COHOMOLOGY OPERATIONS AND
APPLICATIONS IN HOMOTOPY THEORY

Harper's Series in Modern Mathematics
I. N. Herstein and Gian-Carlo Rota, Editors

COHOMOLOGY
OPERATIONS
AND APPLICATIONS
IN HOMOTOPY THEORY
ROBERT E. MOSHER
Department of Mathematics
California State College at Long Beach
MARTIN С TANGORA
Department of Mathematics
University of Chicago
Harper & Row, Publishers
New York, Evans ton, and London

Cohomology Operations and Applications in Homotopy Theory
COHOMOLOGY OPERATIONS AND APPLICATIONS IN
HOMOTOPY THEORY
Copyright © 1968 by Robert E. Mosher and Martin C. Tangora
Printed in the United States of America. All rights reserved. No part of this book
may be used or reproduced in any manner whatsoever without written permission
except in the case of brief quotations embodied in critical articles and reviews.
For information address Harper & Row, Publishers, Incorporated, 49 East 33rd
Street, New York, N.Y. 10016.
Library of Congress Catalog Card Number: 68-11456

CONTENTS
PREFACE ix
1 INTRODUCTION TO COHOMOLOGY OPERATIONS 1
Cohomology operations and Κ(π,η) spaