xo, and because Gq(xo X I) =
(<ro,O), Go ° ?'o Go ° i’i rel Xq. Let H: (X,A) X I —* (Y,B) be an co-homotopy
from ao to ai. Then H ° Gi ° r0 — H ° Go ° Ti rel x0. Clearly, H ° Go ° fi = ao,
and so ao s=: H ° Gi ° Tq rel xq. Similarly, if H': (X,A) X I —* (Y,B) is an
^-homotopy from a'o to ai, then a'o H' ° Gi ° To rel x0. Because
H\(x0 X I U X X 1) = № | (xo X I U X X 1)
H ° Gi °r0 = H' ° Gi ° r0, and so ao ~ a'o rel xq.
(c) First we observe that the inclusion map
(X X t U x0 X I, A X t U x0 X I) С (X,A) X I
is a cofibration. In fact, let h: (I X 1, I X 0 U t X I) ~ (I X I, О X 1) be a
homeomorphism. Then there is a homeomorphism
1 X h: (X X I X I, A X I X I) (X X I X I, A X J X I)
which maps
XXlXOUXxixIUxbXfXl to XxOxIUxoXfXl
and
AxIXOUAxIXlUxoXlXl to A x О X I U x0 X I X I.
Thus we need only show that (X X 0 U xo X T A X 0 U xo X I) X I is a
retract of (X X I, A X I) X I, which follows from the fact that (X X 0 U xo X I,
A X 0 U xo X 1) is a retract of (X X I, A X I).
Now let F, F': (X X I U xo X I, A X I U xo X I) —(Y,B) be defined by
F(x,0) = a0(x)	F(x,l) = И1(х)	F(x,t) = co(t)
F'(x,0) = a6(x)	F'(x,l) = a((x)	F'(x,t) = co'(t)
Because ao ao, ai al, and co co', it follows that F ~ F'. Because ao is
co-homotopic to ai, F can be extended to a map H: (X,A) X I —* (Y,B). By
the cofibration property established above, F' can be extended to a map
H': (X,A) X I —* (Y,B). Then H' is an co'-homotopy from a'o to al. 
It follows from lemmas 2o and 2b that, given co and ai: (X,A,x0)—»
(Y,B,co(l)), the set of all maps a0: (X,A,Xo) —> (Y,B,co(O)) which are co-homotopic
to at belong to a single homotopy class of maps (X,A,Xo) —» (Y,B,co(O)).
It follows from lemma 2c that this set of maps equals a homotopy class of
maps (X,A,Xo) —> (Y,B,co(O)) which depends only on the homotopy class
I«i] € [X,A,Xo; Y,B,co(l)] and the path class [co]. Therefore, if (X,A) has a non-
degenerate base point, there is a map
382	HOMOTOPY THEORY chap. 7
hlaY. [X,A,x0; Y,B,co(l)] -+ [X,A,x0; Y,B,w(O)]
characterized by the property /i|t|[fti| = [a0] if and only if a0 is co-homotopic
to ai. It follows from lemmas la and lb that this map is functorial in (X,A)
and in (Y,B) and from lemma 1c that if (X,A) is a suspension, the map is a
homomorphism.
3 theorem Let (X,A) be a pair with nondegenerate base point xq. For any
pair (Y,B) there is a covariant functor from the fundamental groupoid of В to
the category of pointed sets which assigns to a point yo С В the set
Y,B,//o] and to a path class [co] in В the map h[wj. If (X,A~) is a suspen-
sion, this functor takes values in the category of groups and homomorphism#.
proof We need only verify the two functorial properties. If a: (X,A,xo) —»
(Y,B,yo) is arbitrary and e is the constant path at yo, the constant homotopy
is an e-homotopy from a to a proving that /ip tj — 1.
Given paths co and co' in В such that co(l) — co'(O), an co-homotopy Ц
from ao to ap, and an co'-homotopy IF from ftj to ft2 [where ao, ap, a2 are
maps of (X,A) to (Y,B)], an (co * co')-homotopy H * H' from a0 to a2 is defined by
m rrni \	(H(x^,t)	0 < t <
This shows that /г[и,и'] = h[a] 0 h^. 
4 corollary If В C Y is path connected and (X,A) has a nondegenerate
base point Xq, then for any yo, yi € В the pointed sets [X,A,xo; Y,B,(/o] and
[/V,A,xo; Y,B,(/j] are in one-to-one correspondence. Furthermore, 7rp(B,!/o) acts
as a group of operators on the left on [X,A,xo; Y,B,//o], and the one-to-one
correspondence above is determined up to this action of •jr-r(B,y0).
proof If [co] is any path class in B, h^ is a one-to-one correspondence.
If [co] £ Wi(B,yo), then /i[w] is a permutation of [X,A,Xo; Y,B,i/o], and in this
way 77i(B,yo) acts as a group of operators. If yo and 1/1 are points in B, the
one-to-one correspondence 7i[Kj determined by path classes [co] in Y from yo
to г/р is the same as the set of maps h[Koj ° h^, where [coq] is a fixed path
class from yo to y1 and [co'] £ 77p(B,yo). 
In all of the above, by taking В = Y, we get the corresponding results
for the absolute case. Thus, if X is a space with nondegenerate base point Xq
and yo € Y, then 77p(Y,yo) acts as a group of operators on [X,x0; Y,i/o]- If Y is
path connected and yo, yr € Y, then [X,xo; Y,t/o] and [X,x0; Y,y0] are &
one-to-one correspondence by a bijection determined up to the action of
Tti(Y,yo).
In case Y is an H space and В C Y is a sub-H-space, there is the following
result, which can be regarded as a generalization of theorem 1.8.4.
5 theorem Let (X,A) have a nondegenerate base point Xo and let (Y,B)
be a pair of H spaces. If yo € В is the base point, ir-fB.yf) acts trivially oft
[X,A,x0; F,B,yol
jgq, 3 CHANGE OF BASE POINTS	3g3
proof Let p: (Y X Y, В x B) —-> (Y,B) be the multiplication. Given
i. (X,A,x0) —> (Y,B,y0) and a closed path co: (I,j) —> (B,y0), define an co-
j homotopy H: (X,A) X I-> (Y,B) from a to a by
f .	H(x,t) = /r(a(x),co(t))
j Therefore h[w][a] — [a] for all [a] £ [X,A,x0; Y,B,y0] and all [co] £ rri(B,y0). 
i
i There is an interesting relation between the action of 77i(B,yo) on
j [X,A,xo; Y,B,yo] and the action of 77p(B,yo) as covering transformations on a
I universal covering space of B. We assume that В and Y are path connected
I and locally path connected, that 77i(B,yo) ~ ”i( Y,y0), and that Y is a simply
j connected covering space of Y with covering projection p: Y —-> Y. Then
f. В = p-1(B) is a simply connected covering space of В [because 77p(B,yo) ~
( ffi(Y,(/о)]- Let yo € P1 ('/о)- There is a canonical map
:	(). [X,A,x0; Y,B,yo] [X,A; Y,B]
 from base-point-preserving homotopy classes to free homotopy classes.
! Because В is simply connected, this map is a bijection [recall that two maps
• ao, ai: (^,-A) ~are freely homotopic if and only if there is a path co in
I В from a0(x0) to ap(xo) such that a0 is co-homotopic to aj.
I 6 lemma With the notation above, let g: ( f,B,yo') —> (Y,B,f/i) be a cover-
ing transformation and let co be a path in В from y0 to Cp. There is a commu-
l‘ tative diagram
!	[X,A,x0; Y,B^0]	[X,A,x0; Y,B,y0]	[X,A; Y,B]
-[/to ’ A',.
j	[X,A,x0; Y,B,yo] [X,A,x0; Y,B,y0]	[X,A- Y,B]
: proof Because g is a covering transformation, p = p 0 g and p# = p#° g#.
j The commutativity of the left-hand square follows from this and from
lemma lb. Since в ° h^ = 0, the commutativity of the right-hand side follows
from the trivial verification that 0 ° g# = g# ° 0. 
Recall the isomorphism f: G(B | B) ~ 77i(B,yo) of corollary 2.6.4, which
assigns to g the element [p ° co] C 77i(B,y0). Therefore lemma 6 expresses a
[ relation between the action of G(B | B) ~ Ci(Y | Y) on the free homotopy
[ classes [X,A; Y,B] and the action of 77i(B,yo) on [X,A,xq; Y,B,y0]-
I 7 corollary Let X be a simply connected space with nondegenerate
I base point and let Y be a simply connected covering space of a locally path-
I connected space Y. There is a bijection from the free homotopy classes [X; Y ]
to the pointed homotopy classes [X,xq; Y,yo] compatible with the action of
G(Y| Y) on the former, the action of 77i(Y,yo) on the latter, and the isomor-
phism 1^: G(Y I Y) rri(X,yo).
proof This follows from lemma 6, with В = Y and A = X, and from the
Observation that because X is simply connected, it follows from the lifting
384	HOMOTOPY THEORY chap. 7
theorem 2.4.5, the homotopy lifting property of p: Y —> Y, theorem 2.2.3, and 5
the unique-lifting property, theorem 2.2.2, that p#: [X,x0; Y,j/0]	[X,x0; Y,i/o]
is a bijection. 	j
We now specialize to the homotopy groups. Because	|
ttb(X,Xq) = [S”(i),0; X,Xq] = [S»(i),S«(i),O; X,X,x0]	I
we obtain the following result.
8 theorem For any space X and any n > 1, there is a covariant functor !
from the fundamental groupoid of X to the category of groups and homomor-
phisms which assigns to x £ X the group rrn(X,x) and to a path class [w] in Д’
the map h[al: rrn(X,w(l)) -> 7TB(X,to(0)). In this way, wi(X,x0) acts as a group
of operators on the left on rrn(X,xf), by conjugation if n — 1, and if X is path I
connected and x0, xy € X, then ття(Х,х0) and wB(X,Xi) are isomorphic by an '
isomorphism determined up to the action of wi(X,xo).	I
proof Everything follows from lemma 3 and corollary 4 except for the j
statement that ^(Х.хь) acts on tti(X,x0) by conjugation. For this let |
H: S(t) X I —> X be on to-homotopy from a0 to ab where to, n0, and cq are |
closed paths in X at x0. Define H' : I X I —> X by	<
HftJ) = H([l,t], f)	•'
Then H' |Oxl=ff|lXl = w and H' 11 X 0 = n0 and FT 11 X 1 = «г. I
It follows from lemma 1.8.6 that (to * of) * (to"1 * «о"1) is null homotopic, )
Therefore [n0] = [w][ai][w]-1, and so h[a][ai] = [w][ni][w]-1. 
Theorem 8 shows that the action of tt1(X,x0) on itself by conjugation, as
in theorem 1.8.3, is extended to an action of tt^X.Xq) on ttb(X,x0) for every,
n > 1.	।
A path-connected space X is said to be n-simple (for n > 1) if for some
Xq £ X (and hence all base points x £ X) tt1(X,x0) acts trivially on wB(X,Xo).,|
Thus a simply connected space is n-simple for every n > 1, and a space X is <
1-simple if and only if 77i(X,x0) is abelian. For n-simple spaces there is a unique j
canonical isomorphism ttb(X,xq) ttb(X,xi), any map a: Sn X determines a !
unique element of 77-B(X,x0) (whether or not a maps the base point p0 £ S'" to Xo), (
and 77я(Х,Хо) is in one-to-one correspondence with the free homotopy classes
of maps S" —> X. The latter is a useful property, and for n-simple spaces X we ;
shall usually omit the base point and merely write тгя(Х). From theorem 5 we ;
obtain the following generalization of theorem 1.8.4.	j
Э theorem A path-connected H space is n-simple for every n > 1.  I
Similar consideration apply to the relative homotopy groups.	j
10 theorem For any pair (X,A) and any n > 1 there is a covariant functor |
from the fundamental groupoid of A to the category of pointed sets if n = 1
and the category of groups if n > 1 which assigns 7rnfX,A,rF) to x £ A and to 1
a path class [to] in A the map	1
SEC. 3 CHANGE OF BASE POINTS	335
W ttb(X,A,w(1)) -> irn(X,A,u(O))
In this way, Tr^A,xfj acts as a group of operators on the left on rrnfX,A,xf),
and if A is path connected and x0, Xj £ A, then ttb(X,A,x0) and rrn(X,A,xf) are
isomorphic by an isomorphism determined up to the action of n1(A,xoy). 
If to is a path in A, it follows from lemma la that there is a commutative
square for n > 1,
wB(X,A,to(l))	wB_1(A,to(l))
wB(X,A,to(0))	тгп^А^О))
Thus there is also a covariant functor from the fundamental groupoid of A to
the category of exact sequences which assigns to x £ A the homotopy sequence
of (X,A,x0).
A pair (X,A) with A path connected is said to be n-simple (for n > 1) if
ят(А,х0) acts trivially on ття(Х,А,х0) for some (and hence all) base points x0 £ A.
If A is simply connected, (X,A) is n-simple for every n > 1.
11 theorem Let fX,Aj be a pair of H spaces with A path connected. Then
(X,A) is n-simple for all n > 1.
proof This is immediate from theorem 5. 
If (X,A) is n-simple and x0, xt £ A, then ття(Х,А,х0) and rTn(X,A,xd)
are canonically isomorphic. Therefore any map a: (F'" ,S"~(X,A) deter-
mines a unique element of ttb(X,A,Xq) (whether or not a maps the base point
po £ Sn 1 to x°), and ttb(X,A,xq) is in one-to-one correspondence with the free
homotopy classes	X,A], If (X,A) is n-simple, we shah frequently omit
the base point and write wB(X,A).
The action of wi(A,x0) on 7T2(X,A,xq) is closely related to conjugation, as
shown by the next result.
11 theorem If a, b £ 772(X,A,xq), then
aba1 = hM(b)
proof Let X' = P(X,xo) and let p: X' —> X be the path fibration. Let
A' = p~\A) and let x'o £ A' be the constant path at x0. By theorem 7.2.8,
there is an isomorphism
p#: 7T2(X',A',Xq) ~ rr2(X,A,x0)
Let a' = p#"1(a) and b' = p+J '-(b) and observe that, by lemma lb,
Ь0а(Ь) = p#(h0a'(b'))
Hence it suffices to prove that a'b'a'^1 = h0a,(b'). Because X' is contractible,
it follows from the exactness of the homotopy sequence of (X',A',Xo) that
0: 7T2(X',A',xo) ~ 7Т1(А',хо)
386	HOMOTOPY THEORY CHAP. "j
So to complete the proof we need only prove that
8(a/b'a'~1) = 0(Ziaa'(b'))
The left-hand side equals (8a')(8b')(8a'), and because 8 commutes with haa,
the right-hand side equals h^a’(db'). The result now follows from the fact that
the action of Wi(A',xo) on itself given by h is the same as conjugation. 
This again implies that W2(X,Xo) ~ я-2(Х,{хо}ло) is abelian. Together with
the exactness of the homotopy sequence, it yields the next result.
13 corollary 77ie inclusion map /: (X,xd) C (X,A) induces a homomorphism
/#: 7г2(Х,х0) -> тг2(Х,А,х0)
whose image is in the center of it2(X,A,x0). 
The following result is a generalization of theorem 1.8.7 to the higher
relative homotopy groups.
14 theorem Let f: (X,A,Xq) (Y,B,y0) and g; (X,A,Xo) (Y,B,yf) be
freely homotopic. Then there is a path w in В from у о to y\ such that
f# — hM ° g-: Trn(X,A,x0) Trn(Y,B,y0) n > 2
proof Let F: (X,A) X I (Y,B) be a homotopy from f | (X,A) to g | (X,A)
and let w(t) = F(xo,t). Then w is a path in В from yo to t/1; and if
a: (In,In,p0) (X,A,X(,) represents an element of wn(X,A,x0), then the composite
X 1 -^4 (X,A)	(Y,B)
is an w-homotopy from f ° « to g ° «. Therefore
/#[«] = [/”«] = B[a]([g ° a]) = (7гы ° g#)[a] 
This yields the following analogue of theorem 1.8.8.
15 corollary Let f: (X,A) —> (Y,B) be a homotopy equivalence. For any
x £ A, f induces isomorphisms
f#: ттп(Х,А,х) ~ w„(Y,B,f(x))
proof Let g: (Y,B) —» (X,A) be a homotopy inverse of f. By theorem 14,
there are paths w in A from gf(x) to x and w' in В from fgf(x) to /(x) such that
the following diagram is commutative
Wn(X,A,x) At» Wn(X,A,gf(x))
/Ц
MW(x)) ^4 MWg/W)
Since the maps h[a] and /г[а'] are isomorphisms, all the maps in the diagram
are isomorphisms. 
SEC. 4 THE HUREWICZ HOMOMORPHISM
387
Д tiib hvhewicz намомопршзм
1’here are no algorithms for computing the absolute or relative homotopy
groups of a topological space (even when the space is given with a triangula-
tion). One of the few main tools available for the general study of homotopy
groups is their comparison with the corresponding integral singular homology
groups. Such a comparison is effected by means of a canonical homomorphism
from homotopy groups to homology groups. The definition and functorial
properties of this homomorphism are our concern in this section. A theorem
asserting that in the lowest nontrivial dimension for the homotopy group this
homomorphism is an isomorphism will be established in the next section.
We shall be working with the integral singular homology theory through-
out this section. Let n > 1 and recall that Hq(In,In) = 0 for q / n and
Hn(lnJn) is infinite cyclic. To consider relations among the homology groups
of certain pairs in I", for n > 1 we define
11» = {(tb •  • Л) € In 11„ < У2]
A” = (i^ n in) и {(tb . . . ,t„) e I» I tn =
i2n = {(ti, . . . ,tn) e in I tn > y2]
A” = (I2« n I") U {(tb . . . ,t„) € In 11„ = y2]
Then Ii" U A” = In and (A” U А”) Cl (jin U A”) = tin U A” By the exactness
of the Mayer-Vietoris sequence of the excisive couple (If'- U I2n, tin U I2»},
we have
HQ(hn и An, ii» и А») © HQ(ii« и i2«, А» и А») ~ hq(i«, А» и А»)
By excision, we also have isomorphisms
ад», A") ~ HQ(An и А», А» и A”)
ffQ(i2n,A”) ~ hq(A” u i2«, А» и А»)
Combining these, we see that if we let fi:	С (Iй, A” U A”) and we
let z2: (А”,А”) С (I», А» U A'1), then we have the following result.
1	lemma Иге inclusion maps ii and i2 define a direct-sum representation
*1# © i2* : Hq(Ii«,A’*) © HQ(An,A’1) ~ HQ(I", А» U A'») 
Let iq: (I",I") (If", A1") be defined by vfifi, . . . ,t„) = (tb . . . ,tn-r,tn/^)
and define v2: (Ififi) (I2nj2n) by r2(ti, . . . ,t„) = (tb . . . ,tn~i, + l)/2).
Let г: (!»,!») С (I», A” U A»)-
2	corollary For any z £ Hn(I«,i»)
A = A# b’s 4“	r2;,; z
PROOF Let /1: (I", А” и Аи) c (1», II" U A") and /2: (I«, A” U А») C
(I», A” U A”)- Then /i# ii^. = 0 and fi* i2^ is an isomorphism of HQ(I2”,A”)
388	HOMOTOPY THEORY CHAP, 7
onto HfI", hn U l2n) (induced by the inclusion map, which is an excision)
Similarly, /2*fe* = 0 and	is an isomorphism of	onto
HQ(I«, Ir” U I2"). It follows from lemma 1 that
ker П ker /2* = 0
Therefore, to prove the corollary it suffices to prove that
1*2 fa*vi*z htyVzmZ
is in the kernel of fa* and in the kernel of /2* 
We first prove that fa* (i* z — fa* z — i2* v2*z) = 0. Because i,= 0,
we must show that fa*i*z = fa*iz*v2*z. Clearly /ii is the inclusion map
(In,F) C (I”, Ii” U /2”) and /11'2^2 is the map f: (In,tn) —> (In, fan U I2»)
defined by/(ti, . . . ,tn) = (tb . . . (tn + l)/2). A homotopy Я from
fai to f is defined by
H((ti, . . . ,tn)> t) = (kt> •   >C-i> (ki + t)/(l + fa)
Therefore fa* i* = f* = fa* i2* v2*. A similar argument shows that
j2* fa* z - г1ф vj* z - i2* v2* z) = 0 
For n > 1 the subset I x Iя-1 U 0 X Iя'1 C tn is contractible. There-
fore Hg(In, I X Iя'1 U 0 X Iя-1) = 0 for all q. By exactness of the homology
sequence of the triple (Iй, ln, I X Iя'1 U 0 X Iя'1), it follows that the map
0: HQ(I«>) ->	1 X Iя'1 U 0 X Iя'1)
is an isomorphism for all q. For n > 2 let
/: (Iя'1,j”-1)	(&, I x Iя'1 U 0 X Iя'1)
be defined by /(#i, . . . ,tn-i) = (1, ti, . . . ,tn-i). Then j is the composite of
a homeomorphism from (Iя'1,Iя'1) to (1 X Iя-1, 1 X Iя'1) and the excision
map
(1 X Iя-1, 1 X I"'1) С (Iя, I X Iя'1 U 0 X Iя"1)
Therefore the homomorphism
j*: Яв(1я-1>-1) Hq(t«, I X i"' U 0 X Iя"1)
is an isomorphism for all q.
We define canonical generators Z„ £ Hn(In,tn) for n > 1 by induction on
n as follows:
(a)	Zr € Ifa(lj) is the unique element with dZj = (1) — (0) in H(l(i).
(b)	For n > 2, Z„ £ Hn(In,in) is the unique element such that
0Z?! = /*Z„_i in Hn-fatn, /xi" 1 U OX Iя'1).
Given a map a: (In,ln) (X,A), then a* Zn £ Hn(X,A). If a ~ ft, then
a* Z!t = ft* Zn. Therefore there is for n > 1 a well-defined map
<p: тгя(Х,А,х0) Hn(X,A)
f SEC. 4 THE HUREWICZ HOMOMORPHISM	389
) such that <р[ог] = a* Zn, where ог: (Iя,Iя) —> (X,A) maps p0 to xo and represents
: jgn element of wM(X,A,x0). By identifying ття(Х,х0) with w„(X,{x0},x0), we also
| have a map cp: тгп(Х,х0) —» Hn(X,x0). Some of the basic properties of <p are
| summarized in the next result.
| 3 theorem If n >2 or if n = 1 and A = {xo}, the map cp is a homomor-
j pjiistn. It has the following functorial properties:
I (a) For n >2 commutativity holds in the square
।	rrfaX,A,xf)	>	77и_1(А,Хо)
I	H	J,?
|	Hn(A,A)	Л	Hn^(A,x0)
‘	(b) Given f: (X,A,x0) —> (Y,B,yf), commutativity holds in the square
(	тгй(Х,А,х0)	Я	iTn(Y,B,y0)
j	H	|sp
|	Hn(X,A) f*> Hn(Y,B)
i proof Let cq, a2: (In,tn) —> (X,A) be such that
ai(ti, . . .	1) = a2(t, . . . ,tn_±, 0)
| [any two maps of (In,t”) to (X,A) are homotopic to such maps if n > 2 or if
f П = 1 and A = {x0}]. Then cq * a2 = ft ° i, where i: (In,in) С (Iя, iin U l2n)
I and ft: (In, Iin U i2n) (X,A) is defined by
I	П/.	_ fal(lb   • dn—1> 2tB)	tn < У2
!	   , n> - ta2(ti, •  • 2tn - 1) tn > y2
> Then <p[ai * a2] = ft*i*Zn = ft^fai^i^Zn + fa^Zn), the last equality by
! corollary 2. Since ftfav± = cq and fti2v2 = a2, we see that
j	<p[ai * a2] = a^Zn +- agjZn — <p[cq] + <p[«2]
( which shows that cp is a homomorphism whenever тгп(Х,А,Хо) is a group.
‘ To prove (a), let a: (In,ln) —» (X,A) represent an element of тгп(Х,А) for
' n > 2 and suppose that a(I X Iя-1 U 0 X I'"1) = Xo- Then d[a] = [a'], where
! a': (Iя-1,Iя-1) —> (A,xo) is defined by a' = (a | (Iя, I X Iя-1 U 0 X Iя-1)) 0 j.
i Then
!	<p0[«] = OL'*Zn^ = (a I (Iя; I x Iя”1 U 0 x F-^ faZ^
j	= (a I (Iя, I X Iя"1 U 0 X Iя'1))* 8ZM
;	= Sa* Zn= 0<p[a]
I Finally, (b) follows from the fact that (fa)* = f*a*. 
The map cp is called the Hurewicz homomorphism. The next result follows
from theorem 3.
390
HOMOTOPY THEORY CHAP. 7
4 corollary The Hurewicz homomorphism maps the homotopy sequence
of (X,A,xo) into the homology sequence of (X,A,xo). 
Our next objective is to show that the Hurewicz homomorphism com-
mutes with the actions of the appropriate fundamental group on the homotopy
set. We consider the relative case first.
5 lemma Let [a] £ nn(X,A,Xo) for n > 2 and let [a>] £ iti(A,Xo). Then
<p(Wa]) = ф[а]
proof Let [a] be represented by a:	—» (X,A) and let /i[t.,|[«] be repre-
sented by a': (In,ln) (X,A). Then a and a' are freely homotopic [that is,
a and a' are homotopic as maps of to (X,A)]. Therefore
<p[a] = а^.2п = <4.Z„ = <p[a'] = <р(/гы[а]) 
Next we prove the corresponding result for the absolute case.
6 lemma Let [a] € 7Tn(X,Xo) and [w] € wr(X,Xo). Then
<P(W«D =
proof Let Y be the space obtained from In by collapsing tn to a single
point, this point to be the base point of Y, denoted by y0. The collapsing map
g:	(Y,y0) induces a one-to-one correspondence between [Y,(/o; X,x0]
and [Iя,tn; X,Xo]. Therefore 7rn(X,x0) can be identified with [Y,y0; X,*oJ.
Furthermore, g*: Hn(In,In) Hn(Y,y0), and we let g* Z„ = Z'n £ H„(Y,i/0).
In these terms, if an element of 7тя(Х,х0) is represented by a: (Y,?/o)	(X,xo),
then <p[a] — Z'n. Let h|t.,|[«] be represented by a': (Y,(/o)	(X,xo). Then a
and a' are homotopic as maps of Y to X. Therefore, if Z" C Hn(Y) is the
unique element such that i'%7", = Z'n [where i'\ Y C (Y,(/o)]> then
(a|Y)rtZ" = («'|Y^Z"
Let X C (X,x0). Then
<p[<x] = (Xy. Zn — a* i'y. Zn = f. (a | YZn
Similarly, <p[a'] = j'% (a' | Y Z", and
<p[a] = <?[<*'] = <р(7г[и][а]) 
We define w^(X,A,x0) for n > 2 to be the quotient group of ття(Х,А,Хо)
by the normal subgroup G generated by
{(h[o,j[«])[«]-11 [«] € w!t(X,A,x0), [w] G чт^Хрсо)}
By lemma 5, <[ maps G to 0 and there is a homomorphism
<p': ття(Х,А,х0) —> Hn(X,A)
whose composite with the canonical map y: ття(Х,А,хо) —» 7l',(X,A,xo) is <(<
Note that, by theorem 7.3.12, tt„(X,A,x0) is abelian for all n > 2.
Similarly, we define Tr'n(X,xf) for n > 1 to be the quotient group of
SEC. 4 THE HUREWICZ homomorphism	397
?r„(X,Xo) by the normal subgroup H generated by
{(^[И][«])[«]-11 [«] € wn(X,x0), [w] G tti(X,xo)}
By lemma 6, <p maps II to 0, and there is a homomorphism
<(Х,хь) Hn(X,Xo)
whose composite with the canonical map i): wn(X,xo) —> w(i(X,x0) is <p. Note
that wl(X,Xo) is the quotient group of tti(X,x0) by its commutator subgroup.
In particular, я^(Х,х0) is abelian for all n > 1.
Because the groups Wn(X,A,Xo) and 7T„(X,x0) are abelian, we shall find
them easier to compare with the homology groups (which are abelian) than
the homotopy groups themselves. For the comparison it will be convenient to
replace the triple (In,In,po\ which is the antecedent triple used to define
%я(Х,А,хо), by the homeomorphic triple (Д'",Л’!,Со), where Д” is the standard
n-simplex used in Sec. 4.1 to define the singular complex (vertices of Д” will
be denoted by o0, щ, . . . , u„). To achieve this replacement we need only
choose a homeomorphism of (A’l,A’l,oo) onto (l” jn,Po). Any homeomorphism
Й: (AH,An) —> (In,In) will induce an isomorphism
h»:H„(A»A»)~H„(I»i«)
The identity map £n: &n С Дя is a singular simplex which is a cycle modulo
A” and whose homology class {£„} is a generator of the infinite cyclic group
H„(AM,A”). Since Z„ is a generator of Hn(In,tn) and h* is an isomorphism,
either	= Z„ or h,b {£„} = — Z„. We want to choose h so that the
former holds. If n = 1, the choice of Zi is such that the simplicial homeomor-
phism h: A1 —> I with h(v0) = 0 and h(*u) = 1 will have the desired
property (that is, h^. {£,} = Z,). If n > 1, we choose an arbitrary homeomor-
phism h: (Д’1,A”) —> (In,ln) such that h(vg) = po. If h* {£„} = — Z,„ we
replace h by hX, where A is a simplicial homeomorphism of A” to itself such
that A(o0) = U(j and A.,. {£„} = — {£„} (for example, A is the simplicial map
which interchanges c i and 02 and leaves all other vertices of A’1 fixed). There-
fore, in any event, we can find a homeomorphism h: (Дя,Ая,оо) —> (In,tn,p0)
such that h* {£„} — Zn. Using such a homeomorphism to represent elements
of wn(X,A,Xo) by maps а: (Дя,Дп) (X,A) such that a(o0) — x0, we see that
<p[a] =	= {«}, the latter being the homology class in (X,A) of the
singular simplex a.
For any pair (X,A) with base point Xq G A and any n > 0, let A(X,A,Xq)”
be the subcomplex of A(X) generated by singular simplexes o: Д'' X having
the property that 0 maps each vertex of Д'' to Xq and maps the n-dimensional
skeleton (A'')" of A® into A. Then A(X,A,x0)”+1 U A(X,A,x0)n, and these two
chain complexes agree in degrees < n. Thus we have a decreasing sequence
of subcomplexes Д(Х,А,хо)" (where n > 0) of Д(Х) whose intersection is con-
tained in A(A). If X is path connected and (X,A) is n-connected for some
ti > 0, we shall see that the inclusion map Д(Х,А,Хо)я C Д(Х) is a chain
equivalence. The following lemma will be used for this purpose.
392
HOMOTOPY THEORY CHAP. 7
7	lemma Let С be a subcomplex of the free chain complex A(X) (that is,
C is generated by the singular simplexes of X in it). Assume that to every
singular simplex о: № —> X there is assigned a map P(a): Д" X / -> X such
that
(a)	P(o)(z,0) = o(z) for z £ A«.
(b)	Define d: № —> X by d(z) = P(o)(z,l). Then d is a singular simplex in
C, and if a is in C, d = a.
(c)	If ef-. A®-1 —> Ae omits the ith vertex, then P(o) ° (ef X 1) = P(oW).
Then the inclusion map С С A(X) is a chain equivalence.
proof Let С С A(X) be the inclusion chain map and let т: Д(Х) —» Cbe
the chain map defined by t(h) = a [(c) implies that т is a chain map]. By (b),
т ° / = lc, hence to complete the proof we need only verify that j ° т ~ 1й(Х).
For any space Y let h0, hy Y —> Y X I be the maps h0(y) = (y,0) and
hfiy) = (y,l). In the proof of theorem 4.4.3 it was shown (by the method of
acyclic models) that there exists a natural chain homotopy D: Д( Y) ^ A( Y X I)
from A(Ao) to &(hf). Define a chain homotopy
D'-. L(X) A(X)
by ТУ(а) — A(P(o))(D(£e)), where а: —> X and A® С A<?. By (c), D' is a
chain homotopy, and by (a) and the definition of o, D' is a chain homotopy
from 1Д(х) to т ° /. 
8	theorem Let x0 € А С X and assume that X is path connected and
(X,A) is n-connected for some n > 0. Then the inclusion map Д(Х,А,х0)я C A(X)
is a chain equivalence.
proof For o: Af/ X we define P(a) by induction on q to satisfy the prop-
erties of lemma 7, and to have the additional property that if a is in A(X,A,Xo)’1,
then P(o) is the composite
Де X 1^ №±>X
where p is projection to the first factor.
If q = 0, then о: Д° —> X is a point of X, and because X is path connected,
there is a map P(o); A0 X I —> X such that Р(о)(Д° X 0) = <r(A°) and
Р(о)(Д° X 1) = xo [and if n(A°) = x0, we take P(d) to be the constant map to
Xo]. This defines P(<r) for all a of degree 0 to have the desired properties.
Assume 0 < q < n and that P(o) has been defined for all a of degree < q
to have the properties stated above. Given a singular simplex о: A® X, if о
is in A(X,A,x0)n, define P(o) = о ° p. If a is not in A(X,A,x0)", a and c of
lemma 7 define P(o) on А® X 0 U A® X I, and we let/: X 0 U Д« X
be this map. There is a homeomorphism hr. E(! X I ~ Де X I such that
h(E« X 0) = A« X 0 U A<? X I, hfir1 X 0) = A" X 1
and
h(Sn X I U Еч X 1) = AQ X 1
sfC. 5 THE HUREWICZ ISOMORPHISM THEOREM	393
Let/'1 (E^Sr1)	(X,A) be defined by f '(z) — f(h(z)). Because q < n and
(X,A) is «-connected, there is a homotopy H: (E«,Se-1) X I —> (X’A) from f'
1 to some map of Eq into A (in fact, by the definition of n-connectedness, there
’ is even such a homotopy relative to S'?-1). Then the composite
X I ® X f X
can be taken as P(o).
In this way P(o) is defined for all degrees q < n. Note that a singular
simplex of degree > n is in Д(Х,А,х0)я if and only if every proper face is in
д(Х,А,хо)”. Therefore, if P(o) has been defined for all degrees < q, where
I q n, and if о: Де —> X, then we define P(o) = 0 0 p if о is in Д(Х,А,х‘о)я and
j to be any map Де X I —» X satisfying a and c of lemma 7 (such maps exist by
j the homotopy extension property). Then P(a) will necessarily satisfy b of
> lemma 7, and we have shown that P(a) can be defined for all a to satisfy
 lemma 7. 
, For n > 0 we define
!	HeW(X,A,x0) = Не(Д(Х,А,х0)я, Д(Х,А,х0)я А Д(А))
। There are canonical homomorphisms
|	-----> HeW(X,A,x0) Hfy- i-)(X,A,x0) -+-------> He(O)(x,A,xo) He(X,A)
*. <> cobollary Assume that A is path connected and for some n > 0,
j (X,A) is n-connected. Then the canonical map is an isomorphism for all q
[	He(«)(X,A,x0) ВДА)
j proof For any n > 0, Д(Х,А,Хо)я А Д(А) is generated by the set of singular
I simplexes of A all of whose vertices are at x0. This is independent of n, and
i because A is path connected, (A,{x0}) is О-connected, and it follows from
( theorem 8 that the inclusion map Д(Х,А,х0)я Г1 Д(А) С Д(А) is a chain
( equivalence for all n > 0.
Since (X,A) is n-connected, where n > 0, and A is path connected, X is
I also path connected, and by theorem 8, the inclusion map Д(Х,А,Хо)” С Д(Х)
? is a chain equivalence. The result follows from these facts, using exactness
< and the five lemma, и
i
<
I 5 THE IIUBEWICZ ISOMORPHISM THEOREM
! The main result of this section asserts that if X and A are path connected and
1 for some n > 1, (X,A) is n-connected, then the Hurewicz homomorphism <p
I induces an isomorphism <[/ of :t.'h i(A',A,.to) with PIm+1(X,A). This result is
f equivalent to a homotopy addition theorem which asserts that the sum of the
J (n + l)-dimensional faces of an (n + 2)-simplex is the homotopy boundary of
the identity map of the simplex. We prove both these theorems simultaneously
by induction on n.
394
HOMOTOPY THEORY сНдр; 7 ? sEC. 5
THE HUREWICZ ISOMORPHISM THEOREM
395
In the proof we shall make essential use of the complexes &(X,A,x0)n ail(]
of corollary 7.4.9. Let a: (Дя,А«,(Д«)о)	(X,A,x0) represent an element 0| 1
тгп(Х,А,Хо)- Then a is a singular simplex in A(X,A,x0)”“1 and represents a |
homology class {«} £ Hn&~V(X,A,Xo)- Since any element of nn(X,A,x0) can be I
represented by such a map a, the Hurewicz homomorphism cp': rddX^xd) ’
Hn(X,A) factors into the composite	|
тт'п(Х,А,хо) Hn^X,A,xo) -+ Hn(X,A)	j
and there	is	a commutative diagram	'
ттп(Х,А,Хо) Л	тт'п(Х,А,х0)	j
Hn(X,A)	H„(-i)(X,A,x0)	j
We now formulate the propositions corresponding to the relative and ;
absolute	Hurewicz isomorphism theorems.	।
1 proposition Ф„ (n > 2). Let A be path connected and let (X,A) be I
(n —	V)-connected. Then cp' is an isomorphism	»
ср': гт'п(Х,А,хо) Hn(X,A)	I
2 proposition Ф„ (n > 1). Let X be (n — reconnected. Then cp' is an
isomorphism	I
ср': я4(Х,х0) = Hn(X,x0)	I
We shall prove both these propositions simultaneously by induction on n,
together with a third proposition, which we now formulate. For n > 2, each I
face map eh+i is a map of triples	'
e%+1: (An,An,vo) —> (Дп+1,(А«+1)п-1,о1)	j
eil+i: (A« A’Wo) -> (A«+i,(Ab+i)«-i,o0)	0 < i < n + 1
For vertices v and o' of A’l+1 we use [ooz] to denote the path class of the l‘
linear path in A'"11 from о to o'. We define an element bL £ 7t1(A2,o0) and, for 1
n > 2, an element bn £ 77я(Дя+1,(Дя+1)я-1,оо) by	|
bi = [ooOj] ° [oio2] ° [o2o0]	;
b2 = (Ь[г,€,„1][ез0])[ез2][ез1]“1[е33]—1
Ьз = h[i,oi)1][e^+i] + S (— 1)'[Оп+1] n > 3	*
0<?-<n4-l	1
For n = 1 let j: (A2,o0) C (A2,o0) and for n > 2 let j: (Ав+1,(Дв+1)я~1,Оо) C f
(Дя+1,(Дв+1)я-1,Оо). The following proposition corresponds to the homotopy
addition theorem.	|
3 PROPOSITION Bn (n > 1). /#Ь„ = 0.	j
The simultaneous proof of propositions 1, 2, and 3 will consist of the fol- ।
lowing five parts:
(c) Proof of
(b)	Proof that Bi => Ф1
(c)	Proof that Ф1, Ф2, . . . , Ф?!-1 => Bn for n > 2
(d)	Proof that Bn =^> Ф„ for n > 2
(e)	Proof that Ф„ => Фп for n > 2
(a)	proof of Bi We must prove that j#bi = 0. But /#bi £ wi(A2,o0), and
ffi(A2,oo) = 0 because A2 is contractible. 
(b)	proof that Bi =^> Ф1 Let X be path connected. We must prove that
(p'; 7ri(X,x0) ~ Hi(X,x0)- Because X is path connected, the inclusion map
Д(Х,{хо] ,x0)° C A(X) is a chain equivalence, and we need only show that
cp": 7Ti(X,Xo) ~ Hd°KX,{x0},x0)
If «: (A1,A1)	(X,x0) represents an element [a]' £ 7г1(Х,Хо), then
= {a}, where {a} is the homology class in I/l((J>(X,{xo},X(J) of the
singular cycle a. Given a singular 1-simplex o: (A1,A1) (X,x0) in A(X,{xo},Xo)°,
it determines an element [a] £ tti(X,Xo), and therefore an element
[a]' € w4(X,x0). If о is the constant singular 1-simplex at x0, then clearly,
[n]' = 0. Because ni(X,Xo) is abelian and Ai(X,{xo},xo)° is the free abelian
group generated by the singular simplexes in it, there is a homomorphism
Ai(X,{xo],xo)°/Ai(xo) -> wl(X,x0)
such that = [a]'. We shall show, by using Вг, that the composite
A2(X,{xo},xo)o/A2(xo) Л Ai(X,{x0},xo)o/Ai(x0) ± 7rl(X,x0)
is trivial. Given o: (A2,(A2)0) -> (X,x0), let 0<°>, nd), and be the faces of o,
as usual. Then
^0[n] = [o<2)]' + [n(0)]' - [n(D]' = [(a® * o*0>) * (d1))'1]'
= 4(0 I A2)#([o0Oi] ° [O1O2] 0 [o2o0]) = i)0#/#bi = 0
Therefore defines a homomorphism
V: Н1(°)(Х,{Х0},Х0)	771(X,Xo)
and this is easily seen to be an inverse of cp". 
(c)	PROOF THAT Ф1, . . . , Ф„_1 => Bn FOR n > 2 Consider the commutative
diagram
77n+l(A’t+1,A«+l,O0)	77.!t(A”+1,(A«+1)”-1,O0)
г|	77„(A»+1,(A»+1)’1-1,Oo)
A
7T„(Ab+1,Oo)	7<\_|((A,!+I)'" 1,Uo)
The top row, being part of the homotopy sequence of the triple
(Д’!+1,Аг!+1,(Дп+1)п-1), is exact. The bottom row, being part of the homotopy
396	HOMOTOPY THEORY CHAP, 7
sequence of the pair (Ая+1,(Дп+1)”-1), is also exact. From the exactness of the
homotopy sequence of the pair (An+1,A”+1) and the fact that Д’1+1 is contract-
ible, it follows that 8 is an isomorphism. Therefore
ker = im 8' = im ° 8) = im i# = ker 8"
Thus Bn is equivalent to the equation 8"(b„) = 0. We prove the latter, giving
one proof for n = 2 and another for n > 2.
If n = 2, we have
3'W) = (^о«1]9"[езо])Э"[ез2]а"[ез1]_1а"[ез3]-1
To calculate 8"[ез{], let £: (Д2,А2,о0) С (A2,A2,o0) be the identity map. Then
[fl € W2(A2,A2,Oo), and because w1(A2,oo) is infinite cyclic (since A2 is homeo-
morphic to Si), it follows from $i that <p: 7t1(A2,o0) ~ Hi(A2,t?0). There is a
commutative square
7T2(A2,A2,Oo) A 771(A2,Oo)
4
H2(№^ H^Vo)
and	8<p[fl = 8{£} = {fl2’ + fl°> - fl1’} = {of} = <p[w]
where w: (A1,A1)	(A2,o0) is the path w = (fl2> * fl0’) * (fl1*)-1. (The 2-chain
fl2’ + fl0’ — fl1* is homologous to w because it is easy to find singular
2-simplexes m and o2 in A2 such that
CT1(0) _ £(0)	CT1(1) — £(2) £(0)	ai(2) _ £(2)
a2(0) = fl1*	U2(1> = fl2’ * fl0’	аг*2’ = (fl2’ * fl0’) * (fl1’)-1
Then 8(oi — n2) = fl2’ + fl0’ — fl1’ — w). Because <[ is an isomorphism, it
follows that
8 [fl = [w] = [o0oi] ° [О1О2] ° [u2o0]
To return to the calculation of 8"[ез’], we have
8"[езг] = 8'W)#[fl = Ы I A2)#S[fl
= [ез{(по)езЧг>1)] ° [езг(и1)езг(и2)] ° [^(t^WW]
Using this, direct substitution into the right-hand side of the equation for
8"(b2) shows that 8"(b2) = 0.
For n > 2 note that (A«+i)«“i contains the two-dimensional skeleton of
A”+1. Therefore (A»+i)»-i is simply connected (because Д"+1 is simply con-
nected). Similarly, for q < n — 2, Нв((Дв+1)я-1,г?0) ~ Не(Дя+1,Оо) = 0. By
®i, . . . , Ф,| 2, it follows that (Дя+1)я-1 is (« — 2)-connected, and by Ф«-ь
there is an isomorphism
<p: 7тй_1((Дя+1)я-1,и0) ~ Нй_1((Дя+1)я-1,о0)
Hence, to complete the proof it suffices to show that <рЭ"(Ьи) = 0. This
follows from the equalities
SBC* 5 THE HUREWICZ ISOMORPHISM THEOREM	397
= 8"<р(Ь„) = a"{S (-1)^+1} = a"8'{£n+i) = э'%а{£п+1) = о 
proof that Вп => Фп for п > 2 The argument is similar to the proof of
part (b) above. The map <p' factors into the composite
т/„(Х,Ал) нмад Hn(X,A)
Jf a: (A«,An,v0) —> (X,A,x(l) is a map such that a maps all the vertices to Xo,
then <p"[n]' = {° } € Hntn~1\X,A,Xo). To define an inverse of <p", if
0: (Д’!,Ая,(Дга)°) —> (K,A,Xo) is a singular simplex in An(X,A,x0)n-1, then
[a] € tt„(X,A,xo) and i;[ct] = [ст]' £ тг'п(Х,А,х0). If ст(Дя) C A, then [ст]' = 0,
and because tt]i(X,A,Xo) is abelian, there is a homomorphism
Д.!1(Х,А,Хо)^1/(Дв(ХА,хо)^1 П ДВ(А)) -+ <(X,A,x0)
such that ^(ct) = [ст]'.
We show that the composite
° 8: Дп+1(Х,А,хо)и"1/(Дп+1(Х,А,Хо)?!-1 А Д„41(Л)) тг'п(Х,А,Хо)
is trivial. This follows from Bn, because if
ст: (Д«+1>(Дп+1)»-1)(Ди+1)0) _> (X,A,x0)
then
^Э(ст) = S (-1)’[ст«]' = 7)(ст I (Дв+1,(Дя+1)я~1)#(Ьи))
= vo#j#(b„) = о
Therefore f defines a homomorphism
4/; Hn(«-B(X,A,xo) ^(X,A,xo)
such that ст] = [ст]', and is easily seen to be an inverse of <p". 
(e) proof that Ф„ => Ф„ for n > 2 For n > 2, if X is (n — l)-connected,
then the pair (X,{xo}) is (n — l)-connected and ttb(X,{xo],Xo) is canonically
isomorphic to wB(X,Xo) = тгп(Х,Хо). Then Фя results from Ф„ applied to the
pair (X,{x0}). 
This completes the proof of propositions 1, 2, and 3. From proposition 1
we obtain the following relative Hurewicz isomorphism theorem.
4 theorem Let xo € А С X and assume that A and X are path connected.
If there is an n > 2 such that 77Q(X,A,x0) = 0 for q < n, then Hg(X,A) = 0
for q < n and <p' is an isomorphism
<p'- 7tb(X,A,Xo) ~ Hn(X,A)
Conversely, if A and X are simply connected and there is an n >2 such that
Hq(X,A) = 0 for q < n, then -nq(X,A,xf) = 0 for q < n and <p is an isomorphism
<p'-. 77;(X,A,xo) Hn(X,A) 
Similarly, from proposition 2 we obtain the following absolute Hurewicz
isomorphism theorem.
398	HOMOTOPY THEORY CHAP, 7
5 theorem Let xo € X and assume that there is n > 1 such that
7tq(X,xo) = 0 for q < n. Then HQ(X,x0) = 0 for q < n and q>’ is an isomorphism
(p': 7T.n(X,Xo) Hn(X,x0)
Conversely, if X is simply connected and there is n> 2 such that Hg(X,x0) ~ q
for q < n, then 1Г^Х,Хо) = 0 for q < n and <p is an isomorphism
<p: n„(X,x0) ~ H„(X,x0) 
In the absolute case when X is simply connected and in the relative case
when X and A are simply connected, each of these theorems asserts that the
first nonvanishing homotopy group is isomorphic to the first nonvanishing
homology group.
6 corollary For n > 1 there is a commutative diagram of isomorphisms
rrn+1(En+1,Sn,p0)	7T„(S’l,po)
I?
Hn+1(E«+i,S”) Л Hn(S”,p0)
proof The diagram is commutative, by theorem 7.4.3a, and both horizontal
maps are isomorphisms because En+1 is contractible [and because the homo-
topy and homology sequences of (En+1,Sn,p0) are exact]. The right-hand ver-
tical map is an isomorphism, by proposition 2 and the fact that (in the case
n = 1) wi(S1,p0) is abelian. 
The following useful consequence of corollary 6 is called the Brouwer
degree theorem.
7 corollary For n > 1 two maps f, g: S” -a Sn are homotopic if and
only if f* = g*; Hn(Sn) -A.	Similarly, two maps f, g: (E«+i,S’>)
(En+1,Sn) are homotopic if and only iff* = g*: Hn+i(En+1,Sn) —> Hn+1(En+1,Sn),
proof We consider the absolute case first. Given maps f, g: S” S’1, there
exist homotopic maps f' and g', respectively, such thatf '(p0) = g'(po) = Po
(because S’1 is path connected). Because Sn is n-simple, f' and g' are freely
homotopic if and only if they are homotopic as maps from (S”,po) to (S”,po)-
Therefore f ~g if and only if [/'] = [g/] in 7rn(Sn,p0). By corollary 6,
[f'] = [g'] if and only if <p[f'] = <p[g/], and from the definition of <p,
<p[/'] = <p[g'] if and only if
f* =gi= Hn(S”,p0)	Hn(&,p0)
Since there are commutative squares
H„(S«)	Hn(S",p0)	H„(S”)	H„(S»,po)
]/*	J.®»
Hn(S«) Hn(S",Po)	H„(S«) Hn(Sn,po)
the result follows.
5 THE HUREWICZ ISOMORPHISM THEOREM	ЗЯЯ
Sb.
| For the relative case note that because En+1 is contractible, it follows
« from the homotopy extension property of (E’1+1,S’') that two maps
| g: (En+1,Sn) —> (En+1,Sn) are homotopic if and only if f | S", g | S’1: S'" Sn
I are homotopic. Since there are commutative squares
Hn+i(En+1,Sn) H„(S”)	Hn+1(En+1,S”) Hn(Sn)
Ц Fs,,>*	4 Иs”>*
;	Hn+1(En+1,Sn)	H„(S«)	H;i+i(E’l+1,S”)	H„(S«)
the relative case follows from the absolute case. 
8 corollary For xo € X the map
f: [S\p0; X,x0] Hom (wn(Sn,po), w!t(X,x0))
I sending [a] to a# is an isomorphism.
proof This follows from corollary 6, because the fact that wn(S”,po) is infinite
cyclic implies that there is an isomorphism
/3: Hom (w„(S”,po), тгп(Х,хб)) ~ rrn(X,x0)
[ sending a homomorphism X to X(a), where a £ rrn(Sn,po) is the homotopy
( class of the identity map. Then, (j8 ° ^)[«| = cqfa) = [a], and so f is an
I isomorphism. 
[ The following useful consequence of the relative Hurewicz isomorphism
theorem is known as the Whitehead theorem.
9 theorem Let X and Y he path-connected pointed spaces and let
I f. (X,xo) (Y,?/o) be a map. If there is n > 1 such that
/#: we(XAo) -> iTg(Y,yo)
is an isomorphism for q <3n and an epimorphism for q = n, then
f*.H(fX,xf)^Hg(X,yf)
| is an isomorphism for q < n and an epimorphism for q = n. Conversely, if
X and Y are simply connected and f* is an isomorphism for i[ C n and an
epimorphism for q = n, then f# is an isomorphism for q < n and an epimor-
phism for q = ii.
| proof Let Z be the mapping cylinder of f. There are inclusion maps
У i: X C Z and j: Y C Z and a deformation retraction r: Z Y such that
! f = r ° i. Then r: (Z,y0)	(Y,y0) induces isomorphisms r#: iTg(Z,y0) ~ 7tq(Y,?/o)
j and r* : H(;(Z,i/o) ~ Efe(Y,y0) for all q. Because X and Y are path connected,
I so is Z, and 7rg(Z,x0) zz ng(Z,y0). Therefore r: (Z,xo) —> (Y,y0) also induces
1 isomorphisms r#: rrg(Z,x0) ZZ wQ(Y,y0) and r*: Hg(Z,x0) ZZ Hg(Y,y0) for all q.
I It follows that we can replace (Y,?/o) in the theorem by (Z,xo) and the condi-
| lions un f# and f* by the corresponding conditions on i# and i*. From the
I exactness of the homotopy sequence of (Z,X,x0), it follows that i# is an
400	HOMOTOPY THEORY
isomorphism for q < n and an epimorphism for q = n if and oi
779(Z,X,Xo) = 0 for q < n. Similarly, from the exactness of the homolog
sequence of the triple (Z,X,x0), it follows that i* is an isomorphism for q
and an epimorphism for q = n if and only if Hg(Z,X) = 0 for q < n. Th
result now follows from the relative Hurewicz isomorphism theorem 4. 
6 CW COMPLEXES
For homotopy theory the most tractable family of topological spaces seems to
be the family of CW complexes (or the family of spaces each having the same
homotopy type as a CW complex). CW complexes are built in stages, eacli
stage being obtained from the preceding by adjoining cells of a given dimen-
sion. The cellular structure of such a complex bears a direct connection witt
its homotopy properties. Even for such nice spaces as polyhedra it is useful to
consider representations of them as CW complexes, because such complexes
will frequently require fewer cells than a simplicial triangulation.
In this section we shall investigate CW complexes and related concepts.
In Sec. 7.8 we shall show that any topological space can be approximated by
a CW complex which is unique up to homotopy. We begin with some results
about a space X obtained from a subspace A by adjoining n-cells (defined in
Sec. 3.8).
1 lemma If Xis obtained from A by adjoining n-cells, then X X 0 U A x I
is a strong deformation retract of X X I.
proof For each n-cell ef of X — A let
f: (En,S»-1)	(ef,ef)
be a characteristic map. Let D: (En x I) X I -> En X I be a strong deforma-
tion retraction of En X I to En x 0 U S)r 1 X I (which exists, by corollary
3.2.4). There is a well-defined map Dp (ef X I) X I ef X I characterized
by the equation
Dj((fj(2),t), T) = (fi x li)(Dj(z,t,t')) z € E”; t, t' € I
Then there is a map О': (X X I) X X I such that D’ | (е, X I) X I = Dj
and D'(a,t,tf — (a,I) for a £ A, and t, t' £ I, and D' is a strong deformation
retraction ofXxftoXxOUAxl. 
2	corollary If X is obtained from A by adjoining n-cells, then the
inclusion map А С X is a cofibration. 
3	lemma Let X be obtained from A by adjoining n-cells and let (Y,B) be
a pair such that rrn(Y,B,b) = 0 for all b QB if n > 1 and such that every
point of У can be joined to В by a path if n = 0. Then any map from (X,A)
to (Y,B) is homotopic relative to A to a map from X to B.
Ж / Й CW COMPLEXES	407
СНА»,	f its- °
>nl • ^pjtoOF This follows from theorem 7.2.1 by a technique similar to that in
|el„jnal above. »
A relative CW complex (X,A) consists of a topological space X, a closed
sjjbsp.ace A, and a sequence of closed subspaces (X,A)fc for к > 0 such that
(a)	(X,A)° is obtained from A by adjoining 0-cells.
(b)	For к > 1, (X,A)fc is obtained from (X,A)1—1 by adjoining /c-cells.
(с)	X = U (X,A)L
(d)	X has a topology coherent with {(X,A)fc]fc.
jn this case (X,A),; is called the k-skeleton of X relative to A. If X = (X,Af
{(jr some n, then we say dimension (X — A) < n. An absolute CW complex X
js a relative CW complex (X, 0), and its fc-skeleton is denoted by X,:.
Following are a number of examples.
[ 4 If (K,L) is a simplicial pair, there is a relative CW complex (|K|,|L|),
with (|K|,|L|)* = \K* U L|.
5 If (X,A) is a relative CW complex, for any к the pair (X, (X,A)fc) is a rela-
tive CW complex, with
(X,
q < к
q > к
Similarly, the pair ((X,A)A, A) is a relative CW complex, with

q < к
q Q> к
G As in example 3.8.7, for i = 1, 2, or 4 let F, be R, C, or Q, respectively,
and for q > 0 let Pe(Fj) be the corresponding projective space of dimension q
over Ft. Then Ре(Е{) is a CW complex, with
(ТК/.;](Ег)
I ад)
к < iq
к > iq
Ж* =
7 En is a CW complex, with (En)k = po for к < n — 1, (En)n 1 = S’1 x,
jand (E’l)fc = E'" for к > n.
< 8 I is a CW complex, with (I)0 = 1 and (I)1 = I for к > 1.
[ 8 If (X,A) and (Y,B) are relative CW complexes and either X or Y is locally
^compact, then (X,A) X (Y,B) is also a CW complex,1 with
•	((X,A) x (T,B))fc = Ui+j=fc (X,Af X (Y,B)i
.10 If (X,A) is a relative CW complex, so is (X,A) X I, with
j	(X X I, A X If = (X,Af X t U (X,A)fc-1 X I
1 It is not true that the product of two CW complexes is always a CW complex. For a counter-
I example, see С. H. Dowker, Topology of metric complexes, American Journal of Mathematics,
I vol. 74, pp. 555-577, 1952.
402	HOMOTOPY THEORY СЩ,, у
11 If (X,A) is a relative CW complex, then X/A is a CW complex, v'ith
(X/A)k =
A subcomplex (Y,B) of a relative CW complex (X,A) is a relative C’W
complex such that Y is a closed subset of X and (Y,B)/; = Y П (X,A)/; for all
If (Y,B) is a subcomplex of (X,A), then (X, A U Y) is a relative CW complex
with (X, Л U Y)'1 = (X,A)k U Y for all k. In particular, if X is a CW complex
and A is a subcomplex of X, then (X,A) is a relative CW complex. A CW pair
(X,A) consists of a CW complex X and subcomplex A (hence a CW pair is a
relative CW complex).
The definition of relative CW complex suggests its inductive construction.
We start with a space A, attach О-cells to A to obtain a space Ao, attach 1-cells
to Ao to obtain Ai, and continue in this way to define Ak for all к > ().
I,effing X be the space obtained by topologizing U Ak with the topology
coherent with {A/;}/c.o, then (X,A) is a relative CWcomplex, with (X,A)'‘ = AZi.
12 theorem If (X,A) is a relative CW complex, then the inclusion map
А С X is a cofibration.
proof This follows from corollary 2, using induction and the fact that X X I
has the topology coherent with {(X,A)‘ X I}it- 
13 theorem Let (X,A) be a relative CW complex, with dimension
(X - A) < n, and let (Y,B) be n-connected. Then any map from (X,A) to
(Y,B) is homotopic relative to A to a map from X to B.
proof This follows, using induction, from corollary 7.2.2, lemma 3, and
theorem 12. 
14	corollary Let (X,A) be a relative CW complex and let (Y,B) be
n-connected for all n. Then any map from (X,A) to (Y,B) is homotopic rela-
tive to A to a map from X to B.
proof Let/: (X,A)	(Y,B) be a map. It follows from theorems 12 and 13
that there is a sequence of homotopies
Efi: (X,A) X 1^ (Y,B)	k>0
constructed by induction on к such that
(a)	Ho(x,0) = fix) for x € X.
(b)	Hk(x,l) = Hk+1(x,0) for X e x.
(c)	Hk is a homotopy relative to (XrA)ft-1.
(d)	Hk((X,A)k X 1) С В.
Then a homotopy H: (X,A) X W> (Y,B) with the required properties is
defined by
’ (1/fc) - i/(fc + 1)/ k- - к + 1
15	lemma If X is obtained from A by adjoining n-cells, then for n > h
(X,A) is (n — l)-connected.
sgC. 6 CW COMPLEXES	4Q3
proof For к < n — 1 let /: (E/;,S/;l)	(X,A) be a map. Because fiEk) is
cOmpact, there exist a finite number, say, ek, ... , em, of n-cells of X — A
slicfi that f(Ek) C Ci U •  • U em U A. For 1 < i < m let x, be a point of
e — dj. Each of the sets Y = A U (e, — xy) U    U (e„, — xm) and e; — e-t
for 1 < * < m intersects f(Ek) in a set open in f(Ek). There is a simplicial
triangulation of Ek, say, K, such that (identifying |I<| with Ek) for every
simplex s € К either fi|s|) C Y or for some 1 < i < m, fi\s\) C e, — e{. Let A'
be the subpolyhedron of Ek which is the space of all simplexes s £ К such
that/(|s|) C Y, and for 1 < i < m let Bj be the subpolyhedron which is the
space of all simplexes s of К such that /(|s|) Cq- e,. Then S^-1 CA',
gfr = A' U Bi U   • U Bra, and if i =/= j, then B, — A' is disjoint from
p. — A'. Let В., = В., П A' and observe that (Bj,Bj) is a relative CW complex,
with dim (В,- — В.г) < к < n — 1.
For 1 < i < m the pair ((e{ — e{), (e{ — e,) — x{) is homeomorphic to
1 (/?' — S”-1, (E” — S'"~ l) — 0) and has the same homotopy groups as (E'",Snl)-
gy corollary 7.2.4, (En,SB-1) is (n — l)-connected. It follows from theorem 13
that f | (If,Bi) is homotopic relative to B{ to a map from B, to (e.; — efi — X;,
Because Bi — Bi is disjoint from Bj — Bj for i /, these homotopies fit
together to define a homotopy relative to A' of f to some map /' such that
। f'(Ek) C Y. Clearly, A is a strong deformation retract of Y. Therefore /' is
I homotopic relative to S/l t to a map /" such that f''(Ek) C A. Then
f ~f' ~ f", all homotopies relative to S*-1. Therefore (X,A) is n-connected.  16
16 corollary If (X,A) is a relative CW complex, then for any n > 0,
(X, (X,A)n) is n-connected.
proof We prove by induction on m that ((X,A)ffl, (X,A)n) is n-connected for
I m > n. Since (X,A)w l1 is obtained from (X,A)n by adjoining (n + l)-cells,
1 it follows from lemma 15 that ((Х,А)я+1, (X,A)n) is n-connected. Assume
I m )> n + 1 and that ((ХД)’"1, (X,A)n) is n-connected. By lemma 15, the pair
I ((X,A)m, (XjA)’”^1) is (m — l)-connected, and since n < m — 1, it is also
. n-connected. Then tt0((X,A)?!)	TT0((X,A)m-1) and rr0((X,A)m~r) —> rr0((X,A)m)
j are both surjective, whence 7T0((X,A)!t)	тт0((Х,А)т) is also surjective,
j Furthermore, for any x £ (X,A)n, it follows from the exactness of the homotopy
/sequence of the triple ((X,A)m, (X,A)m-1, (X,A)n), with base point x, that
J 7Г^((Х,А)"!, (X,A)n, x’) = 0 for 1 < к < n. By corollary 7.2.2, ((X,A)m, (X,A)”)
{is n-connected.
| To show that (X, (X,A)n) is n-connected, if 0 < к < n and a: (E^S1^1)
j (X (X,A)’1), then because a(Ek) is compact and X has a topology coherent
( with the subspaces (X,A)m, there is m > n such that <x(Ek) C (X,A)m. Hence
« can be regarded as a map from (Ek,S*-1) to ((X,A)m, (X,A)n) for some m > n.
Because ((X,A)™, (X,A)n) is n-connected, a is homotopic relative to S^-1 to
some map of Ek to (X,A)n. 
Given relative CW complexes (X,A) and (X',A'), a map /: (X,A) —» (X',A')
is said to be cellular if fi(X,A)k) C (X',A')fc for all k. Similarly, a homotopy
В (X,A) X I (X',A') is said to be cellular if F((X,A) X I)k C (X',A’)k for
4Q4	HOMOTOPY THEORY CHAp,
all k. Analogous to the simplicial-approximation theorem is the following ®
cellular-approximation theorem.	t
17 theorem Given a map f: (X,A)	(X',A') between relative CW сощ.. |
plexes which is cellular on a subcomplex (Y,B) of(X,A), there is a cellular map i
g: (X,A) —» (X',A') homotopic to f relative to Y.	}
proof It follows from corollary 16, theorem 13, and theorem 12 that there J
is a sequence of homotopies ffr: (X,A) X I —* (X',A') relative to Y, for к > Ц J
such that
(a) Ho(x,O) = /(x) for x E X.
(h)	Hft(x,l) = Hft+i(x,0) for x E X.
(c)	Hk is a homotopy relative to (Х,А)Л-1.
(d)	Нк((Х,А)* X 1) C (X',A')k.
Then a homotopy H. (X,A) X (X',A') with the desired properties is
defined by
H(x,t) - fc_i(x, _ 1/(fc + T J	k < < k + 1 a
18 corollary Any map between relative CW complexes is homotopictq
a cellular map. If two cellular maps between relative CW complexes an
homotopic, there is a cellular homotopy between them. 
A continuous map fi X Y is called an n-equivalence for n > 1 if J
induces a one-to-one correspondence between the path components of X and
of Y and if for every x E X, /#: ttq(X,x) we(Y,/(x)) is an isomorphism for
0 < q < n and an epimorphism for q = n (the condition concerning the case
q = n is sometimes omitted in the definitions occurring in the literature|
A map/: X —> Y is called a weak homotopy equivalence or co-equivalence d,
/is an n-equivalence for all n > 1. The following results are immediate from
the definition and from corollary 7.3.15.
19	A composite of inequivalences is an n-equivalence. 
20	Any map homotopic to an n-equivalence is an n-equivalence. 
21	A homotopy equivalence is a weak homotopy equivalence. 	!
Let fi. X	Y be a map and let Zf be the mapping cylinder of /. Thea I
/ = r ° i, where r: Zf Y is a homotopy equivalence. Therefore / is -an ;
n-equivalence if and only if i: X C. Zf is an n-equivalence. It follows from the* |
exactness of the homotopy sequence of (Zf,X) and from corollary 7.2.2 that I
is an n-equivalence if and only if (Zf,X) is n-connected.	I
22 theorem Let fi X Y be an n-equivalence (n finite or infinite) and,
let (P,Q) be a relative CW complex, with dim (P — Q) < n. Given maps .
g: Q X and h: P Y such that h\ Q = / ° g, there exists a map g': P-sX i
such that gz | Q = g rind f ° g h relative to Q.	j
proof Let Zf be the mapping cylinder of fi with inclusion maps i: X C Zf 
ggc, 6 CW COMPLEXES	405
arid f:	aiK’ retraction r; Zf Y a homotopy inverse of /. Then in
Q С P
cl
X -G. Zf
a homotopy i ° g ~ j ° h | Q can be found whose composite with r is constant,
py theorem 12, there is a map h':	Zf such that h' | Q — i ° g and such that
!» h' ~r°j°h relative to Q. We regard h' as a map from (P,Q) to (Zf,X).
Since (Zf,X) is n-connected and dim (P — Q) < n, it follows from theorem 13
that h' is homotopic relative to Q to some map g': P X. Then g' | Q = g and
f ° g' = r ° i ° g' ~ r ° h' ~ r ° j ° h — h
all the homotopies being relative to Q. Hence g' has the desired properties. 
2» corollary Let f: X Y be an inequivalence (n finite or infinite) and
consider the map
f#: [P;X] -a [P;Y]
If P is a CW complex of dimension < n, this map is surjective, and if
dim P < n — 1, it is injective.
proof The first part follows from theorem 22 applied to the relative
CW complex (P, 0).
For the second part, we apply theorem 22 to the relative CW complex
(P X I, P X I)- Given g0, gi: P —» X such that / ° g0 ~ / ° gi, there is a map
g: P X I X such that g(z,O) = g0(~) and g(z,l) = gfiz) for z E P and a map
h; P X I X such that h | P X 1 = g. Since dim (P X I) < n, by theorem 22
there is a mapping g': P X 1A> X such that g' | P X I — g- Then g' is a
homotopy from g0 to gi, showing that [go] = [gi]. 
24 corollary A map between CW complexes is a weak homotopy equiv-
alence if and only if it is a homotopy equivalence.
proof It follows from statement 21 that a map which is a homotopy equiv-
alence is always a weak homotopy equivalence. Conversely, if fi X Y is a
weak homotopy equivalence between CW complexes, it follows from corol-
lary 23 that / induces bijections
/#: [Y;X] -> [Y;Y]	/#: [X;X] -a [X;Y]
If g: Y A is any map such that fifg] = [ly], then / ° g ~ ly, and also
° f ] = If ° g ° f ] = [lr ° f ] = [/ ° lx] = /#[lx]
Therefore [g ° /] = [lx] or g ° / ~ 1A, and so / is a homotopy equivalence. 
Thus, for C W complexes the concepts of homotopy equivalence and weak
homotopy equivalence coincide. The following theorem is a direct consequence
pf the Whitehead theorem 7.5.9.
406	HOMOTOPY THEORY CHAP 7	HOMOTOPY FUNCTORS	4(ff
25 theorem A weak homotopy equivalence induces isomorphisms of the	Y?; —> V Y„ and [V 1„]: V Уя-> V Yn
corresponding integral singular homology groups. Conversely, a map between	fBooF Since jn+1 ° in = jn - ln, it follows that {jn} ° V in = {/.„} ° V 1„.
simply connected spaces which induces	isomorphisms	of the	corresponding 1 {:jvcii a map	f:	V Y„^ Z' such that f ° V in ~ f ° V 1„, let fn: Y„ -» Z' be
integral singular homology groups is a weak homotopy equivalence.		j jefined by fn	=	/' | Yn. Then j'l+1 - in ~ fn, and using the fact that Y„ C Y„+1
1 js a cofibration and by induction on n, there is a sequence of maps g„: Yn Z'
} such that gn	~	fn and g?l+1 ° in = gn. Let g: Y Z' be the map such that
7 homotopy functors	|	g I = "^hen g ° 7 — f completing the proof. 
In this section we shall study a general class of functors on the homotopy,
category of path-connected pointed spaces. The main result characterizes, on
the subcategory of CW complexes, those functors of the form ttv for some Y
in terms of simple properties. In the next section we shall apply this result to
prove the existence of approximations to any space by a CW complex.1
In a category S, given objects A and X and morphisms f ,: A X and
A —» X, an equalizer of f0 ancl/i is a morphism j: X Z such that
(o) j ° fo = j ° fi-
fty If f: X Z' is a morphism in G such that f ° f о = f ° fi, there is a
morphism g: Z —» Z' such that f = g ° j.
Note that it is not asserted in condition (ty that g is unique.
We define to to be the homotopy category of path-connected pointed
spaces having nondegenerate base points.
1 lemma The category Cb has equalizers.
proof Let A and X be arbitrary objects of Co and let /о: A X
and/i: A X be maps preserving base points. Let Z be the space obtained
from the topological sum Xv (A X I) by identifying («,()) 6 A X I with
fo(o) € X, (o,l) E A X I with fi(d) € X for all a E A, and (o0,t) E A X I with
(oo,0) (oo the base point of A) for all t E I- Then Z is an object of tb and the
inclusion map /: X C Z has the property that / 0 fo ~ / 0 fi [in fact, the com-
posite AxlCXv (Ax I) -> Z is a homotopy from j ° f0 to j ° fi].
Furthermore, if f: X —> Z' is a map such that f ° fo — f ° fi, there is a map
G: X v (A X I) -» Z' such that G | X = f and G | A X I is a homotopy
from f ° fo to f 0 fi. Then G is compatible with the collapsing map
к: X v (A X I) Z, so there is a map g: Z —» Z' such that G = g 0 k. Then
f = go j, and therefore [/]: X —> Z is an equalizer of [/0] and [/1] in (?o- 
2 lemma Let {Y,(},(>o be objects of Go that are subspaces of a space Y int
such that Yr C Yn+1 is a co fibration for all n > 0, Y = U„ Yn, and Y has
the topology coherent with {Yn}. Let in: Y„ C Yn+i, 1и: Yn C Y,„ and
jn: Yn C Y be the inclusion maps. Then the homotopy class [{/я}]: V Yn Y
is an equalizer in Go of the homotopy classes
JThe techniques of this section are based on E. Brown, Cohomology theories, Annals of ' .
Mathematics, vol. 75, pp. 467-484,1962.	i ls ° ёгоиР multiplicaton on H(SXf which is abelian if Xis a suspension. IfH
| A homotopy functor is a contravariant functor H from tb to the category
V of pointed sets such that both of the following hold:
(a)	If [ / ]: X —» Z is an equalizer of [/0], [/i]: A X and if и G H(X) is
such that H([fo])u = H([ff])u, there is о £ H(Z) such that H(\j]')v = u.
(ty If is an indexed family of objects in cb and iK: X> С V Xx,
there is an equivalence
{Ж]К= H( V Xx) ~ X ВД
If f: X Y is a base-point-preserving map and H is a homotopy functor,
we shall also use H(f) for H([/]). If X С X' and и £ H(X'), we use и | X for
H(i)u, where i: X С X'.
If X is a one-point space and Xi, and X2 are both equal to X, then
Xj v X2 is also equal to X, and the equivalence of condition (b)
{Hli^Hlif)}: H(Xi v X2) H(Xi) X H(X2)
corresponds to the diagonal map of H(X) to H(X) X H(X~). Because this is a
bijection, H(X) consists of a single element.
Following are some examples.
( 3 Let Y be a pointed space. Then the functor тту on Co defined as in
 Sec. 1.3 (that is, wy(X) — [X; Y] for an object X in Eb) is a homotopy functor.
I 4 Fix an integer n ~^> 0 and an abelian group G. Then the functor
i H(X) = Hn(X,xo; G) (singular cohomology) on Gq is a homotopy functor called
j the nth cohomology functor with coefficients G.
I 5 Let G be an arbitrary group (possibly nonabelian). There is a homotopy
[ functor H such that H(X) is the set of all homomorphisms tti(X,xo) G with
the trivial homomorphism as base point.
An important result of this section is that on the subcategory of pointed
f path-connected CW complexes every homotopy functor is naturally equivalent
у to tty for a suitable pointed space Y.
j 6 lemma Let v: SX —> SX v SX be the comultiplication map. If X is in Gq
J and H is a homotopy functor, the composite
!	H(SX) X H(SX)	> H(SX V SX) H(SX)
408	HOMOTOPY THEORY СНДр.
is a homotopy f unctor taking values in the category of groups, the two group ?
structures on II (SX) agree.	$
f
proof Each of the group properties for this multiplication follows from the j
corresponding H cogroup property of v. The final statement of the lemma *
follows from theorem 1.6.8, because the two multiplications in H(SX) are '
mutually distributive. 	i
In particular, for any homotopy functor H, H(So) is a group for q > ]
and abelian for q > 2 and is called the qth coefficient group of H. Thus the |
qth coefficient group of the functor tty of example 3 is rrq(Y). The qth coeffi; [
cient group of the nth cohomology functor with coefficients G of example 4 J
is 0 if q 7^ n and isomorphic to G if q = n. The qth coefficient group of the I
functor of example 5 is G if q = 1 and 0 if q > 1.	i
If Y is an object of (?o and H is a homotopy functor, any element
и € H(Y) determines a natural transformation	j
defined by T„([f]) = H([f])(«) for [/] E [X;Y]. For a suspension SX, Tu is a j
homomorphism from 7>'r(SX) = [SX;Y] to the group H(SX), with the multiple j
cation of lemma 6 (because both group multiplications are induced by the ;
comultiplication v: SX SX v SX). An element и £ H(Y) is said to be I
n-universal for H, where n > 1, if the homomorphism
If. 7-fiXj H(S«)
is an isomorphism for 1 < q < n and an epimorphism for q = n. An element
и E H( Y) is said to be universal for H if it is n-universal for all n > 1, in which
case Y is called a classifying space for H.
7 theorem Assume that H is a homotopy functor with universal elements |
и E H(Y) and и' E H(Y') and letf Y Y' be a map such that H(f)u' = u.
Then f is a weak homotopy equivalence.
proof Since Y and Y' are path connected, this is a consequence of the com- I
mutativity of the diagram (for q > 1)	I
i
[S«;Y] A [Se;Y']	-
i
H(Se)		|
The same kind of argument establishes the next result.	।
8 lemma Let Y be an object of eb and let Y' be an arbitrary path
connected space. A map f: Y Y' is a weak homotopy equivalence if and
only if[f] E [Y; Y'] = wr(Y) is universal for ттХ 
We are heading toward a proof of the existence of universal elements for .
any homotopy functor. The following two lemmas will be used in this proof. I
SEC. 7 HOMOTOPY FUNCTORS	4()<)
p lemma Let H be a homotopy functor, Y an object in (?0, and и € H(Y).
There exist an object Y' in c’o, obtained from Y by attaching 1-cells, and a
^.universal element и' E H(Y') such that u' | Y = u.
proof For each A E ^/(S1) let SJ be a 1-sphere and define Y' = Yv Vx S^1.
Then Y' is an object of Йо obtained from Y by attaching 1-cells. If gx is the
composite S1 Sx' C Y', it follows from condition (b) on page 407 that there
js an element и’ E H(Y') such that u' | Y = и and H(gf)u' = A for A E ^(S1).
Since МЫ = A, ^.([S1; Y]) = kfiS1), and u' is 1-universal. 
Ю lemma Let H be a homotopy functor and и E H(Y) an n-universal
element for H, with n > 1. There exist an object Y' in Co, obtained from Y by
attaching (n + Ifcells, and an (n + l)-universal element u' € H(Y') such
that и' | Y = u.
proof For each A E H(Sn+1) let Sxn+1 be an (n + l)-sphere, and for each map
a: Sn —> Y such that H(a)ti = 0 attach an (n + l)-cell ean+1 to Y by a. Let Y'
be the space obtained from Y v Vx Sx«+1 by attaching the (n + l)-cells {ean+1}.
Then Y' js an object of (?0 obtained from Y by attaching (n + l)-cells.
If g>- Sn+1 —> Y v \/\ SX',J1 is the composite S’l+1 Sxn+1 C Yv Vx SX"H, it
follows from condition (b) on page 407 that there is an element
it E H(Y v Vx S\"H) such that й | Y = и and H(gx)fi = A for A E H(S,i+1).
For each map a: S” Y such that H(a)u = 0 let San be an n-sphere
and let fo: Vtt Sa” Y v VK Sxn+1 be the constant map and let Д: Va San
Y v VK SKn+1 be the map such that San is mapped by a. Then
Yv VxSx"+i c Y'
is a map such that [ / ] is an equalizer of [/o] and [/i]. Since H(/0)f; = 0 =
H(fi)fi, by condition (a) on page 407 there is an element и' E H(Y') such that
H(f)u' = й. Then u' | Y = и and to complete the proof we need only show
that u' is (n + l)-universal.
There is a commutative diagram
TTe+1(Y',Y) Д TTe(Y)	TTe(Y')	,re(Y',Y)
A	A
H(S«)
with the top row exact. Since Y' is obtained from Y by attaching (n + l)-cells,
it follows from lemma 7.6.15 that ttq(Y', Y) = 0 for q < n. Therefore i# is an
Isomorphism for q < n and an epimorphism for q = n. Since и is n-universal,
Tu is an isomorphism for q < n and an epimorphism for q = n. It follows
that TU' is also an isomorphism for q < n and an epimorphism for q = n.
Furthermore, if a E [Sn; Y] is in the kernel of Tu, then a is represented by a
map a: S’1 Y and
a = [a] E 3(wn+i(ean+1,ea«+1)) C 3(t7„+i(Y',Y)) = ker i#
410	HOMOTOPY THEORY CHAP. 7
Therefore, for q = n, ker Tu = ker i#, and so TU’ is an isomorphism from
77„(Y') to H(S«). For any A E H(SM+1) the map j ° gx: S’1+1 —» Y' has the
property that
Tu'(ii ° gJ) = H(&ya = x
showing that TU’ is an epimorphism for q = n + 1, and so u' is (n + ]).
universal. 
11 theorem Let H be a homotopy functor, let Y be an object in Gf, and
let и E H(Y). Then there are a classifying space Y' for H containing Y such
that (Y',Y) is a relative CWcomplex and a universal element u' £ H(Y') such
that u' | Y = u,
proof Using lemmas 9 and 10, we construct, by induction on n, a sequence
of objects {Yn}n>o in Co and elements un £ H(Yn) such that
(</) Yo = Y and t/0 = u.
(b)	Y,f |.i is obtained from Yn by attaching (n + l)-cells, where n > 0.
(c)	t/n+i | Yn = un,
(d)	un is n-universal for n > 1.
It follows from (b) above that Y' = U Yn topologized with the topology
coherent with {Y„} is a path-connected pointed space containing Y such that
(Y',Y) is a relative CW complex. By lemma 2, the homotopy class
[{/,,}]: V Yn Y' is an equalizer of the homotopy classes [V i„]: V Yn V Yn
and [V !„]: V Y-n^> V Yn. By condition (b) on page 407 there is an element
й E H(V Yn) such that й | Y„ = un. It follows from (c) above that H(V i„)6 —
H( V 1„)й, and by condition (o) on page 407 there is an element и' E H(Y*)
such that	= й (that is, u‘ | Y„ = un for n > 0). Then u’ | Y = u, and
it remains to show that u’ is universal.
By the definition of Y' and u', there is a commutative diagram for q > 1
lim., Og(Yn)} 77q(Y')
H(Sa)
Since un is n-universal, Tu„ is an isomorphism for n > q, and so the left-hand
map is an isomorphism. Therefore TU' is also an isomorphism, and u' is
universal. 
12	corollary For any homotopy functor there exist classifying spaces
which are CW complexes.
proof Apply theorem 11 to a one-point space Y, with и the unique element
of H(Y). 
13	corollary Let и E H(Y) be a universal element for a homotopy
functor H. Let (X’,A) be a relative CW complex, where A and X are objects
V SEC- 1 HOMOTOPY FUNCTORS	4Ц
9;
I (-n Qo. Given a map g: A	Y and an element v € H(X) such that v | A = H(g)u,
ihere exists a map g': X —> Y such that g = g' | A and v = H(g’)u.
| proof Let i: X С X v Y and Г: Y С X v Y and let j: X v Y Z be a map
I such that [ / ] is an equalizer of [i ° /] (where f: А С X) and [i' ° g], By condi-
i tion (b) on Page 407, there is an element v £ H(X v Y) such that v | X = v
! gud v | Y = Ч- Since H(f)v = H(if)u, it follows that H(i ° f)v — H(f ° g)c,
t and by condition (a) on page 407, there is an element й £ H(Z) such that
| Н(/)й = V- We now apply theorem 11 to й to obtain a Y' containing Z and a
’ ulljversal element u' C. H(Y') such that й = и' | Z. Let YY' be the
I composite
I	i' : h
I	YCXv z c y
I Then H(f)u' = u, and by theorem 7, j' is a weak homotopy equivalence,
j Since the composite
I	ACXCXvY^ZCY'
is homotopic to f ° g, it follows from the fact that f is a cofibration that there
is a map g: X —> Y' such that g | A = f ° g and g is homotopic to h ° j ° i.
Since /' is a weak homotopy equivalence, by theorem 7.6.22, there is a map
| g': X Y such that g' | A = g and f ° g' ~ g. Then
H(g')u = H(g)H(j')u' = H(i)H(f)H(h)u' = v | X = v
showing that g' has the requisite properties. 
14	theorem If Y is a classifying space and и £ H(Y) is a universal
element for a homotopy functor H, then for any CW complex X in Co, Tu is a
f natural equivalence of тту(Х) with H(X).
proof Given v £ H(X), apply corollary 13, with A — xo and g the constant
i map, to obtain a map g': X Y such that H(gf)u = v. Then Tu[g'J = v,
I proving that Tu is surjective.
! If go, gi: X Y are maps such that Tlt[go] = Tit[gi], let X' be the CW
I complex X X I/xo X I, with (X')e = [(X« X I) U (Хе-1 X l)]/(xo X I) for
| q > 0. Let v C. H(X') be defined by c = H(h)H(go)w, where h: X' X is the
: map Zi([x,t]) = x. Let A = X X 1/xo X 1 and let g: A Y be the map such
! that g([x,0]) — go(x) and g([x,l]) = gi(x). Then H(g)rz = v | A, and by corol-
i lary 13, there is a map g': X' Y such that g' | A = g. Then the composite
’	Xxl^ XxI/x0Xl^Y
is a homotopy relative to xo from go to gi, showing that Tu is injective. 
15	corollary If Y and Y' are classifying spaces which are CW complexes
! and и £ H(Y) and u' £ H(Y') are universal elements for a homotopy functor H,
there is a homotopy equivalence h: Y Y', unique up to homotopy, such
that H(h)u' = u.
412
HOMOTOPY THEORY СНЛр у
proof By theorem 14, there exists a unique homotopy class [g]:	y,
such that H(g)u' = u. By theorem 7, g is a weak homotopy equivalence В ,
corollary 7.6.24, g is a homotopy equivalence. 	}
SB WEAK HOMOTOPY TYPE
In this section we shall show that any space can be approximated by CW
complexes. This leads to an equivalence relation based on weak homotopy
equivalence which is weaker than homotopy equivalence. We shall also con-
sider the same equivalence relation in the categoiy of maps. This will be used
in defining and analyzing the general relative-lifting problem.
A relative CW approximation to a pair (X,A) consists of a relative CW
complex (Y,A) and a weak homotopy equivalence/: Y X such that f(a) = ц-
for all a G A. A CW approximation to a space X is a relative CW approxima-
tion to (X, 0).
1	theorem Any pair has relative CW approximations, and two relative
CW approximations to the same pair have the same homotopy type.
proof First we consider the case where X is path connected. Let x0 G X
and let be the set of path components of A, and for each /GJ choose
a point a-j € Aj. There is a relative CW complex (A',A) with (A',A)0 = A U e°,
where e° is a single point and
A' = (A',A)i = (A',A)0 U U e/
A J
where e/ is a 1-cell such that ё/ = e° U aj for / G J- Let g: A’ X be a map
such that g(a) = a for a G A, g(e°) = xo, and g | e/ is a path in X with end
points xo and aj for each / G J. Then A' is a path-connected space with non-
degenerate base point e° and [g] G irx(A'). It follows from theorem 7.7.11
that there is a relative CW complex (Y,A') [which can be chosen such that
(YjA')1 = A' v V Sa1] and a universal element [g'] G 7>'A(Y) for rrx such that
g' | A' ~ g. Since A' C Y is a cofibration, there is a map /: Y —> X such that
[/] G rrx(Y) is universal for тгх and /1 A’ = g. By lemma 7.7.8, / is a weak
homotopy equivalence. Since (Y,A) is a relative CW complex [with (Y,A)° =
(A',A)0 and (Y,A)« = (Y,A')Q for q > 1] and since/(a) = a for a G A, (Y,A)
and / constitute a relative CW approximation to (X,A).
Next we consider the case where X is not path connected and we let
{Xa} be the set of path components of X. By the case already considered, for
each a there is a relative CW approximation fa: (Ytt, Xa П A) (Xa, Xa П A).
Let Y be the space obtained from the disjoint union A U U У„ by identifying
x G XQ П A C Y„ with x G A for each a and let k: A U U Ya —> Y be the
collapsing map. Then к | A: A Y is an imbedding and (Y,A) is a relative
CW complex with (Y,A)« = k(A U U (Ya, Xa П A)e) for all q > 0. There is a
map /: Y X such that fk(a) = a for a G A and / ° (k | Ya) = fa for all a.
" SfiC- 8 WEAK HOMOTOPY TYPE	413
Sine6 {к(Ти)} is se^ °f path components of Y and f induces a weak
Yiomotopy equivalence of each of these with the corresponding path compo-
nent Xa of X, fis a weak homotopy equivalence from Y to X. Identifying A
with k(A), we see that (Y,A) and f constitute a CW approximation to (X,A).
Given two relative CW approximations to (X,A), say Д: (Yi,A) (X,A)
and/2: (Y2,A)	(X,A), it follows from theorem 7.6.22 that there are maps
:gi= (b,A) —> (T2,A) and g2: (Y2,A) (Yi,A) such that f2 ° gi ~/i and
д ° g2 ~ /2, both homotopies relative to A. Then J2 ° (gi ° g2) ~ /2 ° 1 rel A,
and by theorem 7.6.22 again, gi ° g2 ~ 1 rel A. Similarly, gg ° gi ~ 1 rel A,
and so (Yr,A) and (Y2,A) have the same homotopy type. 
Two spaces Xi and X2 will be said to have the same weak homotopy type
if there exists a space Y and weak homotopy equivalences Д: Y —» Xj and
fa Y X2. By replacing such a space Y with a CW approximation to it, we
see that Xi and X2 have the same weak homotopy type if and only if they
have CW approximations by the same CW complex.
2	lemma The relation of having the same weak homotopy type is an
equivalence relation.
proof The relation is reflexive and symmetric by its definition. To prove it
transitive, let Xi, X2, and Хз be spaces and let Yi and Y2 be CW complexes
such that there exist weak homotopy equivalences
Yi	Y2
V й/
Xi X2 Хз
Then/2: Yi X2 and g2: Y2 X2 are both CW approximations to X2, and
by theorem 1, there is a homotopy equivalence h: Yj Y2 such that
fz — g2 ° h. Then g3 ° h: Yi —» Хз, being the composite of weak homotopy
equivalences, is a weak homotopy equivalence. Therefore Xi and Хз have the
same weak homotopy type. 
We are interested in applying these ideas to weak fibrations. The main
result is that any two fibers of a weak fibration with path-connected base
space have the same weak homotopy type.
3	lemma Let p-. E В be a weak fibration with contractible base space B.
For any bo £ В the inclusion map i: p~1(bo) С E is a weak homotopy
equivalence.
proof Let F = p-fb(>). Since В is contractible,	= 6 for q > 0.
From the exactness of the homotopy sequence of p, it follows that for any
e £ F, i induces an isomorphism i#: ttq(F,r) ~ TTq(E,e) for q > 1 and
i#(v7o(F,e)) = w0(E,<?).
It only remains to verify that i# maps тт0(Е,е) injectively into тт0(Е,е).
Assume that e, e' £ F are such that there is a path c in E from e to e'.
Since В is simply connected and p ° w is a closed path in В at bo, there is a map
414	HOMOTOPY THEORY CHAP, 7
H: I X I В such that H(t,0) = pw(t) and H(0,t') = H(l,f) = H(t,l) ~
Let g: I X 0 U t X I —> E be the map defined by g(t,O) = w(t), g(O,t') — (>
and g(l,f') = e'  By lemma 7.2.5, there is a map G: I X I E such that
p ° G = H and G|IxOUIxI = g. Let co': I —> E be the path defined by
co'(t) = G(l,t). Then co' is a path in F from e to e' [because pcfat) — Ь()]
showing that fa: 7T0(F,e) тт0(Е,е) is injective. 
4	corollary Let p: E В be a weak fibration and let co be a path in B,
Then p-1(w(0)) and р~х(со(1)) have the same weak homotopy type.
proof Let p': E' —» I be the weak fibration induced from p by co: I —> j
Then p-1(co(O)) and p-1(co(l)) are homeomorphic to p'-1(0) and
respectively. By lemma 3, each of the inclusion maps p'"1(0) С E' and
p'^1(l) С E' is a weak homotopy equivalence. The corollary follows from this
and lemma 2. 
This result implies the following analogue of corollary 2.8.13 for weak
fibrations.
5	corollary If p: E В is a weak fibration with path-connected base
space, any two fibers have the same weak homotopy type. 
We now consider the categoiy whose objects are continuous maps
a: P" P between topological spaces and whose morphisms (also called
map pairs') f: a —> ft are commutative squares
P" Lfa Q"
4
P' Q'
In this category a homotopy pair H: fa ~fa, where fa, fa: a ft, is a com-
mutative square
P" X I	Q"
a x 4
P' x I	Q'
such that H":f'6 ~ f'{ and H': f'o c^f) (note that H is a map pair from
a X 1/ to ft). If such a homotopy pair exists, fa is said to be homotopic to fa.
This is an equivalence relation in the set of map pairs from a to ft, and the
corresponding equivalence classes are called homotopy classes. We use [a;/?]
to denote the set of homotopy classes of map pairs from a to ft, and if
f: a ft is a map pair, its homotopy class is denoted by [/]. It is trivial to
verify that the composites of homotopic map pairs are homotopic, so there is
a homotopy category of maps whose objects are maps a: P" P' and whose
morphisms a ft are homotopy classes [/], where f: a ft is a map pair,
A map pair f: a —> ft is called a homotopy equivalence from a to ft if [/] is
an equivalence in the homotopy category of maps. Two maps a and ft are
SgC. 8 WEAK HOMOTOPY TYPE	4/5
said to have the same homotopy type if they are equivalent in the homotopy
category of maps.
Given a map pair g: a' —> a (or a map pair h: ft ft') there is an induced
map g#: [«;/?] -> [«';Z3] (or h#: [a;/?] -» [a;/?']) such that g#[/] = [/ ° g] (or
h#[f] — [h ° /]). Since g# ° h# = h# ° g#, the function which assigns [a;jB] to
d and ft and g# and h# to [g] and [h], respectively, is a functor of two
variables from the product of the homotopy category of maps by itself to the
category of sets that is contravariant in a and covariant in ft.
If a: P" —> P' and ft: Q" Q' are maps, given a map f: P' Q", there
js a map pair p(f): a —> ft consisting of the commutative square
P"	Q"
P	Q'
[that is, (p(/))" = / ° a and (p(/))' = ft ° /]. Given a map pair/: a ft, a
lifting of / is a map f: P Q" such that p(/) = /. Two liftings fo, fr- P Q"
off: a —> ft are homotopic relative to f if there is a homotopy H: P X I—> Q"
from fo to /1 such that H ° (a X 1/) and ft ° H are both constant homotopies
[that is, p(H) is the constant homotopy pair from f to /]. Such a map H is
called a homotopy relative to f, and we write H: fo ~ /i rel /. Homotopy
relative to / is an equivalence relation in the set of liftings of / and the set of
equivalence classes is denoted by [P';Q"[/- The relative-lifting problem is the
study of [P';Q"]/ (for example, do liftings of / exist, and if so, how many
homotopy classes relative to / of liftings of / are there?)
6 example If P” is empty, then a map pair f: a ft consists of a map
/': P Q', and a lifting/: P' Q” of /is a lifting of/' to Q" in the sense
defined in Sec. 2.2. In this case, if ft is a fibration, two liftings f0, fi: P' Q"
of/' are homotopic relative to /if and only if they are fiber homotopic in the
sense of Sec. 2.8. Thus the absolute-lifting problem is a special case of a
relative-lifting problem.
7 example If a is an inclusion map and Q' is a one-point space, then a
map pair /: a ft corresponds bijectively to a map /": P" Q" and a
lifting /: P' —> Q" of / corresponds bijectively to an extension of /" to P. In
this case two extensions fo, fi: P' Q" are homotopic relative to / (as liftings)
if and only if they are homotopic relative to P". Thus the extension problem
is a special case of a relative-lifting problem.
8 example Let fo, fr. P Q" be liftings of a map pair /: a ft. Let
R' = P X I and let R" be the quotient space of the disjoint union of P' X I
andP" X I by the identifications (z",0) £ P" X I equals (a(z"),0) € P' X land
(z",l) E P" X I equals (a(z"),l). Define a map y: R" R' by y(z",t) = (a(z"),t)
for (z",t) £ P" X I and y(z',t) = (z',t) for (z',t) £ P X t- There is a map pair
g: у ft consisting of the maps g": R" Q" and g': R’ —» G' such that
416	HOMOTOPY THEORY CHAP. 7	’
g"(*"/) = /"(s") for (z",t) e P" X I, g"(X,0) = fo(z') and g"(z',l) = /Д/) foP
z' G F, and g'(z',t) = for (z',t) £ F X I. Then /0 and Д are homotopic;
relative to f if and only if there exists a lifting of g.
We are particularly interested in the relative-lifting problem in case a is
the inclusion map of a relative CW complex and f> is a weak fibration. Thus
if i: А С X is an inclusion map and p: E В is a weak fibration, a map pair
f:i^>p consists of a map /': X В and a lifting /": A E of f' | д
A lifting/of/is a lifting of/' to E, which is an extension of/". Two liftings
of / are homotopic relative to / if and only if there is a fiber homotopy rela-
tive to A between them. The following relative homotopy extension theorem
is the main reason for giving particular attention to this case.
9	theorem Let (X,A) be a relative CW complex, with inclusion map
i: А С X, and let p: E В be a weak fibration. Given a map f: X E and
a homotopy pair EE: i X lr —> p consisting of a homotopy H': X X 1 —> В
starting at p ° / and a homotopy El": A X 1 E starting at f ° i, there is a
homotopy H: X X J -> F. starting at f such that IT = p ° H and
H" = H ° (i X lr).
proof Let g: X X 0 U A X I E be the map defined by g(x,0) = fix) for
x С X and g(a,t) = H"(a,t) for a E A and t E I- Then EE' is an extension of
p ° g, and by the standard stepwise-extension procedure over the successive
skeleta of (X,A) (applied to polyhedral pairs in the prooLof theorem 7.2.6 and
equally applicable to any relative CW complex), there is a map H: X X I E
such that p ° H — El" and H | X X 0 U A X i = g- Then H has the desired
properties. 
Let us reinterpret this last result. A map pair /: i —> p is a commutative
square
A	E
4	[p
X	В
Therefore, if we let Bv X' EA denote the fibered product of the map Bx BA
induced by restriction and the map EA BA induced by p, the pair
is a point of Bx x' EA. In this way the set of map pans /: i p is identified
with the fibered product Bx X' EA. The map p corresponds to a map
p: Ex Bx x' EA, and [X;E]^ is the set of path components of p-1(/).
10	corollary Let (X,A) be a relative CW complex, with Inclusion map
i: А С X, and let p: E В be a weak fibration. Then p: Ex —> Bx x' EA is a
weak fibration.
proof Given a map g: Iя Ex and a homotopy El: In x I Bx x' EA, the
exponential correspondence assigns to g a map g: X X Iя E and to El a
homotopy pair Hi from (i X 1г») X L to p, starting with p(g). By theorem 9,
WEAK HOMOTOPY TYPE
417
there is a homotopy Hi: X X In X I E starting with g such that p(Hi) = Hi.
Then the exponential correspondence associates to Hi a map G: Iй x I —» Ex
starting with g such that p ° G = H 
It follows from corollaries 10 and 4 that if f0, fa i p are homotopic
jnap pairs, then [X;E]/-0 and [X;7C]/, are in one-to-one correspondence. Thus
the relative-lifting problem for f0 is equivalent to the relative-lifting problem
for fi-
Given weak fibrations pf. Ej Bt and p2: E2 B2, a map pair
g: pi Pz is called a weak homotopy equivalence if g": 7 '4 E2 and
g': Bi —> B2 are weak homotopy equivalences. We shall show that a weak
homotopy equivalence in the category of maps has much the same properties as
a weak homotopy equivalence in the category of spaces. The following ana-
logue of theorem 7.6.22 is our starting point.
11	lemma Let (X,A) be a relative CW complex, with inclusion map
i: А С X, and let g: pt p2 be a weak homotopy equivalence between weak
Jibrations. Given a map pair f: i pt and a lifting h: X E2 of the map
pair g ° f there is a lifting f: X Ef of f such that g" ° f and h are homo-
topic relative to g ° f.
proof The proof involves two applications of theorem 7.6.22 and then two
applications of theorem 9. We shall not make specific reference to these
when they are invoked.
We have a commutative diagram
д Д E, 4 £2
4 N- >
x Д Bi 4 b2
in which g" and g' are weak homotopy equivalences, and we are given a map
h; X^> E2 such that h ° i = g" ° f" and p2 ° h = g' ° f. Then there is a
map f: X E\ such that f ° i = f" and a homotopy G": g" ° f ~ h rel A.
The maps pi ° f and/' agree on A and p2 ° G" is a homotopy relative to A
from g' ° pi ° f = p2 ° g" ° /to g' ° f — p2 ° h. Therefore there is a homotopy
F': pt ° f ~f' rel A and a homotopy H': g7 ° F' ~ p2 ° G" rel A X I U X X I.
Let F": X X I Ft be a lifting of F' such that F"(x,0) = /(x) for x E X
and F"(a,t) = f"(a) for a E A and t £ I. Define / X Ei by /(x) = F"(x,l).
We show that / has the desired properties. It is clearly a lifting of /.
The maps g" ° F" and G" are homotopies relative to A from g" ° / to
g" ° / and to h, respectively, and H' is a homotopy from p2 ° g" ° F" to p2 ° G"
rel A X I U X X 1- Since there is a homeomorphism of (X X 7 X h A X I X f)
Onto itself taking X X (1 X 1 L 0 X 7) onto X X I X 0, there is a lifting H"
of H' which is a homotopy from g" ° F" to G" rel X X 0 U A X I- Then
the map H: X X I —> E2 defined by H(x,t) = H"(x,l,t) is a homotopy from
g" ° f to h relative to f 
This gives us the following important result.
41 fj	HOMOTOPY THEORY СНдр. у
12 theorem Let (X,A) be a relative CW complex, with inclusion map
i:ACX, and let g: pi pz be a weak homotopy equivalence between weak
fibrations. Given a map pair fi. i pi, the map pair g induces a bijection
gj: [X;Ei]f^ [X;E2]e,f
proof The fact that g# is surjective follows immediately from lemma Ц,
The fact that g" is injective follows from application of lemma 11 to the rela-
tive CW complex (X,A) X (U). 
EXERCISES
A EXACTNESS OF HOMOTOPY SETS
I Assume that j: (X',A') C (X,A) is a cofibration, where A and X' are closed subsets of
X and A' = А П X'. Prove that the collapsing map
(Q,Cr) —> (C/,Cr)/CX' = (X,A)/X' = (X/X', A/A')
is a homotopy equivalence.
2 With the same hypotheses as in exercise 1, let g': (XA) C(X',A') be any map such
that g'(x') = x' for x' £ X' and let g: (X/X',A/A') -> S(X',A') be the map such that the
following square is commutative, where k' and k" are the collapsing maps:
(ХД) 4 C(X',A')
'4	I*"
(X/X',A/A') 4 S(X',A')
Prove that there is a coexact sequence
(X'A') ->-----> S«(X',A')	$»(Х,Л) 44 S"(X/X', A/A') 44 ...
3 If (XA) is a relative CW complex, prove that there is a coexact sequence
A C X-> X/A SA C SX----------------> S»A C S«X-^ • • •
II HOMOTOPY GROUPS
1	If A is a retract of X, prove that there is an isomorphism
wn(X,xo) ~ rrn(A,xo) © w„(X4,xo) n > 2
2	If X is deformable into A relative to x0 £ A, prove that there is an isomorphism
w,t(A,x0) ~ w„(X,x0) © irn+i(XA,xo) n > 2
3	If p: E —> В is a weak fibration such that the fiber F = p-1(bo) is contractible in E f
relative to e0 € F, prove that there is an isomorphism
7T„(B,bo) ~ rrn(E,eo) © 7T„_i(F,eo) n > 2	'
4	If p: E —> В is a weak fibration which admits a section, prove that there is an iso-
morphism for eo £ F = p^fibo)
irn(E,eo) ~ <nn(B,b0) © 77„(F,e0)	n>2	i
I- EXERCISES
5 Let {Xj} be an indexed family of spaces with base points x, £ Xj. Prove that there is
an isomorphism
J	7T„(X Xj,(Xj)) ~ X 1Tn(Xj,Xj) n > 0
6 Given X v Y = X x ift- U го X I С X X к prove that there is an isomorphism
,	rrn(X\ Y, (xo,.//(>)) ~ 7T„(X,x0) © w„(Y,y0) © wn+i(X x Y, Xv Y, (x0,y0)}
5 C BASE POINTS1
j 1 Give an example of a degenerate base point.
j 2 If X and Y have nondegenerate base points, prove that also Xv Y, X x Y, and
t X X Y/Xv Yhave nondegenerate base points.
! 3 If (X,xq) and (Y,y0) have the same homotopy type, prove that xo is a nondegenerate
' base point of X if and only if y0 is a nondegenerate base point of Y.
t 4 Prove that any space has the same homotopy type as some space with a nondegen-
erate base point.
r
15 Let X and Y be path-connected spaces with nondegenerate base points x(1 and y0,
respectively. Prove that X and Y have the same homotopy type if and only if (X,xq) and
(Y,(/o) have the same homotopy type.
D THE WHITEHEAD PRODUCT
| Let p > 1 and q > 1 and let h: (Ip+iJp+q) (Ip,Ip) x (I9,!9) be the homeomorphism
I h(h, . . . ,tp+q) = ((ti, . . . ,tp),(tp+i, . . . ,tp+Q)). Then h determines an element
। И € Wi<X(h'>i7') X (I9,!9), (0,0)) and an element
Vp.q = Э[Л] £ iTp+Q_i(lP x I" U Ip x I9, (0,0))
Given maps a: (Ip,Ip) -> (X,x0) and fi: (I9j9)	(X,x0), define a map
y: (Ip y i9 C Ir x I9, (0,0))	(X,x0) by
- f4Z') e ip x 19
1 -1ДХ) z e Ip, (z,z’) e IP x 19
I Prove that y#(ijM) € wp+Q_1(X,xo) depends only on [a] and [/?]. It is called the
Whitehead product of [a] and [Д] and is denoted by [[«],[/?]] £ TTp+Q_1(X,x0).
2 Prove that if p = q = 1, then [[a],[fi]] =
3	If p > 1 and q = 1, prove that [[«],[/?]] = [o]/i|/!j([«]1).
4	If p + q > 2, prove that [[«],[/?]] = (— l)P5[[/?],[a]].
5	Iff (X,x0) -> (Y,;/o), prove that/#[[«],[£]] = [/#[«],/#[£]].
6	If w is a path in X, prove that /i[„][[a],[j8j] =
7	Prove that [[«],[/?]] = 0 if and only if there is a map f: Ip x I9 X such that
f(tr, . . . ,tp, 0, . . . ,0) = a(ti, . . . ,tp)
and
f(0, ... ,0, tp+l, . . . ,tpfi) — fi(tp+l, . . . ,tp+g)
8	If X is an Я space, prove that [[«],[/?]] = 0 for all [a] and [/?].
1 See D. Puppe, Homotopiemengen und ihre induzierten Abbildungen. I, Mathematische
Zeitschriften, vol. 69, pp. 299-344, 1958.
420	HOMOTOPY THEORY CH Др. rj
9 Prove that S’1 is an H space if and only if [[«],[/]] = 0 for all [a], [/?] C w„(Sn).
E CW COMPLEXES
1	If (Х,Л) is a relative CW complex, prove that X has a topology coherent with the
collection {A, e | e a cell of X — A}.
2	If (Х,Л) is a relative CW complex, prove that X is compactly generated if and only
if A is compactly generated.
3	If (Х,Л) is a relative CW complex and A is paracompact, prove that X is paracompact.
4	If (Х,Л) is a relative CW complex and A has the same homotopy type as a CW com-
plex, prove that X has the same homotopy type as a CW complex.
5	Prove that a CW complex is locally contractible.
6	Prove that a CW complex has the same homotopy type as a polyhedron.
F ACTION OF THE FUNDAMENTAL GROUP
1 Prove that the real projective n-space P" is simple if and only if n is odd.
2 For 1 < n < m show that P2n+1 x S2m+1 and p2m+i x S2,1+1 are sjmpie compact
polyhedra having isomorphic homotopy groups in all dimensions, but are not of the same
homotopy type.
3 Let (Z,Z) be an (n — l)-connected CW pair, with n > 2, such that Z is simply con-
nected. Let (X*,X) be the adjunction space obtained by adjoining Z to a CWcomplexX
by a map/: (Z,a0) -a (X,x0) and let g: (Z,Z,a0) —> (Х*,Х,яь) be the canonical map. Prove
that (X * ,X) is (n — I)-connected and that the map
©	[n-n(Z,Z,z0)][„] -> 1гД*Лл)
[w] C 7Ti (X,Xo)
sending [a][„j to h[w](g#[«]) for [a] £ wn(Z,Z,Zo) is an isomorphism. [Hint: Let X be the
universal covering space of X and let {/[„j: Z -a X}6,(x,^) be the set of liftings of/.
Show that the space X* obtained by attaching a copy of Z to X for each map /|lL j is the
universal covering space of X *. Then use the fact that wQ(X * ,X) ~ ттд(Х * ,X) and com-
pute w„(X*,X) by the Hurewicz theorem.]
4 Let X be the CW complex obtained from S1 v S2 by attaching a 3-cell by a map
representing 2[a] — h[aj(a], where [a] is a generator of w2(S2) and [w] is a generator of
th/S1). Prove that the inclusion map S1 С X induces an isomorphism of the fundamental
groups and all homology groups but not of the two-dimensional homotopy groups.
G CW APPROXIMATIONS
1 If (X,A) is an arbitrary pair, prove that there is a CW pair (X',A') and a map
/: (X',A') —> (X,A) such that / | X': X' —> X and /1 A': A' —> A are both weak homotopy
equivalences.
2 If /i: Xi —> Y] and /2: X2 —> Y2 are weak homotopy equivalences, prove that
fi X /2: Xi X x2 —> Yi x Y2 is also a weak homotopy equivalence.
3 If /i: X! —> Yj and /2: X2 —> Y2 are weak homotopy equivalences, show by an
example that/i v/2: Xi v X2 —> Yi v Yg need not be a weak homotopy equivalence.
4 Show by an example that a weak homotopy equivalence need not induce isomor-
phisms of the corresponding Alexander cohomology groups.
5 If X is simply connected and H* (X) is finitely generated, prove that X has the same
weak homotopy type as some finite CW complex.
gxEHCisES
6 A space X is said to be dominated by a space Y if there exist maps /: X —> Y and
g: Y —> X such that g ° f ~ lx. Prove that a space is dominated by a CW complex if and
only if it has the same homotopy type as some CW complex.
Ц GROUPS OF HOMOTOPY CLASSES
Throughout this group of exercises it is assumed that Y is (n — l)-connected, where
fi > 2, with base point y0, and that X is a CW complex of dimension < 2n — 2.
] Prove that any map X —> Y is homotopic to a map sending X’1-1 to y0 and that if
f g: (X,X'rl) —> (Y,y0) are homotopic as maps from X to Y, they are homotopic relative
toX”-2.
2 Prove that the diagonal map d: X —> X x X is homotopic to a map d' such that
d'(X) С (X x Xй-2) (J (Xй-2 x X). Prove that maps d’, d": X —> (X x Xй-2) (J (Xй-2 x X)
which are homotopic in X x X are homotopic in (X X X'r 1) U (Xй-1 x X). (Hint: Use
the cellular-approximation theorem.)
Let d': X (X x Х”‘г) U (X'" 2 x X) be homotopic in X x X to the diagonal
map. Given f, g: X -> Y, let/', g': (X,X’1-1) -> (Y,y0) be homotopic to/and g, respec-
tively. Then (/' x g') ° d’: X -> Y x Y maps X into Yv Y. Let y: Y v Y —» Y be defined
by у(у,Уо) - У = y(ijo,y)-
3 Prove that [y ° (/' x g') ° d'] depends only on [/] and [g] and that the operation
[/] + И — [Т ° (f' X g') ° d'J is associative, commutative, and has a unit element,
making [X;Y] into a commutative semigroup with unit.
4 Prove that if g: Y —> Y', where Y' is also (n — l)-connected (or if h: X' X, where
X' is a CW complex of dimension < 2n — 2), then g#: [X; Y] —> [X;Y'J is a homomor-
phism (or h#: [X;Y] -a [X';Y] is a homomorphism).
5 The semigroup [X;Y] is a group. (Hint: Use induction on the dimension of X, the
fact that [X^+r/X^jY] is a group for any к and any Y, because Xk+1/Xk, being a wedge
of (k + l)-spheres, is a suspension, and the exactness of the sequence of homomorphisms
[X^+VX^Y] -> [X^Y] - >[X'-;Y] [X';Y]
where X' is a disjoint union of к spheres, one for each (k + l)-cell of X.)
In case Y = S’! and dimension X < 2n — 2, the group [X;S’!] is called the nth
cohomotopy group of X,1 denoted by trn(X).
I MISCELLANEOUS
I Let д’: wn+1(An+1,A’>+1,ti0) -» етг!(Д’1+1,(Дй+1)й-1,с0) if n > 2 and let
д': 7т2(Д2,Д2,г>о)	’Г1(Д2,Оо)
if n — 1. Prove that = b„ for n > 1 (see page 394 for definition of b?!).
2 Let H be a homotopy functor and let /: X —> Y be a base-point-preserving map
between path-connected spaces, with nondegenerate base points. Prove that the sequence
H(Cf) -+НЩ-+ H(X)
is exact.
3 If H is a homotopy functor and (Х,Л) is a CW pair, prove that there is an exact
sequence
H(A) H(X) <- H(X/A) H(SA) <-----------«- H(S«A) <------
1 For more details see E. Spanier, Borsuk’s cohomotopy groups, Annals of Mathematics, vol. 50,
pp. 203-245, 1949.
CHAPTER EIGHT
OBSTRUCTION THEORY
\ IN THIS CHAPTER WE DEVELOP OBSTRUCTION THEORY FOR THE GENERAL LIFTING
[ problem. A sequence of obstructions is defined whose vanishing is necessary
and sufficient for the existence of a lifting. The Ath obstruction in the sequence
is defined if and only if all the lower obstructions are defined and vanish, in
which case the vanishing of the Ath obstruction is a necessary condition for
definition of the (A + l)st obstruction.
We begin by applying the general theory of homotopy functors to study
the set of homotopy classes of maps from a CW complex to a space with
exactly one nonzero homotopy group and we show that a suitable cohomology
functor serves to classify maps up to homotopy in this case. This result is then
used to obtain a solution, in terms of cohomology, of the lifting problem for a
fibration whose fiber has exactly one nonzero homotopy group.
With this in mind, we then consider the problem of factorizing an arbi-
trary fibration into simpler ones each of which has a fiber with exactly one
nonzero homotopy group. We show that such factorizations do exist for a
large class of fibrations, and that when they exist, a sequence of obstructions
Can be associated to the factorization. These obstructions are subsets of coho-
423
424	OBSTRUCTION THEORY CHAP. § Л
mology groups, and we apply the general machinery to some special cases I
where, because of dimension restrictions, the only obstructions which enter 
are either the first one or the first two. For the case of only one obstruction I
we obtain the Hopf classification theorem.	'
Finally, we prove the suspension theorem, which we use to compute the
(n + l)st homotopy group of the n-sphere. Combining this with the technique |
of obstruction theory, we obtain a proof of the Steenrod classification theorem, j
Section 8.1 is devoted to spaces with exactly one nonzero homotopy 1
group. We prove tlyat a suitable cohomology functor serves both to classify 1
maps from a CW complex to such a space and to provide a solution for the ।
extension problem for maps involving a relative CW complex and such a space. '
We use this result to derive the Hopf extension and classification theorems for
maps of an n-dimensional CW complex to S’1. Section 8.2 deals with fibrations
whose fiber has exactly one nonzero homotopy group, and again it is shown J
that a suitable cohomology functor serves to provide a solution for the lifting )
problem and to classify liftings of a given map.	|
In Sec. 8.3 we prove that many fibrations can be factored as infinite '
composites of fibrations each of which has a fiber with exactly one nonzero
homotopy group. The corresponding lifting problem is then represented as an
infinite sequence of simpler lifting problems. In Sec. 8.4 we show how fo i
define obstructions inductively for such a sequence of fibrations, and how to I
apply the resulting machinery.
In Sec. 8.5 we shall study the suspension map and prove the exactness '
of the Wang sequence of a fibration with base space a sphere. This result is
used to prove the suspension theorem, which is applied to compute wn+1(Sn)
for all n. We then prove the Steenrod classification theorem for maps of an
(n + l)-dimensional CW complex to S’1.
1 EILENBEBC-MACLANE SPACES	1
This section is devoted to a study of spaces with exactly one nonzero homotopy
group. Such spaces are classifying spaces for the cohomology functors, and
because of this, there is an important relation between the cohomology of
these spaces and cohomology operations. At the end of the section we shall
apply tire results to derive the Hopf classification and extension theorems.
Then, later in the chapter, we shall study arbitrary spaces by representing
them as iterated fibrations whose fibers are spaces with exactly one nonzero
homotopy group. Thus, these homotopically simple spaces serve as building .
blocks for more complicated spaces.	I
Let 77 be a group and let n be an integer > 1. A space of type (т7,п) is a (
path-connected pointed space Y such that 77,/Y,y(l) =0 for q n and j
77?i( Y,i/0) is isomorphic to 77. An Eilenberg-MacLane space1 is a path-connected ’
pointed space all of whose homotopy groups vanish, except possibly for a
1	See S. Eilenberg and S. MacLane, On the groups H(w,n), 1, Annals of Mathematics, vol. 58,
pp. 55-106, 1953.
SEC. I EILENBERG-MACLANE SPACES	425
single dimension. Thus a space of type (тг,п) is an Eilenberg-MacLane space.
Conversely, if Y is an Eilenberg-MacLane space and 77q(Y,j/0) = 0 for q У n,
then Y is a space of type (t7„(Y,j/o)» «)• Let us consider a few examples.
J It follows from corollary 7.2.12 that S1 is a space of type (Z,I).
2	Let F" be the CW complex which is the union of the sequence
pi C F2 C  • • topologized by the topology coherent with the collection
[P,};>1- Then 77e(F”) ~ lim,	and it follows from application of corol-
lary 7.2.11 to the covering S” P11 that P°° is a space of type (Z2,I).
3	Let Fcr(C) be the CW complex which is the union of the sequence
Pj(C) С F2(C) C •  • topologized by the topology coherent with the collec-
tion {P/C)}j>i. Then 77,;(F„(C)) ~ lim^ (77,/iyC))}, and it follows from
corollary 7.2.13 that Рет(С) is a space of type (Z,2).
Let 77 be an abelian group and Y a path-connected pointed space.
An element v C Hn( Y,y0; 77) is said to be n-characteristic for Y if the composite
77„(Y,y0) Hn(Y,yo) 77
is an isomorphism (where tp is the Hurewicz homomorphism and h is the
homomorphism defined in Sec. 5.5). If Y is (n — l)-connected, it follows from
the absolute Hurewicz isomorphism theorem and the universal-coefficient
theorem for cohomology that there is an n-characteristic element
0 £ Hn(Y,i/o; w) if and only if 77	77?i(Y,yo). Such an element is unique up to
automorphisms of 77. In particular, a space Y of type (77,n) with 77 abelian has
n-characteristic elements v E Hn(Y,y0; tt).
4	lemma Let и E H”(Y,i/0; G) be a universal element for the nth coho-
mology functor with coefficients G, where n > 1. Then Y is a space of type
(G,ri) and и is n-characteristic for Y.
proof By theorem 7.7.14, there are isomorphisms
Tv: trg(Y,y0) H”(Se,p0; G) q > 1
Therefore 77e(Y,i/o) = 0 if q n, and Tu-. rrn(Y,y<f) Hn(Sn,po; G). If
a: (S”,po)	(T,i/o)> then Tu([a]) = a* (u), and there is a commutative diagram
^n(S’!,po)	H„(S”,p0)
\11Л(а*(и)) = Л(Т„[а])
“4	4 g
My,j/o) HnCfyf)
Let v: Hn(Sn,p0; G) ~ G be the isomorphism defined by
p(n) = h(o)(<p[ls"]) v E Hn(Sn,p0; G)
From the commutativity of the diagram above,
(h(u) ° <p)[a] = (h(u) ° <p ° a#)[ls"] = h(T„[a]) (<jp[ls"]) = (p ° T„)[a]
426	OBSTRUCTION THEORY CHAP, fj
It follows that h(u) ° <p equals the composite
H"(Sn,p0; G) G
and so is an isomorphism. Therefore Y is a space of type (G,n) and и is
«-characteristic for Y. 
5	corollary Given n > 1 and a group w (abelian if n > I), there exists!
a space of type (77,n).
proof If 77 is abelian, it follows from lemma 4 that any classifying space for
the nth cohomology functor with coefficients 77 is a space of type (77,n).
If n = 1 and 77 is arbitraiy, it is easy to see that a classifying space for the
homotopy functor of example 7.7.5 which assigns to a pointed path-connected
space X the set of homomorphisms 77i(X,Xq) —> 77 is a space of type (w,l). In
either case, since any homotopy functor has a classifying space by corollary
7.7.12, the result follows. 
6	corollary Let {т7п}„>1 be a sequence of groups which are abelian for
n > 2. There is a space X, with base point x0, such that 7rn(X,xf) ~ тгп
for n > 1.
proof By corollary 5, for each n > 1 there is a space Y„, with base point yn>
such that 77Q(Y„,i/n) = 0 for q n and 77„(Y„,y„) ~ 77„. Then the product
space X Y„ with base point ((/„) has the desired properties. 
The last result can be strengthened so that if 771 acts as a group of oper-
ators on 77n for every n > 2, then the sequence is realized as the sequence of
homotopy groups of a space X in such a way that the action of 771 on 77„ cor-
responds to the action of 77i(X,Xo) on 77n(X,Xo) of theorem 7.3.8.
7	lemma Let F: H —» H' be a natural transformation between homotopy
functors which induces an isomorphism of their qth coefficient groups for
q < n and a surjection of their nth coefficient groups ( where 1 < n < 00). For
any path-connected pointed CW complex W the map
F(W): H(W) -> H'(W)
is a bijection if dim W < n — 1 and a surjection if dim W < n.
proof Let и £ H(Y) and u' £ H'(Yr') be universal elements for H and H',
respectively, and let/: Y —> Y' be a map such that H'(/)(i/) = F(Y)(u). For
any CW complex W there is a commutative square
[W;Y] -A-> [W;Y']
H(W) H'(W)
in which, by theorem 7.7.14, both vertical maps are bijections. Since
F(S'i): H(Se) {{'(Si) is an isomorphism for q < n and a surjection for q = n,
it follows that/#: 77e(Y) —» 77e(Y') is an isomorphism for q < n and a surjec-
tion for q = n. Since Y and Y' are path-connected pointed spaces, the map /
sgc. I EILENBERC-MACLANE SPACES	^27
; j$ an «-equivalence. The result follows from corollary 7.6.23 and the commu-
tativity of the above square. 
We use this last result to obtain the following classification theorem,
which is a converse of lemma 4.
jl theorem Let w be an abelian group, Y a space of type fn,ri), and
i € Hn(Y,yo; it) an n-characteristic element for Y. Let f: tty Hn( • ;w) be the
natural transformation defined by >/[/] = f*ifor [/] E [X; Y ]. Then f is a
natural equivalence on the category of path-connected pointed CW complexes.
proof By lemma 7, it suffices to verify that f induces an isomorphism of all
coefficient groups of the two homotopy functors wy and H"( • ;w). The only
nonzero coefficient groups are wn(Y,yo) and Hn(Sn,po; it), and we need only
verify that
rrn(Y,y0) Hn(Sn,p0; tt)
is an isomorphism. If v: Hn(Sn,po; w) w is defined by /’(с) =	(as
in the proof of lemma 4), then v ° i//(S") = h(t) ° q>. Because t is n-characteristic
for Y, v ° i^(S”) is an isomorphism, and thus so is ^(S"). 
9 theorem Let Y be a space of type (tt,1) and let H be the functor which
assigns to a pointed space X the set of homomorphisms from TrfiX,xf) to
57i(Y,t/o)- Let f: tty —> H be the natural transformation defined by ф[/] = f#
for [/] € [X; Y ]. Then f is a natural equivalence on the category of path-
connected pointed CW complexes.
proof By lemma 7, it suffices to verify that
^(S1): wi(Y,(/0) -> H(Si,p0)
is an isomorphism. Let i>: H(S1,po) ~ TrfiY,y0) be the isomorphism defined by
v(y) = y([lsi]) for y: TrfiSfpo) 77i(Y,yo)- Then v is an inverse of ^(S1),
showing that ^(S1) is an isomorphism. 
Note that if rrfiY,yf) is abelian in theorem 9, the set of homomorphisms
from 77i(X,x0) to wi( Y,(/o) is in one-to-one correspondence with the group
Hom (wl(X,x0), wi(Y,y0)) ~ Hom (Hi(X,x0), wi(Y,y0)) ~ НЦХ,х0-, wi(Y,yo))
and so theorems 8 and 9 agree in this case.
We now consider the free homotopy classes of maps from X to Y. Since
any О-cell Xo of a CW complex X is a nondegenerate base point (because, by
theorem 7.6.12, the inclusion map xq С X is a cofibration), it follows from
corollary 7.3.4 that there is an action of wi(Y,(/o) on the set [X,x0; Y,y0].
Furthermore, if Y and X are path connected and this action is trivial, then
the map from base-point-preserving homotopy classes to free homotopy classes
[X,x0; Tyo] [X;Y]
is a bijection. In case Y is a space of type (w,n), with n > 1, then 57j( Y,y0) = 0,
and so there is a bijection
428	OBSTRUCTION THEORY CHAR, g
[X,xo; Y,y0] ~ [X;Y]
In case Y is a space of type (77,1), the action of wi(Y,(/o) on [X,x0; Y,y0] corre-
sponds under the bijection f of theorem 9 to the action of 77i( Y,yo) on H(X,Xq)
by conjugation. Thus, if 77 is abelian, there is a bijection
[X,x0; Y,y0] ~ [X; Y]
IO theorem If w is an abelian group, Y is a space of type (т7,п), and
i £ H'!(Y,y0; w) is n-characteristic for Y, then for any relative CW complex
(X,A) the map
[X,A; Y,y0]	H»(X,A; w)
is a bijection.
proof In case A is empty and X is path connected, it follows from theorem 8
and the observation above that there is a commutative square
[X,x0; Y,y0]	[X;Y]
4=
Hn(X,xo; 77)	Hn(X;rr)
and so f: [X;Y] Нп(Х,тт}. In case A is empty and X is not path connected,
let {Xx} be the set of path components of X. The result follows from the first
case on observing that [X;Y] ~ X [Xx;Y] and Ня(Х;7т) ~ X Нп(Хх;т7). In
case A is not empty, let k: (X,A) —> (X/A,x'o) be the collapsing map. Then the
result follows from the already established bijection f: [X/A;Y] ~ H?1(X/A;7r)
and the commutative diagram
[X,A; Y,y0] < [X/A,x0; Y,y0]	[X/A;Y]
H»(X,A; 77)	H«(X/A,x0; 77)	H»(X/A;t7) 
11 theorem Let Y be a space of type (77,1). For any path-connected CW
complex X the set of free homotopy classes of maps from X to Y is in
one-to-one correspondence with the set of conjugacy classes of h omomorphisms
77i(X,x0)	77i(Y,y0) under the map [/]	f#.
proof This follows from theorem 9 and the remark above covering the
action of 77i(Y,y0) on [X,xq; Y,y0]. 
12 theorem Let Y be a space of type (гтрг), with n > 1 and 77 abelian,
and let 1 C H"(Y,yo; 77) be n-characteristic for Y. If (X,A) is a relative
CW complex, a map f: A —> Y can be extended over X if and only if
8f*(t.') = 0 in H'1+1(X,A;t7)
proof Assume f = g 0 i, where i: А С X and g: X Y. Then 8f* (t) =
8i*g*(i) = 0, because 8i* =0. Hence, if f can be extended over X, then
8f*(f) = 0.
SEC. 1 EILENBERG-MACLANE SPACES
429
Conversely, assume 8/*(t) = 0. To extend/over X we need only extend
/ over each path component of X, and therefore there is no loss of generality
!jn assuming X to be path connected (and A to be nonempty). Let Y' be the
space obtained from the disjoint union X U Y by identifying a C A with
! f(a) € Y for all a £ A. Then Y is imbedded in Y', the pair (Y',Y) is a relative
| CW complex, and there is a cellular map j: (X,A)	(Y',Y) which induces an
j isomorphism ;*: H*(Y',Y) ~ H*(X,A) such that there is a commutative
! square
j	H«(Y,y0) -A H«+i(Y',Y)
I	H«(A) А №+1(ХД)
! Since 8/*(i) = 0, it follows that 8(t) = 0, and there is v C Hn(Y',y0; 77) such
I that v | (Y,(/o) = ' Since X and Y are path connected and A is nonempty, Y'
। is path connected.
I Let Y = Y' v I (that is, y0 £ Y1 is identified with 0 £ 1) and let
y0 = 1 C Y. Then Y is a path-connected space with nondegenerate base
point yo. Let r: (Y,I) —» (Y',yo) be the retraction which collapses I to yo and
let v = r*(o) | (Y,y0) € Hn(Y,y0; it). By theorem 7.7.11, there is an imbedding
I of Y in a space Y" which is a classifying space for the nth cohomology functor
with coefficients 77 and which has a universal element й C Hn(Y",y0; it) such
that й | Y = 6. Then Y" is a space of type (т7,п), and there is a unique
n-characteristic element и C H!l(Y",yo; rr) such that и | Y" = й | Y". Then
и | (Y,y0) = i, and it follows from theorem 8 and the commutativity of the
diagram
[S<?,p0; Xj/o] [S«,po;
~|/ф«
I	Нп(Бч,ро; if)
[ that Y C Y" is a weak homotopy equivalence. Since the composite
I X у' c Y" is an extension of the composite A 4 Y C Y", it follows from
theorem 7.6.22 that / can be extended to a map X Y. 
• We now show that cohomology operations are closely related to the
cohomology of Eilenberg-MacLane spaces. Let O(n,q; 77,G) be the group of
all cohomology operations of type (n,q; 77,G). Thus 77 and G are abelian
groups and an element О C O(n,q; 77, G) is a natural transformation from the
singular cohmology functor Hn( • ;w) to the singular cohomology functor
I №(-;G).
I 13 theorem Let 77 be an abelian group and let Y be a space of type (тт,п),
। with an n-characteristic element i £ Hn(Y,y0; it). There is an isomorphism
y: Q(n,q-, tt,G) №(Y,y0; G)
defined by y(0) = 6(i) for в £ O(n,q; 77,G).
430	OBSTRUCTION THEORY CHAP, §
proof Since, by theorem 7.8.1, every pair has a relative CW approximation
a cohomology operation corresponds bijectively to a cohomology operation on
the category of relative CW complexes. To define an inverse to y, given
it £ №(Y,y0; G), let 6U be the cohomology operation of type (n,q; tt,G) defined
for a relative CW complex (X,A) by
0„(r) =/*(«)	v£H«(X,A)
where /: (X,A) —> (Y,y0) is a map such that f * (t) = v (fv exists and is unique
up to homotopy, by theorem 10). Then
У (Ou) = 0u(l) = 1*(«) = U
showing that the map и O.„ is a right inverse of y. To show that it is also a
left inverse of y, let (X,A) be a relative CW complex and let v C Hn(X,A; it).
We must show that 07(в}(о) = 0(v). Let fv: (X,A)	(Y,y0) be such that
f * (1) = v. Then we have
0(v) = 6(f* (c)) = /* (0(c)) = f* (y(0)) = 0Y(e)(o) -
We present one application of this result.
14 corollary Let 0 he a cohomology opera tion of type (n,q; rr,G). For any
relative CW complex (X,A) the map
6: №((X,A) X (1,1); тт) №((X,A) X (I,I); G)
is a homomorphism.
proof The collapsing map
k: (X X I, A X I U X X i) X X I/(A X I U X x i)
induces isomorphisms in cohomology. Furthermore, X X I/(A X I U X x 1)
is homeomorphic to S(X/A) (where X/A is understood to be the disjoint
union of X and a base point x(l in case A is empty). Thus it suffices to show
that if X' is any pointed CW complex, then the map
6»: H»(SX',x£); tt) №(SX',x'o; G)
is a homomorphism.
Let У be a CW complex of type (тт,п), with n-characteristic element t,
and let Y' be a space of type (G,q), with q-charac ten’s tic element t'.
Let/: У —» У' be a map such that /*t' = 0(i.). There is then a commutative
diagram
[SX',Xq; Y,t/o] [SX',xq; У',</6]
4=
H\SX',xh-, tt) 4 №(SX',x'o; G)
It is trivial that f# is a homomorphism when the top two sets are given group
structures by the H cogroup structure of SX'. By lemma 7.7.6, it follows that
SEC. 1 EILENBERG-MACLANE SPACES
431
both vertical maps are homomorphisms. Hence the bottom map 0 is a
homomorphism. 
Let I C №(!,!; Z) be a generator and define an isomorphism
т: ЩХД; G') ~ №+i((X,A) x (U); G')
by t(w) = и X L Given a cohomology operation 0 of type (n,q; tr,G), its
suspension SB is the cohomology operation of type (n — 1, q — 1; 77,G)
defined by (S0)(h) =	for и C №-1(X,A; 77). Then corollary 14 implies
that the suspension of any cohomology operation is an additive cohomology
operation.
We now extend theorems 10 and 12 to other spaces Y by restricting the
dimension of the relative CW complex (X,A). Let Y be an n-simple (n — 1)-
connected pointed space for some n > 1 [if n = 1 then 7?j( Y,y(l) is abelian].
]f i C №!(Y,(/o; w) is an n-characteristic element for Y, an argument similar to
that in theorem 12 shows that Y can be imbedded in a space Y' of type (w,n)
having an n-characteristic element и £ Hn(fl',yo’, tt) such that и | Y = t. It
follows that the inclusion map Y C Y' is an (n + Inequivalence. Then
theorems 7.6.22 and 10 yield the following generalization of theorem 10.
15 theorem Let i C №(Y,j/o; 77) be n-characteristic for an n-simple (n — 1)-
connected pointed space Y and let (X,A) be a relative CW complex. The map
[X,A; Y,(/o] H\X,A; 77)
defined by i/4/] = /*(t) is a bijection if dim (X — A) < n and a surjection
if dim (X — A) < n +1. 
For the special case Y = S" let s* £ Hn(Sn,p0; Z) be a generator. Then
s* is an n-characteristic element of S”, and we obtain the following Hopf
classification theorem.1
16 corollary Let (X,A) be a relative CW complex, with dim (X — A) < n,
where n > 1. If s* C Hn(Sn,p0; Z) is a generator, there is a bijection
fs*: [X,A; S« po] №(X,A; Z)
defined by fs* ([/]) = f* (s*). 
Similarly, we obtain the following generalization of theorem 12.
17 theorem Let i C Hn(Y,y0; it) be n-characteristic for an n-simple (n — 1)-
connected pointed space Y and let (X,A) be a relative CW complex, with
dim (X — A) < n + 1. A map f: A Y can be extended over X if and only
if 8f*(f) =0in Hn+1(X,A; tt). 
This specializes to the following Hopf extension theorem.
1 See H. Hopf, Die Klassen der Abbildungen der n-dimensionalen Polyeder auf die n-dimen-
sionale Sphare, Commentarii Mathematici Helvetic!, vol. 5, pp. 39-54, 1933, and H. Whitney,
The maps of an n-complex into an n-sphere, Duke Mathematical Journal, vol 3, pp. 51-55,1937.
432	OBSTRUCTION THEORY CHAP, g
18 corollary Let (X,A) be a relative CW complex, with dim (X — A) <
n + 1, and let s* £ Hn(S,l,p0; Z) be a generator. A map fi ASn can be
extended over X if and only if	= 0 in Hn+fiX,A-, Z). 
2 PRINCIPAL FIBRATIONS
This section is concerned with fibrations whose fiber is an Eilenberg-MacLane
space. We shall develop an obstruction theory for the lifting problem of maps
of relative CW complexes to such fibrations. In the next section we shall show
that many maps can be factored up to weak homotopy type as infinite com-
posites of such fibrations. In this way the obstruction theory for these special
fibrations leads to an obstruction theory for arbitrary maps.
For any pointed space B' there is the path fibration PB' B', where PB'
is the space of paths in B' beginning at the base point b'o. Under the expo-
nential correspondence there is a one-to-one correspondence between homot-
opies H: X X I —> B' such that H(x,O) = b'o and maps H': X PB', the cor-
respondence defined by H'(x)(f) = This easily implies the following result
(which is dual to lemma 7.1.1).
1	lemma A map X B' is null homotopic if and only if it can be lifted
to the path fibration PB' B'. 
If в: В В' is a base-point-preserving map, there is a fibration pe: Ee —» В
induced from the path fibration PB' B'. This induced fibration is called the
principal fibration induced by 0 and has fiber рв л(Ъо) — bo X ЯВ'. A
straightforward verification shows that there is a covariant functor from the
category of base-point-preserving maps between pointed spaces to the suh-
category of fibrations which assigns to в the principal fibration induced by 0.
Let (X,A) be a pair and let i: А С X be the inclusion map. Let pe: Ee —> В
be the principal fibration induced by 0: В B'. Recall that a map pair
fi i—t pe (defined in Sec. 7.8) is a commutative square
A^Ee
x -А в
The set of homotopy classes [i;pe] of map pairs from i to p6 is the object function
of a functor of two variables contravariant in pairs (X,A) and covariant in base-
point-preserving maps 0. We are interested in studying in more detail the
relative-lifting problem (that is, the map p: [X;Ee] [i;pe]) for this situation.
Because pe is an induced fibration, the relative-lifting problem is equivalent
to an extension problem, as shown below.
Let pe: Ee —> В	be	induced by 0: В B'. For	any space W a	map
fi W—> Ee consists	of	a pair fi: WВ and /2:	W PB' such	that
p' ° fi = 0 0 /1. By	the	exponential correspondence,	fi corresponds	to a
homotopy F: W X I —» B' from the constant map to 0 0	fi. Thus, given a	map
fi: W B, there is a one-to-one correspondence between liftings fi W —» Ee
/ jgC. 2 PRINCIPAL FIBRATIONS	433
of fi and homotopies F: W x I B' from the constant map to 0 ° fi.
Let (X,A) be a pair with inclusion map i: А С X and let f: i pg be a
«niap Pah consisting of maps f ": A —> Ee and f': X В such that pe ° f" =
fi' ° i. We define a map
0(f): (A X I U X X I, X X 0) (B',bfi)
by the conditions 0(f)(x,0) = b'o, 0(f )(x,l) = 0f'(x), for x С X, and
0(f) I -A X I is the homotopy from the constant map A b'o to the map
О ° f' ° i corresponding to the lifting f" off' ° i. There is then a one-to-one
correspondence between liftings of f and extensions of 0(f) over X X I-
We now specialize to the case where B' is a space of type (77,n), with
n > 1 and 77 abelian, and we let t С Нп(В',Ъ'о; tt) be n-characteristic for B'.
Jn this case, if 0: В B' is a base-point-preserving map, the induced fibration
рв: Eg —> В is called a principal fibration of type (77,n). If (X,A) is a relative
CW complex, then (X,A) X (1,1) is also a relative CW complex, and given a
map g: A X I U X X 1	B', it follows from theorem 8.1.12 that g can be
extended over X X I if and only if 8g* (1) = 0 in Hn+1((X,A) X (1,1); rf-
In particular, given a map pair f:i^> pe, there is a lifting of f if and only if
80(f)* (t) = 0. The obstruction to lifting f, denoted by c(f) £ Hn(X,A; rr),
is defined by
80(f)* (t) = (-lWc(f))
where t: Hn(X,A; 77) qs f/"+l((X,A) X (1,1)', w) is the map t(u) = и X I, de-
fined in Sec. 8.1 [I £ ff1(I,j; Z) is the generator such that if б C H°({0}; Z)
and 1 € П°({1}; Z) are the respective unit integral cohomology classes, then,
identifying №(I;Z)	H<>({0};Z) © H°({1};Z), we have 81 = 1 = -80].
2	example In case A is empty, a map pair fi i —> p6 is just a map
f': X —> B. In this case 0(f): X X 1 B' is such that 0(f )(x,0) = b'o and
0(f)(x,l) = 0f'(x). Then 0(f)* (1) = f' *(1) X 1, and so, by statement 5.6.6,
80(f)* (t) = (-1)Т*(0 X I = (-1W'*W
Therefore, in this case c(f) = f' * (1).
It is clear from the definition that the obstruction to lifting f is functorial
in i and 0 and that it vanishes if and only if there is a lifting of fi We obtain
a similar cohomological criterion for the existence of a homotopy relative to f
of two liftings of f.
Let f: i pe be a map pair, where (X,A) is a relative CW complex, with
i: А С X, and pe is a principal fibration of type (77,n). Given two liftings
fo, Д: X E of f let g: i' —» p6 be the map pair consisting of the commuta-
tive square
AxIUXxi	Eg
4	>
xx i	в
434	OBSTRUCTION THEORY CHAP. §
where g' is the composite X X I —> X В and g" is the map such that
g"(x,O) = f0(x) and_g"(x,l) = fi(x) for x £ X and g"(a,t) = f "(o) for a £ A and
t £ I. Then f0 and Ji are homotopic relative to f if and only if g can be lifted
The obstruction to lifting g is an element c(g) £ Hn((X,A) x (1,1); w)> and we
define the difference between fо and ft, denoted by d(/0,/i) £ H’!-1(X,A; By
c(g) = (-l^fo/l))
[so 80(g)* (t) = T2(d(/o,/i))]- Then/о and/i are homotopic relative to/if and
only if d(/o,/i) = 0. The difference d(f0,ff) is functorial and has the following
fundamental properties.
3 lemma Given a map pair f: i pe and liftings fo, fi, Jv'- X —» Ee, then
d(f0,fy = d(f0,ff) + d(fi,f2)
proof Let Ii = [0,i/2], Ii = {0,bi}, I2 = [1/2Д], and I2 = {bs,l} and define a
map pair G: i pe consisting of the commutative square
A X I u X X (ii и I2) Eg
4	>
Xxl
where G'(x,f) = f'(x), G"(a,t) = f"(a), G"(x,0) = f0(x), G"(x,W) = fi(x), and
G"(x,l) = /г(х). Then c(G) £ I/,!((X,A) x (/, Ii U Zz); w), and by the naturality
of c(G) and the definition of d, we see that
c(G) I (X,A) X (I,I) = (-l)«r(d(fo,f2))
C(G) I (X,A) X (IiJi) = (-l)MWi))
c(G) I (X,A) X (I2,Z2) = (-lY^d^ffi)
where	T1: H’!-1(X,A) ~ H«((X,A) X (IiJi))
and	t2: Hn”1(X,A)	H«((X,A) X (/2Л))
are defined analogously to t. From these properties, an argument similar to
that used in proving that the Hurewicz homomorphism is a homomorphism
(cf. theorem 7.4.3) shows that
Ш-Л)) = T(d(/o,/l)) + r(d(fbf2))
Since т is an isomorphism, this is the result. 
4 theorem Given a map pair f: i —> pe, a lifting fo'- X E6 of f, and an
element v £ Hn~1(X,A; w), there is a lifting fi: X Ee of f such that
<Wi) = a
proof The map 0(/): AxIGXxt^>B' used in defining c(/) admits an
extension h0: X X I B' which corresponds to the lifting fo: X Ee. We
seek another extension of 0(/) which will correspond to the desired lifting/1
of f. Let F: (A X I X I U X X (0 X I U I X I), X X I X 0)	(B',bo) be the
map defined by F(a,t,t') = 0(f )(o,t) for a £ A and t, t' £ I, and
F(x,0,t) = ho(xf), F(x,t,0) = b'o, and F(x,t,l) = li0(x,l) for x £ X and t £ I
! . sEC. 2 PRINCIPAL FIBRATIONS	435
I Because X X I X 0 is a strong deformation retract of the space
A X I X I U X X (0 X I U I x I), there is a homotopy relative to X X I X 0
I from F to the constant map F' from A X 1 X I U X X (0 X I U I X I) to b'o.
! Let G: (X X 1 X 1, AxlXlUXxlXl)—» (B',b'o) be a map such
j that G*(i) = ( — I)”-1!? X 1 X I £ H»((X,A) X {1} X (I,t); w) [such a map
! exists, by theorem 8.1.10, because (X,A) X {1} X (1,1) is a relative CW com-
; plex]. There is a well-defined map
!	H': (A X F U X X I2, A X I X I U X X (0 X I U I x I))	(B',%)
j such that H' | X X 1 X I — G. Then
!	H' IA XI X I и X X (0 X I и I X I) = F
i and because (X,A) X (I X I, 0 X I U I X I) is a relative CW complex, the
 homotopy F' ~ F rel X X I X 0 extends to a homotopy H' ~ H rel X X I X 0,
j where
I H: (Ax I Xiu Xxixic Xxlxi, XxIxO)^ (B',b'o)
is an extension of F. Let hy. Xxl -> B' be defined by hi(x,t) = H(x,l,t).
Since H is an extension of F, hi is an extension of 0(f), and hence hi corre-
j sponds to a lifting Д of f.
* We now show that Д has the desired properties. The definition of the
I map pair g: i' —> pe used to define d(fo,jf) is such that 0(g) = H. Therefore
Wo>/i)) = 8H*(i) = 8H'*(i)
H' is a map from (A X I2 U X X P, A X I2 U X X (0 X I U I X I)) to
(B',b'o) whose restriction to X X 1 X I is G. From the commutativity of the
diagram [where the map p is given by p(w X I X I) = w X I for
| w£H*(X,A)]
4	Hn(A X I2 u X x i2, A X p u X X (0 X I и I X t))
Hn(A X p и X X i2, x X I X 0) №>(X X1X I, Axlxiuxxlxi)
I	8l	4=
.1	H«+1((X,A) X (P,i2))	<^?1T	№((X,A) X (I,i))
I it follows that
I	8H'*(i) = (~l)«-iTpG*(i) = r(v X I) = t2(u)
Since t2 is an isomorphism, d(f0,fi) — v. 
 5 theorem Let (X,A) be a relative CW complex and let (X',A) be a sub-
i complex, with inclusion maps i: А С X, i': А С X', and i": X' С X. Given a
i map pair f: i pe (consisting of f": X Ee and f': X —» B) and two liftings
go, gi: X' E off | i': i' pe, let go, gi: i” Pe be the map pairs consisting,
lespectively, of the commutative squares
436
OBSTRUCTION THEORY
MOORE-POSTNIKOV FACTORIZATIONS
437
X'	Ee	X'	-S^	Ee
'"I	•?"	'"I	lA-
x А	в	x	А	в
TTien	8d(go,gi) = c(go) - c(gt)
where 8: Hn~1(X',A; d) Hn(X,X'; if).
proof Let 7г: i pe be the map pair defined by the commutative square
л x i ux' x / i;e
4	>
r x i и x x i Д в
where fi"(a,t) = f "(a) for a £ A and t £ 1, h"(x',O) = go(x') and h"(x',l) ~
gi(x') for x' £ X', and h'(x,t) = /'(x) for (x,t) £ X' X I U X X 7. Then
c(K) £ Hn(X' X I U X X t, A X I U X' X I; vr). There is an isomorphism
H«(X' X I U Xx I, A x I U X'
№((X',A) X (1,1); tt) © 77n((X,X') X t; тг)
induced by restriction. By the naturality of the obstruction, c(/i) corresponds
to (— l)!tTd(g0,gi) = (— l)nd(g0,gi) X I in the first summand and to
c(go) X 0_+_c(gi) X 1 in the second summand.
Let h: i pe be the map pair defined by the commutative square
We compute the obstruction c(/) explicitly for the case of a fibration
* p': QB' b'o, where B' is a space of type (w,n), with n > 1. Then QB' is a
< {pace of type (w, n — 1), and if t' £ H?i-1(S2B',wo; w) is (n — ^-characteristic
J |or QB' and i £ Hn(B',b'o; w) is n-characteristic for B', then 8t' and p* i [where
8: H^iQB',wo) ~ Hn(PB',QB’) and p: (PB',QB') -> (B',b'o)] are both elements
J of Ня(РВ',ЯВ'; ir). The characteristic elements t and i' are said to be related
>,	8/.' = p* Given one of t or t', it is always possible to choose the other one
• (uniquely) so that the two are related.
6 theorem Let i £ H«(B',b'o; and l' € Нп~1(ЯВ',со6; it) be related
I characteristic elements. Let (X,A) be a relative CW complex, with inclusion
! щар i- А С X. Given a map pair f. i p', where p': QB' —» b'o, then
c(J") = — 8f"*(i'), where f": A QB' is part off.
i proof Let/: (A X 7, A X 7) —> (РВ',ЯВ') be the map defined by f(a,t)(f) =
\ f\a)(tf). Then
i	0(/): (A X I U X x I, X X 0)	(B',b'o)
| is the map such that P(/) | A x I = p ° f and P(/)(X X I) = b'o. Let
! /: (A X I U X X I, X X 7) -> (B',b6) be the map defined by P(/) and let
i (A X t, A X 0) (QB',co'o) be the map defined by f. There is then a com-
i mutative diagram [in which / and f are appropriate inclusion maps and
A (X X 1, A X 0) is defined by /ii(o) = (й,1)]
H»(A xiu Xxi, x X 0)
Ax ic X' xi Ee

4	4"
X x I	В
where //(x,t) = f'(x) for x £ X and t £ I. Then
c(7i) £ 77”(X x I, A x I U X’ X I; d)
and by the naturality of the obstruction again,
c(h) | (X' x I U X x I, A x I U X' x 7) = c(h)
From the exactness of the sequence
№(X XI,AXIU X' X I) Hn(X' XIUXx/AxlUX'xI)
Л H»+i(x x i, X' x i u x x i)
it follows that 8c(h) = 0. Therefore, in Hn+1((X,A) X (7,7); w) we have (using
theorem 5.6.6)
0 = 8[(~lMgo,gi) X I + c(go) X 6 + c(gi) X 1]
= ( — l)n8d(g0,gi) XI- (~l)nc(go) X I + (-l)nc(gr) X I
Therefore T(8d(go,gi) — c(go) + c(gi)) = 0, and since т is an isomorphism, the
result follows. 
Hn(B',b'o) К Hn(A xlUXxtXxi) Н«+1((ХЛ) X (1,7))
"I ,
j Я»(РВ',ЯВ') -A H”(A xI,Axt)	Hn(X,A)
I 4	4	Is
I №-i(QB',<do^	X 7, A X 0)	H^~\A)
Furthermore, 8 ° t-1 ° f* = t1 ° 8: Hn(A X I bl X X I, X X I) —* 77Я(ХД).
Since f" = f' ° hi, then /" * = hf ° /' *, and we have
(_l)n-iT-W(/)r(l) = 8f''*(,.')
By definition, the left-hand side above equals —df). 
О MOORE-POSTNIKOV FACTORIZATIONS
This section is devoted to a method of factorizing a large class of maps up to
j, weak homotopy type as infinite composites of simpler maps, the simpler maps
438	OBSTRUCTION THEORY < Н.Др, £
being o£ the same weak homotopy type as principal fibrations of type
for some 77 and n. The cohomological description of the lifting problem fOt •
these fibrations, given in the last section, will lead us ultimately to an iterative |
attack on general lifting problems.
Given a sequence of fibrations Eo Ift , we define	f
Ет — Iim«_ }Eg,pg} = {(^e) £ X Ee| Pqft'q) — &q— 1}	J
and we define ag: E„ Eg to be the projection of EM to the qth coordinate, 
Then each map ag is a fibration and ag = pg+i ° ae+i for q > 0. For any i
space X a map f: X Em corresponds bijectively to a sequence of maps: ;
{ fq'- x Eq}q>0 such that fg = pg+1 ° fg+1 for q > 0 (given fi the sequence :
{ fg} is defined by fg = ag ° f). In particular, given a pair (A, A) with inclusion'
map i: А С X and a map pair fi i a0 consisting of the commutative square
A^Ea	‘
'l	Ь	1
X	Eo	|
a lifting f: X —» EM corresponds bijectively to a sequence of maps ‘
{ fg: X Ед}д>о such that	i
(0)fo=f':X^E0 _	।
(b) For q > 1 the map fg: X Eg is a lifting of the map pair from г to pg
consisting of the commutative square
A-^Eq
i . b
X Eq^	I
In this way the relative-lifting problem for a map pair f: i—+ ao corresponds
to a sequence of relative-lifting problems for map pairs from i to pg. In many
cases the relative-lifting problems for the fibrations pg may be simpler to deal
with than the original relative-lifting problem for the fibration a0.
A sequence of fibrations Eo <£?... is said to be convergent if for i
any n <C oo there is Nn such that pg is an n-equivalence for q > Nn.	j
Let /:} "-> У lie a map. A convergent factorization of f consists of a
sequence {pg,Eq,fq}g>i such that
(a)	For q > 1, pg: Eg Ee_i is a fibration, and for q = 1, pi: Ei —> Y I
is a fibration.	|
(b)	For q > 1, fg: Y' —> Eg is a map, fg = pg+1 ° fg+i for q > I, and ।
f = Pi ° fi-	-j
(c)	For any n oo there is Nn such that fg is an n-equivalence for I
q > Nn.	‘
Conditions (a) and (b) imply that for q > 1, f equals the composite
SEC. 3 MOORE-POSTNIKOV FACTORIZATIONS	43g
pl0 •   ° Pq ° fa- The convergence condition (c) implies that, in a certain
sense, the infinite composite pi ° p2 °   • exists.
И {Pa’Eqfq} g>i is a convergent factorization of a mapf: Y' Y, then
pie sequence of fibrations Y E, . . . is convergent. The following
theorem shows that any convergent sequence of fibrations is obtained in this
way from a convergent factorization of some map.
j theorem If Eq E^   • is a convergent sequence of fibrations,
then {рд,Ед,ад}д>1 is a convergent factorization of the map ao- E„ —> Eo.
proof Conditions (a) and (b) for a convergent factorization are clearly
satisfied. To prove that the convergence condition (c) is also satisfied, given
j < n < 00, choose N so that pq is an (n + Inequivalence if q > N. We
prove that aq is an n-equivalence for q > N. Because ag = pg+i ° ag+i, and
pff+i is an n-equivalence for q > N, it suffices to prove that ajyis an n-equivalence.
Let (P,Q) be a polyhedral pair such that dim P < n and let a: Q —» E„
and /ifo P Eh be maps such that ftft | Q = an ° « We now prove that
there is an extension ft: P —> E,^ of a such that an ° P = ft к- The map a
corresponds to a sequence ag = ag ° a: Q —> Eg such that ag = pg+1 ° ae+i,
and to define a map ft: P —» EOT with the desired properties, we must obtain
a sequence of maps pg: P Eg such that fig | Q = ag, ftg = pe+i ° fte+i, and
ftn — ftft. Such a sequence of maps {pq} is defined for q < N by ftg =
pg+i 0   • 0 Pn ° f N, and for q > N it is defined by induction on q as follows.
Assuming fq defined for q > N, we use theorem 7.6.22 to find a map
P'q+r- E Eq+1 such that ftg+1 | Q — ae+i and such that ftg ~ pe+i ° ft'g+1
rel Q. We use the fact that pe+i is a fibration (and theorem 7.2.6) to alter ftg+1 by
a homotopy relative to Q to obtain a map ftg+i: P Eg+1 such that
pQ+11 Q = aQ+i and such that fig = pg+1 ° ftg+i. Thus the sequence {ftg} can
be found, and hence a map ft: P E„ with the requisite properties exists.
Taking P to be a single point and Q to be mpty, we see that an is
surjective, and so an maps 77о(Еет) surjectively to 77q(E^. Taking (P,Q) = (1,1),
we see that an maps rro(Em) injectively to 77о(Едг). Then an induces a one-to-
one correspondence between the set of path components of Ex and the set of
path components of En-
Let e^ = (ee) C E„ be arbitrary and let 1 < к < n. Taking (P,Q) = (Sk,Zo)
it follows that aN# maps rr^fE^e^) epimorphically to rqfEn,en). For 1 < к < n,
taking (P,Q) = (E/c+1,S/c), it follows that aN# maps •nj((Em,ess.') monomorphically
to TT]t(En,en)- Hence an is an n-equivalence. 
2 corollary Let {pq,Eq,fq}g>i be a convergent factorization of a map
ft Y' Y and let f': Y' E„ be the map such that ag° f’ = fg for q > 1
and a0 ° f' = f. Then f' is a weak homotopy equivalence.
proof For any 1 < n < oo there is q such that ag and fg are both
n-equivalences (by theorem 1). Then f' is also an n-equivalence (because
aq ° f' = fq). Since this is so for all n, f' is a weak homotopy equivalence. 
440	OBSTRUCTION THEORY СНЛР. fj
In particular, given a convergent factorization {pe,Ee,/Q}e>i of a weak
fibration p: E B, there is a weak homotopy equivalence g: p a0 consist-
ing of the commutative square
Iй"
В -1» В
If (X,A) is a relative CW complex, with inclusion map i: А С X, it follows
from theorem 7.8.12 that the relative-lifting problem for a map pair h: i—»p
is equivalent to the relative lifting problem for the map pair g ° h: i flo
We shall now add hypotheses which will ensure that the sequence of fibra-
tions into which the fibration ao is factored (namely, the fibrations {/),/}) leads
to relative-lifting problems, which can be settled by the methods of the last
section.
A Moore-Postniko v sequence of fibrations Eo E1 . . . js a convergent
sequence of fibrations such that py: Eg Er/_t is a principal fibration of type
(Ge,ny) for q > 1. A Moore-Postniko v factorization of a map f: Y' —> У is a
convergent factorization {pe,EQ,/Q}e>i of f such that Eo Ei • • • is a
Moore-Postnikov sequence of fibrations. A Postnikov factorization of a space
Y' is a Moore-Postnikov factorization of the map f: Y' Y, where Y is the
set of path components of Y' topologized by the quotient topology and f is
the collapsing map. Thus, if Y' is path connected, a Postnikov factorization of
Y' is a Moore-Postnikov factorization of the constant map Y' —» y0.
A Moore-Postnikov factorization of a map is a factorization of the map
(up to weak homotopy type) as an infinite composite of elementary maps.
The relative-lifting problem associated to this sequence is thereby factored
into an infinite sequence of elementary relative-lifting problems. We shall
show that Moore-Postnikov factorizations exist for a large class of maps
between path-connected spaces.
Let f: Y' Y be a map between path-connected pointed spaces. For
n > 1 an n-factorization of f is a factorization of/as a composite	1'41
such that
(a)	E' is a path-connected pointed space, p' is a fibration, and b' is a
lifting of / (that is, f — p' ° //)
(b)	b'#: nq(Y') nqfE') is an isomorphism for 1 < q < n and an epimor-
phism for q = n (that is, // is an n-equivalence)
(c)	pf: iTtfE') we(Y) is an isomorphism for q > n and a monomorphism
for q — n
A map /: Y' Y between path-connected pointed spaces is said to be
simple if/#(wi(Y')) is a normal subgroup of wi(Y) and the quotient group is
abelian, and if (Zf, Y') is n-simple for n > 1 (as defined in Sec. 7.3). We are
heading toward a proof of the result that a simple map admits Moore-Postnikov
factorizations. We need one more auxiliary concept.
Jgc. 3 MOORE-POSTNIKOV FACTORIZATIONS
441
Given a pointed pair (X,A) of path-connected spaces, a cohomology
class v € Hn(X,A; 77) is said to be n-characteristic for (X,A) if either of the fol-
lowing conditions hold:
(g)	n = 1 and i#(wi(A)) is a normal subgroup of 77i(X) whose quotient
group is mapped isomorphically onto 77 by the composite
77i(X)/i#(771(A)) Л H^Xj/i* (ЕЦА)) Л H^X,A) rr
(b) n > 1 and the composite
77?i(X,A) Л Hn(X,A) 77
is an isomorphism
In case A = {Xo}, the concept of n-characteristic element for the pair
(X,{xo}) agrees with the concept of n-characteristic element for the space X
as defined in Sec. 8.1.
3 lemma Let i: А С X be a simple inclusion map between path-connected
pointed spaces such that the pair (X,A) is (n — l)-connected, where n > 1.
Then there exist cohomology classes v £ Hn(X,A; tt) which are n-characteristic
for (X,A), where 77 = 77i(X)/i#(771(A)) for n = 1 and 77 = 77?i(X,A) for n > 1.
proof If n = 1, it follows from the absolute Hurewicz isomorphism theorem
applied to A and to X that there are isomorphisms
77i(X)/i#(77i(A)) £ Hi(X)/i;i!(Hi(A)) ~ H1(X,A)
By the universal-coefficient formula for cohomology, there is also an
isomorphism
lr. H\X,A-, rr) ~ Hom (Hi(X,A),77)
Hence, if 77 = 77i(X)/i#(771(A)), there exist 1-characteristic elements
v e нух,А; w).
If n > 1, it follows from the relative Hurewicz isomorphism theorem and
the universal-coefficient formula for cohomology that there are isomorphisms
<p: 77B(X,A)	H„(X,A) and /г: H»(X,A; 77) Hom (Hb(X,A),t7). Therefore, if
77 = 77b(X,A), there are n-characteristic elements v £ Hn(X,A; 77). 
4 lemma Let (X,A) be a pointed pair of path-connected spaces (n — 1)-
connectedfor some n > 1 and such that the inclusion map i: A C Xis simple.
Then there is an n-factorization A E' A, X of i such that p' is a principal
fibration of type (rr,ri), where 77 = 77i(X)/i#(771(A)) if n = 1 and 77 = 77?i(X,A)
ifn> 1.
proof By lemma 3, there is a class v £ Hn(X,A; 77) which is n-characteristic
for (X,A). Let CA be the cone (nonreduced) over A and observe that {X,CA}
is an excisive couple in X U CA. Therefore there is an element
v' C Hn(X U CA; 77) corresponding to v under the isomorphisms
H»(X U CA; 77) №fX U CA, CA; 77) №fX,A; rr)
442	OBSTRUCTION THEORY CHAP g
It is possible to imbed X U CA in a space X' of type (w,n) having an
«-characteristic element t' such that i' | X U CA = v'. Let p': E' X be the
principal fibration induced by the inclusion X С X' and let pj_: EA —» a be
the restriction of this fibration to A. There is a section s: A Ед such that
s(a) = for a C A, where is the path from x0 to the vertex of CA
followed by the path from the vertex of CA to a (that is, wa(t) = [x0,1 — 2t]
for 0 < t < Vz and w„(t) = [a, 2,t — 1] for ¥2 < t < 1). We define b': A£*
to be the composite А А Ед С E' and shall prove that А Д E' A X is an
«-factorization of i.
The fiber of p' (and hence also of pA) is ЯХ', and we define g: Ед —» QX-
by g(o,w) = co * (s(o))-1. Then g | ЯХ': ЯХ' SIX' is homotopic to the identity
map. If i": RX' С Ед is the inclusion map, it follows from the exactness of the
homotopy sequence of the fibration pj: Ед A that there is a direct-sum
decomposition
я-е(Ед) г#7те(ЯХ') © s#we(A) q > 1
(This is a direct-product decomposition for q = 1, but we shall still write it
additively.) We define a homomorphism X: wQ(X,A) we_i(S2X'), where
q > 1, to be the composite
тте(Х,А) %q(E',E1) А wQ_1(EA) ^_т(ЯХ')
We show that the following diagram commutes up to sign:
^(A) -A ^(X) Л тте(Х,А) ^_X(A)
^(Е')Е^^(Х) -Л^ях^Л^ДЕ')
In fact, the left-hand and middle squares are easily seen to be commutative.
We shall show that b# ° d = —i'# 0 X.
For q = 1 this is so because wq(A) = 0 implies that b# ° d is the trivial
map and the fact that is suqective implies that i'# 0 X 0 /# = i'# ° Э is also
the trivial map. For q > 1 we have
a = i#g#a + s#pA#a a £
Since the composite wQ(E',EA) wQ-i(Ea) А we_1(E/) is trivial, it follows that
for ft £ 7Tg(E',EA')
0 = гА#Э/1 = iA#i#g#9/l + ij^p^ft
= *#g#9/l + b'#dp#ft
By definition of X, we see that Xp#j6 ?= g#9j6. Therefore
i'^p#P + b'#Sp#P = 0
Since p#; tt^E'^a) ~ wQ(X,A), this proves b# ° Э = — i'# 0 X.
A straightforward verification shows that X is also the composite
; gEC. 3 MOORE-POSTNIKOV FACTORIZATIONS	443
'	7тй(Х,А) rrn(X U CA, CA) vn(X U CA) rrn(X') Ь ^-i(flX')
The construction of X' and t' С Нп(Х',тт) shows that there is a commutative
diagram
ттй(Х,А)	vn(X U CA, CA)	rrn(X U CA) чтп(Х’)
H„(X,A)	Hn(X U CA, CA)	Hn(X U CA) -+ Hn(X')
Therefore X: ттй(Х,А) wM_i(flX').
In case n = 1, Э: wi(X) —» 770(fiX') is surjective [because ^(А) = 0], and
so E’ is path connected. If n > 1, E' is path connected because тто(ЯХ') = 0.
Therefore E' is a path-connected pointed space. Since 77,/fiX') = 0 for q > n,
it follows from the exactness of the homotopy sequence of the fibration
p': E' X that p'#: ^(E') we(X) is an isomorphism for q > n and a
monomorphism for q = n.
Because X: we(X,A) —» 77e_i(fiX') is a bijection for q < n (the only non-
trivial case in these dimensions being q = n), it follows from the five lemma
and the commutativity up to sign of the diagram on page 442 that
b#: we(A) —» w0(E') is an isomorphism for 1 < q < n and an epimorphism for
q = n. Therefore b' and p' have the properties required of an n-factorization
of i. 
5	corollary Let g: X' X be a simple map between path-connected
pointed spaces such that for some n 1 the map g#: rrq(X') —> we(X) is an
isomorphism for 1 < q < n — 1 and an epimorphism for q — n — 1. Then
there is an n-factorization X' E' X of f such that p' is a principal
fibration of type (rr,n) for some abelian group <n.
proof Let Z be the reduced mapping cylinder of g (that is, the mapping
cylinder of g | Xq: x'o x0 has been collapsed to a point). Then (Z,X') is a
pointed pair of path-connected spaces (n — l)-connected and with simple
inclusion map i: X' C Z. By lemma 4, there is an n-factorization X' E" Z
of i such that p" is a principal fibration of type (w,n). Let p': E' X be the
restriction of p" to X. Then E' С E" is a homotopy equivalence, so there is a
map b": X' E' such that b" is homotopic to the composite X' E' С E".
Then p' ° b" is easily seen to be homotopic to g. By the homotopy lifting
property of p', there is a map b': X’ E' homotopic to b" such that
p' ° b' = g. Then X' E' x is easily verified to have the requisite
properties. 
444	OBSTRUCTION THEORY CHAR. §
We are now ready to prove the existence of Moore-Postnikov factoriza-
tions of a simple map between path-connected pointed spaces.
6	theorem Let f: Y' Y be a simple map between path-connected
pointed spaces. There is a Moore-Postnikov factorization {pg,Eg,fg}g>i of f
such that for n > 1 the sequence
Y' Д. E„	у
is an n-factorization of f
proof By induction on q, we prove the existence of a sequence {pg,Eg,fQ}g>1i
such that
(a)	For n — 1 the sequence Y' Ei	Y is a 1-factorization of f
(b)	For n > 1 the sequence Y'	En	En_i is an я-factorization of/„_1(
(c)	For n > 1, pn is a principal fibration of type (rrn,n) for some rrn.
Once such a sequence {pg,Eq,fq} has been found, it is easy to verify that
it is a Moore-Postnikov factorization of f with the desired property. Therefore
we limit ourselves to proving the existence of such a sequence.
By corollary 5, with n = 1, there is a 1-factorization Y' A Ei Y of f
with pi a principal fibration of type (Xi.,1) for some tti. This defines pi, Ei,
and /i. Assume {pg,Eg,fg} defined for 1 < q < n, where n > 1, to satisfy (a),
(b), and (c) above. By corollary 5, there is an n-factorization Y' En E?l_i
of /„_! such that p„ is a principal fibration of type (тг.п,п) for some тгп. Then
pn, E?!, and fn have the desired properties. 
7	corollary Let Y' be a simple path-connected pointed space. Then Y'
has a Postnikov factorization {pg,Eg,fg}g>i in which тгв(Ея) = 0 for q > n
pnd fn: Y' En is an n-equivalence.
proof If Y' is a simple space, the constant map Y' //o is a. simple map.
The result follows from theorem 6. 
In the above the spaces E„ approximate Y' in low dimensions. We now
present an alternate method of approximating a space in high dimensions by
killing low-dimensional homotopy groups.
8	corollary Let Y be a simple path-connected pointed space. There is a
Moore-Postnikov sequence of fibrations Y Ei . . . such that En is
n-connected arid pi °  • • ° pn: En Y induces isomorphisms ттд(Еп) ~ 7tq( Y)
for q > n.
proof If Y is a simple space, the inclusion map yo C Y is a simple map.
The result then follows from theorem 6. 
In the last result the fibration px: Ej —» Yhas the homotopy properties of a
universal covering space of Y. The fibration pi ° • • • ° pn: E„ -^ Y is a kind
, of “n-covering space.”
4 OBSTRUCTION THEORY
445
Д OBSTnUCTION THEORY
Jn this section we show how to use Moore-Postnikov factorizations to study
the relative-lifting problem. A sequence of obstructions to the existence of a
lifting (or to the existence of a homotopy between two liftings) is defined
iteratively, and we apply the general machinery to the special case where
either the first one or the first two obstructions are the only ones that enter.
Let p: E В be a fibration between path-connected pointed spaces and
assume that p is a simple map. By theorem 8.3.6, there exist Moore-Postnikov
factorizations {p<;,F,;,/,;},;>i of p. By corollary 8.3.2, there is a map p': E-^ E„
which is a weak homotopy equivalence. Since p = Oq ° p', where Gq: Erj._ B,
if (X,A) is a relative CW complex, with i: А С X, it follows from theorem 7.8.12
that the relative-lifting problem for a map pair from i to p is equivalent to the
relative-lifting problem for a corresponding map pair from i to Gq. Thus we
are led to consider the relative-lifting problem for a map pair from i to g0.
Let Eq Ei • • • be a sequence of fibrations with limit E^ and maps
ag: Em Eg and let (X,A) be a relative CW complex, with inclusion map
i: А С X. A map pair f: г ao is a commutative square
A Ex
X-^Eo
where f" corresponds to a collection {fg- A —> L;}<;>o such that pg+r ° fg+i = fg
for q > 0. For q > 1 lel /,;: i p, ° • • • ° pg be the map pair consisting of
the commutative square
A	E,;
X	Eo
If fg: X —> Eg is a lifting of fg, then pQ ° fg is a lifting of fg~i for q > 1 and a
lifting f: X Ea of f corresponds to a sequence (fg: X —> Eq}q>i such that
(g) fg is a lifting of fg for q > 1.
(b)	p,;+1 ° fg+i = fg for q > 1.
Given a lifting fg: X —» Eg of fg for q > 1, let g(/e): i pg+i be the map
pair consisting of the commutative square
Д A+1 p
Ps+i
X Eg
446	OBSTRUCTION THEORY CHAR, §
A map/e+i: X Eg+i is a lifting of g(fe) if and only if it is a lifting of
such that pQ+i ° fg+i = fg. Thus a sequence of maps {fg: X	satisfies
conditions (a) and (b) above if and only if it has the following properties:
(c)	fi is a lifting of f±.
(d)	For q > 1, fg+1 is a lifting of g(fe).
We now add the hypothesis that Eo A E| <AZ . . . js a Moore-Postnikov
sequence of fibrations. For each q > 1, pg is then a principal fibration of type
(7rQ,nQ). It follows from Sec. 8.2 that Д can be lifted if and only if
c(/i) C Hn'(X,A', 77i) is zero. The class c(ff) is called the first obstruction to
lifting f.
Assume that for some q > 1 there exist liftings fg-i: X Ед_г of the
map pair fg_r. i pi °  • • ° pg~i- We then obtain map pairs g(/e-i): i —» pg
and corresponding elements c(g(/Q_i)) g	77q). The collection
{c(g(/Q_i))} corresponding to the set of all liftings X Eq_! of fg_± is
called the qth obstruction to lifting f. It is a subset of H"'i(X,A; 77q) and is de-
fined if and only if fQ~i can be lifted. It is clear that there is a lifting of fQ if
and only if the qth obstruction to lifting f is defined and contains the zero
element of H'"'i(X,A; тгд).
Corresponding to a Moore-Postnikov sequence of fibrations we have been
led to a sequence of successive obstructions. The first obstruction is a single
cohomology class, while the higher obstructions are subsets of cohomology
groups. In some cases these obstructions can be effectively computed in terms
of the given map pair f:i—> ao, and this computation provides a solution of
the lifting problem in these cases. In general, however, the determination of
the successive obstructions involves an iterative procedure of increasing com-
plexity and has not been effectively carried out in each case.
We illustrate this technique by applying it to the Postnikov factorization
of a simple path-connected pointed space Y, given in corollary 8.3.7. There is
a Postnikov factorization {pg,Eg,fg}g>i of У in which 77Q(EBi) = 0 for q >m and
fm: У E,„ is an m-equivalence. We call this the standard Postnikov factori-
zation of У By corollary 8.3.2, there is a weak homotopy equivalence
f'; У —> E^, and so we consider the lifting problem for a map i ao, where
i: А С X and «о: Ex yo- Since у о is a point, this is equivalent to the exten-
sion problem for a map f": A Em.
Thus we seek a sequence of maps fg: X Eg such that Д: X Fi is an
extension of cq ° f" and fg+r. X Eg+1 for q > 1 is a lifting of the map pair
g(fe): i Pv+i consisting of
A	Eg+i
b+i
X	Eg
Since pg is a principal fibration of type (т7д(У,г/о), У + 1)> the obstruction to
lifting g(/Q) is an element of H«+1(X,A; т7в(У,г/о)). Hence there is defined a '
t SEC- 4 obstruction theory
447
I sequence obstructions to extending A Y, the f/lh obstruction being a
* subset of №+1(X,A; wQ(Y,yo))- If Y is (n — l)-connected for some n > 1, the
< fowest-dimensional nontrivial obstruction is in H"1 '(X,A; 77я(Т,у0)). If
; t g //"(Yyo; w) is n-characteristic for such a space Y, it follows easily from
л феогет 8.2.6 that this lowest obstruction is ±8f"*i. This gives us the fol-
1 jewing generalization of theorem 8.1.17.1
1 theorem Let i £ Hn(Y,yo; тг) be n-characteristic for a simple (n — 1)-
; connected pointed space Y, where n > 1, and let (X,A) be a relative
i CW complex such that №+1(X,A; 4TqfY,yo)) = 0 for q n. A map f: A—>Y
( can be extended over X if and only if 8f* («) = 0 in Hn+1(X,A; чт).
j
 proof We use the standard Postnikov factorization of Y. This leads to a se-
quence of obstructions to extending /'which are subsets of №+1(X,A; 4Tq(Y,yoj).
• Since these are all zero except Hn+±(X,A-, 4Tn(Y,yoj) ~ Hn+1(X,A; чт), the only
; obstruction to extending f is an element of Hn+\X,A-, чт). By the remarks
\ above, this obstruction vanishes if and only if 8f* (t) = 0. 
; Let fo, fr. X—>Y be maps and define g: X X I —> Y by g(x,0) = ffx)
; and g(x,l) = fi(x). For any и £ №(Y), 8g*(u) = (-l)qr(ffu - ffiu) in
 №+1(X,x4). Therefore 6g*(u) = 0 if and only if f$(u) = f*(u), alK' we obtain
; the following partial generalization of theorem 8.1.15 by applying theorem 1
J to the pair (X X I, X x 7).
I 2 theorem Let 1 £ Hn(Y,yo; чт) be n-characteristic for a simple (n — 1)-
1 connected space Y, where n > 1, and let X be a CW complex such that
Hq(X-, 4T,j(Y,yo)) = 0 for qf>n. Thenfo, /1: X—> Y are homotopic if and only
-
j This last result gives a condition that the map [X;Y] —> Нп(Х,чт)
, be injective. The condition that be surjective is that if {p<;,L,;,/'(/},;> । is the
standard Postnikov factorization of Y, then any map X —» En+1 can be lifted.
The obstructions to lifting such a map lie in №+1(X; 4Tq(Y,yo)) for q > n.
Therefore, by combining these, we have the following result.
J 3 theorem Let 1 C. Hn(Y,yo; чт) be n-characteristic for a simple (n — 1)-
I connected space Y, where n > 1, and let X be a CW complex such that
I Hq(X;4rq(Y)) = 0 and Hq+1(X;4rq(Y)) = Qforallq f> n. Then there is a bijection
;	[X;Y] Нп(Х;чт) 
j These last results have been derived by assuming hypotheses which ensure
I that the lowest-dimensional obstruction is the only nontrivial one. In this case
I we are essentially studying maps to a space of type (чг,п). The case where the
|l two lowest-dimensional obstructions are the only nontrivial obstructions is
essentially the study of maps to a fibration E В of type (G,q), where В is a
•’	1 See S. Eilenberg, Cohomology and continuous mappings, Annals of Mathematics, vol 41,
pp. 231-251, 1940.
448	OBSTRUCTION THEORY СНдр, g
space of type (тг,п). Before we consider this, let us establish some cohomology i
properties of X X I-	j
Define inclusion maps
A X I U X x 1 С A X I U X x t С (A X I U X X t, A x I U X X 1>
There is a weak retraction г: A x I U X X I A X I U X X 1 defined by
r(x,t) = (x,l) for (x,t) d A Xi U X xl (that is, r ° q is homotopic to the
identity map of A X I U X X !)• Using the exactness of the cohomology
sequence of (A X I U X X I, A X I U X X 1), it follows that for an arbitrary
element и £ №(A X I U X X I) there is an associated unique element '
u' E №(A x I X) X x I, A X f U X x 1) such that
и = jfu' + i*ifu
Let h: (X,A) (A X I U X X I, A X I U X X 1) be defined by
/i(x) = (%,()) for x E X. Then h induces an isomorphism
h*: №(A XIU Xxt, A X 1 U X X 1) X №(X,A)
and we define an epimorphism
Д: №(A X I U X X /'H №(X,A)
by Д(м) = h* u', where и’ E HfiA X I U X X L A X I U X X 1) is the
unique element associated to u. Then Д is a natural transformation on
the category of pairs (X,A).
4 lemma Commutativity holds in the triangle
№(A X I и X X j) A H«+1((X,A) X (1,7))
zp-1)5+4
№(X,A)
proof Let г: X X I —> A X. I С X X, 1 be defined by f(x,t) — (x,l). Then
f | (A x I U X x i) = Л and so r * i J и = (f * i * и) | (A X I U X X I) for
и E №(A X IU Xx f). For any v E №(X x l),8(c|(A X 1 U X x I)) = 0.
Therefore, 8r * i J и = 0, and to complete the proof it suffices to show that for
и' E №(A X I U X x 4 A x I U X x 1), S/T(«') = (- 1)«+М1* (id). This
follows from the commutativity of a diagram analogous to the one used in the
proof of theorem 8.2.4. 
5 corollary Let (X,A) he a relative CW complex, with inclusion map
i: А С X, and let p': QB' —» b'o he the constant map, where B' is a space of
type (rr,n + 1). Given a map pair f: i —» p' and two liftings fo, fr. X —> W
off, let g": AxIX)Xxi—> QB' be defined by g"(x,0) = fo(x), g"(x,l) =
fi(x), and g"(a,t) = fo(d)- If d C. Hn(QB',u'o; tt) and i € Hn+1(B',b'o; tt) are
related characteristic elements, then d(fo,ffi =
sec. 4 obstruction theory	44g
proof Let g: i' —> p' be the map pair consisting of the commutative square
AxlUXxI^W
4 к
Xxl	b'o
prom the definition of d(fo,fi) we have d[f0,fi) = ( — l)n+1r^1(c(g)). By theo-
rem 8.2.6 c(g) = — Sg"*(«'), and therefore d(f0,fi) = (— l)nr-18g"* («').
The result follows from this and lemma 4. 
6 lemma Let h0, hp. (X,A) -а (А X I U X X t, A X I) be defined by
ji0(x) = (x,0) and hfix) = (x,l). For any и £ №(A x I U X x t, A X I)
Д(» I (A X I и X X i)) = h%(u) - h* (u)
proof There are inclusion maps
(A x I U X X 1, A X I) C (A x I U X x t A X I) C
(A X I U X x t, A X I U X X 1)
and a weak retraction r't (A X Г U X X t, A X I) (A X I U X X 1, A X /)
defined by /(x,/) = (x,l). For v £ №(A X I U X X I, A X I) there is an
associated unique element v' E №(A X I U X X I, A X I U X X 1) such that
v = /1 * v’ + r' * il * v
If к: A X I U X X t С (A X I U X X I, A X 4), we then have
k*v — k*ji*v' + к*/*й*« = /*v' + i*ifk*v
Therefore Дк*и = h*v'. Since h = /1 ° ho and 7q =	°/° ho, we have
Дк*и = hfij'i*v' = h% (v — / * i'i * v) — hfiv — h$v 
7	corollary Given a map pair g; i' —> p, where (X,A) is a relative CW
complex, i': A x I С A X I G X X t, and pt E В is a principal fibration
of type (G,q) induced by a map в: В В', let f0, fit i —> p be the map pairs
from i-. А С X to p defined by restriction of g to (X,A) x 0 and (X,A) x 1,
respectively. Then
= c(fo) - c(fr)
where g': A X I U X x I В is part of the map pair g.
proof The obstruction c(g) £ №(A X 11) X xl A X I; rr) has the prop-
erty that c(g) | (A X I U X X t) is the obstruction to lifting g'. Therefore
c(g)|(A X I U X X i) = g'*0*(t)
By the naturality of the obstruction, h^c(g) = c(/0) and h* c(g) — c(fj). The
result now follows from lemma 6. 
OBSTRUCTION THEORY CHAP, fj T S| ( . 4 OBSTRUCTION THEORY
j proof Let Fo: i' p be the map pair consisting of
451
450
Let 6 be a cohomology operation of type (n,q-, tt,G). Given a cohomology
class и E H"(X;77), we define a map &(6,u): Hn(X,A-, it) —> №(X,A; G) by
Д(б»,и)(о) = A0(;*h*-1(n) + B*«) v E H«(X,A; it)
where к: A X I U X X I X is defined by k(x,t) — x. In case 6 is an addi-
tive cohomology operation, we have
X(F,u)(v) =	+ k*0(w)) = F(v)
Therefore A(0,u) = в if в is additive.
Given a cohomology operation 6 of type (n,q; rr,G) and a cohomology
class и F Hn(X-,rr), we define a map SA(fi,u):	rr) —> H'; I(X,A; G) by
the equation SA(fi,ii) = t’1 ° &(0,u') ° t, where и' E Hn(X x I; тг) is the
image of и under the homomorphism induced by the projection X
If 0 is an additive operation, then SA(0,u) — SO. In any case, we have the
following analogue of corollary 8.1.14.
8	lemma If 0 is a cohomology operation of type (n,q; rr,G) and
и F №!(X;tt), the map
SX(0,u): H«~\X,A-, it) -> №~i(X,A-, G)
is a homomorphism.
proof Let I] = [0,14], Ij = {0,%}, I2 = [14,1], and Z2 = {14,1}, and let
Vi, v2 F FP^ifiXA; тт). Let vi = E Hn((X,A) X (ZiJi)) and let
«2 = т2(и2) E НИ((Х,А) x (I2Дг))> and let v F Hn((X,A) X (I, L U Z2)) be the
unique class such that v | (X,A) X (Lji) = vi and v | (X,A) X (Z2,Z2) = t>2.
Then о | (X,A) X (I,t) = r(t>i) + t(u2). Since 0 and Д are both natural,
X(6,u')(v) | (X,A) x (Ц) = tSA(0,h)(o1 + v2)
and	Д(0,u')(v) | (X,A) X (Iiji) = TiSA(e,u)(t>i)
A(0,u')(v) | (X,A) x (I2,Z2) = r2SL(6,u)(v2)
Therefore, as in the proof of lemma 8.2.3,
tSX(6, u)(vi + v2) = rSA(0,n)(t>i) + tSA(0,w)(c2)
Since т is an isomorphism, this gives the result. 
Let В be a space of type (тт,п) and let p: E В be a principal fibration
of type (G,q) induced by a map в: В —> В'. Let O' = 0*(t') E Нч(В,Ь0; G)
correspond to a cohomology operation 0 of type (n,q; rr,G) (that is, 0(i) — O').
Given a CW complex X, a map f: X В can be lifted to E if and only if
0(f * («)) = 0. For any element и F Hn(X;if) such that 0(ii) = 0 it follows that
there are liftings f: X —> E such that (p ° f) * (i) = u. We shall determine how
many homotopy classes of such liftings there are.
9	lemma Let fo, fy. X E be maps such that p ° fo = p ° fi (that is,
fo and fi are liftings of the same map X —-> B). Then f о if and only if
there is d F Н^ЦХ;^) such that d(f0,fi) = SA(0,u)(d), where и = (p ° fo)*(0-
where Ео(х,О) = fifx), Во'ДД) = fi(x), and F&(x,t) = pf0(x). Then tl(/o,/i) =
It is clear that /0 ~/i if and only if there is a homotopy
Fi: X X I —> В from p ° f0 to p ° fi such that for the corresponding map pair
Fy i' p we have c(Fi) = 0. Let G': (X X I) XlU (X X I) X I В be
defined by G'(x,0,t) = G'(x,l,t) = pf0(x), G'(x,t,0) = F'0(x,t) and G'(x,t,l) =
Fi(x,t). By corollary 7,
AG'*(^) = c(Fo) - c(Fi)
'fhus fo — fi if and only if there is a map Fi: X x I В such that for the
corresponding map G' we have
d(fo,fi) = (-l)«T-i(AG'*(0'))
It is easily verified that G' * (() = / * h* -1AG' * (t) + k*u', where
и' E Hn(X X I; w) is the image of it = (p° fo)* («) under the projection
X X I X. By definition,
AG'* (6»') = AG'*0(<) = A0G'*(i) = A(0,h')(AG'*(i))
Since F'o, Fi: X X 1^- В are two liftings of the map pair
Xx t -+ В
j	X X I b0
j it follows from corollary 5 that d(Fo,Fi) = — XG' * (i), and by theorem 8.2.4,
( given d F EF^X;^), there is a homotopy Fi: X X I В from p ° fo to p ° fi
I such that AG' * (i) = (— l)«r(d). Combining all of these, we see that f0 = fi
! if and only if there is d F Н^ЦХ^) such that
j	d(fo,fi) = r^(e,u')r(d) = sx(e,u)(d) 
f We summarize these results in the following classification theorem.
IO theorem Let p-.E^Bbe a principal fibration of type (G,q) over a
i space В of type (rr,n) induced by a map 0: В —> В' such that 0*(i') = 0(t).
i Given a CW complex X, there is a map f: [X;E] Нп(Х;тт) defined by
 ’ИЛ = (P ° /)*(«)• Then im = (u Нп(Х;тт) | 6(u) = 0}, and for every
I и F im f the set f^fit) is in one-to-one correspondence with
1	He-l(X;G)/SA(6l,t(.)H«-1(X;7r)
proof We have already seen that im f is as described in the theorem.
Given и E im f, let f0: X E be such that ^[ /o] = u. Given any map
4S2	obstruction theory chap. 8
fr. X—> E such that ^[fi] — u, there is a map fi: X E homotopic to f
such that p ° f i = p ° fo (by the homotopy lifting property of p). To such a
map fl we associate the element tl(fo,f 1) E №-1(X;G). In this way the set of’
maps X —> E which are liftings of p ° f0 is mapped into H«-1(X;G), and bv
theorem 8.2.4, this map is surjective.
Two maps fi,f2: X E such that p ° Д = p ° f0 - p ° f2, are homotopic
by lemma 9 if and only if dlfi,f2) € SA(0,u)Hn~1(X;7r). By lemma 8.2.3
d(fo,f2) = d(fo,fi) + dlfafz), and so fi ~f2 if and only if d(f0,ff) and
d(f0,f2) belong to the same coset of S'A(0,u)H"1(X;77) in H«“1(X;G). Hence
the function which assigns the coset tl(f0,fi) + SA(0,ri)Hn-1(X;7T) to a map
ft: X —> E with p ° fi = p ° f0 induces a bijection from f-1(ti) to
№-1(X;G)/SA(6l,w)H’’-i(X;77) 
We now apply this to the complex projective space Pm(C) for m > 1.
There is a map Pm(C) -r> P0O(C) and PJC) is a space of type (Z,2), by
example 8.1.3. Furthermore, if t is a characteristic element for РДС) andB'
is a space of type (Z, 2m + 2), there is a map 0: P„(C) —» B' such that
0 * (t') — (()’"+'. For the principal fibration p: E P0O(C) induced by 0 there is
a map Pm(C) —> E which is a (2m + 2)-equivalence. In this case the operation
0 is the (m + l)st-power operation, and therefore
SA(0,m)(o) =	Ь*-1(т(о)) + /<*n']’"+1
= т-1Д[(т + l)/< *(;/)’" о j * h* -1(т(ю))] = (m + Y)um о v
because t(v) о t(v) = 0. This gives the following application of theorem 10.
11 theorem Let i f H2(Pm(C);Z) be 2-characteristic for Pm(C) and letX
be a CW complex. Define ф: [X;Pm(C)] -> №(X-,Z) by if[/] = f * (t). If
dim X < 2m + 2, then im f = { // E H2(X;Z) | um+1 = 0}. If dim X < 2m + 1,
then f is surjective, and for a given и € H2(X;Z), f-1(«) is in one-to-one
correspondence with НЯт+ЦХ^/^т + l)um о HX(X;Z)]. 
5 THE SUSPENSION MAP
One of the most useful tools for the study of the homotopy groups of spaces
is the suspension homomorphism from ttq(X) to 7rg+i(SX). Iteration of this
homomorphism yields a sequence of groups and homomorphisms
77e(X)	77e+1(SX)	77e+2(S2X)	
This sequence has the stability property that from some point on, all the
homomorphisms are isomorphisms. For a fixed X and q, therefore, there are
only a finite number of different groups in the above sequence.
In this section we shall study the suspension map in some detail and
establish the stability property. This will enable us to compute 7Tn+i(Sn) for
all n. Knowledge of these groups, combined with obstruction theory, will lead
SEC. 5 THE SUSPENSION MAP	453
10 the Steenrod classification theorem, which closes the section.1
We consider the category of pointed spaces and maps. There is a
functorial suspension map S: [X;Y] [SX;SY] such that S[/] = [S/]. The
exponential correspondence defines a natural isomorphism
[SX;SY] [X;fiSY]
and we define S: [X;Y] [X;fiSY] to be the functorial map which is the
composite of S with this isomorphism. The following result shows that S is
induced by a map Y KSY.
] lemma Let p: Y fiSY be the map defined by p(ijfit') = [y,t] for у E Y
and t E I. Then for any space X
S = p#-. [X;Y] -a [X;fiSY]
proof The exponential correspondence takes the identity map SY C SY to
the map p: Y KSY. Because of functorial properties of the exponential
correspondence, it takes the composite
SX SY C SY
to the composite
X 4 Y 4 fiSY 
Thus, to study the suspension map S, we study the map p. To do
this we use the fibration PSY SY, which has fiber KSY With this
in mind, let us investigate homology properties of fibrations over SY.
We assume that yo E Y is a nondegenerate base point. We define
ELY = {[y,t] E SY | 0 < t < %} and C+Y = {[y,t] £ SY \ Vz < t < 1}. Then
SY = C _ Y U C+Y, and there is a homeomorphism Y ~ C_Y A C+Y (sending
у to [y, La]) by means of which we identify Y with C_Y П C+Y. Let S'Y be
the unreduced suspension defined to be the quotient space of Y X I in which
Y X 0 is collapsed to one point and Y X 1 is collapsed to another point and
let CLY,C±Y be analogous subspaces of S'Y (so CLY П C'+Y = Y). The map
collapsing S'yo in S'Y is a collapsing map k; S'Y —» SY such that k(CLY) = C_Y
and k(C;Y) = C+Y.
2	lemma If yo is a nondegenerate base point, the collapsing map
k: S'Y —> SY defines a homotopy equivalence from any pair consisting of the
spaces S'Y, CLY, C'+Y, and Y to the corresponding pair consisting of SY, ELY,
C+Y, and Y.
proof Because yo is a nondegenerate base point of Y, it follows, as in the
proof of lemma 7.3.2c, that Y X 1 U yo X 1 C Y X 1 is a cofibration. Let
[y,t]' E S'Y denote the point of S'Y determined by (x,t) E Y X I under the
quotient map k': Y x I —> S'Y. Let H': (Y X 1 U yo x 1) X 1 a S'Y be the
homotopy defined by H'(y,0,t) = [yo,t/2]', H'(y,l,t) = [yo, (2 — t)/2]', and
H'(yo,d,f} = [yo, (I — t)t' + t/2]'. Then H' can be extended to a homotopy
1 The first detailed study of the suspension map appears in H. Freudenthal, Uber die Klassen
der Spharenabbildungen I, Compositio Mathematica, vol. 5, pp. 299-314, 1937.
454	OBSTRUCTION THEORY CHAP, fj
H": Y X I X I S'Y such that H"(y,t,Q) = k'(y,t). Since H"(y,0,t) = //"((/',()/)
and H"(y,l,t) = H"(y',l,t) for all у, у' £ У, it follows that there is a
homotopy H: S'Y xl~> S'Y such that H([y,t]’, t') = H"(y,t,f). Then His a
homotopy from the identity map of S'Y to a map which collapses S'y0 t0 a
single point such that H(S'y0 X I) C S'y0. Since H(B X I) С В if В = CLY
C'+Y, or Y, the result follows from lemma 7.1.5. 
3	corollary If Y is a path-connected space with nondegenerate base
point, then SY is simply connected,
proof By lemma 2, S'Y and SY have the same homotopy type, so it suffices
to prove that S'Y is simply connected. It is clearly path connected, being the
quotient of the path-connected space Y X I.
Let U- = {[y,#]' 6 S'Y | t < 1} and I7+ = {[y,t]' E S'Y | 0 < t}. Then
U~ and U+ are each open and contractible subsets of S'Y. If w is any closed
path in S'Y at [//о, L]', there is a partition of I, say, 0 = to < < • • • < = 1
such that for each 1 < i < n either	C LL or w([hi.h]) C U+,
Furthermore, it can be assumed that ti £ IL A U+ for all 0 < i < n (if some
h is not in U- A U+, it can be omitted from the partition to obtain another
partition of I satisfying the original hypothesis, and iteration of this procedure
will lead to a partition’ having the additional property demanded). Since
[7_ A U+ is homeomorphic lo Y X R, it is path connected. For each i let щ
be a path in I7_ A U+ from w(/j_i) to «(h) and let w' be the closed path at
[yod/z]' defined by w'(t) = w{((t —	— h_i)) for t^ < t < tt.
Because U- and U+ are each simply connected, w | is homotopic to
a' | [ti-i,ti] relative to {hi,/;}. Therefore w ~ w' rel I. Since w' is a closed
path in U+, it is null homotopic. Therefore w is null homotopic and S'Y is
simply connected. 
4	corollary Let Y have a nondegenerate base point and let p: E SY
be a fibration. Then {p-1(C_Y),p-1(C+Y)} is an excisive couple in E.
proof Let pL E' -» S'Y be the fibration induced from p by k: S'Y —> SY and
let к: E' —> E be the associated map. It follows from lemma 2 that к induces
vertical isomorphisms in the commutative diagram
H#(p'-i(C;Y),p'-i(Y))	HJE',p'~fiCLY))
4	1=
H#(p-i(C+Y),p-i(Y))	H^(E,p~\C_Y))
Since C'+ Y is a strong deformation retract of U+ (with U+ as defined in
corollary 3) and Y is a strong deformation retract of [7+ A CL Y, it follows that
p'fiCfY) and p'~1(Y) are strong deformation retracts of p'-1({7+) and
p'^fL A CLY), respectively. This implies that {p'-1(CEY),p'-1(C4Y)} is an
excisive couple. From the commutative diagram above, the result follows. 
Because C+Y and C_Y are contractible relative to yo, it follows, as in
Sec. 2.8, that for any fibration p: E —» SY with fiber F — p~\yo) there are
4f.C. 5 THE SUSPENSION MAP	455
I gber homotopy equivalences/L: C_Y X F p~1(C^Y) and g+: p-1(C+Y)
I Ср’ X 7 such that | y0 X F is homotopic to the map (y0,z) z and
J g4 I F is homotopic to the map z (yo,z). The corresponding clutching
; function p: Y X F Y is defined by the equation
j	g+f-M = (y, M(y>z)) У E У, z E F
I Then p I yo X F is homotopic to the map (yo,z) z.
5	theorem Let p: E SY be a fibration with F = p^yo), where yo is
d nondegenerate base point of Y. If p: Y X F —» F is a clutching function
for p, there are exact sequences (any coefficient module)
I --------> Hq(E) -a H,;(C Y x F, Y X F)	H^fiF)	Hq^(E)	• • •
_______> №(E)	№(F) IF’LCLY x F, Y X F)№+1(E)	 • •
proof Consider the exact homology sequence of (E,F)
-----> Hq(F) Hq(E) Hq(E,F) A Нд-fiF) -+   
Using homotopy properties and corollary 4, there are isomorphisms induced
by inclusion maps
Hq(E,F) Hq(E,p~\C+Y))	Hq(p~1(C_Y),p~1(Y))
There is also a homotopy equivalence
I	f_: (C_Y X F, Y x F) -+ (p-^C.Y), p~\Y))
and a commutative diagram
Hq(E,F) Hq(E,p~\C+Y))	Hq(p~\C^Y),p^(Y)) Hq((C_Y,Y) X F)
al al	al	a!
j Hq_fiF) Hq_fip-\C+Y)) Hg^fiY))^--!^^. Нд-fiY x F)
There is also a homotopy equivalence g+: p-1(C+Y) C+Y X F and
isomorphisms
HQ_!(p-i(C+Y)) Hq_fiC+Y xF)^ Hq_fiF)
where the right-hand homomorphism is induced by projection to the second
factor. Because g+ | Fis homotopic to the map z —» (yo,z), the above composite
equals -1. By definition, p is the composite
Y x F Л|Гх<> p-i(Y) C p-l(C+Y) C+Y X F^ F
Therefore there is a commutative diagram
Hq(E,F) Hg((C_Y,Y) X F)
al la
Hq^(F) ф Hq^(YxF)
456	OBSTRUCTION THEORY CHAP, §
The desired exact sequence for homology follows on replacingHQ(E,E) by
H,/(GY,Y) X F) and c by у.,, й in the homology sequence of (E,F). A similar
argument establishes the exactness of the cohomology sequence. 
Specializing to the case where У = S”-1, by lemma 1.6.6, S(S’1-1) is
homeomorphic to S’1, and we obtain the following exact Wang sequence of a
fibration over Sn.
6	corollary Let p: E Sn be a fibration with fiber F. There are exact
sequences
-----> H^F) Hq(E)	Hq_n(F) Hq_fiF)	• • •
-----> №(E) A №(F)	№~n+\F)	№+1(E)	• • •
If the second sequence has coefficients in a commutative ring with a unit,
then
6(ll О V) = 6(u) О V + ( — deg«t/ 0^
proof Letting У = Sn-1 in theorem 5, we have (СУ,У) homeomorphic to
(En,Sn^1). Therefore
Я;((СУ,У) X F)	X Fl = Hq_n(F)
and the exact sequences result from the exact sequences of theorem 5 on
replacing HfC. Y X F, Y X F) and №(С_У X F Y X F) by Hq~n(F) and
Hrn(F), respectively. The additional fact concerning 6 results from the obser-
vation that for the map p*: №(F)	x F) the definitions are such that
/'* (u) = 1 x и + s* X 6(u)
where s* E Hn~1(Sn~1) is a suitable generator. Then, since s* о s* = 0,
1 x (и о v) + s* X 6(u о v)
= /j* (и о v)
= [1 X U + s* X 0(uj] о [1 X V + s* X #(«)]
= 1 X (HO tj + .F X [6(u) О V + ( —deg«ti
This implies the multiplicative property of в. 
We now specialize to the path fibration p: PSY —> SY with fiber KSY.
In this case there is the following simple expression for a clutching function.
7	lemma Let C_Y p^^CJY) and s+: C+Y —» p-1(C+Y) be sections
of the fibration p: PSY —» SY such that s_(yo) and s+(yo) ore both null
homotopic loops. Then the map p: Y x SSY fiSY defined by
p(y,u) = (U * s_(y)) * s+(y)-J
is a clutching function for p.
proof Such sections exist because C_Y and C+Y are contractible relative
to yo- We define fiber-preserving maps
5ЁС. 5 THE SUSPENSION MAP
C_Y X QSY p-'iCLY)
f+-. C+Yx®SY^p-4C+Y)
457
g_: р'(С'_У)	C_Y X Й8У
g+: рЛС+У)	С+У X fisy
fiy/_(z,w) = w * s_(z) andg_(w) = (p(w),w * (s_p(w))-1) and/+(z,w) = w * s+(z)
and g+(w) = (p(w), w * (s+p(w))-1), respectively. It is easy to verify that
g_ ° /_ is fiber homotopic to the identity map of С. У x OS'У and/ ° g_ is
fiber homotopic to the identity map of р л(С_У). Therefore /_ is a fiber
homotopy equivalence. Similarly, g+ is a fiber homotopy equivalence. Further-
more, /_(yo,w) = w * s_(yo) is homotopic to the map (yo,w) —> w because
s_(y0) is null homotopic. Similarly, for и £ OS'У, g/w) = (yo, w * sfyo)f is
homotopic to the map w (y(),w). Therefore the composite
У X OSY Л» p-^Y) У X OS'y osy
is a clutching function for p. This composite is the map
(y,w) (w * sjiy)) * s+(y)"1 "
Let s_ and s+ be sections as in lemma 7 and let p/: У OSY be defined
by y'(y) = s-(y) * si (?/)'• is called a characteristic map for the fibration
p: OS'y^ SK
8	cobollaby Let у': У —> OS'}7 be a characteristic map for the fibration
p: PSY —> SY. The map Y X OSY —> OSY sending (y,u) to w * y'(y) ™ homo-
topic to a clutching function for p.
pboof This follows from lemma 7, because the map
(y,w) -» (w * s_(y)) * s+(y)-1
is clearly homotopic to the map (y,w) w * (s-(y) * s+(l/)-1) — w * Р'(У)- "
The following theorem is the main part of the proof of the suspension
theorem.
9	theobem Let У be n-connected for some n > 0 and let yo be a non-
degenerate base point of Y. If у': У —> OSY is a characteristic map for the
fibration p: PSY —> SY, then y' induces an isomorphism
p'*: HfY) = Hq(Q,SY) q < 2n + I
pboof By corollary 3, SY is simply connected. By corollary 4, (С. У,С+У }
is an excisive couple, and from the exactness of the reduced Mayer-Vietoris
sequence, He(SY) ~ H,;_|(Y). Combining these with the absolute Hurewicz
isomorphism theorem, SY is (n + l)-connected. Therefore OSY is n-connected.
Because PSY is contractible, it follows from the version of theorem 5, using
reduced modules, that there is an isomorphism
y*d: Hq((C_Y,Y) X OS'Y)
If Wo is the constant loop, then because S2SY is n-connected and (C_Y,Y)
is (n + I)-connected, it follows from the Kiinneth theorem that the inclusion
458	obstruction theory chap. 8
map (C_Y,Y) X “о C (C_Y,Y) X LS'Y induces an isomorphism
HQ((C_Y,Y) X Wo) ~ Hq((C_Y,Y) X fiSY) q < 2n + 2
Let p: Y X SSY —> LS'Y be a clutching function which is homotopic to
the map (y,w) —> w * //(y) (such a /.( exists, by corollary 8). Since p(y,Wo) js
homotopic to the map у —> //(у), there is a commutative diagram
Н(/(С У,У)	Hq((C-Y,Y) X WO)	H,/(CY,Y) X fiSY)
"I
He_i(Y) HQ_t(Y x w0) Hq_i(Y X fiSY)
HQ_r(fiSY)
The result follows from the commutativity of this diagram. 
IО corollary Let Y have a nondegenerate base point. If Y is n-connected
for n > 0, the map p: Y —> QSY induces an isomorphism
P* :H^Y) Hq(flSY) q < 2n + 1
proof Let : C_Y —> p-1(C_Y) and s+: C+Y p-1(C+Y) be the sections
defined by s_[y,t](t') = [y,tt'] and л ([у,/](/') = [у, 1 — t' + tt'J. The corre-
sponding characteristic map is equal to the map p: Y fiSY. The result
follows from theorem 9. 
We are now ready for the following suspension theorem.1
11 theorem Let Y be n-connected for n > 1 with a nondegenerate
base point and let X be a pointed CW complex. Then the suspension map
S: [X;Y] [SX;SY]
is surjective if dim X < 2n + 1 and bijective if dim X < 2n.
proof Because Y and OS Y are simply connected, it follows from corollary 10
and the Whitehead theorem that p is a (2n + Inequivalence. The result
follows from corollary 7.6.23 and lemma 1. 
Let Y be a space with a nondegenerate base point. Then SY also has a
nondegenerate base point and is path connected, S2Y is simply connected,
and SmY is (m — l)-connected. If X is a CW complex, so is SmX, and
dim (S’"X) = m + dim X. Hence, if X is finite dimensional and m > 2 + dim X,
it follows from theorem 11 that S: [S’"X; S’"Y] ~ [S’"+IX; S’"+IY ]• Therefore,
for any finite-dimensional CW complex X the sequence
[X;Y] A [SX;SY] A • • • A [SffiX;S'«Y] A •  
1 For a general relative form of this theorem see E. Spanier and J. H. C. Whitehead, The theory
of carriers and S-theory, in “Algebraic Geometry and Topology” (a symposium in honor of
S. Lefschetz), Princeton University Press, Princeton, N.J., 1957, pp. 330-360.
5 SEC. 5 THE SUSPENSION MAP	45g
' consists of isomorphisms from some point on. Taking X = Sn+k and Y = S’1
' and recalling that the suspension of a sphere is a sphere, we see that there is
; a sequence
!	^n+k(Sn) Д 7гя+7с+1(8и+1) Д • • •
consisting of isomorphisms from some point on. The direct limit of this
sequence is called the к-stem. It follows from theorem 11 that the k-slein is
isomorphic to rT2k+2(Sk+z). In particular, the О-stem is infinite cyclic. The fol-
lowing result determines the 1-stem.
12 THEOREM 774(S3)	Z2.
proof Let u0 € H°(S2S3) be the unit integral class and define generators
t(j £ H2i(fiS3), by induction on i from the exactness of the Wang sequence in
corollary 6 for the fibration PS3 S3, by the equation
0(«i+i) = ut i > 0
Because в is a derivation, 0(rq о rq) = 2«i, whence lit о tq = 2ii2- We
know 772(fiS3)	77b(S3) is infinite cyclic. It follows that OS'3 can be
imbedded in a space X of type (Z,2) such that the inclusion map OS3 С X
induces an isomorphism ^(OS3) ~ t72(X). Since Pra(C) is also a space of type
(Z,2), it follows that H* (X) ~ H* (P0O(C)) line (H* (P/C))} is a polynomial
algebra with a single generator v C H2(X), and v can be chosen so that
V I OS3 = tq.
An easy computation using the exact cohomology sequence of (X,0S3)
establishes that №(X,QS3) = 0 for q < 5 and H5(X,OS3) ~ Z2. By the
universal-coefficient formula, He(X,0S3) = 0 for q < 4 and H4(X,0S3) ~ Z2.
By the relative Hurewicz isomorphism theorem, 774(X,0S3) ~ Z2. Because
tr3(X) = 0 = tt4(X), we have Tt4(X,fiS3) zz 7t3(fiS3)	w4(S3). 
The (n — 2)-fold suspension of a generator of 77g(S2) is a generator of
trn+i(Sn) (because S: 7q(S2)	Tt4(S3) is an epimorphism, by theorem 11).
Attaching a cell to Sn by this map must, therefore, kill 7tn+1(Sn). The resulting
CW complex has the same homotopy type as the (n — 2)-fold suspension of
the complex projective plane P2(C). Therefore we have proved the following
result.
13 COROLLARY 77,l+i(Sn 2(Рг(С))) = О n > 2 
We want to classify maps of an (n + l)-complex into Sn. For n = 2 this
is given by the case m = 1 of theorem 8.4.11. By using the standard
Postnikov factorization of S”, we are reduced to classifying maps of an
(n + l)-complex into E, where p: E В is a principal fibration of type
(Z2, n + 2), with base space В a space of type (Z,n). This fibration determines
a cohomology operation of type (n, n + 2; Z,Z2).
14 lemma For n > 2 the cohomology operation 6n is Sq2 0 p*, where
ft*: Hn(X;Z) —> Hn(X;Z2) is induced by the coefficient homomorphism
fi: Z —> Z2*
460	OBSTRUCTION THEORY CHAP, fj>*4 pXEBCISES	461
proof Sn C Sn~2(P2(C)) is not a retract, by theorem 12 and corollary 13 J jj exact sequences containing g#
Therefore в„: Hn(Sn~2(P2(C));Z)	Hn+2(Sn~2(P2(C));Z2) is nontrivial (if pet g: (Y®) -> (Y'.B4 * * 7) be a base-point-preserving map and let g' = g [ Y: Y -> Y' and
were trivial, there would be a map f: Sn-2(P2(C)) S” such that	'	; g" = g I В B'.
f*: H"(S«;Z)	/P1+2(Sn-2(P2(C));Z)
is inverse to the restriction map Hn(Sn~2(P2(C));Z) Hn(Sn;Z), and such a
map f would be homotopic to a weak retraction). Since Sq2 ° p* is also non*
trivial, it follows that 6„ = Sq2 ° p.,. in the space Sn~2(P2(C)).
The rest of the argument follows by showing that Sn-2(P2(C)) is universal
for вп and Sq2 0 p*. Let X be any CW complex of dimension < n + 2 and let
и C Hn(X;Z). Because 77n+i(S’!-2(P2(C))) = 0, there is a map/: X Sn-2(P„(C))
such that/*!? = u, where v is a generator of /Р'($'г2(Р,,(€/)). By the natural-
ity of f)n and Sq2 0 p*, it follows that
= #»/* v = /* 0nv = /* Sq2p* v — Sq2p* (u)
Since this is true for every CW complex of dimension < n + 2 and 0n and
Sq2 0 p* are operations of type (n, n + 2; Z,Z2), it is true for every CW
complex. 
Combining lemma 14 with theorem 8.4.10 yields the following Steenrod
classification theorem.1
15 theorem Let s* E Hn(Sn;Z) be a generator, where n )> 2, and letX
be a CW complex. Then the map f: [X;Sn] —> Hn(X;Z) has image equal to
(и C Hn(X;Z) I Sq2pit. (u) = 0} if dim X < n + 2, and if dim X < n + 1,
/^(w) is in one-to-one correspondence with Hn+1(X;Zf)/Sq2p^Hn~1(X;7^. 
j Prove that Eg” is a subspace of Eg- and p,,- = pB- | Eg--.
2 Define p: (EB-,Eg--) —> (Y,B) so that p | Ep- = pg- and /: (S2Y',f2B') —> (Eg-.Eg--) so that
= (yo,co). Prove that there is an exact sequence
(OY.OB) (QY,QB’) -fi (EB-,Eg-) -fi (Y,B) A (Y',B')
3 Prove that there is an exact sequence
;	...	f2"(Ep,,Er)	Q”(Y,B)	I2/Y',B') -> •   A (Y',B')
1 4 Define a map (£2Y' x Eg-, QB' X Eg--) -» (Eg-,Eg--) sending ы X (уо,и') to (y0, w * u')
j and use this to define an action a T b of [X,A; 12Y',SIB'] on the left on [X,A; EB-,Eg„].
; prove that p#(hi) = p#(b2) for blt b2 £ [X,A; Eg-,Eg-] if and only if there is
! Д £ [X,A; S2Y',S2B'] such that bi = a T b2.
! 5 Prove that j#(ax) = /#(az) for fli, fl2 € [X,A; S2Y',S2B'] if and only if there is
j c € [X,A; QY,QB] such that <21 = c2(S2g)#(c).
C EXAMPLES
1 Find an example of an n-dimensional polyhedron X, with n > 1, and a map
/: X - > S" such that f* : H* (X) —> Й* (Sn) is trivial but / is not homotopic to a constant
map.
2 Let X be an n-dimensional polyhedron. Prove that f, g: X —> Sn are homotopic if and
only if/ = g* :Hn(X;G) -a- Hn(Sn;G) for G = Z;„ with p a prime, and for G = R.
3 Compute the cohomotopy group w2m-1(Pm(C)) for m > 2.
4 Let (Y,B) be a pair which is (n — /-connected for n > 2, with a simple inclusion
map В C Y, and let 1 £ Hn(Y,B; 77) be n-characteristic for (Y,B). If (X,A) is a relative CW
complex and/: (X,A) -> (Y,B), prove that f*(i) £ Hn(X,A; it) is the first obstruction to
deforming/relative to A to a map from X to B.
EXERCISES
A SPACES OF TYPE (т7,п)	n
I For p an integer let Ln(p) be the generalized lens space Ln(p) — L(p, 1, . . . ,1).
Show that Ln(p) C Ln+1(P) and that L„(p) = UnLn(p) topologized with the topology
coherent with {L,.(p)} is a space of type (Z;„l).
2 If X is a CW complex of type (ir,n) for n > 1 and Y is a CW complex, prove that
77n(XvY)~Wn(Y)® © 77X
xc'jtiW
where itx = rr for each X £ wi(Y).
3 Given a sequence of groups	with abelian for <7 > 1, and given an action
of wi as a group of operators on for q > 1, prove that there is a space Y which realizes
this sequence (that is, 77e(Y)	and tti(Y) acting on 77e(Y) corresponds to the action
of 771 on 77q).
1 See N. E. Steenrod, Products of cocycles and extensions of mappings, Annals of Mathematics,
vol. 48, pp. 290-320, 1947.
D SUSPENSION
1 Let X be an (n — /-connected CW complex of dimension < 2n — 1. Prove that
there is a CW complex Y such that SY has the same homotopy type as X. [Hint; Show
j that X has the same homotopy type as a CW complex X', with (X')n-1 a single point.
; Construct Y inductively by desuspending the attaching maps of the cells of X'.]
I 2 Let A and В be closed subsets of a space X such that X = A U B. Assume that
I f> g: X —» Y are such that /(A) = yo = g(B) and define h: X —» Y so that h | A = g | A
and h I В = f\B. Prove that, in [SX;SY],
[S/][Sg] = [S7i]
3 Let X and Y be path-connected pointed CW complexes. Prove that a map /: X -> Y
has the property that S'/: S'-X —> S'- Y is a homotopy equivalence for some к > 0 if and
J' only if S/: SX —> SY is a homotopy equivalence. [Hint: Show that either condition is
equivalent to the condition /: Hi: (X) ~ H* (Y).]
I 4 Let X and Y be path-connected pointed CW complexes and let pi: X X Y —» X and
f p2: X X Y —> Y be the projections and k:XxY—>X/tY=Xx Y/X v Y the collaps-
j ing map. Regard all three as maps into Xv Yv (X / Y) and prove that
462	OBSTRUCTION THEORY CHAP, (Ji :
((Spt) * (Sp2)) * (Sfc): S(X x Y) -» S(Xv Y v X # Y)
is a homotopy equivalence.
5 Show that there exist CW complexes with different homotopy type whose suspensions
have the same homotopy type.
E THE SUSPENSION CATEGORY
Let {X,A; Y,B} = firn . [S/'X,StA; S^Y^B], and for q an integer (positive or negative) let
{X,A; Y,B}q = lin-L, [Sfc+eX.Sfc+eA; S^YS^B], If a: S'=+«(X,A)	S*(Y,B), then {«} wffl
denote the corresponding element of {X,A; Y,B}e.
I Prove that there is a pairing
{Y,B; Z,C}P ® {X,A; Y,B}Q-> {X,A; Z,C}p+q
sending {«} ® {/?} to {a ° /?}, where
S?+q+a-(X,A) Д S>+'-'( Y.B) А» Sfc(Z,Q
2 If A is closed in X and (X,A) has a nondegenerate base point, prove that
{(CX,C.A), (C+X,C+A)} is an excisive couple of subsets. Let S: He(X,A) HQ+l(SX,SA)
and S: №(X,A)	№+1(SX,SA) be the isomorphisms of the corresponding relative
Mayer-Vietoris sequences.
3 Prove that there are -pairings
{X,A; Y,B}P ® HQ(X,A) -» Hp+e(Y,B)
{X,A; Y,B}P ® Hr(Y,B) -> H’-p(X,A)
sending {a} 8 г to S_/c(as. (Sft+Pz)) and {a} ® и to S''"(a * (S'-’iv)) for z £ f/,;(X,A),
и € Hr(Y,B), and a: S'c+p(X,A) -» S*(Y,B).
4 If (X,A) is a pointed pair, with А С X a cofibration, and Y is a pointed space, prove
that there is an exact sequence
-----> {X;Y]q—> {A;Y}e—> {X/A;Y}g_i—> {X;Y}e_j—> ...
5 Let X be a pointed space and (Y,B) a pointed pair, with В C Y a cofibration.
If/: X —» Y is such that the composite X -4 Y A Y/B is null homotopic, prove that Sf
is homotopic to the composite SX A SB C SY for some Deduce the existence of an
exact sequence
-----> {X;B}e-^ {X;Y}g-> {X;Y/B}g^ {X;B}Q^• • 
F DUALITY IN THE SUSPENSION CATEGORY1
In this group of exercises all spaces are assumed to be finite CW complexes with base
points. An n-duality is an element и £ {X* # X; S,f’}_ „ such that the map sending
{«} E {S°;X* }Q ~ {S’;X* } to и ° ({«} # {lx}) € {S« # X; S0}_„	{X;S'%_„ is an
isomorphism
D„: {SO;X*}g=;{X;SO}g-„
and the map sending {/?} € {S°;X}g ~ {S«;X} to и ° ({lx*} # {/!}) £ {X* # S«; S°}_n S
{X*;S°}g „ is an isomorphism
1 See E. Spanier, Function spaces and duality, Annals of Mathematics, vol. 70, pp. 338-378,
1959, for a different development of this topic. The one given in the text is based on a sugges-
tion of P. Freyd and has also been considered by D. Husemoller.
EXERCISES
463
Du- {SO;X)g^{X*;S°}e_n
1 If /: Sp # S° —> S?+4 is a homeomorphism, prove that { f} £ {Sp # So-, S°}_p^q is a
(p + q)-duality.
2 If « € {X* # X; S,(l}„ is an n-duality, prove that the element «' £ {X # X*; S°}_„
corresponding to и under the homeomorphism X * X* -> X* #Xis also an «-duality.
3 И и € {X* £ X; S(l} ,, is an n-duality, prove that for any Y and Z there are
isomorphisms
D„: {Y;Z#X*}g^{Y#X; Z}g_„
{Y;X#Z}g^{X* # Y; Z}g_„
such that £>,,{«} = ({lz} # u) ° ({«} # {lx}) for {«} £ {Y; Z # X* }g and D“{fl =
(ti # {lz}) ° ({hr*} # {/'}) for {Л“} £ {Y; X # Z}g. (Hint: If Y and Z are spheres, this
is true by definition of n-duality. For arbitrary Y and Z use induction on the number of
cells and the five Iemma.)
Given n-dualities и £ {X* # X; and i: £ {Y* # I; S°)_M, define an isomorphism
D(u,v): {X;Y},;~ {Y*;X*}g
so that the following diagram is commutative:
{X;Y}e {Y*;X*}e
{Y*#X; S0}g_„
4 Prove that D(v',w') = (D(k,v))-1: {Y*;X*}g ~ {X;Y}g.
5 If и £ {X* # X; v £ {Y* # Y; S0}_, and w £ {Z* # Z; S°}_„ are
n-dualities and {a} £ {X;Y}P and {/?} £ {Y;Z}g, prove that, in {Z*;X*}r+g,
D(t(,w)({/?} « {«}) = (D(H,«){«}) ° (D(v,w){/?})
j Assume that /: X* # X —> S’1 and g: Y* # Y —> Sn are such that {/} and {g} are
j n-dualities and let a: X —> Y and P: Y* —> X* be maps such that
f °(P# lx) ~ g ° (ly* # a): Y* # Х-» S"
[which implies jD({/},{g}){a} = {/?}]. Let Ca and Cp be the mapping cones of a and p,
respectively, and consider the coexact sequences
6 Prove that there is a map h: Cp # C„-> Sn+1 such that the following squares are
homotopy commutative:
X* # Ca X* # SX < , S(X* # X)
Cp # ca	S»+1
Cp # Y	Cp# Ca
j?
SY* # Ye, S(Y* # Y) S"+i
Deduce that [h] £ {C/; # Ca; S°}_M_i is an (n + l)-duality.
7 For any X there is an integer n for which there exists a space X * and an n-duality
и £ {X* # X; S(l} _.„. (Hint: Prove this by induction on the number of cells of X, using
exercises 1 and 6 above.)
CHAPTER MINE
SPECT1AL SEQUENCES
and нометегт gioups
OF SPHERES
J THE TECHNIQUE OF OBSTRUCTION THEORY DEVELOPED IN THE LAST CHAPTER
I focuses attention on the computation of homotopy groups. In this chapter we
'obtain some results about the homotopy groups of spheres. The method we
follow is due to Serre1 and uses the technical tool known as a spectral sequence.
This algebraic concept is introduced for the study of the homology and coho-
mology properties of arbitrary fibrations, but it has other important applica-
tions in algebraic topology, and the number of these is constantly increasing.
“Some indication of the power of spectral sequences will be apparent from the
results obtained by its use here.
j Section 9.1 contains the definition of a spectral sequence, and in Sec. 9.2
> the homology spectral sequence of a fibration is established. This is used in
Sec. 9.3 to prove generalizations of the Gysin and Wang exact homology
sequences. There is also a proof of the homotopy excision theorem, which is
•used in connection with the Hopf invariant to study in more detail the sus-
! pension map for homotopy groups of spheres.
1 See two basic papers, J.-P. Serre, Homologie singulicrc des espaces fibres, Annals of Mathematics,
vol. 54, pp. 425-505, 1951, and Groupes d’homotopie et classes de groupes abeliens, Annals of
Mathematics, vol. 58, pp. 258-294, 1953.
466	SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES CHAP <J
Cohomology spectral sequences are considered in Sec. 9.4, and the
cohomology spectral sequence of a fibration is established. The multiplicative
property of cohomology spectral sequences is applied in Sec. 9.5 to obtain
stronger results that were obtained with the homology spectral sequence
Serre classes of abelian groups are introduced in Sec. 9.6, and some technical
results, based on spectral sequences, are derived for isomorphisms of groups
modulo a Serre class.
In Sec. 9.7 the machinery based on Serre classes is used to prove that all
the homotopy groups of spheres are finitely generated and are finite except
for stated exceptions. There are also some results concerning p-primary com-
ponents of homotopy groups of spheres. Further information about homotopy
groups of spheres appears in the exercises at the end of the chapter.
1 SPECTRAl SEQUENCES
Corresponding to a subcomplex of a chain complex, there is an associated
exact sequence of homology modules. Hence, corresponding to an increasing
sequence of subcomplexes of a chain complex, there is an associated sequence
of exact sequences of homology modules. This sequence of exact sequences
constitutes a new algebraic object, known as a spectral sequence, which pro-
vides information about the homology of the chain complex in terms of the
homology of the quotient complexes of the sequence of subcomplexes. A
spectral sequence consists of a sequence of chain complexes each of which is
the homology module of the preceding one. There is an associated limit
module, and the spectral sequence itself is viewed as a sequence of approxi-
mations to this limit module. In this section we shall define the algebraic con-
cepts involved. In the next section we shall apply these concepts to study the
homology of a fibration.
Let us consider modules over a fixed principal ideal domain R. A
bigraded module E (over R) is an indexed collection of R modules ESit
for every pair of integers s and t. A differential d: E —» E of bidegree
(— r, r — 1) is a collection of homomorphisms d: Esj —> Es-r,t+r-i, for all s and
t, such that d2 = 0. The homology module IKE) is the bigraded module de-
fined by
Hs,t(E) = [ker Iff: Es,t Es	)]/d(Es+r,t—r+i)
Note that if Eg is defined to equal @S+(=Q Es,t, the differential d defines
a homomorphism 8: Eq —> Eq_i such that {EQ,8} is a chain complex. Further-
more, the (/th homology module of this chain complex equals @s+<=e
An Ek spectral sequence E is a sequence {Er,dr} for r > к such that
(a)	Er is a bigraded module and dr is a differential of bidegree (—r, r — 1)
on Er.
| SF*- 1 SPECTRAL SEQUENCES	467
jj integer к's to sPecify where the spectral sequence starts. In our applications
f jt will usually turn out that к = 1 or 2. Clearly, any Ek spectral sequence de-
? fines an Ek' spectral sequence for every k' > k.
A homomorphism <p: E —» E' from one Ek spectral sequence to another is a
• (collection of homomorphisms <pr: EJj E'srtt for r > к and all s and t commut-
jng with the differentials and such that </..(.: H(E'r) H(E'r) corresponds to
7 ф(+1: Er+1 —> E'r+1 under the isomorphisms of the spectral sequence. The
' composite of homomorphisms is a homomorphism, and so there is a category
! of Ek spectral sequences (for fixed k) and homomorphisms.
' To define the limit term of a spectral sequence, for r > к we regard Er+1
t as identified with H(Er) by the isomorphisms of the spectral sequence. Let Zk
i be the bigraded module Zg>f = ker (dk: Eg>( E^_jett+le_1) and let Rk be the
? Ungraded module B£;( = dk(E^+kit^k+1). Then Bk C Zk and Ek+i = Zk/Bk.
• Let Z(Ek+1) be the bigraded module Z(E7c+1)Sjf = ker (dk+1: Ej/P—> Esffffltt+k)
i and let B(Efc+1) be the bigraded module B(Ek+1)s>t = dk+1(Es+k+1^k). By the
Noether isomorphism theorem, there exist bigraded submodules Z/H l and
; gk+1 of Zk containing Bk such that Z(E7c+1)Sif = Zksff/Bktt and B(E/£+1)Sj( =
for aU s and к It follows that Bk+1 C Zk+i, and we have
Bk c Bk+i c 7/. + 1 c zk
Continuing by induction, we obtain submodules for r > к
Bk c Bk+i c    с вг c    c zr c  • • c zfcH c zk
! such that Er+1 = Zr/Br. We define bigraded modules Z" = ПZr, B~ = U,. Br,
(and E“ = Z“/B“. The bigraded module E“ is called the limit of the spectral
sequence E, and the terms Er of the spectral sequence are successive approxi-
mations to E".
(	The spectral sequence E is said to con verge if for every s and t there exists an
j integer r(s,t) > к such that for r > r(s,t), dr: Eiit —» EJ_r>(+r_i is trivial. Then
lEjJ1 is isomorphic to a quotient of E£( and E£( is isomorphic to the direct
limit of the sequence
_> £V(s,t)+l _> . . .
| It is frequently the case that the spectral sequence converges in the
j strong sense that for given s and t there exists r(s,t) such that Ej>( ~ E£( for
! r > r(s,t). For example, if E has the property that for some r there exist
j integers N and N' such that EJ>( = 0 for s < N or t < N', the same is true
for Ej(f for / > r. Then, given s and t, if r' > r is chosen so that
, / > sup (s — N, t — N' + 1), we have
Eg't —> Es-r',t+t'~i
The first module equals 0 because t — / + 1 < N' and the last module
i equals 0 because s - / < N. Therefore, if / is large enough,
(b)	For r > к there is an isomorphism H(Er) ~ Er+1.
E^( = Es73 ~	~ E“,
O jl'	О j L	jt
’’ Note that the spectral sequence begins with Ek, and the only role of the
and E is convergent in this stronger sense.
4(jfi	SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES	. 1 SPECTRAL SEQUENCES
A particular example of such spectral sequences is a first-quadrant | finite in the sense that FSA =
spectral sequence, which is defined to be a spectral sequence E having v ' • AX J -A 4
property that for some r, Ef, = 0 if s < 0 or t < 0. Such a spectral sequence
is convergent in the strong sense, and for any q there are only a finite nurnbei
of nontrivial modules ETj with s + t = q. A first-quadrant spectral sequence
is conveniently represented by attaching Eft to the lattice point (s,t) in tfie
first quadrant of the plane and representing the differential dr by oblige
arrows:
Then Er+1 is the quotient of the kernel of the arrow which originates at (s,t)
by the image of the arrow which terminates at (s,t).
A homomorphism <p: E —> E' between Ek spectral sequences induces a
homomorphism <p“: E“ —> E'“ between their limit terms. Therefore there is a
covariant functor from the category of Ek spectral sequences to the category
of bigraded modules which assigns to every spectral sequence its limit. The
following useful result is an easy consequence of the fact that a chain trans-
formation which is an isomorphism induces an isomorphism of the corresponding
homology modules.
I theorem Let <p: E —> E’ be a homomorphism of Ek spectral sequences
which is an isomorphism for some r > k. Then <p is an isomorphism for all
r' > r. Furthermore, if E and E' converge, ([ '" Is an isomorphism of their
limits. 
An (increasing^ filtration F on an В module A is a sequence of submodules
FSA for all integers s such that FSA C Fs+iA. If A is a graded module (that is,
A = {At}), the filtration F is required to be compatible with the gradation
(that is, FSA is graded by {FsAt}). Given a filtration F on A, the associated
graded module G(A) is defined by G(A)S = FgA/F^A. If A is a graded
module, the associated module G(A) is bigraded by the modules G(A)s,t =
FsAs+t/Fs^iAs+t. In this case, s is called the filtered degree, t the complemen-
tary degree, and s + t the total degree of an element of G(A)S(f. The
sequence
• • • C FsJA C FSA C FS+1A C • • •
is an infinite composition series for A, and the associated module consists of
the quotients of this composition series.
A filtration F on A is said to be convergent if Fls FSA = 0 and U s FSA — A.
For convergent filtrations the associated module G(A) is more closely tied to
A than in the case of an arbitrary filtration. However, even if the filtration is
469
,	___ „ = 0 for some s and FS'A = A for some s', it is not
| true that G(A) determines A. In the latter case F(A) determines A up to a fin-
| jte number of module extensions.
A filtration F on a chain complex C is a filtration compatible with the
gradation of C and with the differential of C (that is, FSC is a chain subcom-
i p|ex of C consisting of {FsCt}). The filtration F on C induces a filtration F on
defined by
;	FSH* (Q = im [H* (FSC)	н* (C)]
> Because the homology functor commutes with direct limits, if F is a
•’ convergent filtration of C, it follows that Us FSHS.(C) ~ Н^(С); however, it is
; not generally true that Ds FSH^ (C) =0. Thus, to ensure that F be a conver-
! gent filtration on Л.;. (C) we need a stronger assumption about the original fil-
; (ration on C.
, A filtration F on a graded module A is said to be bounded below if for
! any t there is s(t) such that FS(t)At = 0. It is clear that if F is a filtra-
j tion bounded below on a chain complex C, then the induced filtration
I on H* (C) is also bounded below. Thus, if F is convergent and bounded below
(on C, the same is true of the induced filtration on H* (Q.
; The following theorem associates a spectral sequence to a filtration on a
I chain complex.
1 2 theorem Let F be a convergent filtration bounded below on a chain
j complex C. There is a convergent E1 spectral sequence with
|	E^H^C/F^C)
I [and d1 corresponding to the boundary operator of the triple (FSC,FS^C,FS^2C)\
and E“ isomorphic to the bigraded module GH* (C) (associated to the filtra-
tion FSH^(C) = im [H*(FSQ H*(C)]).
proof For arbitrary r we define
Z( = {ceFsC|8ceFs_rC}
Z“ = {c e FSCI 8c = 0}
These are graded modules with Zj>( = {с C: FsCs+t | Sc £ Fs_.rC} and
Z“j. = {c £ FsCs+t | 8c = 0}. We then have a sequence of graded modules
•  • c azri c azo c 8Zi+1 c ... с ac n fsc c zs“ c • • 
|	 •  C ZJ C Z° = FSC
I We define
E'- = z»-/(Z’“i + az»-* j
.<? S' ' S—l 1 Sf-r~ 1'
E“ = Z“/(Z“_1 + ac n FSC)
| The map 8 sends Zj to Zi ,- and Z?~[ + aZJT,Li to 8Z(“|. Therefore it
j induces a homomorphism
i	(F:Er~^Ers_r
470
SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES
CHAP. I)
Then Er is a bigraded module and dr is a differential of bidegree (— r, r
on it. For r < 0, dr = 0 and £)' = FSC/FS iC. Therefore
E°t = FsC^t/Fs-iC'+t = G(C)s>t
and d°: FsC^t/Fs-iC^t -> FsCs+t-i/Fs-iCs+t-i is just the boundary operator
of the quotient complex FgC/F^C. Furthermore,
e^= zi>(/(z°_1>(+1 + 0Z°(+1),
where Z}t = {c G FsCs+t | 0c £ FsiCui). Therefore Z|f/Z9_i,f+i is the
module of (s + t)-cycles of FSC/FS_ LC and (Zo_1;+1 + 0Z?.z+1)/ZO_l>f+1 is the
module of (s + f)-boundaries of FsC/Fs_iC. By the Noether isomorphism
theorem, E}j ~ HS+((FSC/FS_1C). The fact that under this isomorphism fi
corresponds to the boundary operator of the triple (FSC, Fg_iC, FS_2C) js
proved by direct verification, using the definitions.
We prove that E = {£’},.> । is a spectral sequence by computing the
homology of Er with respect to dr. We have
{c e z»:| de e zcLi + 0ZrJ}
= {c e Z?I 0c G F^-iC) + {c e ZJ I 0c G 0Z£1)
= Zr+1 + (Z;:i + zs“) = Zf+1 + Zjzj
Therefore ker (dq E? -> ЕГг) = (Z^1 + Zjzj)/(Zjzj + 0Z^|_i). By defini-
tion, im (dq Ers+r -> E;) = (0Zj+r+ Z^/tZ^l + dZ?-}^). Hence, by the
Noether isomorphism theorem, in Ej we have
ker d7im rF ~ (Zf11 + Zjz')/(0Z?,, + Zjzj) ~ ZrVK+1 П (0Zj+r + Z£j)]
= Zr+i/(dZl+r + Zj_i) = Ef+i
Therefore there is an isomorphism Л... (Er) ~ Er+1, and E is a spectral
sequence.
We now compute the limit of this spectral sequence. By definition and
the Noether isomorphism theorem,
Er = Z’-/(Zy I + 0ZJ+L1) ~ (ZI + FS_1C)/(FS_1C + 0ZSZ^1)
In the last expression the numerators decrease as r increases and the denom-
inators increase as r increases. By definition, the limit equals
Dr(Zf + Fs_1C)/Ur(Fs_1C + 0Z£ti) =
(n.rZ£ 4- FS_!C)/(FS_1C + Ur0Zlzb)
Since Us FSC = C, then U,. 0ZjzLi = 8C П FSC. For given t, П,. Zj., = 2£,,
because FsCt = 0 for s small enough. Therefore the limit term equals
(Zs“ + FS„1C)/(FS_1C + 0C П FSC) = Z“/(Z“_t + 0C П FSC) = Er
To show that the spectral sequence converges, note that because the filtration
is bounded below, for fixed s + t, E\i — 0 for s small enough. Therefore, for
fixed s and t there exists r such that for / > r, Eff1 is a quotient of E£t, and
the spectral sequence converges.
*'sEC< 1 SPECTRAL SEQUENCES	471
'! To complete the proof we interpret the limit E“ as GH* (C). By defini-
; tion, GH*(C)s,t = FsHs+t(C)/Fs~iHs+t(C), where
!	FsHs+f(C) = im [Hs+t(FsC) -> Hs+f(C)]
: therefore the graded module FSH^(C) = Zs/dC П FSC, and
; FSH* (Cj/F^H* (C) = (ZZ/0C П F.5C)/(Zr_i/0C n Fs_iC)
i	~ Zr/(Z^i + 0C n F,C)
*	= Er 
i
:	In theorem 2 note that even in the most favorable circumstances E“ does
•	nOt determine H* (C) completely, but only up to module extensions. Note that
J vve have, in fact, defined an E° spectral sequence. The theorem was stated in
; terms of the corresponding E1 spectral sequence because the E1 term con-
•: tains more information than the E° term.
;	It should be observed that the spectral sequence of theorem 2 is functorial
' on the category of chain complexes with a convergent filtration bounded
I below. Combining this with theorem I, we obtain the following result.
i 3 theorem Let т: C —» C' be a chain map preserving filtration between
i chain complexes having convergent filtrations bounded below. If for some
f	-«	»7 _ • _ J_1 ___ —V- T7-r . TZf-r G.	1 г.гллл.,	4-1'1 rr •j’lTZT.l IZ'ZW ZTTT
j > 1 the induced map rZ Er —> E'r is an isomorphism, then r induces an
isomorphism
t*: H*(C)	H*(C)
proof By theorem 1, r“ is an isomorphism. We have a commutative diagram
j with exact rows
;	0	^-^„(С) FsHn(C) E“„_s	0
Ч Ч к
0 Fs^Hn(C') FsHn(C') -+ E'f,n-s -> 0
i For fixed n, Fs^iHn(C) and Fg_iH„((7) are both 0 for s small enough (because
I the filtrations are bounded below). It follows by induction on s, using the five
j lemma and the fact that r" is an isomorphism, that r*: FsHn(C) FsHn(C')
) for all s. Because the filtrations are convergent, ЕЦС) = Us FsHn{C) and
 H^C') = Us FsHn(C'), and so : Hn(C) 7^ Hn(C). 
•	We present some examples of spectral sequences.
4 Let C and C" be free nonnegative chain complexes with boundary oper-
ators 0' and 0", respectively, and let С = С' ® C" be their tensor product,
with boundary operator 0. There is a convergent filtration bounded below on
j C defined by FSC = Cg ® C". For the corresponding spectral sequence,
f	Ei (= Cs ® ЕЬ(С")
472
SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES CHAp g 
and for r > 2 Ejj ~ US(C' ® JE(C")). A similar result is obtained by filterin
the tensor product by the gradation of the second factor.	°
5 An (increasing) filtration of a topological pair (X,A) is a sequence of sub-
spaces Xs containing A such that Xs C Xs+i. Such a filtration on
induces a filtration F on the chain complex A(X,A) by FS(A(X,A)) = Д(Х8,Д}‘
If Xs = A for some s, the induced filtration is bounded below. If X = U J
and every compact subset of X is contained in some Xs, then Us Fs(i\(X,A)) —
A(X,A). Therefore, if the filtration {Xs} has both the above properties, the in-
duced filtration on A(X,A) is convergent and bounded below so there is a
convergent E1 spectral sequence with EJ,( ~ Hs+((Xs,Xs_i) and in which di
corresponds to the boundary operation of the triple (XS,XS_1,XS_2). The limit
term of the spectral sequence is the bigraded module associated to the corre-
sponding filtration of H(X,A).
In particular, if (X,A) is a relative CW complex, Xs = (X,A)S is the
.s'-dimensional skeleton for s > 0, and Xs = A for s < 0, then Efi W 0 if and
only if t = 0 and EJj0 Hs(Xs,Xg_i). Therefore, for r > 2, Ej0 is the
homology of the chain complex C = {Cs,3}, where Cs = UfiX^X^fi and
Э: Cs —» (>s- i is the boundary operator of the triple (Xg,Xg_i,Xg_2).
Our next example is an alternate description of the spectral sequence of
example 5 whose construction does not involve the chain modules. It can also
be applied to obtain a spectral sequence corresponding to any sequence of
functors having the exactness properties of the homology functors.
6 Let {Xs} be an increasing filtration of a pair (X,A). For each s there is an
exact homology sequence of (Xs,Xs„i) (with some coefficient module)
-----> He(Xg_i) H4(XS) 4 HQ(Xs,Xs_i) 4
Therefore we have a sequence of exact sequences. These combine to form a
commutative diagram
i	i
..  4 He(Xs_r) 4 H^X^X^) 4 He_r(Xg_2) 4 • • •
4	1'*
  • 4 HQ(XS) 4 HQ(XS,XS^) 4 H^x^) 4 • • •
'4	1'*
.  л HQ(XS+1) Д Hg(xs+1,xs) Hq_r(xs) 4 • • -
1	!
in which any sequence consisting of a vertical map fi followed by two hori-
zontal maps fi and Э and then a vertical map fi followed again by fi and d
and iteration of this (one possible such sequence is indicated by heavy arrows
SEC. 2 THE SPECTRAL SEQUENCE OF A FIBRATION
473
jn the diagram) is exact. From this diagram there is obtained an E1 spectral
sequence in which EJ,t = Hs+t(Xs,Xs_i) and for r > 2, Efy is defined to be
the quotient Z?,f/Bj;(, where
Zlt =
Blt = fe(ker [i!3.’~1: HS+«(XS) -+ Hs+((Xg+r_i)])
This spectral sequence converges if for fixed q, Hg(X) 7Z lim- {HQ(XS)}, and for s
small enough HQ(XS,A) = 0. In this case the limit is the bigraded module associ-
ated to the filtration of H(X,A) defined by FSH(X,A) = im [H(XS,A) H(X,A)].
It is not hard to verify that the E1 spectral sequence of example 6 is the
same as the E1 spectral sequence of example 5. The same process can
be applied to obtain a spectral sequence from any diagram having the exact-
ness properties of the diagram in example 6. These properties have been for-
malized in the concept of exact couple,1 but we omit the precise definition.
2 THE SPECTRA!. SEQUENCE OF A FIBHATION
One of the most fruitful applications of spectral sequences is to the homology
of fibrations. With a suitable orientability assumption on the fibration, there is
a spectral sequence converging to the bigraded module associated to a filtra-
tion on the homology of the total space and whose E2 * term is isomorphic to
the homology of the base space with coefficients in the homology of the fiber.
This section is devoted to a construction of the spectral sequence. It depends
on a study of the homology of the total space of a fibration whose base space
is a relative CW complex utilizing the filtration of the total space consisting of
the inverse images of the skeleta of the base space. By using CW approxima-
tions, an E2 spectral sequence is defined for a fibration over any path-
connected base space. Some applications of this spectral sequence will be
given in the next section.
Let p: E Б be a fibration. For a subspace А С В let Ед = p"1(A) С E.
Then p maps (Е,ЕЛ) to (B,A). Assume that (B,A) is a relative CW complex
and let Es = p-1((B,A)s) be the part of E lying over the ^-dimensional skeleton
of В for л > 0 and Es = EA if s < 0. Then Es C Es+i, so {Es} is an increas-
ing filtration of (E,Ea). Furthermore, E_ | = Ел, Us Es = E, and every com-
pact subset of E is contained in Es for some s. By the method of example 9.1.5, we
have the following result.
1 theorem Let p: E В be a fibration over a relative CW complex
(B,A). For singular homology with any coefficient module G there is a con-
vergent E1 spectral sequence with Efit ~ Hs+t(EsEs~i; G), d1 the boundary
1 See W. S. Massey, Exact couples in algebraic topology, parts I and II, Annals of Mathematics,
vol. 56, pp. 364-396. 1952, and parts III, IV, and V, Annals of Mathematics, vol. 57, pp. 248-
286, 1953.
474
SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES CHAP 9
operator of the triple (Es,Es^i,Es^2), and Em the bigraded module associated
to the filtration of Ifi (E,EA; G) defined by
FSH* (Efifi, G) = im [H* (ES,EA; G) H* (E,EA-, G)] 
To apply this result we need to compute the module Hn(Es,Es-i; G). We
let {ex} be the collection of s-cells of В — A.
2 lemma The inclusion maps f: (p 1(ex),p x(ex)) C (Es,Es-i) induce a
direct-sum representation
K* }: ®Hn(p^(ef), p-i(ej) ~ Hn(Es,Es_f)
X
and a direct-product representation
{i?}: Hn(Es,Es-i) ~ x
proof For each X let ef be a simplex of dimension s contained in ex _ eK.
Then the inclusion maps ((B,A)3 * S, (B,A)S-1) C ((B,A)S, (B,A)S — Uf (e/ — e/))
and (ex,ex) С (ex, ex — (e/ — ex')) are homotopy equivalences. Therefore the
corresponding inclusion maps (Es,Es„i) C (Es, Es — U x — efi) and
(p-1ex,p'1ex) C (p"1ex,.p~1(ex — (e/ — ex'))) are homotopy equivalences. There
is a commutative diagram induced by inclusion maps
©ЩГЧ.ГЧ)	H.n(Es,Es^)
л i	!
©H„(p-1ex,p-1(ex - (ex - ex))—> Hn(Eg,Es - U p~i(e{ - ex))
A	T	T
© Hfip-^p !ех) ---------> H„(Up-4Up-4)
x	лл
in which all the vertical maps are isomorphisms, the top two because they are
induced by homotopy equivalences and the bottom two because they are
induced by suitable excision maps. Since ex' is disjoint from e/ if A p, die
bottom map is an isomorphism because it is induced by a chain isomorphism.
This proves the first part of the lemma. A similar argument proves the result
for cohomology. 
Before proceeding further with the computation of Hn(Es,Es_i) and the
boundary operator of the triple (Es,Es_i,£g_2), we introduce a subcomplex of
the singular chain complex of a relative CW complex which is chain equivalent
to the singular complex itself.
3 theorem Let {Xs} be an increasing sequence of subspaces of a space
X and let Д(Х) be the subcomplex of Д(Х) generated by singular simplexes
а: Де —» X such that ofi'fi- C Xi-for all k. If (X,XK1) is (s — reconnected for
all s, then the inclusion map Д(Х) С Д(Х) is a chain equivalence.
SEC. 2 THE SPECTRAL SEQUENCE OF A FIBRATION	475
proof We shall use lemma 7.4,7. We associate to every singular simplex
tr: Де X a map P(a): A’X X such that the hypotheses of lemma 7.4.7
are satisfied [with С = A(X)]. This is done by induction on q. If q = 0 and
а(Д°) C Xo, define P(a) to be the composite Д° X I	X. If а(Д°) ([ Xo,
there is a path w: I X from а(Д°) to some point of Xo [because (X,X0) is
О-connected]. Then P(a): Д0 x I X is defined by Р(а)(со>#) = «(t).
Let q > 0 and assume inductively that P(a) has been defined to satisfy
lemma 7.4.7 for all singular simplexes a of dimension < q. Let a: Де —> X be
a singular (/-simplex of X. If о(Д®)л C for all k, define P(o) to be the com-
posite Де X I —> А® X. If а(Д®)/£ ф X/- for some k, then a and c of lemma
7.4.7 define P(a) on Д® X 0 U A« X I and b of lemma 7.4.7 ensures that on
До x 1 the resulting map sends (A®)fc у ] into X/- for all k.
It is clearly possible to find a homeomorphism Д® X I ~ Eq X I which
takes (Д® X 0 U А® X I, A® X 1) onto (/"XS®1) X 0 and Д’ X 1 onto
Sr1 X I U Eq X 1- Because (X,X(;) is (/-connected, it follows that the given
map from (Де X 0 U А® X I, A® X 1) to (X,Xf;) extends to a map
P(a): Д’ X X such that Р(а)(Д® X 1) C Xg. Then P(a) | Д® X 1: A® X 1 X
is a map such that (A®)fc is sent into Xfc for all k, and so P(o) can be defined
for all a to satisfy the hypotheses of lemma 7.4.7. 
Note that theorem 3 applies to the filtration defined by the skeleta of a
relative CW complex (X,A). Hence, if A(X) C Д(Х) is the subcomplex of
cellular singular simplexes, then A(X) C Д(Х) is a chain equivalence. Further-
more, if (X',A) is a subcomplex of (X,A), then A(X') = Д(Х') П A(X). Using
theorem 4.6.10 and the five lemma, we see that the inclusion map A(X,X') C
Д(Х,Х') is a chain equivalence. In particular,
A((X,A)*, (X,A)s-1) С Д((Х,А)Л, (X,A)S-1)
is a chain equivalence for any s.
4 corollary Given a relative CW complex (X,A), let C(X,A) = {Cs,8}
be the chain complex, with
Cs = HS(A((X,A)°, (хлгч)
and with d: Cs —» Gt the boundary operator of the triple (fX,A f, (X,A)8-1,
(X,A)S-2). Then HM^A)) = Н*(Х,А).
proof Let F be the filtration on Д(Х,А) defined by FsA(X,A) = A((X,A)s, A).
Then the corresponding spectral sequence has the property that E}t zr:
HS+(((X,A)S, (X,A)sl) and d1 corresponds to the boundary operator of the
triple ((X,A)S, (X,A)S-1, (X,A)S‘ 2). By application of lemma 2 to the trivial
fibration X ~> X, it follows that there is an isomorphism
®Hfe^ - Hg((X,Af, (X,A)-i)
x
where {e> } is the collection of s-cells of (X,A). Then H5((X,A)S, (X,A)S -1) = 0
if q yA s, and so Ejj = 0 if t yA 0 and ~ Cs. This implies that = 0
476	SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES CHAP. 8
if / -/ 0 and Ejfi ~ HS(C(X,A)). Therefore, by induction on r, we see that
Ei.t = 0 for t #= 0 and El>0 = E|o for r > 2. Hence Efy — 0 for t yt 0 and
E?,o ~ HS(C(X,A)). Since E“ is the bigraded module associated to a filtration
on H(X,A), we have HS(C(X,A)) ~ HS(X,A). 
Recall that, by theorem 2.8.12, there is a contravariant functor from the
fundamental groupoid of В to the homotopy category which assigns to b £ j
the fiber F(, over b and to a path class [w] in В the homotopy class
h[w] £ [Fw(o);Fw(i)]. Therefore, for fixed R there is a contravariant functor
from the fundamental groupoid of В to the category of graded R modules
which assigns to b £ В the module H* (F(,;B) and to a path class [w] the
homomorphism /г[и]# : H* (FL,(0);B)	II... (F^;R). The fibration is said to be
orientable over R if for any closed path co in B, /г[о:].,. = 1. This is a general-
ization of the concept of orientability of a sphere bundle. (In fact, a sphere
bundle £ is orientable as a sphere bundle if and only if pp В is orientable
as a fibration.)
5 theorem (я) A fibration over a simply connected base space is orient-
able over any R.
(b) A fibration induced from a fibration orientable over R is itself
orientable o ver R.
proof The first statement is immediate from the definition. For the second,
let p': E' —> B' be induced from p: E —» В by a map f:B'—>B and let
g': E' E be the associated map. For any path class [w'] in B' let
go: F''(0) Fgb,’(ff) and gi:	—> Fg^{1) be the homeomorphisms defined by g'.
It is then easy to verify that [gl] ° h[w'] = h[g ° w'] ° [go], and this implies (h). 
A map f: р~1(Ъ0) р^ЦЪ^) between fibers of a fibration p: E —» В is said
to be admissible if there is some path class [w] from &o to hi such that
[f] = h[w]. The following facts are immediate from the definition.
6	An admissible map is a homotopy equivalence. 
7	The composite of admissible maps is an admissible map. 
8	A homotopy inverse of an admissible map is an admissible map. 
» If В is path connected, there is an admissible map between any two
fibers over B. 
1© If P- F, В is orientable over R, any two admissible maps from p 1(ho)
to p-1(bi) induce the same homomorphism from H;i. (p~l(bo);R) to
H*(p~4bf)-,R). -
Let b0 £ В be a base point and let F = p~1(B0)- Given a map a: X B,
an admissible lifting of fis a map a: X X F E such that pd(x,z) = a(x) for
X £ X and z £ F and such that for any x £ X the map fx: F —> p^fifix))
defined by ffiz) = a(x,z) for z £ Z is an admissible map. The following result
is a useful criterion for the admissibility of a lifting.
SEC. 2 THE SPECTRAL SEQUENCE OF A FIBRATION
477
jl lemma Let p: E В be a fibration and let X be a path^connected
space. Given maps a: X —> В and d: X X F —> E "nch that pd(x,z) = a(x),
then d is an admissible lifting of a if and only if there is some To € X such
fhatfxo- E P-1(a(Jo))Is admissible.
proof The necessity of the condition is obvious. To prove the sufficiency, let
Х1 £ X and let w: I X be a path from x0 to xy. Since /Жс is admissible, there
is a path w' in В from bo to a(xo) such that [/Жо] = h[a>']. It is then easy to
verify that [/J = h[u' * (a ° w)], and so /Ж1 is admissible. 
We want to prove the existence of admissible liftings in certain cases.
The following is an alternate version of corollary 7.2.7 valid for a nonpolyhedral
pair (X,A).
12 lemma Let p: E —> В be a fibration and let X be a space and A a strong
deformation retract of X. Given maps fi A —> E and g: X—» В such that
p о f — g | A, there is a map g: X E such that p ° g = g and g | A is fiber
homotopic to fi
proof Let D-. X X I X be a homotopy relative to A from some retraction
г: X A to lx (D exists, because A is a strong deformation retract of X).
Then g ° D: X X I В and f ° г: X —> E are maps such that gD(x,O) =
gr(x) = pfr(x). By the homotopy lifting property of p, there exists a map
F: X x I —> E such that p ° F = g ° D and F(x,O) = ffifi). Let g: X -a E be
defined by g(x) = F(x,l). Then
pg(x) = pF(x,l) = gD(x,l) = g(x)
and F | A X I is a fiber homotopy from / to g | A. Therefore g has the requi-
site properties. 
Let p: E —» В be a fibration over a path-connected base space and let
(B,A) be a relative CW complex. Let bo € В be a base point and F = p-1(bo).
For any s, Vq is a strong deformation retract of Д®, and so v0 X F is a strong
deformation retract of As X F. It follows from lemma 12 that, given a singular
simplex
m (Д®,Д«) -> ((B,A)S, (B,A)S-1)
in A((B,A)S), there exist maps a: (AS,AS) X F (Es,Es-i) such that pd(x,z) — o(x)
for x £ As and z £ F and such that d | Vq X F: !-< P 1(°r(l’o)) is admissible.
By lemma 11, d is an admissible lifting of a.
If d0, di: (AS,AS) X F (Es,Es~l) are two admissible liftings of a, there
is an admissible map fi F -a F such that o0 | v0 X F ~ (<ii | v0 X F) ° / Let
g: As X F X 0 U o0 X F X I U As X F X 1 —> Fs be the map defined by
g(x,z,O) = d0(x,z), g(x,z,l) = dfixfiz)), and g | f0 X F X F. o0 | v0 X F ~
(bi | oo X F) ° f. Since p ° g can be extended to a map As X F X I —» В [by
sending (x,z,t) to o(x)] and AsxFxOUfoxFxIUAsxFx1 is a
strong deformation retract of A® X F X I, it follows from lemma 12 that
478	SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES CHAP ()
g I Д8 X F X o g I Д® x F X 1.. Therefore a0 is homotopic to the composite’
(Д®,Д«) X F (AS,AS) X F (Es,Es_p)
In case p: E —> В is orientable over R, then fi.: Ifi (F)	(F) is the
identity map, and so the composite
H* ((Д®,А8) X F) —H* ((Д®,Д8) X F) H* (ES,ES^)
equals ар*. Therefore O():;. = d|... , and we have proved the following result.
13 theorem Let p: E —» В be an orientable fibration over a path-connected
relative CW complex (B,A), with F — p-1(b0). For o: (Д»,Д«) —» (fB,A)s,
(B,A)S ') there is a well-defined homomorphism
6* : H* ((Д«,Д«) x F) H* (ES,ES^)
defined to be the homomorphism induced by any admissible lifting
д: (Д8,Д«) X F (Es,Es-i) of a. 
The identity map fi: Д8 СДК is a cycle modulo Д8, and its homology
class {£s} generates Н8(Д»,Д®; R). Given iv £ Hn(F;G), then {£s} X w £
H„+S((AS,AS) X F; G) and d* ({G} X w) £ Hn+s(Es,Es-i', G). It is clear that fop-
fixed о the map iv —> fi. ({G} X w) is a homomorphism from Hn(F;G) to
Hn+s(Es,Es~i; G). Because the cellular a’s form a basis of the free module
AS(B) = ^(Bfity), there is a homomorphism
f: ДДВ,АУ) ® Hn(F;G) -> H)l+s(Es,Es_p; G)
defined by f(o 0 w) = d* ({£s} X w). If а(Д«) C (B,A)sl, then й(Д8 x F) C
Lsl for any admissible lifting of o. Therefore d:;. ({£s.} x ie) = 0, and so f
defines a homomorphism
f: As((B,Af, (B,Af-^ ® H„(F;G) -> H.n+s(Es,E^i; G)
Next we show that f induces a homomorphism from the module
Я8((ВД)«, (B,A)*-i; НЛ(Е;С)).
14 lemma The composite
As+fi(BAf, (B,A)«-i) ® H„(F;G)
0 ®1j,
As((B,A)«, (ВЛ)«-1) ® H„(F;G)
4
Hn+s(Es,E^p;G)
is trivial.
proof Let o: (As+1,(As+1)s-1) —» (fB,A)s,(B,A')s~1') be a cellulai’ (s + l)-simplex
of (B,A)S and let
а: (Дз+^Д^1)8-1) X F^ (Es,Eg_p)
SEC. 2 THE SPECTRAL SEQUENCE OF A FIBRATION	479
be an admissible lifting of a. For G < i < s + 1 let ei+i= Д8 —> As+1 omit the
zth vertex. Then a® = a ° ei+i, and the composite
А8 X F As+i x F-^ Es
where o' = d | As+1 X F, is an admissible lifting of o(i). Therefore
® w) = <4(4+iX !)*({&} X 10) = ^({e’+i} X w)
where {ei+i} £ Hs(As+1,(As+1)s-1). It follows that
\l(8o X w) = o'({2 (-l)M+i} X w)
However, in A(As+1) we have the relation 3£s+i = 2 ( — 1)*е'+1. Hence, if
/: (As+1,(As+')sl) С (A8*1,^8*1)8"1), then /*{2 ( — l)ie’+1) = 0. Because a'
equals the composite
Д8+1 x/q As+1 X F-^ Es
it follows that
^(0O ® to) = d*(j X	X w) = о*(0) = 0 
Every element of AS((B,A)8, (B,A)fiH) ® Hn(F;G) is an s-dimensional
cycle of the chain complex A((B,A)8, (B,A)8-1) ® Hn(F;G), and the s-dimen-
sional boundaries are the elements in the image of
0 ® 1: AS+1((B,A)8, (B,A)S^) ® H„(F;G) AS((B,A)S, (B,A)^i) ® H„(F-G)
It follows from lemma 14 that ip induces a homomorphism
HS((B,A)8, (B,A)8-1; H„(F;G)) -+ Hn^Eg,Eg~r, G)
The computation of the E1 term of the spectral sequence is completed by the
following result.
15 theorem (a) For all s >0 there is an isomorphism
HS((B,A)8, (B,A)8-T Hn(F;G)) ~ Hn+S(ES,ES^-, G)
(b) For s > 1 there is a commutative square
HS((B,A)S, (B,A)^; Hn(F;G)) Hn+g(Es^g_i; G)
"I	1'
HS1((B,A)8	(B,A)8-2; Hn(F;G))	Hii+s_i(Es_bEs_2; G)
proof (o) Because of direct-sum properties of both modules (the right-hand
one by lemma 2) and an obvious naturality property of , it suffices to prove
that for an s-cell e of В — A the map
Hs(e,e; H„(F;G))	Hn+s(p-\e),p-\e)- G)
is an isomorphism. Let f: (Es,S8-1)	(e,e) be a characteristic map for e and
let p': E Es be the induced fibration over E8 with corresponding map
(E',p'”1(S8“1)) —» (р^1(е),р"1(ё)). Then there is a commutative square
480
SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES CHAP. 9
Hn(F;G)) Л Hs(E',p'~\S^- G)
4
life,A Hn(F;G))	Hs(pfe),p~i(e); G)
in which the vertical maps are isomorphisms (by excision and homotopy
properties). Therefore it suffices to prove the result for a trivial fibration over
Es, and for such a fibration is an isomorphism by the Kiinneth theorem,
(b) Given о: (Д®,Д®)	((B,A)S, (B,A)S1), let {a ® w] be the element of
Efs((B,A)s, (B,A)S-1; 77,,(F;G)) determined by the cycle a ® w. Then in
Hs^fB,A)^, (B,A)S~2; Hn(F;G)), we have 8{o ® w) = (S	® w).
Let а: (Д8,Д«) X F —> (Es,Es_i) be an admissible lifting of a. For 0 < i < «
the composite
(Д«-\Д«-1) x F (Д«,(Д®)«~2) x FZ> (Es_1;Es_2)
where o' = д | (Д®,(Д®)8-2), is an admissible lifting of o®. Therefore
^8{a ® w) = S (-l)%(e8' x l)*({£s-i) X w) =	(~1)W) X w)
=	X w) = o*8({£s} x w) = So*({£s} x w)
— {a ® m) 
Because Efs((B,A)s, (B,A)®-1) is a free module, it follows from the universal-
coefficient theorem that
Efs((B,A)s, (ВД)«-1; H„(F;G))	Н8((ВД)«, (В,A)8'1) ® H„(F;G)
= Cg(B,A) ® H„(F;G))
Under this isomorphism, it is easy to see that the boundary operator of the
triple ((B,A)S, (B,A)S-1, (B,A)S-2) corresponds to the map
8 ® 1: CS(B,A) ® H„.(F;G) CS^(B,A) ® H„(F;G)
Therefore theorem 15 can be interpreted as asserting that f induces an
isomorphism of the bigraded chain complex C^(B,A) ® H*(F;G), with the E1
term of the spectral sequence of theorem 1. This, together with corollary 4,
gives the following result about the E2 term.
16 theorem Let p: E В be an orientable fibration over a path-
connected relative CW complex (B,A) and let F — p^1(bo). There is a conver-
gent E2 spectral sequence with Efj ~ HS(B,A; Ht(F-,G)) and Г" the bigraded
module associated to the filtration of H.,. (Е,Ед-, G) defined by
FSH* (E,EAi G) = im [H* (Es,Ea; G) H* (E,EA; G)] -
Note that the spectral sequence of theorem 16 is a first-quadrant spectral
sequence and is functorial on the category of orientable fibrations p: E В
over a path-connected relative CW complex (B,A) and fiber-preserving maps
f: £' —» E such that the base space pair is mapped by a cellular map
f. (B\A')	(B,A).
SEC. 3 APPLICATIONS OF THE HOMOLOGY SPECTRAL SEQUENCE
481
To extend the spectral sequence to fibrations with more general base
spaces, let p: E —» Б be an orientable fibration over a path-connected base
space В and let A С B. Let f: (B',A) —» (B,A) be a relative CW approximation
to (B,A) (which exists by theorem 7.8.1). Let p': E' —> B' be the induced
fibration and/': E' E the fiber-preserving map induced by/. It follows
from the exactness of the homotopy sequence of a fibration and the five
Iemma that/'is a weak homotopy equivalence. Therefore/' induces an iso-
morphism of the homology sequence of (E',E4) with the homology sequence
of (E,EA). Because Б' is path connected and p': E' —> B' is orientable, there is
a convergent E2 spectral sequence with
E2,f ~ HS(B',A; Ht(F;Gj) ~ HS(B,A; H^G))
and E“ associated to a filtration of H* (E',EA; G) ~ Jf. (E,EA; G). If
g: (B",A) —» (B,A) is another relative CW approximation to (B,A), there is a
cellular map h: (B",A)	(B',A) such that/0 h ~ g rel A. The map h induces
an isomorphism of the E2 spectral sequences of p": E" —» B" and p't E' —> B'
(but not an isomorphism of the E1 terms). It follows that the filtration of
H* (E,Ea; G) induced by the isomorphisms If. (E',Е_д; G) ~ If. (E,EA; G) and
H* (Е",Ед; G) ~ H* (E,EA; G) correspond, and we have the following main
result.
17 theorem Let p: E В be an orientable fibration with В path
connected and fiber F over bo £ B. Given A С B, there is a convergent E2
spectral sequence, with E^-t ~ JL:(B,A; Jlfil'jG)) and Ex the bigraded module
associated to some filtration of Н*(Е,ЕА; G). This spectral sequence is a first-
quadrant spectral sequence functorial on the category of orientable fibrations
and fiber-preserving maps. 
3 APPLICATIONS OF THE HOMOLOGY SPECTHAL SEQUENCE
hi this section we shall consider applications of the spectral sequence of a
fibration and show that it leads to generalized Gysin and Wang homology se-
quences in case the fiber or base is a homology sphere. We shall also use the
spectral sequence in the proof of the homotopy excision theorem. The section
concludes with a definition of the Hopf invariant homomorphism and an
exact sequence connecting it and the suspension homomorphism of homotopy
groups of spheres.
1	theorem Let p; E В be a fibration which is orientable o ver a field
with path-connected base and with fiber F. Assume that the Euler character-
istics x(F) and x(B) are defined (over the field). Then x(E) is defined, and
X(E) = x(B)x(F).
482	SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES CHAP. 0
proof We use the spectral sequence of theorem 9.2.17. For a finitely gen.
erated bigraded module Er we define the Euler characteristic x(Er) ==
Ss>f( —l)s+t dim E'G- Because we are considering a field as coefficients, ft
follows from the Kunneth formula that
E?>f ~ HS(B;H((F)) = HS(B) ® Ht(F)
Therefore x(E2) = x(B)x(F). Because Er+1 ~ H(Er), it follows (as in theorem
4.3.14) that
X(E2) = x(E3) = • •  = x(E-)
Because E|( = 0 if s and t are large enough, the same is true of Es/ for any
r. Therefore E“ = Er for large enough r, and so x(^“) = x(B)x(h)- By a
standard property of dimension,
dim [H„(E)] = S dim E%t
S+t—П
and so x(E) = X(E“) = x(B)x(E)- "
We now compute the homomorphism induced by i: F С E in terms
of the spectral sequence. For r > 2, E^1 is a quotient of Eg^ (because
= 0 in a first-quadrant spectral sequence). Therefore there is an
epimorphism Eg t Eqj. Because В is path connected, there is an isomor-
phism Ht(F;G) ~ Efo(B; Ht(F;G)). By using the spectral sequence of the
fibration F ~^> bo and the functorial property of the spectral sequence, it fol-
lows that i* : Efi(F;C) -^> Ht(E;G) is the composite
Ht(F;G) = Ho(B; Ht(F;G)) ~ E^ Ejft = F0Ht(E;G) C Ht(E;G)
This leads to the following generalized Wang homology sequence.
2	theorem Let p: E —» В be a fibration, with fiber F and simply con-
nected base В which is a homology n-sphere (over B) for some n > 2 [that is,
Нд(В) = 0 if q or n and H0(B)	~ Hn(B)]. Then there is an exact
sequence
-----> Ht(F;G) S Ht(E;G) H(_„(F;G) Ht-fiF-G}
proof Because Н*(В) has no torsion, E?>( ~ HS(B) ® Ht(F;G) in the spec-
tral sequence of p. Therefore E^t = 0 unless s = 0 or n, and the only non-
zero differential is dn: E&j —> E^t+n_1. Hence there are exact sequences
0-»E“;/ -»E2>f Eo.^n-i —> £o,t+n-i —> 0
and	0	E^j-» Ht(E;G) —>	0
These fit together into an exact sequence
-----> Ht(E;G) -> E21_„	Е§>(_! Ht_fiE-,G) •
The result follows on observing that
SEC. 3 APPLICATIONS OF THE HOMOLOGY SPECTRAL SEQUENCE
483
El^n ~ Hn(B) ® Ht_n(F;G) = Ht_n(F;G)
~ H0(B) ®	~ Ht^(F-,G)
and that on replacing Eg t l by J7/_|(F;G) in the exact sequence, the resulting
map EGi(F;C) Ht_ffE;G) is i*. 
Let p: E —» В be an orientable fibration with path-connected base and
let В' С В and E' = //'(В'). We now show how the homomorphism induced
by p. (E,E') —» (B,B') is determined from the spectral sequence. For r > 2,
EfJ1 is a submodule of Ej;0 (because Ej+r_r+1= 0). Therefore there is a
monomorphism E^o E^o. The augmentation homomorphism Hq(F,G} —» G
induces a homomorphism HS(B,B'; Ef0(F;G)) —» HS(B,B'; G). By using the
spectral sequence of the fibration В G В and the functorial property of the
spectral sequence, it follows that p*: HS(E,E'; G) —> HS(B,B'; G) is the
composite
HS(E,E'; G) =
FSHS(E,E'; G) E-o -> E20 ~ ВДВ'; H0(F;G))	H^B,B'; G)
This leads to the following generalized Gysin homology sequence.
3	theorem Let p: E В be an orientable fibration with path-connected
base space and with fiber F a homology n-sphere (over B), where n > 1. If
В' С В and E' — p-1(B'), there is an exact sequence
-----> HS(E,E'; G) HS(B,B'- G) HS^(B,B'-, G) -+ Hs_i(E,E'; G) •  
proof Because, in the spectral sequence of p,
Eh ~ HS(B,B'; Ht(F;G)) = 0 t =/= 0 or n
the only nonzero differential is dm+1: Efi0 Effn~lin. Hence there are exact
sequences
0 E“o -> E?,o	Es“_n_i.„	0
and
0 E^n,n HS(E,E'; G) -» Es“0 -+ 0
These fit together into an exact sequence
-----> HS(E,E'; G) -> Es20 ELn-i.n -> Hs^i(E,E'; G) —> •  •
The result follows on observing that
E2,o ~ HS(B,B'; Ho(F;G)) ~ HS(B,B'; G)
Eb,-!.,, ~ Hs_n^(B,B'- Hn(F-G)) = Hs_n_fiB,B'-, G)
and that on replacing Ef 0 by HS(B,B'; G) in the exact sequence, the resulting
map HS(E,E'; G) H8(B,B'; G) is p*. 
4	lemma Let p: E —» В be an orientable fibration with path-connected
base space and with path-connected fiber F. Assume that He(B,B') = 0 for
q <fn and Hq(F) — 0 for t) q <fm (all coefficients B). Then the homomor-
484
SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES CHAP. 9
phism p*: Hg(E,E')	is an isomorphism for q < n + m — 1 and
an epimorphism for q = n + m.
proof For the spectra] sequence we have
Elt ~ HS(B,B'-, HtfF)) ~ H8(B,B’) ® Ht(F) © Hg_i(B,B') * Ht(F)
By the hypotheses, E|jf = 0 if s < n or 0 < t < m. Therefore, if
q <_ n + m — 1, then E^>(?s = 0, except possibly for the term E|,o. It fol-
lows that Ers,g^s = 0, except for the term Ej.o, and Efo ~ E$fi. Therefore
E“o ~ E20 and E",e-s = 0 if s =/= q. Hence
HQ(E,E') = HQ(B,B'; H0(F)) = Hq(B,B')
and the isomorphism is induced by p*.
If q = n + m, then E^n+m-s = 0 except for the terms Е?1+т,о and E^,„.
Since Eg+m-r.r-i = 0 for r > 2, it follows that
E£+m.o ~ E2+m.o Hn+m(B,B'; H0(F)) ~ H„+m(B,B')
Therefore p* (Нп+т(Е,ЕУ) = Hn+m(B,B'). 
We use this to prove the following homotopy excision theorem.1
5	theorem Let A, B, and А П В be path-connected subspaces of a space
X such that
(a)	Either X = int A U int B, or X = A U В where A and В are closed
subsets of X such that А П В is a strong deformation retract of some
neighborhood in A (or in B).
(b)	А П В, A, B, and X have isomorphic fundamental groups.
(с)	(А, А П B) is n-connected and (В, А П B) is m-connected, where
n, m > 1.
Then the homomorphism
/#: Wg(A, А П B) 77g(X,B)
induced by the excision map j: (А, А П В) C (X,B) is an isomorphism
for q < n + m — 1 and an epimorphism for q = n + m.
proof First we reduce consideration to the case X = int A U int B. If A
and В are closed in X and А П В is a strong deformation retract of some
neighborhood U in B, let A' = A U U and observe that A is a strong defor-
mation retract of A'. Furthermore, А' П В = U, and the inclusion map
(А, А П В) С (А', А' П B) is a homotopy equivalence, so that (А', А' П B) is
n-connected. By the exactness of the homotopy sequence of the triple
(В, А' П В, А П B) and the fact that (А' П В, А П B) is fc-connected for all
k, we see that (В, А' П B) is m-connected. Note that
X = A U (В — A) C int A' U int B,
XA more general form of this theorem can be found in A. L. Blakers and W. S. Massey,
The homotopy groups of a triad, II, Annals of Mathematics, vol. 55, pp. 192-201, 1952.
г
? SEC. 3 APPLICATIONS OF THE HOMOLOGY SPECTRAL SEQUENCE	485
,j and so A' and В satisfy conditions (a), (b), and (c). Since there is a commuta-
( tive triangle
I	779(A, A n B) nq(A', A' П B)
f	\	/
!	/7=
I
779(X,B)
we are reduced to proving that has the desired properties.
Similarly, if A A В is a strong deformation retract of some neighborhood
| V in A, let В' = V U В and observe that В is a strong deformation retract of
j B'. Then А П В' = V, and it follows, as in the case above, that (А, А П B')
I is n-connected and (В', А П B') is «(-connected. Since X = (A — B) U В is
i contained in int A U int B', we see that A and B' satisfy conditions (a), (b),
i and (c). From the commutativity of the square
TTg(A, A A B) ttq(A, А П B')
I	TTQ(X,B)	Kg(X,B')
we are reduced to proving that j# has the desired properties.
In either case we have shown that it suffices to prove the theorem under the
hypothesis that X = int A U int B, and we make this assumption now. By
I corollary 8.3.8, there is a fibration p: E —> X such that E is simply connected
I and p#: 7Tq(E) ~ ttq(X) .for q > 1. Let EA and Ев be the parts of E over A and
B, respectively, and note that EA П EB is the part of E over А П B. From
theorem 7.2.8 it follows that (Ел, Ел Fl EB) is n-connected and (£в, EA П EB)
is in-connected. Using (b) and the exactness of the homotopy sequence of a
fibration, it is easy to see that EA П EB, EA, and EB are all simply connected.
I Since it is obvious that E C p x(int A) U p '(int В) C int EA U int £B, we
have reduced the theorem to the case where all the spaces in question are
[ simply connected by virtue of the commutativity of the square
7Гд(ЕА,ЕА n EB) ттд(А, A n B)
I	7tq(E,Eb)	wQ(X,B)
j Thus, assume X = int A U int В and that А П В, A, B, and X are all
!' simply connected. We replace the inclusion map А С X by the homotopically
equivalent mapping path fibration p: В X as in theorem 2.8.9. Then P is
the space of paths w: (£0) —> (X,A) in the compact-open topology, and
p(w) = w(l). The fiber F of p over a point a0 € А П В is the space of paths
in X which start in A and end at a0. If p': PX —> X is the path fibration of all
paths in X which end at a0 and p'(6J) = w(0), then F = p'“x(A). Since PX is
contractible, there are isomorphisms
, -	"'^q(X,A) <4- vg(PX,F) Trq^F)
486	SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHEjRES CHAP. У
Because X = int A U int B, the excision map j': (В, A Cl В) C (X,A) induces !
isomorphisms in homology. It follows from the relative Hurewicz isomorphism ''
theorem and the nr-connectedness of (B, A Cl B) that (X,A) is also m-connected.
Therefore F is (m — l)-connected, and so Hq(F) — 0 for 0 < q < m.
Let E' = p-1(B) and observe that since X is simply connected, the
fibration p: EX is orientable. Since у* : Hq(A, A Cl B) ~ HQ(X,B), it follows
that Hq(X,B) = 0 for q < n + 1. By lemma 4, the homomorphism
p*:HQ(E,E')-^flQ(X,B)
is an isomorphism for q < n + m and an epimorphism for q — n 4- m -j- 1,
The map /: (A, A Cl В) C (X,B) has a lifting j: (A, A Cl B) —> (E,EZ), where
j(a) is the constant path at a for all a € A. There is a commutative triangle
Hq(A, A Cl B) H9(E,Ez)
A?\j	^P*
Therefore j* is an isomorphism for q < n + m. Since j | A: A —> E is a
homotopy equivalence, it follows from the five lemma that the homomorphism
is an isomorphism for .q < n + m — 1.
Because tti(E') ~ 771(B) zz tz2(X,A), and the latter group is a quotient
group of яг(X) since 771(A) ~ 77i(X), we see that E' has an abelian fun-
damental group. Since А С В is simply connected, it follows from the absolute
Hurewicz isomorphism theorem that E' is also simply connected. By the
Whitehead theorem, the homomorphism
(/|A П B)#: Wq(A C B)^ 77Q(E')
is an isomorphism for q < n + m — 2 and an epimorphism for q = n + m — 1.
Since; | A: A E is a homotopy equivalence, it follows from the five lemma that
the homomorphism
/#: ^e(A, A Cl B) -» 77q(E,E')
is an isomorphism for q < n + m — 1 and an epimorphism for q = n + tn.
The result follows from this and the commutativity of the triangle
77q(A, А П В) Д 77Q(E,E')
'2/p#
77e(XB)	
It should be noted that the main argument above involved the case
where A and В satisfy c, satisfy b in the stronger form that all the spaces in
question are simply connected, and satisfy the condition that {A,B} is
an excisive couple of subsets of X, which is a weak form of a. It should also
SEC. 3 APPLICATIONS OF THE HOMOLOGY SPECTRAL SEQUENCE
487
|)e observed that if A and В satisfy condition a of theorem 5, then if
0, А Л B) is n-connected [or (В, A П B) is m-connected], it is easy to show
that (X,B) is also n-connected [or (X,A) is m-connected]. Furthermore, if A
and В satisfy a and c and А Г) В is simply connected, then it follows that A
and В are each simply connected and also that X is simply connected. Hence
condition b is also satisfied, and theorem 5 is valid in this case.
6 corollary Let (X,A) be an n-connected relative CW complex, where
л > 2, such that A is m-connected, where m > 1. Then the collapsing map
(X,A) (X/A,xo) induces a homomorphism
k#: foX,A) тгд(Х/А)
which is an isomorphism for q < m + n and an epimorphism for
q = m -f- n -f- 1.
proof Let CA be the unreduced cone over A and regard it as a space whose
intersection with X is A. Since A is пг-connected and CA is contractible, it follows
that (CA,A) is (m -|- l)-connected. We shall apply theorem 5, with A and В
replaced by X and CA, respectively. Since X Cl CA = A is a strong deforma-
tion retract of some neighborhood in CA, a of theorem 5 is satisfied. Since A
is simply connected and c is also satisfied, it follows, as in the remarks above,
that b is satisfied too. Hence the hypotheses of theorem 5 are satisfied, and it
follows that /: (X,A) С (X U CA, CA) induces a homomorphism
fo wq(X,A) rfoX U CA, CA)
which is an isomorphism for q < n + m and an epimorphism for
q = n + m + 1. It follows from lemma 7.1.5 that the collapsing map
к': (X U CA, CA) —> (X U CA, CA)/CA is a homotopy equivalence. The
result follows from the commutativity of the triangle
ТТ9(ХЛ) foX U CA, CA)
—77q(X/A)	
7 corollary Let f: (X',A') —> (X,A) be a relative homeomorphism between
relative CW Complexes both of which are n-connected, with n > 2, and such
that A' and A are m-connected, with m > 1. Then f induces an isomorphism
fo: тгд(Х',А') ~ 7tq(X,A) q <n + m
proof Let k':	(X'/A'fo) and k: (X,A) (X/A,xo) be the collaps-
ing maps. Then f induces a homeomorphism fo. X'/A' X/A such that
f' ° к' = к ° f. Since f' induces isomorphisms of the homotopy groups in all
dimensions, the result follows from corollary 6. 
We use this last result to study the suspension map
S: 775(S«) irQ+i(S»+i)
488
SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES
CHAP. 9
in more detail. Since S”+1 = S(S”), there is a characteristic map /j/: S'* —> fiS«+x
for the path fibration PSn+l Sn+1. From the commutativity of the triangle
779(S«)	779(RS«+1)
s\ -A
77e+1(S"+1)
it suffices to study the map p#.
Let X2n be the space obtained from S” X S" by identifying (z,z0) with
(zq,Z) for all z £ S” (where z0 is a base point of S'*). We regard S'* as imbedded
in X2n as the set of points corresponding to S” X z® in S'* X S’1. Then X2n is a
CW complex consisting of S” and a single 2n-cell attached by a map
S2”1 -> S'*.
8 lemma There is a map g: XZn —> RSn+1, where n > 2, which is a
(3n — Inequivalence such that g | S’1 = p'.
proof Let p: S” X SIS”'1 —> RSn+1 be the map defined by g,(z,w) — w * b'(z).
By corollary 8.5.8, p is homotopic to a clutching function for the fibration
PS’*+1 —> S”+1. Let f: Sn X S’* —> flS’l+1 be defined by f(z,z') = pXz') * p'(z).
There is a commutative diagram
Hn+i(C-S’*,S”) ® H,t(fiS’*+i) H2n+1((C_S« S«) x fis*-1-1)
H«(S”) ® H„(BS»+1)	H2„(S” X fiS"+i) H2„(f2S»+i)
(lxrt\	//,
H„(S«) ® H„(S")	H2„(S" X s«)
Therefore f*: Нгп(8” X S”) ~ f/2n(flS"+1). Since f\ S” v S” is homotopic to
the map sending (z,Zo) to p'(z) and (zo,z) to p'(z), fis homotopic to a map f'
such that/'(z,Zo) = p'(z) = f'(zQ,z). Then/' defines a map g: X2” —> fiS'"11
such that g ° к = f, where k: S’* X S” X2n is the quotient map. Then
g | S” = p’, and since H„(Sn) тг Ня(Х2и), g;i.: Hn(S") ~ H„(S2Sw+1). Since
k* : H2n(S” X S’*) ~ H2n(X2”), it follows that g*: H2n(X2rl) zz H2n(BS”+1). The
only nontrivial homology groups of X2’’ are in degrees, 0, n, and in, and in
degrees < Sn the only nontrivial homology groups of S2S”+1 are in degrees 0,
n, and in. Therefore g*: HQ(X2”) flQ(fiS’*+1) for q < Sn. Since n > 2, X2’*
and BS’*+1 are both simply connected. By the Whitehead theorem, the
homomorphism
g#: 77q(X2”)	^(fiS»+i)
is an isomorphism for q < Sn — 1 and an epimorphism for q = Sn — 1. 
Let an: (E2n,S2n~1) (X2m,Sn) be the characteristic map for the 2n-cell
of X2” corresponding to the attaching map an: S2”-1 Sn. Then an is a
SEC. 3 APPLICATIONS OF THE HOMOLOGY SPECTRAL SEQUENCE	4^9
relative homeomorphism between (2n — l)-connected pairs such that S2n~1
and S" are both (n — l)-connected. It follows from corollary 7 that
«я#: wQ(EZn,SZn-1) —> 7Tq(X2n,Sn) is an isomorphism for q < 3n — 2.
The Hopf invariant1 is the homomorphism
H. TT9+1(S«+1)	77q_1(Sz»-1) q < 3n - 2
defined so that the following diagram is commutative [where /: X2n C (XZ7i,S”)]:
^+1(S"+1) >	-^4 7Tq(X2^
Hl
^l(SZ'rt) 4 77q(EZ«,SZ’>-1) 44- TTqtX2”,^
The Hopf invariant plays an important role in the study of the suspension
homomorphism by virtue of the following exactness property.2
® theorem For n > 2 there is an exact sequence
^(S") Л-----------> 77,/S«) 4 ^(S^l) 4 779_1(S2«-1)	•
proof The result follows from the exactness of the homotopy sequence of
(X^.S") and the commutativity of the following diagram:
^(S«)
~j	4
^+b’!+1) > Kq(flS"+1) TTq^
НГ
77Q_1(S2’'-1) 4 ^(EZ’SS2»-1)	wQ(X2’l,S«)
a,41
77Q_1(S'!)	
If G is an infinite cyclic group, we define a function | • | from G to the
set of nonnegative integers by the condition |g| = m if and only if there is a
generator g' C G such that g = mg'. Since a2ft i(S2,r l) Z, we can define
|H[a|| for [a] € W2n+i(Sn+1)- The following is an interpretation of |H[«]|
IO theorem Let a: SZn+1 —> S?1+1 be a base-point-fireserving map and let
Ea —> sz,1+1 be the principal fibration induced by a. Then |И[а|| = m if and
only if the integral homology group H2n(Ea) is isomorphic to Zm (where
Zo = Z).
1 See H. Hopf, Uber die Abbildungen von Spharen auf Spharen niedrigerer Dimension,
Fundamenta Mathematica, vol. 25, pp. 427-440, 1935, and G. W. Whitehead, A generalization
of the Hopf invariant, Annals of Mathematics, vol. 51, pp. 192-237, 1950.
2 See G. W. Whitehead, On the Freundenthal theorems, Annals of Mathematics, vol. 57,
pp. 209-228, 1953.
490	SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES CHAP. 9
proof From the definition of H and the naturality of the Hurewicz homo-
morphism tp, it is easily seen that |И|о]| = |(p3[a]|, where
<рЭ[«] €	Z
Since a induces a map a: (Ea,f2S’t+1) —> (7?S'i+1,!LS"+I), there is a commutative
diagram
^n+l(S2’i+1) .
^+1(S«+1) 5
Therefore |<p8[a]| = |<p3a#[lS2>i+i]| = |<p8[lS2»+i]|. There is also a commutative
diagram
W2n+l(S2,l+1)	772n+1(Ett,fiS»+l)	A	772?1(fiS«+l)
Ф	4	I4’
H2»+1(S2n+1)	H2?l+1(Ea,fiS«+i)	Л	H2„(fiS«+i)
from which it follows that |<рЭ[1в2П+1]| = |8(z)|, where z is a generator of
H2n+i(EH,fiS«+i). By lemma 4, H2?i(E„,QS« + l)	= 0, and so
H2n(E„)	H2n(fiSn+1)/8H2,l+1(E(nf2S!'+1)
and this gives the result. 
4 MULTIPLICATIVE PROPERTIES OF SPECTRAL SEQUENCES
This section is devoted to pairings from two spectral sequences to a third. This
will be applied, by means of the cross product, to pair the homology spectral
sequences of two fibrations to the spectral sequence of their product. We shall
also consider cohomology spectral sequences. There is a cohomology spectral
sequence for a fibration and a cross-product pairing of the cohomology spec-
tral sequences of two fibrations to the cohomology spectral sequence of their
product. The diagonal map then endows the cohomology spectral sequence
with a multiplicative structure, which will be applied in the next section.
Let p: (Е,Ед) —> (B,A) and p': (E',EA')	be fibrations over relative
CW complexes and let p": E X E' —> В X В' be the product fibration (that is,
p" = p X p'). There is a filtration of the pair (E X E', EA X E' U E X E^)
defined by (E X Е')л = EA X E' U E X Ei' U Ui+J=fcEi X Ej, where {£; }
and {EJ} are the filtrations of (E,EA) and (E’,Ea) corresponding to the skeleta
of (B,A) and (B',A'), respectively. Then E X E' = Ufc(E X E')fc, and every
compact subset of E X E' is contained in (E X E')k for some k. By the method
W example 9.1.5, there is a convergent E1 spectral sequence with
SEC. 4 MULTIPLICATIVE PROPERTIES OF SPECTRAL SEQUENCES	491
El,t ~ Hs+t(JE X E')s, (E X E'^s-l; G) and £“ the bigraded module associated
to the filtration of II* = H* ((Е,ЕЛ) X (E',EA'); G) defined by
FSH* = im [H* ((E X E')s, E X Efi U EA X E'-, G) H*]
We relate this spectral sequence to the cross product of the spectral
sequences of p and of p'. If E, E', and E" are Ek spectral sequences, a pairing
from E and E' to E" is a sequence of homomorphisms
h‘: E^t ® E'£r -> E'&S',t+t'
for all r > к such that for x € EJ;t
drhr(x ® y) = hr(drx ® y) + (— l)s+fhr(x ® d'Ty)
and such that hr+f-is the composite
E’-+l' ® E'^+i ?zJEEr) ® H(E’r) -> H(ET ® E'r) H(E"’) ~ E"riL
For the sequence of submodules used to define E"
Bk c /y+1 C • • • C Br C • • • C Zr C • • • C Zfc+1 C Zk
it is clear that hk pairs Zk and Z'k to Z”k in such a way that ZT ® Z'r is
mapped to Z"T and B' ® Z'T + ZT ® B'r is mapped to B"T for all r > k.
It follows that hk maps Z" ® Z'“ = (Г) Zr) ® (Pl Z'r) to П Z"T = Z"“ and
maps B“ ® Z'“ + Z“ ® B'“ = UJB’’ ® П7- Z'f) + U,.(n;Z' ® B"‘) to
U B"r = B"“. There is induced a pairing
7i“: E” ® E'“ —» £"”
which is compatible with the pairings {hr}.
1 theorem Let p: E В and p': E' —> B' be orientable fibrations over
path-connected relative CW complexes (B,A) and (B',A') with fibers
F = p^lfio) and F' = p'^1(bo), respectively. There is a pairing {/ir} from the
E1 spectral sequences of p and p' to the E1 spectral sequence of p X p' such
that hx is induced by the cross-product pairing
H*(E,Ea; G) ® H*(E',E^ G') H*(JE,Ea') X (E',Elfi G ® G')
proof An Eilenberg-Zilber chain map
Д(Е,£Л) ® Д(Е',Е^) Д((Е,ЕЛ) X (E',E1))
induces a map from FsA(E,£a) ® FSA(E',E1') to Fs+S'A((E,Ej4) X (E',Ef)) for
all s and s'. Therefore it induces in a natural way a pairing of the corresponding
spectral sequences. Since an Eilenberg-Zilber chain map induces the homology
cross product, the result follows. 
To interpret this result on the E2 level, let C* (B,A) and C* (B',A') be the
chain complexes of the relative CW complexes (B,A) and (B',A'), respectively,
defined as in corollary 9.2.4. If о £ Д8((В,А)«, (В,A)8"1), then
{a} e HS(A((B,A)S, (B,A)8-1)) = CS(B,A)
492	SPECTRAL SEQUENCES AND HOMOTOPE GROUPS OF SPHERES CHAP. 9
and these elements {о} C CK(B,A) generate CS(B,A). We define a
homomorphism
CS(B,A) ® CS,(B',A') ® H„(F X F; G")
* Н$+в'+п((Е X В )s+ss (E X E)S4-s'— 1? G")
by	® {<} ® w) = (®X 0% F*({£s} X {&} X w)
where o: (A®,A®) X F	(ES',E^X) and o': (A®',A®') X F (E’^,E'S^ ) are
admissible liftings of a and o', respectively, and
T: (A®,A®) x (A®',A®') X F X F' (A®,A«) X F x (A®', A®') X F
is the map which interchanges the second and third coordinates. The fact
that i//' is well-defined follows by an argument similar to that of lemma 9.2.14.
2	lemma The map x[/' induces an isomorphism
fQ (B,A) <8> C* (B',A')]S ® H„(F X F; G") = E"*
such that ° (3 <8 1) = d1 ° and such that there is a commutative square
CS(BA) ® H((F;G) ® CS<B,A) ® HV(F;G) El,t ®
4	I7'1
CS(B,A) ® CS<B',A') ® Ht+t,(F X F; G")	Es"V,f+r
where <p(c ® w ® d ® w) = ( — V)ts'c ® c' ® (w X m'), with G and G
paired to G".
proof The first part follows by an argument similar to that of theorem 9.2.15.
For the second part we have
</'<p({o} ® w ® {o'} ® w') = ( — l)fs'V/”({o} ® (o'} ® (w X m'))
= (-I)fs'(d X o')* T* ({£s} X {£/} X (w X w'))
= (6 X d')*(({£s} X w) X ({&'} X w'))
= h1(^'((o} ® w) ®	® w')) “
It follows from theorem 1 and lemma 2 that
E's;2 ~ HS((B,A) X (B',A'); Ht(F X F'; G"))
and that the pairing h2 from Ef>t, E's?r to Е"Д',(+г corresponds to ( —l)f®' times
the pairing given by cross product
Н8(ВД; Hf(F;G)) ® HS-(B',A'; Hr(F;G'))
HS+S,((BA) X (B',A'); Ht+r(F X F'; G"))
where the coefficients are themselves paired by the cross product. That is, the
left-hand side is isomorphic to
Н8((В,А)®,(ВД)®-1) ® Hf(F;G) ® HS^B',A'F, (B',Ay- ^ ® Hr(F';G')
the right-hand side is isomorphic to
SEC. 4 MULTIPLICATIVE PBOPEBHES OF SPECTBAL SEQUENCES	493
Hs+sfi(B,A) X (B',A')f, ((В,A) X (В',A')/-’) ® Ht+r(F X F'; G")
and the map sends x ® у ® x' ® y’ to (— l)ts'(x X ж7) ® ((/ X ?/').
3	theorem Let p: E В and p'-. E' —> B' be orientable fibrations with
path-connected base spaces and with fibers F and F', respectively. Let А С В
and A' С B' and assume that {В x A', A x B'} is an excisive couple in
В X B' and [Ea x E', E x Efi] is an excisive couple in E X E'. Given a
pairing G	—> G", there is a pairing of the E2 spectral sequences of p
and p' tp the E2\ spectral sequence of p X p', which on E2 corresponds to
(— l)fs' times the cross-product pairing
HS(B,A; HdffiG)) ® HsfB',A’; Ht,(F'-G'))
Hs+s,(fB,A) X (B7,A7); Hf+r(F x F; G"))
and on Em is compatible with the cross-product pairing
Hn(E,EA; G) ® Hn,(E',EA>; G') Hn+n,((E,EA) X (E',Efi); G")
proof Let/:(X,A) (B,A) and/7: (X',A7)	(B',A’) be relative CW approx-
imations to (B,A) and (B',A'), respectively. Let Ex and Efi be the induced
fibrations over X and X', respectively, with corresponding maps fi. (Ex,Ел) —>
(E,EA) and /': (EfiJEfi) (E'Jffi). The excisiveness hypotheses ensure that
the Kiinneth formula can be applied to deduce isomorphisms
(/ X Г)*- H*((X,A) X (X',A')) X Н^((ВЛ) X (В',A'))
(f X ff*: H* ((EX,EA) X (Efi,EA')) H* ((E,EA) X (E',EAf)
The result now follows from application of theorem 1 and lemma 2 to
the fibrations Ex X and Ifi X' and from the remarks made above about
the pairing induced on the E2 terms (the resulting E2 spectral sequence being
independent of the choices of X and X'). 
The pairing of theorem 3 has properties analogous to those of the cross-
product pairing. In particular, it is functorial on fiber-preserving maps and
commutes up to sign with the homomorphism induced by interchanging the
factors of p X p'-
We next consider cohomology spectral sequences. Let С* — {Ci,<5} be
a cochain complex. A (decreasing) filtration F on C* is a sequence of subcom-
plexes FC* such that FSC* D Fs+lC* for all s. The filtration is convergent
if U FSC* = C* and A F’C* = 0. It is bounded above if for each n there is
s(ri) such that pWC” = 0. Given a convergent filtration bounded above on a
cochain complex C*, there is an analogue of theorem 9.1.2 which asserts the
existence of a convergent Ej spectral sequence {Er,dr}, where Er is bigraded
by EF and dT is a differential on Er of bidegree (r, 1 — r). Furthermore, we
have E®4 ~/E+f(FsC*/Fs+1C*) and dt corresponds to the coboundary
operator of the triple (FSC*,Fs+1C*,Fs+2C*). The limit term E„ is the
bigraded module associated to the filtration on H*(C*) defined by
494
SPECTRAL SEQUENCES AND HOMOTOPT GROUPS OF SPHERES CHAP. 9
FSH*(C*) = ker [H*(C*)	Н*(Р~1С*)]
(that is, E& ~ker	Н«+*(р-1С*)]Дег [Hs+t(C*)	Н««(Е*С*)]_)
4	example Let [Xs] be an increasing filtration of a pair (X,A) and
let Д(Х,А) be the subcomplex of Д(Х,А) generated by singular simplexes
о: Д® —> X such that o((C X/- for all k. Let C * = Hom (Д(Х,А), G). A
decreasing filtration on C * is defined by
F«C* = {c £ C* | c | Д(Х8_ЬА) = 0}
where A(Xs_r,A) = Д(Х,А) П A(Xs_i,A). Since Д8(Х,А) = As(Xs,A), it follows
that Fs+1C* = 0, and so the filtration is bounded above. In case the original
filtration on (X,A) is bounded below (that is, Xs = A for some s), then
U FSC* = {с E C*| c | Д(А,А) = 0} = C*. Hence, in the latter case there
is an associated convergent Ei spectral sequence. In case the inclusion maps
Д(Х,А) С Д(Х,А) and Д(Х8,А) С Д(Х,А) are chain equivalences, this spectral
sequence has the property that E|>f Hs t /(X.s,Xs l; G) and E,,_ is the bigraded
module associated to the filtration on H* (X,A; G) defined by
FSH*(X,A; G) = ker [H*(X,A; G) -a H*(XS_M; G)]
In particular, if (X,A) is a relative CW complex, Xs = (X,A)S if s > 0,
and Xs = A if s < 0, it follows from theorem 9.2.3 that the hypotheses are
satisfied and that
Es,t ~ Hsl'((X,A)-\ (Х,А)8Д G) = 0 t ф 0
Therefore the spectral sequence collapses and HS(X,A; G) is isomorphic to
E|>° ~ H*(C*), where C* = {Cf/,S} is the cochain complex
Ci = №((X,A)e, (XA)r-i; G)
and 8 is the coboundary operator of the triple ((Х,А)«, (Х,А)ч~г, (X,A)o-z). By
the universal-coefficient theorem for cohomology, C* = Hom (G,. (X,A), G).
Hence we have proved that H* (X,A; G) ~ H* (C* (X,A); G).
5	theorem Let p: E —> В be a fibration over a relative CW complex
(B,A). There is a convergent E\ cohomology spectral sequence, with
E^ ~ Hs+t(Es,Es~i; G) and Em the bigraded module associated to the filtra-
tion of H* (Е,ЕЛ; G) defined by
FSH*(E,EA; G) = ker [H*(E,EA; G) H*(Es_1,EAi G)]
proof Since (B, (B,A)«) is s-connected for all s, it follows easily from
theorem 7.2.8 that (E,ES) is s-connected for all s. By theorem 9.2.3, the chain
complex Д(Е,Еа) is chain equivalent to Д(Е,Ел) and Д(Е8,Ел) is chain equiv-
alent to A(Es,Ea)- The result now follows by the method of example 4. 
To compute Eff we assume that В is path connected and that p: E В
is an orientable fibration. Let F = pWb(>) and let о: (Д®,Д®) —» ((B,A)S, (B,A)S-1)
SEC. 4 MULTIPLICATIVE PROPERTIES OF SPECTRAL SEQUENCES	495
be a singular simplex in A(B,A). If a: (AS,AS) X F (Es,Es^i) is an admissible
lifting of o, the hobiomorphism
a* H«(Es,Es_i; G) Н"((Л8Д8) X F, G)
depends only on о and not the particular choice of the lifting d (because the
fibration is orientable). Let {&}* € №(&,&) be the generator characterized
by the condition <{£«}*>{&}> = 1- ft follows from theorem 5.6.1 that the
map о —> {£s}* X is an isomorphism
№(F;G)	As) X F; G)
As in theorem 9.2.13 and lemma 9.2.14, it can be shown that there is a well-
defined homomorphism
ip=: H«(Eg,Es_i; G) №({B,Af, (B,A)8-1; H«~S(F;G))
characterized by the equation
{&}* X	- d*(u)
where a: (Д8,Д8) -^> ({B,Af, (B,A)sl) and o: (Д8,Д8) X F (Es,Es~i) is an ad-
missible lifting of o, and (ip' (u),{a)) £ Hn~s(F,G). Analogous to theorem 9.2.15
is the result that fi* is an isomorphism (this uses the second part of lemma
9.2.2 instead of the first part) and that it commutes with the differentials d|
and the coboundary operator of the triple ((B,A)S, (B.A)8-1, (B,A)S~Z). Using
the technique of relative CW approximation, we have the following analogue
of theorem 9.2.17.
6	theorem Let p: E—^Bbean orientable fibration over a path-connected
base and let F = p-^bo). Given A С B, there is a convergent E% cohomology
spectral sequence, with Efi ~ HS(B,A; Hf(F;G)) and EM the bigraded module
associated to some filtration of H* (E,EA; G). This spectral sequence is a first-
quadrant spectral sequence functorial on the category of orientable fibrations
and fiber-preserving maps. 
For the multiplicative properties of cohomology spectral sequences we
shall use the following result about pairings of cohomology spectral sequences.
7	theorem Let p-. E —» В and p': E' —> B' be orientable fibrations over
path-connected relative CW complexes (B,A) and (B',A'), with fibers F and F',
respectively. There is a pairing {hT} from the Ei cohomology spectral sequences
of p and p' to the E± cohomology spectral sequence of px p' such that h? is
induced by (— l)te' times the cross-product pairing
EF(B,A-, HfiF-Gfi ® HS'(B',A'-, №'(F'-G'))
№+s'((B,A) X (B',A'); Ht+t'(F X F; G"))
and h„ is induced by the cross-product pairing
H*(E,EAi G) ® H*(E',E^; G')	H*((E,EA) X (E',Efifi G")
where G and G' are paired to G".
496
SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES CHAP. 9
proof There are chain equivalences
Д(Е,ЕЛ) ® Д(В'Лл-) С Д(Е,ЕЛ) ® Д(Е',Е1)	Д((Е,Е4) x (E',Efi))
and therefore an isomorphism
н*((№) X (E',EP); G") ~ Н*(Д(Е,Еа) ® Д(Е',Е.Р); G")
We define a filtration on C* = Hom (Д(Е,Ел) ® A(E',E1'), G") by
FSC* = (c € C* | c | Д(ЕЬЕЛ) ® Д(Е), E1-) = 0, i + / = s)
Then the cross product
Hom (Д(Е,ЕЛ), G) ® Hom (Д(Е'Л£,), G') C*
maps Es ® F8' to Fs+s'C *. It follows easily that there is an induced pairing of
the corresponding cohomology spectral sequences and that hx has the stated
property.
To prove the statement about the pairing h2, let C* and C* be the chain
complexes of (B,A) and (B',A'), respectively, and let C.f = С* ® C*. We
define a homomorphism
>)"*:	Hom (C", №(F x F'; G"))
by the condition
{&}* X {fij}* X <>/'*(«), {o} X {o'}> = (b X o')*(u)
where о: (Д«,Д«) -> ((B,A)«,	a': (#,&)	(_(B',A')j, (В'Л'у-i), with
i + / = s, д: (Д’, Д’) X E —> (EirE;. i), and o': (&,№) X E' —> (E),E)__i) are
admissible liftings, and where и g Hs+t((E X E')s, (E X E')s_i; G ® G'). Then
f" is an isomorphism taking d4 into the coboundary operator of the cochain
complex Hom (C£ H*(F X F'; G")).
Furthermore, if v £ Ep' and v' £ Els'>f', then v X v' £ Ej+s'.j+tp an(j
from the definitions we have
{&}* X {&'}* X (f*(v x v'), {a} x {a'}>
= (a X o')* (v X r') = °* (г) X o' * (o')
= ({&}* x <^(0),{o}>) X ({&}* X <«r *(</),{o'})
= (_!)*'{&}* x {&'}* X <«P=(o) X <A'*(o'), {o} X {o'})
Therefore i/' * (о X o') = (— l)fsV* (о) X </ * (o'), and this implies the result
about the pairing h2. 
This gives the following important multiplicative property for the
cohomology spectral sequence of a fibration.
8 theorem Letp: E-^ В be an orientable fibration over a path-connected
base, with fiber F. Let {Ai,A2} be an excisive couple of path-connected sub-
spaces of В such that {Ед,,Г.ау } is an excisive couple in E. Then there is a
functorial pairing of the E2 cohomology spectral sequences of (Е^Ел^ and
(E,Ea2) to the E2 cohomology spectral sequence of (E, Елг U E^2), which on
E.2 is isomorphic to ( —l)fs' times the cup-product pairing (G and G' paired
rtoG")
IjSEC. 4 MULTIPLICATIVE PROPERTIES OF SPECTRAL SEQUENCES	497
J HS(B,AX; Hf(F;G)) ® №'(B,A2; Hf'(F;G')) -> №+°'(В, Ax U A2; H«+''(F;G"))
t!iid on Em is induced by the cup-product pairing
H*(E,EA1; G) ® H*(E,EA.2; G') -+ H*(E,EA. и Ел2; G")
proof We begin by showing that there exists a CW complex X, with sub-
complexes Xx and X2, and a weak homotopy equivalence f: X —> В such that
у | Xx: Xx —> Ax and /I X2: X2 —> A2 are also weak homotopy equivalences. In
fact, let g: Y B, gx: Yx —> Ax, and g2: Y2 -> Л2 be CW' approximations.
Then there exist maps gx: Yx Y and g2: Y2 Y (which can be taken to be
cellular) such that g ° gi: Yx —> В and g ° g2: Y2 —> В are homotopic, respec-
tively, to the composites Yx Д Ax С В and Y2 Д A2 С B. Let X be the
; CW complex obtained from the disjoint union Yx X I U Y U Y2 X I by
identifying (i/x,0) with gx(yx) € Y for all yi € Yx and (y2,0) with g'2(i/2) € Y
for all y2 £ Y2. Let k: Yx X I U Y U Y2 X I ~> X be the collapsing map and
define a map f: X В such that (f ° k) | Y = g, (f ° k) | Yx X I is a homot-
opy from g ° gi to i ° gx, and (f ° k) | Y2 X I is a homotopy from g ° gi to
j' ° g2. Let Xx = k(Yx X 1) and X2 = k(Y2 X 1) and observe that Xx and X2
are subcomplexes of X such that f\ Xx: Xx —> Ax and /| X2: X2 —> A2 are
weak homotopy equivalences. Furthermore, k(Y) is a strong deformation re-
tract of X, and since /1 k(Y): k(Y) —x В is a weak homotopy equivalence, so
is /: X —» B. Therefore the map J. X —> B has the desired properties.
The excisiveness assumption about {AX,A2} implies that f induces
an isomorphism
f*-. H*(B,At U A2) ~ H*(X, Xx U X2)
Let p': Ex X be the induced fibration over X and let f : Ex ~ h be the
corresponding map. Then f induces isomorphisms
H* (E,EA.) ZH* (Ex,Ex.) and H* (E,EA2) H* (Ex^
The excisiveness assumption about {ЕЛ1,Еа,} ensures that/also induces an
isomorphism
H*(E, EA1 U EA2) H*(EX, Ex. U E^)
By theorem 7, there is a pairing of the E2 cohomology spectral sequences
of (Ex,Ex.) and (Ex,Ex2) to the E2 cohomology spectral sequence of
(Ex,Ex.) X (Ex,Ex2), which corresponds to cross product on the E2 and Era
terms. There is a commutative square (whose horizontal maps are diagonal
maps)
Ex —> Ex X Ex
4
p’ xp'
X —> X X x
Let (I: X-> X X X bea cellular approximation to the diagonal map having
the property that d(Xx) С Xx X Xx and d(X2) С X2 X X2 (such maps exist).
It follows that there is a lifting d: Ex—>ExX Ex oi d ° p': Ex^ X X X
498	SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES CHAP <j
which is homotopic to the diagonal map Ex -+ Ex X Ex. Then d maps the
filtration of (EX,EX1 U £x2) into the filtration of (Ex,Exf X (Ex,Ex„) and so
induces a homomorphism from the E2 cohomology spectral sequence of
(EX,EX1) X (Ex,EXs) into the E2 cohomology spectral sequence of (Ex, EX1 U Exf
Since d takes cross products in Ex X Ex into cup products in Ex, the compos-
ite of this homomorphism with the pairing above is a pairing from the spec-
tral sequences of (Ex,Ext) and (EX,EX,) to the spectral sequence of
(Ex, Ex, U £x2), which is induced by ± cup product on the E2 and E„ terms
By means of the isomorphisms induced by f and f, this gives a pairing
from the E2 cohomology spectral sequences of (Е,ЕЛ,) and (Е,ЕЛг) to the E2
cohomology spectral sequence of (E, £л, U ЕЛг), which is induced by -+- cup
product on the E2 and Era terms. The resultant pairing is independent of the
choice of X. 
9 corollary Let p: E —» В be an orientable fibration with path-connected
base B, with fiber F. For any А С В there is a convergent E2 cohomology spec-
tral sequence of bigraded algebras with E$f ~ HS(B,A; HL(F;R)) and EK the bi-
graded algebra associated to some filtration of Н*(Е,ЕЛ; R). This spectral
sequence is functorial on the category of such fibrations and fiber-preserving
maps. 
5 APPLICATIONS OF THE COHOMOLOGY SPECTRAL SEQUENCE
Because the cohomology spectral sequence of a fibration has a multiplicative
structure, it is a more powerful tool than the homology spectral sequence. We
shall use it in deriving the generalized Wang and Gysin cohomology sequences
and then apply the cohomology spectral sequence to obtain another descrip-
tion of the Hopf invariant in a particular dimension. The section closes with
some results about the homology and cohomology of spaces of type (tt,1).
Let p: E В be an orientable fibration over a path-connected base and
with fiber F. First we shall determine i*: H*(E;G) —> H*(B;G), where
i: F С E, in terms of the cohomology spectral sequence of E. Because this is
a first-quadrant spectral sequence, there is a monomorphism Eg>f —> EJJ-t
Since В is path connected, there is an isomorphism H°(B-, Hl(F;Gf) ~ Hl(F;Cf
Using the fact that the cohomology spectral sequence is functorial, it follows
that i* maps the spectral sequence of E —> В to the spectral sequence
of F —a bo- Therefore i *: H* (E;G) H* (F;G) is the composite
Hf(E;G) = F°Hf(E;G) -> Eg-' -> Efyt ~ H°(B; Ht(F;G) ~ H«(F;G)
This leads to the following generalized Wang cohomology sequence.
1 theorem Let p: E —» В be a fibration, with fiber F and simply
connected base B, which is a cohomology n-sphere (over R) for some n > 2.
There is an exact sequence
-----> ff(E;G) №(F-,G) \ №^(F-,G) -a №+\E-,G) ^ ...
SEC. 5 applications of the cohomology spectral sequence	499
: in which 0(u v) = 0(u) о о + (- l)<n+1>“ и 0(v), the coefficients
t being suitably paired.
’ proof Since В has no torsion, for the cohomology spectral sequence of
E —> В we have
F%f zz №(B) ® Hf(F;G) =0 s 0, n
As in the proof of theorem 9.3.2, this leads to an exact sequence
----> ff(E;G) -a £°>f	№+\E-G)	• • •
Let 1 £ H°(B) be the unit class and let w € H’!(B) be a generator of Hn(B).
The map и —> 1 ® и is an isomorphism of fF(F;G), with and the map
v w <g) о is an isomorphism of Hf“n+1(F;G), with £”>f-?l+1. Define
0-. FF(F;G) —» H/ ,|+I(F;G) by the condition
	d„(l ® u) = w ® 0(ы)
Then the desired exact sequence is obtained from the exact sequence above
' on replacing E^ by Hf(F;G) and E’-.'«+l by Ht~«+1(F-,G) and interpreting
j the resulting homomorphisms. To verify that 0 has the stated multiplicative
' property, we use the fact that dn is a derivation. Then we have
‘ w ® 0(u ^v) = dn(l ® (и о о)) = dn(l ® и о 1 ® о)
!	= dn(l ® u) о 1 ® v + (~l)deg« I ® u о dn(l ® v)
')	= w ® [0(w) О V + (_l)(«+1)deg,< l/<> #(„)] 
; Let p: E —> В be an orientable fibration with path-connected base ana
j let В' С В and E' = p'(B'). We show how the homomorphism
i	p*: H* (B,B'- G) -a H* (E,E'-, G)
= can be interpreted in terms of the cohomology spectral sequence of (E,E').
I. Because the spectral sequence is a first-quadrant spectral sequence, there is
an epimorphism Eg>° —> E^°. The augmentation G —> H°(F;G) induces a
'• homomorphism HS(B,B'-, G) —» HS(B,B'; H°(F;G)'). Using the spectral sequence
of the fibration В С В and the functorial property of the cohomology spectral
sequence, it follows that p*: H*(B,B'; G) -> H*(E,E'; G) is the composite
| HS(B,B'; G) -> HS(B,B'; H°(F;G))
I	~ £s,o £s,o ~ FSHS(E,E'; G) C №(E,E'-, G)
i	2
This leads to the following generalized Gysin cohomology sequence.
!	2 theorem Let p: E В be an orientable fibration with path-connected
I base space and with fiber F a cohomology n-sphere (over B), with n > 1. If
j В' С В and E' = p~r(B'), there is an exact sequence
i ... ^ №(E,Ef G) -a H^(B,B'; G) №+1(B,B'; G) №+1(E,E'- G)	• • •
;	in which Ф(и) = и о fi for some fi £ H’i+1(B;B). If n is even, 2fi = 0.
proof For the cohomology spectral sequence of (E,EZ) we have
500	SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES CHAP. 9
E^ ~ HS(B,B'; H'(F;G)) =0 t 0, n
As in the proof of theorem 9.3.3, this leads to an exact sequence
• • • -» №(E,E'; G) Е°~п-п E®+1’°	№+\E,E'-, G) ...
Let 1 E H°(F;B) be the unit class and let w £ H^FiR) be a generator of
H”(F;B). Corresponding to these generators are isomorphisms G ~ //°(F;G)
and G ~ H"(F;G). Thus we have isomorphisms
HS(B,B'; G) HS(B,B'; H°(F;G)) = Ei>°
whose composite will be denoted by a: №(B,B': G) ~ E-fi}, and
HS(B,B'; G) H«(B,B'; H»(F;G)) ~ Е|-я
whose composite will be denoted by fi: HS(B,B'; G) E«>n. Define the
homomorphism Ф: 1F~"(B,B'; G) —> Hs+1(B,B'; G) by the equation
«t(u) = ( — l)deg« dn+1fi(ii)
The desired exact sequence is obtained from the exact sequence above on re-
placing E^~n’n by Hs~n(B,B': G), Eg+1>° by Hs+1(B,B'; G) and interpreting
the resulting homomorphisms.
In the spectral sequence of E with coefficients В there are similar
isomorphisms a: №(B,R) ~ E»>° and fi: 1F(B;B)	Eg.,!. Let 1 also denote
the unit class of H°(B:R) and define fi £ H'rtl(B;B) by the equation
«(fi) = d„+1/3(l)
To verify that Ф(н) = и fi, we use the cup-product pairing from the
spectral sequence of (E,EZ) with coefficients G, and the spectral sequence of
E with coefficients B, to the spectral sequence of (E,EZ) with coefficients G.
Then
«Ф(и) = (-l)des«dn+1/?(w) =	/3(1))
= а(«) о dn+i/l(l) = а(ы) a(fi) = а(г/ fi)
Therefore ’il'(u) = u о fi. Since w w = 0, /3(1) fi(l) = 0 in the spectral
sequence of E. Therefore, if n is even,
0 = dn+fifi(l) /3(1)) = «(fi)	j8(l) + j8(l)	«(fi) = /3(2fi)
showing that 2fi = 0. 
We use the cohomology spectral sequence to give another interpretation
of the integer |H[«]|, where [a] £ w2„+i(S,1+1) and H: w2,i+1(S,1+1) —> w2.,l_1(S2?l-1)
is the Hopf invariant defined in Sec. 9.3.
3 theorem Let «: S2n+1 —> S',+1 be a base-point-preserving map and let
Ya be the CW complex obtained by attaching a (2n у 2)-cell to Sn+1 by the
map «. Then Hn+1(YQ) and	are both infinite cyclic, and if и and v
are generators, respectively, then и и = _t|Hj«||c.
SEC. 5 APPLICATIONS OF THE COHOMOLOGY SPECTRAL SEQUENCE	501
pboof If Z is the mapping cylinder of a, then Ya = Z/S2n+1, and so
H*(YJ ~ H*(Z,S2»+i). Let и € H»+i(Z,S2"+1) and v € H2"+2(Z,S2«+i) be
respective generators. It suffices to prove that и о u = ±|H[«]|c.
Let Z -a S’i+1 be the retraction and let E —> Z be the principal fibra-
tion induced by r. Since r is a homotopy equivalence, the induced map
E —> PSn+1 induces isomorphisms of homology. Therefore Й^(Е) = 0 and
Й* (E) = 0. The restriction of E to S2n+1 is the principal fibration E« —> S2’i+1
induced by a. By theorem 9.3.10, |H[«]| = m if and only if H2n(E„) ~ Z?n.
From the following portion of the Wang homology sequence of Ea
0	Я2;1+г(Еи)	н2к(Еа.) 0
it follows that if m 0, Я2я+1(Еа) = 0, and if nz = 0, then H2)i+i(E„) ~ Z.
By the universal-coefficient formula for cohomology, H2n+1(Ea) Zm no
matter whether m = 0 or not (recall that we have adopted the convention
that Zo = Z).
Since H* (E) = 0, there is an isomorphism
5: H2»+1(E,r)	H2» ' 2(E,EO)
and so H2n+2fE,Ef) Zjn, where m — |H[«]|. We compute the order of
H2n+2(E,Ea) by using the cohomology spectral sequence.
For s + t = 2n + 2 the only nonzero term Eg-* is the term
E2»+2,0 ~ #2n+2(Z;S2«+l) ® H°(f2Sn+1),
and for s + t = 2n + 1 the only nonzero term Eg' is the term
Eg+1>«	Hn+1(Z,S2«+i) ® Hn(flSn+1f
It follows that
H2’l+2(E,EQ.)	E2»+2.0 ~ E2n+2,0/Jn+1(En+l,n)
Let w' £ H,l+1(Z) be the generator defined by и' = и | Z. Then, since
H*(E) = 0, there is a generator w £ Hn(S2Sn+1) such that in the spectral
sequence of E we have dn+i(I ® w) = и' ® 1. Using the pairing of the
cohomology spectral sequences of (E,Ea) and Ea to that of (E,E„), we see that
® w) = d„+i(« ® 1 о 1 ® w) = ±u ® 1 о dn+i(l ® w)
= ±u ® 1 va' ® 1 = ±(zz о и') ® 1
= ±(u о u) ® 1
Therefore H2n+2(E,Ef) is infinite cyclic if and only if и о и = 0, and
H2n+2(E,E,r) has order m if and only if и о и = ± mv. Comparing this with
the earlier calculation of H2n+2(E,Ea) gives the result. 
4	совoi .i ,abv For any integer m > 1 the Hopf invariant
H: 774HI+1(S2»»+1)	TT^S4»-1)
is the trivial homomorphism.
502	SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES CHAP, <j
proof For any a: S'1'"'1 -a S2m+1, if Ya is the CW complex obtained by
attaching a (4m + 2)-cell to S2m+1 by the map a and if и £ H2ot+1(Y„)
is arbitrary, then it о и — —и и, and so и u = 0. By theorem 3,
|H[«]| = 0, and so H[a] = 0 for all [a] £ 7r4m+i(S2m+1). 
5 corollary For any m > 1, if a2m: S4m 1 —> Sz™ is the map used in
forming the CW complex X4m, then |H[a2m.]| = 2,
proof Recall the definition of X4™ = Y„2m in Sec. 9.3. There is a collapsing
map k: S2m X S2ra X4m with the property that if u' £ H2m(S2m) is a genera-
tor, there are generators и £ H2m(X4m) and v £ jf4”l(X4m) such that
k*u = и' X 1 + 1 X n' and k*v = и' X a'- Then
A* (n о u) = («' X 1 + 1 X uf о (it' x 1 + 1 X uf — 2u' X u'
Since k*: H4m(X4m) ~ H4m(S2m у S2m), it follows that и о и = 2v, and the
result follows from theorem 3. 
If я- is a group, we define H* (77) [and H* (77)] to be the integral homology
[and cohomology] groups of a space of type (77Д). Since any two spaces of
type (77,1) are easily seen to have the same weak homotopy type, these groups
are independent (up to canonical isomorphism) of the space of type (77,1)
chosen. Furthermore, any homomorphism 77 -a 77' induces homomorphisms
/£5.(77) -а Н*(7т') and H*(t7z) —» H*(t7). We use the cohomology spectral
sequence to obtain information about these groups.
6 theorem For n > 1 there are isomorphisms
0
z
№CZ,f
q odd
<7 = 0
q even, q f> 0
proof Let X be a CW complex of type (Z,2) and let PX -a Xbe the path
fibration. Then the fiber SIX of this fibration is a space of type (Z,l). There-
fore SIX is a cohomology 1-sphere, and since PX is contractible, it follows from
theorem 2 that H*(X} is a polynomial algebra on a generator S2 of degree 2,
characterized by the equation S2 ® 1 = d2(l ® w) [where w is a generator of
П1(ЙХ) and d2 is the differential operator in E2 of the cohomology spectral
sequence of the fibration PX —» X]. Let /: X —> X be a map such that f* 1 — nt
for some 2-characteristic element 1 £ H2(X) (such a map exists, by theorem
8.1.10). It follows thatf*(u) = nu for any и £ H2(X) and/#: 772(X)	t?2(X)
is the homomorphism /#[«] = n[«]. Let p: E —> X be the principal fibration
induced by f. Then p has fiber SIX, and from the functorial property of the
cohomology spectral sequence, we have dz(l ® w) = f * SI ® 1 = nil ® 1 in
the spectral sequence of p: E —> X. Therefore, in the Gysin sequence of p the
homomorphism
Ф: HS1(X) -> №+4(Xf
equals the cup product by nS2, and so
SEC. 5 APPLICATIONS OF THE COHOMOLOGY SPECTRAL SEQUENCE
ker 'P = 0 coker SP Z„
503
? Therefore №(E) = 0 unless q is even, and H°(E) Z and №(E) sr ZB if q is
‘ even and q > 0.
It merely remains to verify that E is a space of type (Z„,l). This follows
from the following commutative diagram with exact rows:
• • • -A 77Q(X)	77Q_4(fiX) —A 77(/ I(E) 77Q_X(X) —A - • -
'•! . 1 1
• • • —> 77q(X) -^A 77e_1(S2X) —A 77Q_1(PX) -A 77Q_4(X) —A • • • 
7 corollary For n > 1 there are isomorphisms
HgCZn)
0
z
ZB
q even, q^0
q = °
q odd, q 0
proof This will follow from theorem 6 and the universal-coefficient formula
5.5.12 once we have verified that H* (ZB) is of finite type. We use the partic-
ular space E of type (Z„,l) constructed in the proof of theorem 6. Since the
fiber RX of the fibration p: E X is a homology 1-sphere, there is, by
theorem 9.3.3, an exact Gysin homology sequence
HS(E) HS(X) Hs.2(Xj -a Hs_r(E) —a •  
: Since H* (X) is of finite type [in fact, Hq(Xf — 0 if q is odd and Hq(Xf ~ Z if
q is even and q > 0, as can be seen by using the Gysin sequence of the
• fibration PX —a X], it follows that Ft* (E) is of finite type. 
1 Because S1 is a space of type (Z,l), we now know H* (rr) if 77 is a cyclic
1 group. The groups (77) for 77 a finite direct sum of cyclic groups can
I be computed by induction, using the Kiinneth formula and the following
result.
8 lemma If Y and Y' are spaces of type (77,1) and (rr',1), respectively,
; their Y X Y' is a space of type (тг x rr', 1).
j proof It is easily verified from the definitions of the homotopy groups that
I 77Q(Y X Y')	77ff(Y) X 57q(Y'). 
:	In this way we can determine H* (77) if 77 is a finitely generated abelian
( group. The following result gives information about Н^(гт) for an arbitrary
I abelian group 77.
) 9 theorem Let {тга} be the family of finitely generated subgroups
j directed by inclusion of an abelian group 77. Then
j	H* (77) ~ lim , fitfi (77„)}
I proof For each element A C 77 let S/ be a 1-sphere. Let X1 — V Sx4 and
? define a homomorphism fi: w1(X1) —> 77 by the condition /?[wA] = X, where
504	SPECTRAL SEQUENCES AND HOMOTOPY CROUPS OF SPHERES CHAP <)
[<^x] E ^i(X') is determined by the inclusion map С X1 (tt±(X') is the free
group generated by the collection {[wx]}x). For every base-point-preserving
map w: S1 -a X1 such that /3[<о] = 0, attach a 2-cell to X1 and let X2 be the
space obtained by adjoining all these 2-cells to X1. Continue inductively, de-
fining X™ for m > 3 to be the space obtained from Xm-1 by attaching m-cells
for every map S’"-1 -> X™-1. Let X be the CW complex whose m-skeleton is
Xm for all m > 1 and whose 0-skeleton is the base point of X1. Then X is a
space of type (%,1).
For any finite subset a of tt let Xa be the largest subcomplex of X such
that X„ ' = V Xe a S£'. Then it is clear from the construction of X that Xa is a
space of type (wo,l), where тта is the subgroup of tt generated by the set a.
Since every compact subset of X is contained in Xa for some finite subset a
of tt, it follows that
н* (тт) ~ H* (X) ~ lim , {//* (Xa)} ~ lim , {If. (wtt)}
Since тта is a finitely generated subgroup of tt and every finitely generated
subgroup of tt is of this form, the right-hand side above is isomorphic to
lim^	
These results on If. (tt) will be used in the next section in the proof of
the generalized Hurewicz isomorphism theorem.
6 SEBRE CLASSES OF ABELIAN GROUPS
The spectral sequence of a fibration is well suited for inductive arguments
based on the lowest (or highest) dimension in which a particular phenomenon
occurs. Such arguments can be simplified further by systematically neglecting
certain abelian groups in order to carry along just that portion of a given
group which is relevant to the phenomenon in question. For example, in
studying the p-primary components of finitely generated abelian groups, it is
convenient to neglect finite summands whose order is not divisible by p. The
process of neglecting certain groups will be formalized in this section by
means of a study of groups “modulo a Serre class of abelian groups.” The
machinery will be applied, by means of the spectr al sequence of a fibration,
to the study of the homotopy groups of a space. In particular, the section closes
with interesting generalizations of the Hurewicz and Whitehead theorems.
A Serre class of abelian groups is a nonempty class of abelian groups
having the property that for any exact three-term sequence of abelian groups
A —* В —у C, if A, С E E, then В f £
I theorem A class G of abelian groups is a Serre class if and only if it
has the following properties:
(a)	G contains a trivial group.
(b)	If A E S and A ~ A', then A' £ (?.
(c)	If А С В and В E E, then Л f £
1 SEC. 6 SERRE CLASSES OF ABELIAN CROUPS	505
(d)	If А С В and В E E, then B/A E E.
(e)	If 0 —> A В C 0 is a short exact sequence, with A, C £ £, then
В f t.
proof If E is a Serre class, it is nonempty, and if A E E, then (a) follows
from the exactness of A 0 A. Properties (b), (c), and (d) follow from (a)
and the exactness of the sequences 0 A' —> A, 0 A B, and
В -> B/A -> 0, respectively, while (e) follows from the defining property of a
Serre class.
Conversely, if E satisfies properties (a) to (e), then E is nonempty, by (a).
[f Л В Д C is an exact sequence, there is a short exact sequence
0 im a В coker a 0
and isomorphisms A/ker a im a and coker a zz im (3 С C. If A E E,
it follows from properties (d) and (b) that im a f £ If C £ E, it follows from
properties (c) and (b) that coker a E E. If A, С E E, it follows on using (e) that
В E £ Hence, G is a Serre class. 
Note that it follows from a and b of theorem 1 that a Serre class does
not form a set. We list some examples of Serre classes.
i
2 The class of all abelian groups
3 The class of trivial groups
; 4 The class of finitely generated abelian groups
|	5	The class	of finite abelian groups
1	®	The class	of torsion abelian groups
•	T	The class	of p-groups for a given prime	p
।	fi	The class	of groups having no element	with order a positive power of a
! given prime p
j Given a class E, we are interested in computing modulo groups in e.
1 Thus a homomorphism <p: A । A2 is defined to be a G-monomorphism (or
J G-epimorphism) if ker (f E £ (or coker tp E E) and is a (^isomorphism if both
i conditions are satisfied. It is easily verified that the composite of E-isomorph isms
! is a E-isomorphism. Two abelian groups Aj and A2 are said to be G-isomorphic,
: denoted by AT s: A2, if there exists an abelian group A and ^-isomorphisms
A Ai and A -r> A2. Note the similarity between the definition of E-isomorphic
I abelian groups and the definition of spaces of the same weak homotopy type.
lemma The relation of being G-isomorphic is an equivalence relation.
) proof The relation is clearly reflexive and symmetric. To show that it is
transitive, assume Ai ~ A2 and A2 ~ A3. There exist abelian groups В and
e	e	r
?' and E-isomorphisms <p±: В —* Ai, <p2-. В —> A2, <p2: B' —+ A2, and <рз: В —+ A3.
506	SPECTRAL SEQUENCES AND HOMOTOPY CROUPS OF SPHERES CHAP &
Let C = {(b,b') E в © В' I <p2(b) = ф2(Ь')} and let P- c В and p'-. C-> B'
be the projections (C is the fibered product of <p2 and <p2 in the category of
abelian groups). Because there is an exact sequence
ker <p2 —> С -If В —> coker <p2
it follows that p is a ^-isomorphism. Similarly, p’ is a ^-isomorphism. There-
fore the composites С Д В Д] and С A В' Д, are G-isomorphisms,
showing that Ai ~ A3. 
A topological space X is said to be Q-acyclic if its integral homology groups
He(X) E Gfor q > 0. In order to ensure that the product of two G-acyclic spaces
be t-асу die, we need G to have the additional property that А, В E G imply
A ® В, A * В E G. A Serre class with this additional property is called a ring
of abelian groups.
A pair (X,X') with X' nonempty is said to be ё-acyclic if the integral groups
Hq(X,X') E G for all q. In order to ensure that the product of a G-acyclic pair
and an arbitrary space is a G-acyclic pair, we need G to have the property that
A E G implies A © В, A * В E G for arbitrary B. A Serre class G with this
additional property is called an ideal of abelian groups. Obviously, an ideal of
abelian groups is a ring of abelian groups. Examples 2, 3, 6, and 7 are ideals
of abelian groups, while examples 4, 5, and 8 are rings of abelian groups
which are not ideals of abelian groups.
In the sequel some of the results will be valid for a ring of abelian groups
and somewhat stronger results will be valid for an ideal of abelian groups.
The results will usually be stated in pairs, one for a ring of abelian groups,
and the other for an ideal of abelian groups. The proofs of the two pairs will
usually differ only in minor details. The following generalization of lemma 9.3.4
is the main result obtained from a spectral-sequence argument.
10 theorem Let p: E —» В be an orientable fibration with path-connected
fiber F and path-connected base В and let B' be a nonempty subspace ofB.
Define E' — p-1(B') and let G be a Serre class. We assume that HfB,B'; R) E G
for- 0 < i n and HfF;G) E G for 0 <j < m. We define an integer r > 0 as
follows:
(a)	If G is a ring of abelian groups and HfiB,B'; R) = 0, let
r = inf (n, m + 1).
(b)	If G is an ideal of abelian groups, let r = n + m — 1.
Then the homomorphism p*: HfE,E'; G) Hq(B,B'; G) is a ё-monomorphism
for q <r and a ё-epimorphism for q < r + 1.
proof We use the spectral sequence of (E,E') and show first that E^t E G if
s + t <r and t > 1. We know that
Elt ~ HS(B,B'-, R) © Ht(F-G) © Hs_i(B,B'; R) * Ht(F;G)
< In case (o), because H0(B,B'; R) = 0 = HfiB,B'; IT), it follows that kft E G if
Л SEC. 6 SERRE CLASSES OF ABELIAN CROUPS	5Q7
). s = 0 or 1. If s > 1, then t < m (because s + t < m + 1). Therefore
4 ffi(F;G) E G, and because it is also true that s <2 n> HS(B,B'; R) and
Hs_i(B,B'; R) are both in G. Because G is a ring of abelian groups, Ej>t E G.
In case (b),
s + t < r — n + m — 1
implies s < n — 1 or t < m — 1. Because G is an ideal of abelian groups, it
again follows that E|i( E G.
To complete the proof, note that the spectral sequence gives a composi-
tion series
0 C Do C Di C • • • C Dq = Hq(E,E'; G)
where Do, D±/Do, . . . , Dq/Dq~i are the limit terms of the spectral sequence
(that is, Di/D^i ~ E£Q_f). Therefore
HQ(E,E'; G)/DQ_r ~ E“o C E|,o = Hq(B,B'; G)
and the kernel of the homomorphism p*: Hq(E,E'; G) Hq(B,B'; G) equals
the kernel of the map Dq Dq/Dq_t. To show that p* is a G-monomorphism,
therefore, we must show that Dq_t E G. By a simple induction (on for
0 < к < q — 1), it suffices to show that Es,t E G for s + t < r and t > 1.
This follows from the corresponding property of F.f already established.
i To prove that p* is a G-epimorphism, we must show that E^,o/Eq,o E G
' if q < r + I. However, there is a sequence
E20 □ £3 0 □ . .. □ Eq+1 = E“o
and again by a simple induction, it suffices to show that E^o/E^1 E G
! for q < r + 1 and к > 2. By definition,
j	E^^ker^E^^EU-s-r)
, Therefore EJq/E^1 is isomorphic to a submodule of Е^^х, and it suffices
to show that E§_fc>fc_x E G for q < r + 1 and к > 2. This follows from the
j fact (already established) that	E G for q < r + 1 and к > 2. 
 By specializing to the case where B' is a point, we get the following
interesting applications of this last result.
<
11	corollary Let p: E -у В be a fibration with path-connected fiber F
i	and	simply connected base B. Assume	that E is	ё-acyclic	and Hi(B)	E	G for
0	<G	i <f n. Then HfF) E G for 0 < i	n — 1, and
j	(a) If G is a ring of abelian groups,	Hn_fiF)	Hn(B).
’	(b) If G is an ideal of abelian groups, ЩЕ)	Ht+fiB)	for	i < 2n	—	2.
।	c
proof Let B' = {b0} and E' = p-1(bo) = F and use induction on n.
 Inductively we can assume HfF) E Gfor 0 < i < n — 1. We apply theorem 10,
; with m =ii — l. In case (a) r = n and in case (b) r = 2n — 2, and
j HfE,F) ~ Hj(B,b0) for i < r. Because E is G-acyclic, HfE,F) Hj. j(F) for
508	SPECTRAL SEQUENCES AND HOMOTOPY CROUPS OF SPHERES CHAP 9
i > 2, and because b0 is a point, ЩВ") ~ HfiB,b0) for i > 0. The result
follows by combining these G-isomorphisms. 
12	cobollaby Let p: E В be a fibration with path-connected fiber F
and simply connected base B. If G is a ring of abelian groups and two of the
three spaces E, B, and F are G-acyclic, so is the third.
pboof If В and F are 6-acyclic, let B' = {b0} and E' = p-1(b0) = F and
apply theorem 10a, with n = m = oo. We find that (E,F) is G-acyclic, and,
since F is G-acyclic, E is G-acyclic. If E and В are G-acyclic, it follows from
corollary Ila that F is G-acyclic. If E and F are G-acyclic, let n > 2 be the
smallest integer such that Hn(B) £ G (if such integers exist). By corollary
Ila, Hn~i(F) ( G, which is a contradiction. Therefore В is G-acydic. 
The following special case of this last result is worth explicit mention.
13	cobollaby Let X be a simply connected space and let G be a ring of
abelian groups. Then X is G-acyclic if and only if its loop space ИХ is
G-acyclic. 
For our next application of the spectral sequence of a fibration (namely,
to prove the generalized Hurewicz isomorphism theorems) we need another
property of Sene classes of abelian groups. A Sene class G of abelian groups
is said to be an acyclic class if any space of type (77,1) with тт E G is (.’-acyclic.
Thus Gis an acyclic class if and only if тт E G implies Hq(tt) E G for q > 0.
From the remarks and results at the end of Sec. 9.5, it follows that each of
examples 2 to 8 is an acyclic class.
The loop space of a space of type (u,n), with n > 2, is a space of type
(тт, n — 1). Hence we have the following result by induction on n from
corollary 13.
14	lemma If G is an acyclic ring of abelian groups, any space of type (77,1г),
with n > 1 and 77 E G, is G-acyclic. 
Using this and a Postnikov system for X, it can be shown that if X is a
simply connected space whose homotopy groups belong to an acyclic ring G
of abelian groups, then X is U-acyclic. This is also a consequence of the fol-
lowing generalized absolute Hurewicz isomorphism theorem.
15	theobem Let G be an acyclic ring of abelian groups and let X be a
simply connected space. The following are equivalent:
(a)	77j(X) E G for 2 < i <( n.
(b)	Hj(X) E Gfor 2 < i < n.
Furthermore, either implies that the Hurewicz homomorphism
<p: 77|(X) Hi(X)
is a (^-isomorphism for i < n.
' pboof It clearly suffices to prove that (b) implies <p: 77„(X) ~ H„(X). For
SEC. 6 SERRE CLASSES OF ABELIAN CROUPS	509
n = 2 this is a consequence of the absolute Hurewicz isomorphism theorem.
We assume n > 3 and prove the result by induction on n. It follows that
77j(X) E G for i < n.
By corollary 8.3.8, there exists a sequence of fibrations
E?l_i Tti» . .. E± = X
such that Ej is /-connected and pp Ej —* Ej.i has a fiber Fj which is a space
of type (t7}-(X), / - 1).
It follows from the acyclicity of G that Fj is G-acyclic for 2 < / < n — 1
[because 77/X) E G for / < n]. By induction on /, for 2 < / < n — 1, we prove
that pj *: Hf Ej) HifEj-f) for i < n. Assuming it for / — 1, where / > 1, we
see that Hj(Ej_i,e0) E G for i < n and Hf F) E G for 0 < i. We deduce from
theorem 10a that /у»: Hi(E;,Fj) =5 Hi(Ej~f) for 0 < i < n. Since Fj is G-acyclic,
this implies that Н,(Е;) H^Ej^f) for i < n. This completes the induction.
Therefore the composite f = pz ° • • • ° p„_i: E„ i —> X has the property
that
: Hj(En_t) ~ Hj(X) i<n
We have a commutative diagram
77?1(E„_1)H?l(E„_i)
4
rrn(X) л ад
and, by the absolute Hurewicz isomorphism theorem, the top homomorphism
is an isomorphism. Since both vertical maps are G-isomorphisms, the result
follows. 
This theorem clearly implies that a simply connected space is G-acyclic
(for an acyclic ring G of abelian groups) if and only if all its homotopy groups
are in G. Taking G to be the acyclic ring of all finitely generated abelian
groups, we obtain the following result.
16 cobollaby A simply connected space has finitely generated homology
groups in every dimension if and only if it has finitely generated homotopy
groups in every dimension. 
In particular, it follows from corollary 16 that any sphere S” has finitely
generated homotopy groups. Corollary 16 is not true if 77i(X) is assumed to be
finitely generated (instead of 0), as shown by the following example.
17 example Let X = S2 v S1. Then 77i(X) ~ Z is finitely generated and
Hj(X) is finitely generated for all i, but 77г(Х) is a free abelian group on
a countable set of generators and so is not finitely generated.
Nevertheless, there is a generalization of corollary 16 valid for spaces
which are not assumed to be simply connected. If X is a path-connected
510	SPECTRAL SEQUENCES AND HOMOTOPY CROUPS OF SPHERES CHAP. <)
space, let j: X C IS be an imbedding of X in a space В of type (t7i(X),1) such
that /#: 7Ti(X) ~ u/B). Let p: P}- В be the mapping path fibration corre-
sponding to j, as defined in Sec. 2.8 (so X and Pj have the same homotopy
type). The space X is said to be strongly simple if TrfiX) is abelian and
if p: P j —> В is orientable over Z.
18 example Assume that X is a space such that for every element a E tt1(X)
there is a map <5: S1 X X X, with co | S1 X x0 representing a. Then X is
strongly simple (because Pj also has the same property as X). In particular,
any //-space is strongly simple.
19 lemma Let & be an acyclic ring of abelian groups and assume that X is
a strongly simple space such that tti(X) £ G and ЩХ) £ G for 0 < i < n,
where n > 2. If F is the fiber of the fibration p: Pj B, then Hq(F) Hq(Pj)
is a ^isomorphism for q < n.
proof Since X and Pj have the same homotopy type, H;(Pj) E G for
0 < i < n. Let m < n and assume inductively that HQ(F) zs HQ(Pj) for q < m.
Then Hq(F) E G for 0 < q < m.
We now prove that Hm+fiF) H?n+1(P;-). From the spectral sequence of
the fibration (the fibration being orientable, since X is strongly simple), there
is a composition series
0 C Do C Dr C • • • C Dm+1 = Hm+1(Pj)
where Ds/Ds—i ~ Ls,m+r—s and Do = im [Hm+i(P) —> Hm+fiPj)]. To show that
Hm+i(F) —> Hm+1(Pj) is a (3-epimorphism, it suffices to show that E£TO+i_s E 6
for .s > 0. This will be so if Ef>m+i_s E 6 for s > 0. However,
If,ni+l~S	Hin+l—slF) © /Is— 1(h) * /Ij»‘sl_,s(/J)
Since G is an acyclic Serre class, HS(B) = 0 for s > 0 [and, of course,
H0(B) ~ Ч- Since, by the inductive hypothesis, Hm+1_S(F) E G for s > 0, we
see that E^m+i_s E G for s > 0 because G is a ring of abelian groups.
To show that Hm+i(F) Hm+1(Pj) is a S-monomorphism, wc have a
sequence of homomorphisms
Eot)n+± —> Eo,m+i —>•••—> Eo,m+i ~ E>o
and it suffices to prove ker (Eo,m+1	Eg/|+1) E G for r > 2. This is equiva-
lent to showing that dr(E^m+2„r) C Eo.m+i is in G for r > 2. This will be true
if Efn+г-г E G for r > 2. However,
E2 m+2_r	Hr(B) ® Hm+Z—fifi') © Hr—1(B) * Ня1+2—r(E)
and because m + 2 — r < m, Hm+2_r(F) E G, by the inductive assumption.
The result follows because G is an acyclic ring of abelian groups. 
We now have the following strengthened version of theorem 15.
20 theorem Let Xbe a strongly simple space and let G be an acyclic ring
of abelian groups. The following are equivalent:
'[ SEC. 6 SERRE CLASSES OF ARELIAN CROUPS	gjj
:	(a) 77i(X) E G for 2 < i < n.
(b) HfiX) LSfor2 < i <n.
Either implies that <p: rrfX) HfX) is a ^isomorphism for i < n.
proof It suffices to prove that (b) implies <p: rrn(X) Hn(X). Let F be the
fiber of the fibration p: Pj B. Since X and Pj have the same homotopy type,
there is a map f: F X, equivalent to F C Pj. Since rrfiF) ~ w»(Pj) for
i > 2, it follows that/#: tt^F') ~ w/X) for i > 2. By Icinma 19, /. ://;(/’)—>//;(X)
is a G-isomorphism for i < n. Since F is simply connected, it follows from
theorem 15 that <p: TTn(F) Hn(F). Since <p ° f# ~ f* ° <p, this gives the result. 
We use this result to establish the following generalized relative Hurewicz
isomorphism theorem.
21 theorem Let G be an acyclic ideal of abelian groups, let А С X, and
assume that A and X are simply connected. The following are equivalent:
(а)	тт, (X,A) E G for 2 < i < n.
(b)	Hj(X,A) E G for 2 < i <X n.
Either property implies
(с)	<p: %7i(X,A) ~ Hn(X,A).
proof It suffices to prove that (b) implies (c) by induction on n. For n = 2
this follows from the relative Hurewicz isomorphism theorem. Therefore we
assume n > 3 and -n-fiX,A) E G for i < n. Let x0 E A, let PX be the space of
paths in X with origin x0, an(i let P: P^ —> X be the fibration sending a path
to its terminal point. The fiber p^(xo) is the loop space ЙХ. By theorem 7.2.8,
p#: TTifPXqj-^A)) ~ 77fc(X,A) for к > 1. Because PX is contractible,
. ^fc(PX,p-i(A)) ~ TT^fip-ЦА)) for к > 2 and Hft(PX,p-i(A)) = Нц(р~1(А))
for к > 2. For i > 2 there is a commutative diagram
rrwfip-^Af)	^(PX,p-\A))	тт/ХЛ)
4	4	I45
Н^р-ЦАУ) 4 Н4(РХ,р-1(А))	HfXA)
Applying theorem 10b, where H{(X,A) E G for i n, and taking m = 1
(KX is path connected because X was assumed to be simply connected), we
see that p^.: Hi(PX,p-1(A))	H;(X,A) for i < n. Therefore, for i = n all the
horizontal maps in the above diagram are G-isomorphisms, and to complete
! the proof it suffices to prove cp: wn_i(p-1(A))	HM-i(p-1(A)). This will follow
\ from theorem 20 once we have verified that p”1(A) is strongly simple.
5 Because p-1(A) is a principal fibration with fiber fiX, there is a continu-
! ous map fiX X P-1(A) —> p"x(A). Since тт2(Х) —у	is an epimorphism,
j so is tti(RX) —a 77i(p-1(A)). Therefore the existence of the map
{	fiX X p-1(A)	p~\A)
512
SPECTRAL SEQUENCES AND HOMOTOPY CROUPS OF SPHERES CHAP. 9
implies that p 1(A) is strongly simple, as in example 18. 
By using the mapping cylinder (as in the proof of theorem 7.5.9), the
following generalized Whitehead theorem can be deduced from theorem 21.
theorem Let Q be an acyclic ideal of abelian groups and let f: X у
be a map between simply connected spaces. For n > 1 the following are
equivalent:
(a) f#: TrfX) —-> 77,-(У) is a ^-isomorphism for i < n and a G-epimorphism
for i = n + 1.
(b)f^: HfX) -s ITfY) is a в-isomorphism for i < n and a G-epimorphism
for i = n + 1. 
0 HOMOTOPY GROUPS OF SPHERES
The results of the last section were obtained by using the homology spectral
sequence of a fibration. In this section we shall use the multiplicative proper-
ties of the cohomology spectral sequence to obtain some specific results about
the homotopy groups of spheres. These homotopy groups are finitely gener-
ated, and we shall obtain information about their p-primary components. The
first main result is that the only homotopy groups of 8я which are infinite are
77n(S”), and if n is even, 772,,-1(8”). The next main result concerns the double
suspension. It will be shown that for odd n the double suspension
S2; 77m(S”)	77„,+2(S”+2)
induces an isomorphism of the p-primary components of these groups for a
wider range of values of m and n (depending on p) than the range for which
it is an isomorphism between the groups. Combining this with specific com-
putations of p-primary components of 77m(S3 *), we determine the lowest
dimension m > n for which 77m(S”) for n odd has a nontrivial p-primary
component.
We begin with the following useful technical result about the cohomology
spectral sequence of a fibration.
1 lemma Let X be a simply connected space and assume that there is an
element и E Hn(X;R), with n > 2, such that td”-1	0 for some m > 2 and
{1,«,«2, . . . form a basis for /7*(A';7?) in degrees inn. Then there is
an element v 6 H”-1(RX;jR) such that {I,?;} form a basis for H* (RX;B) in
degrees <7 inn — 2.
pboof We use the spectral sequence of theorem 9.4.7, with A empty, for
the fibration PX -» X. Because PX is contractible, E*>' = 0 if (s,t) 7^ (0,0),
and because X has no torsion in degrees < mn, Etf ~ /7s (X) ® П'(ЙХ) for
s <7 mn (all coefficients B). Then we have E§>* = 0 if s <f mn and
SEC. 7 HOMOTOPY GROUPS OF SPHERES
513
s =f= 0, n, 2n, . . . , (m — l)n. Because dr has bidegree (r, 1 — r), it follows that for
s < mn, dr- Ep* E«+r,t-r+i is zero unless r = n, 2n, . . . , (m — l)n.
Therefore E»>* ~ E§-* for s < mn. If t < n — 1 Ef1 ~ Ef1, and if 0 < t < n — 1,
we see that
H*(RX) Zi Ef1 = 0
and so H*(RX) — 0 for 0 < t < n — 1. Furthermore, there is an exact sequence
0	Яо(х) ® Я«-1(ПХ)	H»(X) ® H°(RX) E»>°	0
Because E0/"' 1 = 0 = Eg’0, it follows that there is an element v E Hn“1(RX)
such that d„(l ® v) = и ® 1. Because dn is a derivation, dnfd ® o) =
(_1)Ляил+1 0 £ Tire assumption about the cohomology of X ensures that for
s <9 mn the map dn: Ef"’’11 Efr° is an isomorphism. Because dn is a
differential, the composite
£»—2n,2n—2	j?s—n,n—l d"-, £S,0
is trivial. Therefore dn- E%~2t-2n~2 *	Ey’1’"4 is trivial for s- < mn and
En+”’n l = 0 = ^n+if01’ s < mn- Hence
E.®>* = 0 s < mn, t < n — 1, r > n + 1
p0,2n-2 _ Г0.2Я-2
ЛИ|1 — -^2
Assume the lemma false and let q be the smallest integer such that
n — 1 < q < mn — 2 and №(RX)	0. We shall show that
E°>e - E°-« Zi №(ttX),
which is a contradiction. We know that E£’« ~ E°’e- Furthermore,
dn; Efi—> E^-rn+1 is trivial, because if q — n +1 n — 1, then //'r""+l(RX) = 0
and E.S’Q"’,+1 = 0 for all r and s, and if q — n + 1 = n — 1, then E®-2^2 =
£0,271-2 Therefore E^Zi Е??>®. From the assumption that q is the smallest
degree larger than n — 1 for which №(RX)	0, it follows that E«-f = 0 if
s < mn, t < q, and r > n + 1 (in case t < n — 1 this was noted above).
Therefore d.r: E.p-e Epe-r+1 is trivial for all r > n + 1 and E&«
Hence we have the isomorphisms
E&e ~ E0^^ E°.e z; E°’« 
By using the generalized Gysin sequence of theorem 9.5.2, it is easy to
show that if RX is a cohomology n-sphere for some odd n > I, then H*(X)
is a polynomial algebra on a generator of degree n + 1. The following con-
verse is an immediate consequence of lemma 1 for the case m — co.
2 cobollaby Let X be a simply-connected space such that H* (X;ft) is a
polynomial algebra on a generator of degree n (n is then necessarily even).
Then the loop space RX is a cohomology (n — l)-sphere. a
We shall also need the following consequence of the generalized Wang
sequence of theorem 9.5.1.
514	SPECTRAL SEQUENCES AND HOMOTOPY CROUPS OF SPHERES CHAP. 9
3 lemma Let X be a simply connected space which is a cohomology
n-sphere for some odd n > 1. Then the cohomology ring H*(RX) of its loop
space RX has a basis consisting of elements {I,«i,«2>  • with degree
uk = k(n — 1) and up о uq = [fp + q)\/p\q[\up+g.
proof We use the Wang exact sequence of the fibration PX X. Because
PX is contractible, the map
0: jEP(RX) Н*-«+1(ЙХ)
is an isomorphism for t =f= 0. Define uk £ 1P-(,,|)(RX) for к > 0 by induction
by the equations
Uo = 1
6(uk) = ик~к /< > 0
Then the set {l,ui,u2,   .} is a basis for H* (RX), and we verify that it has
the stated multiplicative property by double induction on p and q. If i = 0 or
/ = 0, then и, Uj = ul+j. Let p > 0 and q > 0 and assume that
щ о tq = [(i + /)!/i!/!]«i+3- if i + / < p + q, i > 0, and j > 0. Because n
is odd,
0(llp O Uq) = 0(up) O Uq + Up О 0(l(Q) = Up l О Uq + Up О Uq^r
- Г (P + ~ T)! , (p + q - 1)! 1_ (p + q)l „
I (p - 1)!q! + pl(q - 1)! J P+Q~1 ~ p\q\ Up+Q-1
Because 0 is a monomorphism,
(p + q)l
«₽ 4q = — i«p+q 
p’q!
It follows from lemma 3 that (»i)p = p\up. Over a field of characteristic
0, the elements {I,Wi,tt|, . . .} also form a basis of H* (RX), and so we obtain
the next result.
4 corollary Let X be a simply connected space which is a rational
cohomology n-sphere for some odd n > 1. The rational cohomology algebra
of the loop space RX is a polynomial algebra with one generator of degree
n — 1. 
Let X be a space of type (Z,3) and let f: S3 X be a map such
that/#: it3(S3) ~ tt3(X). Let p: E —> S3 be the principal fibration induced by
f. Then the fiber F of p: E S3 is a space of type (Z,2). We shall need the
following computation of the homology groups of E.
5 lemma Let p: E —> S3 be a fibration with fiber F a space of type (2,2)
such that S: ^(S3) ~ t72(F). Then the integral homology of E is given by
0
z
Hg(E)
q is odd
q — 0
q — 2n f> 0
SEC. 7 HOMOTOPY GROUPS OF SPHERES
515
proof We know that H*(F) is a polynomial algebra with one generator и of
degree 2. Because 3: ттз(83)	772(F), it follows that E is 3-connected, and so
H2(E) = 0. By the exact Wang sequence of the fibration, 8: H2(F) ~
Without loss of generality, we can assume that и has been chosen so that
fi(u) = 1. Then fi(u”) = ntt»-1. By the exact Wang sequence again, №(E) = 0
if q is even and q < 0, and EPn+1(E) ~ Z„ if n > 1. The result then follows
from the universal-coefficient theorem. 
If G is the Serre class of groups having no element with order a positive
powei’ of a given prime p, then HfiE) E G for 0 < i < 2p. By theorem 9.6.15,
7Т.,.(Е) E & for i < 2p and ~ Zp. Because ттг-(Е) 77j(S3) for i > 3, we
obtain the following result.
6 corollary The p-primary component of 77j(S3) is zero if 3 i <f 2p
and is Zp if i = 2p. 
We are now ready to prove the finiteness of the higher homotopy groups
of odd-dimensional spheres.
7 theorem If n is odd, rrm(Sn') is finite for m n.
proof We use induction on n, If n = 1, we know that tt^S1) = 0 if m 1,
and the result is valid in this case. For n = 3, if E is the space of lemma 5
and G is the Serre class of finite groups, then E is G-acyclic. By theorem 9.6.15,
TTi(E) E G for all i. Because тг{(Е) 77j(S3) for i > 3, tz^S3) is finite for i > 3.
Assume n > 3 and that 77m(Sn-2) is finite for m =/= n — 2. We compute
the rational cohomology algebras of RS” and R2S”. By corollary 4, RS" is a
polynomial algebra with one generator of degree n — 1. By corollary 2,
R2S’! is a rational cohomology (n — 2)-sphere. By the universal-coefficient
theorem, the integral group Z4-(R2S») is a torsion group if 0 < i n — 2, and
H?l_2(R2Sn) is isomorphic to a direct sum of Z and a torsion group. Further-
more, 77fc(fl2S2!) ~ 77fc+2(S’!) for all k. Therefore R2Sn is (n — 3)-connected and
<p: 77n_2(fl2S”) ~ Hn_2(fl2S,!)- If «: S«-2 R2Sn is a generator and G is the
Serre class of torsion groups, it follows that a*: /h(S'"~2)	H;(R2S”) for all i.
Because 6 is an acyclic ideal of abelian groups and Sn+2 and 122S'" are both
simply connected, we can apply the generalized Whitehead theorem 9.6.22 to
deduce that вд 7Л;(8"~2)	tz.;(R2Sh). By the inductive assumption тл;(8'"'2) is
finite for i > n — 2. Therefore тг;(1228") ~ 774+2(8") is a torsion group for
i > n — 2. Because 77m(S”) is known to be finitely generated, 77m(S") is finite
for m n. 
We want to establish a result similar to theorem 7 for even-dimensional
spheres. This will be done by considering a suitable (n — l)-sphere bundle
over S". Let W-nl be the subspace of Rn X Rn consisting of pairs of unit
vectors (zi,zf) which are orthogonal and let p: W2n-1 S” map (^1X2) to z±.
Then p: W2'11 —S" is a fiber bundle with fiber S'"1 (it is the unit tangent
516
SPECTRAL SEQUENCES AND HOMOTOPY CROUPS OF SPHERES CHAP. 9
bundle of S'1), as can be verified by constructing an explicit homeomorphism
// '(!’) ~ U X S"-1 for any proper open subset U C S”.
8 lemma If n is even, the integral homology groups of W2’1^1 are all
finite except for HofW2"-1) and H2n^i(W2n~1), which are infinite cyclic.
proof Because n is even, there is no map f: S'n —> S” which sends each
point of Sn to an orthogonal point of S” (by corollary 4.7.11). It follows that
p. \y2n-1 g” has no section. If [a] £ 7T.„(S’') is a generator, [ci] is not in the
image of
P#. ^(W2"-1)	7r.„(S’1)
(because there is no section). Therefore, 8[a] 0 in u,1_|(.$’,l~'). Because
^i(S,1~1) is infinite cyclic, 8: u?l(S’1)	7rB„i(Sm-1) is a monomorphism and
^ru/il 2'"1) ~ w?l_i(S?!“1)/8(77Ji(Sn)) is a finite group. Because W2nl is
(n — 2)-connected, 7rm_i(W2n-1) ~ Hn_fiW2n~1). Therefore Hn_i(W2,i“1) is a
finite group, and by the exact Wang sequence
0 -> H^W2""1) -A H(1(S''i) Л	H.^1(W2«-1) -> 0
we see that H?l(W2il^1) = 0. From this exact sequence it also follows that
H^W2”"1) — 0 for n < i < 2n — 1 and H2n-i( W2’1-1) is infinite cyclic. 
9 theorem If n is even, 7rm(Sn) is finite for m n and m 2n — 1, and
”,2n-i(S”) is the direct sum of an infinite cyclic group and a finite group.
proof Let t’ be the acyclic ideal of abelian groups consisting of the torsion
groups. By lemma 8, //.;(W2”1) £ 6 for 0 < i < 2n — 1. By theorem 9.6.15,
^2)1 -1(W2,r l) H2?i_i( W2”-1). Because 77-2?l_i( W2nl') is finitely generated
(by corollary 9.6.16) and VV2111) is infinite cyclic, U271i(VV'2i1“') is
a direct sum of an infinite cyclic group and a finite group. If a: S2”-1 —-> W2nl
represents a generator of the infinite cyclic summand of U27i-i( VI/2’'1), then
a*: Hi(S2n l) ~ Hi(W2”-1) for all i. By the generalized Whitehead theorem,
ct#: ^(S2”-1)	^(W2" ') for all i. Using this and theorem 7, irf W2"-1) is
finite for i 2n — 1. The theorem now follows from the exact homotopy
sequence of the fibration W2'11 —2 S” and the fact that, by theorem 7,
Wi(S’1^1) is finite for i n — 1. 
We now consider the double suspension
77,(S«) A 77i+1(S"+l) Д 77i+2(S’1+2)
where n is odd. This involves a study of the composite
S« Л S2S«+1 ft(RSn+2)
We begin with the following partial computation of the Zp homology of R2Sn+2.
IO lemma Let n be odd and p be prime. Then HQ(S2S’,+2;ZP) = 0 for
n < q < p(n + 1) — 2.
SEC. 7 HOMOTOPY CROUPS OF SPHERES
517
PROOF By lemma 3, the set of elements {1,i/i,ri2, . . .	forms a basis
for H* (£2Sn+2;Zp) in degrees < p(n -f- 1). By Iemma I, there is an element
v E H«(fi2S"+2;Zp) such that {l,n} forms a basis for H* (R2S’,+2;ZP) in degrees
< p(n + I) — 2. The lemma then follows by the universal-coefficient
theorem. 
This implies the following result about the double suspension.
11 theorem Let n be an odd integer, p a prime, and E the acyclic ideal of
torsion groups with trivial p-primary component. Then
S2: 7TfSn) 77i+2(S«+2)
is a ^-isomorphism for i < pin + 1) — 3 and a ё-epimorphism for
i = p(n + 1) - 3.
proof The composite
S« A RS’|+1 fi2Sn+2
induces an isomorphism of Wn(S”) with irn(fl2Sn+2') and, by the Whitehead
theorem, an isomorphism of Hn(Sn) with H„(fl2Sn+2). From this and lemma 10
it follows that the above composite induces an isomorphism of HfSn;Zp),
with Hff(fi2S’l+2;Z7,) for q < p(n + 1) — 3. By the universal-coefficient theorem,
it induces a в-isomorphism of HQ(S”), with HQ(R2S’i+2) for q < pin + 1) — 3.
From the generalized Whitehead theorem, it induces a E-isomorphism of
77Q(Sn) with 77Q(K2Sil+2) for q < p(n + 1) — 3, and a E-epimorphism of
яу(И+1)-з(8п) to %/,(!!+i)_3(K2S,1+2). The theorem follows from the fact that S2
corresponds to the above induced homomorphism under the isomorphism
77q(fl2Sn+2) ~ We+2(S',+2) 
12 corollary Let n > 3 be odd and p prime. Then TrfS’1) and я,|_л+з(83)
have isomorphic p-primary components if i 4p -f- n — 6.
proof We use induction on n. If n = 3, there is nothing to be proved.
If n > 5, we need only prove that S2: 'п';-2(8’1’2) —> wJ(S’1) induces an isomor-
phism of p-primary components. By theorem 11, this will be true if
i — 2 <f pin — 1) — 3. Hence we need only verify that
4p + n — 6 < p(n — 1) — 1
But this is equivalent to (p — l)(n — 5) > 0. 
Combining corollary 12 with corollary 6, we have the following result.
13 corollary Let n > 3 be odd and p prime. For 0 m < 2p — 3,
77?l+)n(S’!) has trivial p-primary component and яу+гр-з^”) has Zp as p-primary
component. 
518
SPECTRAL SEQUENCES AND HOMOTOPY CROUPS OF SPHERES CHAP. 9
exercises
A SPECTRAL SEQUENCES AND SUSPENSION
In this group of exercises all spaces will be assumed to be finite pointed CW complexes
and all pairs will be finite pointed CW pairs.
I Prove that {X;Y } is finitely generated.
2	For spaces X and Y prove that there is a convergent E2 spectral sequence {E’’} with
E2t HS(Y; {X;S0)t)
and with E" isomorphic to the graded group associated to the increasing filtration on
{X; Y defined by
FS{X;Y}* = im ({XjU}*	{X;Y}B)
3	For spaces X and Y prove that there is a convergent E2 spectral sequence {Er} with
E|.t №(X- {S°;Y)_t)
and with E„ isomorphic to the graded group associated to the decreasing filtration of
{X;Y}^. defined by
^{X;Y}^ = ker ({X;Y}* -+ {X’-i;Y}J
В THE TRANSGRESSION HOMOMORPHISM
Let p: E —> В be a fibration with path-connected base and path-connected fiber
F = p~l(b0). Consider the homomorphisms
№(F;G) A H"I1(E,F; G) №+ЦВ,Ь0-, G) -A №+1(B;G)
The transgression т is the homomorphism [from a subgroup of H4(F;G) to a quotient
group of №!I(B;G)[
t: 51(iin p*)	G)//* (ker p*)
defined by t(w) = j*p* -18(u), where и £ H«(F;G) is such that 8(u) £ p* (HQ+1(B,b0; G)).
1	Prove that т commutes with the Steenrod squaring operations Sq’ and with induced
homomorphisms for induced fibrations.
2	Assume that В is (n — l)-connected for n > 2 and consider the path fibration
p: PB -> В with fiber QB. Prove that t: Hn^1(QB;G) ~	and that i £ H’1-1(£1B;G)
is (n — ^-characteristic for QB if and only if t(i) is n-characteristic for B.
For the remainder of this group of exercises we assume that the fibration is orientable
over В and the coefficient module is R.
3	For the spectral sequence of the fibration prove the following:
(я) S-i(im p*) Е»л C E« 5 ~ №(F).
(	b) №+1(B)//*(ker p*)	E«+L° and a quotient of E«+1-0 ~ i/''!1(B;G).
(	c) Under these isomorphisms т corresponds to d,;+i: EP^—* E^B°.
4 If H* (E) = 0, prove that H’(F) = 0 for 0 <£ i <£ q if and only if H‘(B) = 0 for
0 < i < q + 1 and, in this case, t: №(F) ~ №+1(/i).
EXERCISES
519
S Assume that H’(B) = 0 for 0 <( i < s and H>(F) = 0 for 0 < j < t. Prove the exact-
ness of the following Serre cohomology sequence:
------> H«(F) №+i(B) -A H"'1(E) A. №+1(F)	-----> №+t~4F)
Consider the homomorphisms
ад Л Hq(B,b0) Hq(E,F) A Hq^(F)
and define the homology transgression
r*  1* 1(im P*) HQ-i(B)/a(ker ps)
by t^(z) = 8p*-1fe(z), where z £ HQ(B) is such that fe(z) £ p*(Hq(E,F)).
6 If Hi(B) = 0 for 0 < i<i s and HfF) = 0 for 0 <£ j <£ t, prove the exactness of the
following Serre homology sequence:
H^F) ------------>Hq(F)±* Hq(E) HJB) Hq_i(F) •
C SERRE CLASSES OF ABELIAN GROUPS
A chain complex modulo (/is a graded group C — {Q,} and a sequence of homomorphisms
Cq —> Cg~i} such that (0Q_i ° 0Q)(CQ) £ (? for all q. The homology group of C is the
graded group H(C) = (Hq(C)), where
Hq(C) = ker 0Q/(ker dq Cl im 0Q+i) ~ (ker 0Q U im 0Q+i)/im 0Q+i
A three-term sequence of groups and homomorphisms
(7 L g 4> G"
is said to be ё-exact if (im a U ker jB)/im «ft1 and if (im a U ker /j)/kcr /?£(/. Longer
sequences are Q-exact if every three-term sequence is t’-exact.
1 Let C be a chain complex modulo (/, let C be a subcomplex of C (that is, C'q C Cq and
0q = 0,; | C,. for all q), and define the quotient complex C/C = (Cq/Cq,dq }, where dq is
induced from Prove that there is a E-exact sequence
-----> HQ(C) Hq(C) -+ Hq(C/C) Hq^(C)
2 Let 0 —> C7 —> C Cz —> 0 be a short (/-exact sequence of chain complexes
modulo (/ and chain maps (a and /J commute with the boundary homomorphisms of the
chain complexes). Prove that there is a (/-exact sequence
-----, He(C) Hq(C) llq(C")	. 
3 Prove the five lemma modulo (/. That is, given a commutative diagram
Gs '^> G.t -% G, '-> G2 Gi
Vsl Y<|	Y.'.l	Y?| Yi I
q/	v	ж	v
H5 Hi Hs H2 Hi
with (/-exact rows such that y2> ?•(> and y;> are (/-isomorphisms, prove that y3 is also a
(/-isomorphism.
For the rest of this group of exercises assume that p: E В is a fibration with path-
connected fiber and simply connected base space and that (/ is an ideal of abelian groups.
520
SPECTRAL SEQUENCES AND HOMOTOPY CROUPS OF SPHERES CHAP. 9
4 If H;(B) С C for 0 < i, prove that H^F) H;(E) for all i.
5 Vietoris-Begle mapping theorem modulo G. If F is (?-acyclic, prove that Hj(E) Н^В)
for all i.	L
Ю HOMOTOPY GROUPS OF SPHERES
1 If Sn is an H-space, prove that there is a short exact sequence
0 -+ TT^S») 779+1(S«+1) Д ^(S2"-*) - 0 q < 3m - 2
2 Prove that 777(S4) ~ 77e(S3) © 777(S7) and that the order of 77g(S5) is twice the order
Of 77e(S3).
3 Let X/ be a CW complex consisting of an n-sphere with an (n + l)-cell attached
by a map of degree p. If n > 2 and p is a prime, prove that (X£) is a finite p-group for
all q, and if q < 2n — 2, prove there is an exact sequence
o ,TQ(S") © zp 77,/x;;)	^(s*) * zp о
4 Prove that S(X;'') has the same homotopy type as X'l+1 and {X'JjX/} ~ Zp if p =y= 2
and (Xg;X^'} is a group of order 4.
5 Let p: E -> S3 be л fibration, with fiber F a space of type (Z,2), such that
Э: 1Гз(83) ~ ^(F), as in lemma 9.7.5. Let/: X/ —> E be a map such that/#: tt4(X^) ~ 174(E)
[such a map exists, because 774(E) ~ Z2], Prove that /#: 775(X4) ~ 775(E) and /# is a
monomorphism of 776(X4) onto the 2-primary component of 77e(E). [Hint: Show that
/B: HQ(X4) Hq(E) is an isomorphism of 2-primary components for q < 8 and use the
generalized Whitehead theorem.]
6 Prove that 77,!+2(Sn) ~ Z2 for n > 2.
7 Prove the following:
(a)	775(S2) sr Z2.
(b)	77g(S3) is a group of order 12.
(c)	T77(S4) ~ 776(S3) © Z.
(d)	77,1+3(S”) is a group of order 24 for n > 5.
INDEX
Abelian H cogroup, 40
Abelian H group, 35
Absolute CW complex, 401
Absolute homology group, 172, 175
Absolute Hurewicz isomorphism theorem, 397-398
generalization of, 508-509
Absolute n-circuit, 148
Absolute neighborhood retract, 56
tautness of. 290-291
Absolute retract, 56
Action of the fundamental group, 382-385, 420
Acyclic chain complex, 163
Additive cohomology operation, 270
Adem relations, 276
Adjoint functor, 41
Adjunction space, 56
Admissible lifting, 476
Admissible map, 476
Affine variety, 10
Affinely independent set, 10
Alexander cohomology, 308
with compact supports, 320
universal-coefficient formula, 336
for compact supports, 337-338
Alexander duality theorem, 296
Alexander-Spanier cohomology (see Alexander
cohomology)
Alexander-Whitney diagonal approximation, 250
Associated graded module, 468
Associative coproduct, 266
Associative product, 263
Attaching map, 146
Augmentation, 167, 213
of cochain complex, 237
Augmentation-preserving chain map, 168
Augmented chain complex, 167, 213
Augmented cochain complex, 237
Axiom of compact supports, 203
Axioms, of cohomology theory, 240-241
of homology theory, 199-200
Back i-face, 250
Ball, 9
Barycenter, 117
Barycentric coordinate, 111
Barycentric subdivision, 123-124
interation of, 124
Base of cone, 116
521
522
INDEX
Base point, 15
nondegeneracy of, 380, 419
Base space, 62, 66, 90
Base vertex, 110, 136
Basis, 6, 8
Betti number, 172, 176
Bidegree, 466
Bigraded module, 466
Bijection, 2
Bijective function, 2
Binormal space, 56
Block dissection, 277
Bockstein cohomology homomorphism, 240
Bockstein cohomology operation, 270
Bockstein homology homomorphism, 222
Bockstein homomorphism, 280-281
Borsuk homotopy extension theorem, 57
Borsuk-Ulam theorem, 104, 266
Boundary, 157
of homology manifold, 277
of pseudomanifold, 150
Boundary operator, 156
Bounded subset, 319
Brouwer degree theorem, 398
Brouwer fixed-point theorem, 151, 194
Brouwer theorem on invariance of domain, 199
Brown representability theorem, 411
Bundle projection, 90
в-acyclic pair, 506
(/-acyclic space, 506
(/-epimorphism, 505
в-exact sequence, 519
ci-isomorphic abelian groups, 505
(/-isomorphism, 505
(/-monomorphism, 505
Canonical free resolution, 219
Canonical map, 152
Canonical projection, 152
Cap product, 254-255
Carrier, 112
Cartan formula, 271
Category, 14
of chain complexes, 158
of connected covering spaces, 79
of contiguity classes, 130
of fibrations, 69-70
of maps, 414
with models, 164
of morphisms, 16
opposite, 17
of pairs, 16
of simplicial pairs, 10
small, 14
of a topological space, 279
Cech cohomology, 327, 358-359
Kiinneth formula, 359-360
universal-coefficient formula, 335-336
Cellular-approximation theorem, 404
Cellular homotopy, 403-404
Cellular map, 403
Chain, 157
Chain complex, 157
acyclicity of, 163
chain contractibility of, 163
modulo (/, 519
over a ring, 213
/'Chain contraction, 163
Chain equivalence, 163, 192
Chain homotopy, 162
Chain homotopy class, 205
Chain map, 158
Chain transformation, 158
Characteristic class, 261, 281
Characteristic map, 91, 457
for a cell, 146
Classification theorems, 427, 431, 451, 452, 460
Classifying space, 408
Closed edge path, 135
Closed path, 46
Closed simplex, 111
Clutching function, 102, 455
Coboundary, 236
Coboundary operator, 236
Cobounded subset, 319
Cochain complex, 236
Co chain homotopy, 236
Cochain map, 236
Cocycle, 236
Coexact sequence, 366, 418
Cofibered sum, 99
Cofibration, 29, 33, 57
induced, 99
(See also Homotopy extension property)
Coherent topology, 5
for simplicial complex, 111
Cohomology, of a group, 502-503
Kiinneth formula 247-249
with local coefficients, 282-283
universal-coefficient formula, 243, 246, 283
Cohomology algebra, 264
of projective space, 264-265
Cohomology extension of fiber, 256
Cohomology functor, 407
Cohomology module, 236
with coefficients, 237
Cohomology operation, 269, 429-430
additivity of, 270
of Bockstein, 270
induced by coefficient homomorphism, 270
suspension of, 431
Cohomology theory, 240
Cohomotopy group, 421, 461
Collaring, 297
Comb space, 26
Commutative coproduct, 266
Commutative diagram, 16
Commutative product, 263
Compact-open topology, 5
Compact pair, 203
Compact supports, 203
Compactly generated space, 5
Compatible family, of cohomology classes, 261
of homology classes, 299
of orientations, 207
in a presheaf, 324
Complementary degree, 468
Completion of presheaf, 325
Component, of groupoid, 45
of simplicial complex, 138
Composite, of functions, 2
of morphisms, 14
Comultiplication, 39
homotopy associativity of, 39
homotopy identity for, 39
homotopy inverse of, 40
INDEX
523
Cone, 56, 116
Conjugacy class, 15
Connected groupoid, 45
Connected Hopf algebra, 267
Connected simplicial complex, 138
Connecting homomorphism, 182, 213, 222, 236, 240
Constant presheaf, 323
Contiguity category, 130
Contiguity class, 130
Contiguous simplicial maps, 130
Continuity property, 319, 358
Contractible space, 25, 56
Contraction, 25
Contravariant functor, 18
Convergent factorization, 438
Convergent filtration, 468, 493
Convergent sequence of fibrations, 438
Convergent spectral sequence, 467
Coordinate neighborhood, 292
Coproduct, 266
Counit, 266
Covariant functor, 18
Convex body, 10
Convex set, 10
Covering projection, 62
induced, 98
Covering space, 62, 103
of projective space, 80
of sphere, 80, 103
of topological group, 104
Covering transformation, 85
Cross product, 231-232, 234-235
in cohomology, 249-250
Crosscap, 148-149
Cube, 9
Cup product, 251-253
CW approximation, 412, 420
CW complex, 401, 420
Cycle, 157
Cyclic module, 9
Deformable subspace, 29
Deformation, 29
Deformation retract, 30
Deformation retraction, 30
Degree, 157
of homomorphism, 157
of map, 54, 196, 207
DG group, 157
Diagonal approximation, 250
of Alexander-Whitney, 250
Diagram, commutativity of, 16
homotopy commutativity of, 25
Diagram chasing, 186
Difference between two liftings, 434
Differential, 156
Differential graded group, 157
Differential group, 156
Dimension, of CW complex, 401
of simplicial complex, 109
of simplex, 108
theory of, 152
of topological space, 152
Dimension axiom, 200
for cohomology, 240
Direct limit, 3
of chain complexes, 162
of direct system, 18
Direct system, 3
in a category, 18
Directed set, 3
Discontinuous group, 87
Dissection, 276-277
Dominated space, 421
Double suspension, 516-517
Duality in suspension category, 463
Duality map, 462
Duality theorems, of Alexander, 296
of Lefschetz, 278, 297-298
for manifolds, 296, 351, 357
of Poincare, 296-297
in suspension category, 463
of Whitney, 281, 355
Edge, 134
end of, 134
origin of, 134
Edge path, 134
Edge path group, 136, 151, 208
of graph, 140-141
Edge path groupoid, 136
Eilenberg-MacLane space, 424
Eilenberg-Zilber theorem, 232-234
Endomorphism, trace of, 9
Epimorphism, 6
Equalizer, 406
Equivalence, 15
of edge paths, 135
of fiber bundles, 92
in homotopy, 25
nahirality of, 22
Equivalence class, 2
Equivalence relation, 2
Euclidean space, 9
Euler characteristic, 172, 205
of fibration, 481-482
Euler class, 347-348
Euler-Poincare characteristic (see Euler
characteristic)
Evaluation map, 6
Evenly covered subset, 62
Exact couple, 472-473
Exact sequence, 179
of pointed pairs, 366, 461
Exactness axiom, 200
for cohomologj7, 240
Exactness theorem, 182-183
Excision axiom, 200
for cohomology, 240
Excision map, 188
Excision property, 188-189
Excisive couple, 188-190
Exponential correspondence, 6
Exponential law, 6
Exponential map, 53
Extended lifting function, 93
Extension, 2
of groups, 179
of modules, 242
Extension problem, 20, 28
Exterior algebra, 264
Face, of simplex, 108
of singular simplex, 160
Fiber, 66, 90
Fiber bundle, 90
524
INDEX
Fiber bundle, fibration property of, 96
induced, 98
triviality of, 92
Fiber-bundle pair, 256
Fiber homotopy, 99
Fiber homotopy equivalence, 100
Fiber pair, 256
Fiber space (see Fibration)
Fibered product, 70, 98
Fibration, 66, 104
cohomology spectral sequence of, 493-498
homology spectral sequence of, 480-481
homotopy sequence of, 377
induced, 98
local, 92
with unique path lifting, 68
regularity of, 74
universality of, 84
Filtered degree, 468
Filtration, 276, 468, 493
bounded, above, 493
below, 276, 469
convergence of, 276, 468
of topological pair, 472
Fine presheaf, 330
Finite presentation, 7
Finite simplicial complex, 109
Finite type, 246
Finitely generated graded group, 172
First obstruction to lifting, 446
First-quadrant spectral sequence, 468
Five lemma, 185-186
modulo 519
Fixed point, 151, 194-197
Fixed-point theorem, of Brouwer, 151, 194
of Lefschetz, 195
Flow, 197
Free approximation, 225-226
Free basis, 6, 8
Free chain complex, 157
Free functor, 164
Free generating set, 6
Free group, 6, 145
Free module, 8
Free resolution, 219
Freely homotopic maps, 379
Front i-face, 250
Full subcategory, 16
Full subcomplex, 110
Fully normal space, 316
Functor, 18-20
adjointness of, 41
contravariance of, 18
covariance of, 18
of loop spaces, 39
of several variables, 22
of suspensions, 41-42
Fundamental class, 303
Fundamental family, 301
Fundamental group, 50, 58-59
action of, 382-385, 420
of graph, 141-143
of lens space, 88
of orbit space, 88
of projective space, 80, 147
of sphere, 53, 58
of surface, 149
Fundamental groupoid, 48
Fundamental presheaf, 326
Fundamental theorem of algebra, 59
G structure, 90
Generalized lens space, 88
Graded algebra, 263
Graded coalgebra, 266
Graded group, 157
Graph, 139
Group, finite presentation of, 7
without fixed points, 87
free generation of, 6
generators, 6
gradation of, 157
of homotopy classes, 421
relations, 6
Groupoid, 45
component of, 45
connectivity of, 45
Gysin cohomology sequence, 260
generalization of, 499-500
Gysin homology sequence, 260
generalization of, 483
H cogroup, 39
commutativity of, 40
homomorphism of, 40
H group, 35
commutativity of, 35
H space, 34
homomorphism of, 35
Handle, 148
Height, 264
Higher obstructions to lifting, 446
Homogeneously n-dimensional simplicial complex, 150
Homologically locally connected space, 340
Homologous cycles, 157
Homology, of fiber bundles, 258, 279-280
of a group, 503-504
Kiinneth formula, 228-232
with local coefficients, 281-282
universal-coefficient formula, 222-226, 248, 282
Homology class, 157
Homology group, 157
of lens space, 206
of retract, 193
of simplex, 171-174
of sphere, 190
of surface, 206
Homology manifold, 277-278
Homology module, 213
of bigraded module, 466
Homology operation, 269
Homology sequence, of Mayer-Vietoris, 187-190
of pair, 184-185
of triple, 184-185, 201
Homology tangent bundle, 294
Homology theory, 199-200, 208
with coefficients, 218
with compact supports, 203-205
Homorphism, of H cpgroup, Q9
of H space, 35
of homology theories, 201
of presheaves, 324
of sequences, 180
of spectral sequences, 467
Homotopy, 23
associativity, 35, 39
INDEX
525
Homotopy, of paths, 46
relative to a subspace, 23
Homotopy abelian, 35, 40
Homotopy axiom, 200
for cohomology, 240
Homotopy category, 25
of maps, 414
of pairs, 25
of pointed topological spaces, 25
of topological spaces, 25
Homotopy class, 24
of map pairs, 414
Homotopy commutative diagram, 25
Homotopy equivalence, 25
of maps, 414
Homotopy excision theorem, 484-486
Homotopy extension property, 28, 57, 118
for polyhedral pairs, 118
Homotopy extension theorem of Borsuk, 57
Homotopy functor, 407
Homotopy group, 43-44, 371, 418-419
finite generation of, 509
of spheres, 398, 459, 520
finiteness of, 515-516
p-primaiy components of, 515-517
Homotopy identity, 35, 39
Homotopy inverse, 25, 35, 40
Homotopy lifting property, 66
(See also Fibration)
Homotopy pair, 414
Homotopy sequence, of pair, 374
of triple, 378
of weak fibration, 377
Homotopy type, 25
Hopf algebra, 267, 280
structure of, 268
Hopf bundle, 91
Hopf classification theorem, 431
Hopf extension theorem, 431
Hopf invariant, 489-490, 500-502
Hopf theorem on H spaces, 269
Hopf trace formula, 195
Hurewicz fiber space (see Fibration)
Hurewicz homomorphism, 388-390
Ideal of abelian groups, 506
Identity map, 2
Identity morphism, 14-15
Imbedding of projective spaces, 356, 361
Inclusion map, 2
Index of manifold, 357-358
Induced cofibration, 99
Induced covering projection, 98
Induced fiber bundle, 98
Induced fibration, 98
Induced homomorphism, 158
Induced orientation of simplex, 207
Induced subdivision, 122
Induced weak fibration, 375
Inessential map, 23
Initial object, 17
Injection, 2
Injective function, 2
Integral homology theory, 218
Interval, 9
Invariance of domain, 199
Inverse of morphism, 15
Inverse limit, 3
of chain complexes, 162
of inverse system, 18
Inverse system, 3
in a category, 18
Join of simplicial complexes, 109
Jordan-Brouwer separation theorem, 198
Jordan curve theorem, 198
k-stem, 459
Kernel of induced map, 365
Klein bottle, 149
Kiinneth formula, 228-232, 282
for Cech cohomology, 359-360
for cohomology, 247-249
for singular homology, 234-235
Lefschetz duality theorem, 278, 297-298
Lefschetz fixed-point theorem, 195
Lefschetz number, 194-195
Lens space, 88
generalization of, 88
Leray-Hirsch theorem, 258
Leray structure theorem, 268
Lift (see Lifting)
Lifting, of map, 66
of map pair, 415
Lifting function, 92
Lifting problem, 65
Lifting theorem, 76
Limit, direct, 18
inverse, 18
Line, 10
Line segment, 10
Linear metric, 124
Linear singular simplex, 176
Linking number, 361
Local connectedness, 103
Local fibration, 92
Local homeomorphism, 62
Local homomorphism, 104
Local isomorphism, 104
of presheaves, 329
Local system, 58, 360
Locally constant function, 309
Locally constant presheaf, 360
Locally contractible space, 57
Locally finite simplical complex, 119
Locally isomorphic groups, 104
Locally path-connected space, 65
Locally zero cochain, 307
Locally zero presheaf, 328
Loop, 46
Loop functor, 39
Loop space, 37
Manifold, 292, 356-357
with boundary, 297
without boundary, 292
index of, 357-358
Map, degree of, 54, 196, 207
free homotopy of, 379
homotopy of, 23
lifted, 66
of pairs, 22
Map pair, 414
526
INDEX
Mapping cone, 365
of chain map, 166, 191
Mapping cylinder, 32-33
of pairs, 33
Mapping path fibration, 99
Mayer-Vietoris sequence, 186-189, 218
in cohomology, 239
Mesh, 125
of singular chain, 177
Method of acychc models, 164-166, 169
Metric topology of simplical complex, 111
Model, 165
Module, of extensions, 241-242, 278-279
of homomorphisms, 8, 238-239
rank of, 8
structure theorem, 9
Monomorphism, 6
Moore-Postnikov factorization, 440
Moore-Postnikov sequence of fibrations, 440
Moore-Postnikov tower, 440
Morphism, 14
composite of, 14
identity, 14-15
inverse of, 15
Multiplication, 34
homotopy abelian, 35
homotopy associativity of, 35
homotopy identity for, 35
homotopy inverse for, 35
Multiplicity, 73
n-characteristic cohomology class, 425, 441
n-circuit, 148
n-connected pair, 373
n-connected space, 51
n-duality map, 462
n-equivalence, 404
n-factorization, 440
n-plane bundle, 90
over complex field, 90
n-simple pair, 385
n-simple space, 384
n-sphere bundle, 91
n-universal element, 408
Natural equivalence, 22
Natural transformation, 21
Nerve, 109, L52
Noether isomorphism theorem, 7
Nondegenerate base point, 380, 419
Nonnegative chain complex, 157
Nonorientable pseudomanifold, 206-207
Normal Stiefel-Whitney classes, 354
Null homotopic map, 23
Number of sheets, 73
Numerable covering, 93
Object, 14
initial, 17
terminal, 17
Obstruction to lifting, 433
cc-homotopic maps, 379
co-homotopy, 379
Open covering of pair, 311
Open simplex, 112
Opposite category, 17
Orbit, 87
Ordered chain complex, 170
Orientable fibration, 476
Orientable homology manifold, 278
Orientable manifold, 294-297
Orientable pseudomanifold, 206-207
Orientable sphere bundle, 259
Orientation, of manifold, 294-297
of pseudomanifold, 207
Orientation class, 259
Orientation presheaf, 326
Oriented chain complex, 159
Oriented cohomology, 238
Oriented homology functor, 160
Oriented homology group, 159
with coefficients, 214
Oriented pseudomanifold, 207
Oriented simplex, 158
Oriented sphere bundle, 259
Origin of edge, 134
Pair-, homology sequence of, 184-185
homotopy sequence of, 374
of spaces, 22
tautness of, 289
Pairing, of modules, 250
of spectral sequences, 491
Partial order, 2
Partially ordered set, 2
Path, 46
closedness of, 46
end of, 46
origin of, 46
product of, 46
Path class, 46
category of, 48
Path component, 48
Path connected, 48
Path space, 75, 99
Poincare duality theorem, 296-297
Pointed pair, exact sequence of, 366, 461
Pointed set, 15
Pointed simphcal complex, 110
Pointed topological space, 15
Polar coordinates in a simplex, 117
Polyhedral pair, 114
Polyhedron, 113, 149-150
Polynomial algebra, 264
Postnikov factorization, 440
Postnikov sequence, 440
Presentation of group, 6
Presheaf, 323
fineness of, 330
Principal fibration, 432
of type (t7,h), 433
Principal simplex, 208
Product, 263
in category, 17
of categories, 17
of chain complexes, 161
of edge paths, 135
fibered, 70, 98
of pah s, 234
of paths, 46
of sets, 2
topological, 4
Product bundle, 90
Pr oduct-bundle pah', 256
Projection chain map, 158
INDEX
527
Projection map, 3
Projective space, 9, 146-147
cohomology algebra of, 264-265
covering space of, 80
fundamental group of, 80,147
imbedding of, 356, 361
Proper face, 108
Proper map, 319
Properly discontinuous group, 87
Pseudomanifold, 148, 150, 206-208
without boundary, 148
pfh-power cohomology operation, 270
Quotient chain complex, 158
Quotient set, 3
Quotient space, 5
Rank of module, 8
Realization, 120
Reduced chain complex, 168
Reduced cochain complex, 237
Reduced cone, 365
Reduced homology group, 168, 200
Reduced homology sequence, 184-185
Reduced Mayer-Vietoris sequence, 187-189
Reduced squares, 270-271
Reduced suspension, 41
Regular fibration, 74
Relative Alexander presheaf, 323
Relative CW approximation, 412
Relative CW complex, 401, 420
Relative homeomorphism, 202
Relative homology group, 172, 175
Relative homotopy of liftings, 415
Relative homotopy group, 372
Relative Hurewicz isomorphism theorem, 397
generalization of, 511
Relative lifting problem, 415
Relative manifold, 297
Relative Mayer-Vietoris sequence, 187-190
Relative singular presheaf, 324
Resolution, 219
Restriction, 2
in presheaf, 323
Retract, 28
absolute, 56
absolute neighborhood, 56
deformation, 30
strong, 30
of pairs, 33
weak, 30
weak, 28
Retracting function, 97
Retraction, 28
deformation, 30
strong, 30
weak, 28
Ring of abelian groups, 506
Section, 77
Self-equivalence of fibration, 85
Semilocally 1-connected space, 78
Serre class of abelian groups, 504
Serre cohomology sequence, 519
Serre fibre space (see Weak fibration)
Serre homology sequence, 519
Sheaf, 324-325, 360
Short exact sequence, 179
of chain complexes, 180
Shrinking of an open covering, 331
Simple map, 440
Simple order, 2
Simplex, 108
Simphcal approximation, 126-128
Simplical-approximation theorem, 128
Simplical complex, 108
finer than covering, 124
homogeneously n-dimensional, 150
Simplical map, 109, 150-151
Simplical mapping cylinder, 151
Simplical pair, 110
Simplical product, 359
Singular chain complex, 161
Singular cohomology, 238-239
with compact supports, 323
with local coefficients, 282-283
Singular homology functor, 161
Singular homology group, 161
with coefficients, 214
Kiinneth formula, 234-235
with local coefficients, 281-282
Singular simplex, 160
Six-term exact sequence, 224, 278-279
Skeleton, 109
of relative CW complex, 401
Slant product, 287, 351
Small cell, 299
Solenoid, 358
Space, of adjunction, 56
compactly generated, 5
of loops, 37
obtained by adjoining cells, 145
of orbits, 87
of paths, 75
of simplical complex, 111
of type (w,n), 424, 460
Spanier-Whitehead duality theorem, 463
Spectral sequence, 466
convergence of, 467
of fibration, 480-481, 493-498
of filtered chain complex, 469
in suspension category, 518
Sperner lemma, 151
Sphere, 9
fundamental group of, 53, 58
homology of, 190
homotopy groups of, 398, 459, 515-520
Split short exact sequence, 216
Standard Postnikov factorization, 446
Star refinement, 316
Starlike subset, 53
Steenrod algebra modulo 2, 276
Steenrod classification theorem, 460
Steenrod squares, 270-271
Stem, 459
Stiefel-Whitney characteristic classes, 281
of manifold, 349
Strong deformation retract, 30-33
Strong deformation retraction, 30
Strong excision property, 317-318
Strongly single space, 510
Structure, of finitely generated module, 9
of Hopf algebra, 268
Structure group, 90
528
Subcategory, 16
fullness of, 16
Subcomplex, of chain complex, 158
of relative CW complex, 402
of simplical complex, 110
Subdivision, 121
induced, 122
of singular chains, 178
Subdivision chain map, 192
Subpair, 22
Sum, in category, 17
of chain complexes, 161
cofibered, 99
of sets, 2
Surface, 148
fundamental group of, 149
homology of, 206
Surjection, 2
Surjective function, 2
Suspension, 41, 461-462
of cohomology operation, 431
reduced, 41
Suspension category, 462-463
Suspension functor, 41-42
Suspension theorem, 458
Tangent bundle, 91
Taut pair, 289
Tautly imbedded pair, 289
Tautness, for absolute neighborhood retracts, 290-291
of Alexander cohomology theory, 316-317
Tensor product, 7, 213-217
of chain complexes, 228
of presheaves, 324
Terminal object, 17
Thom class of sphere bundle, 283
Thom-Gysin sequence, 260
(See also Gysin cohomology sequence; Gysin
homology sequence)
Thom isomorphism theorem, 259
Topological pah, 22
Topological sum, 5
Topology, coherent with collection, 5
coinduced, 4
induced, 4
Torsion coefficient, 172, 176
Torsion-free module, 8
Torsion product, 219-221, 224-225, 278-279
of chain complexes, 228
Torsion submodule, 8
Torus, 148
Total degree, 468
Total order, 2
Total pair, 256
INDEX
Total space, of fiber bundle, 90
of fibration, 66
Totally ordered set, 2
Trace of endomorphism, 9
Transgression, 518-519
Tree, 139
Triangulation, 113
finer than covering, 124
Triple, homology sequence of, 184-185, 201
homotopy sequence of, 378
Trivial fiber bundle, 92
Truncated polynomial algebra, 264
Unique-lifting property, 66-67
Unique path lifting, 68
Unit n-sphere bundle, 92
Unit tangent bundle, 92
Universal-coefficient formula, for Alexander
cohomology, 336
with compact supports, 337-338
for Cech cohomology, 335-336
for cohomology, 243, 246, 283
for homology, 222, 226, 248, 282
Universal covering space, 80
Universal element, 408
Universal fibration, 84
Van Kampen’s theorem, 151
Vector bundle, 90
Vertex, of cone, 56, 116
of simplical complex, 108
Vietoris-Begle mapping theorem, 344
modulo (P, 520
Wang sequence, 456
generalization of, 482, 498-499
Weak continuity property, 318, 358
Weak deformation retract, 30
Weak fibration, 374
homotopy sequence of, 377
Weak homotopy equivalence, 404
of maps, 417
Weak homotopy type, 413
Weak retract, 28
Weak retraction, 28
Weak topology, 5
for simplical complex, 111
Whitehead product, 419-420
Whitehead theorem, 399
generalization of, 512
Whitney duality theorem, 281, 355
Wu formula, 350
Zorn’s Iemma, 3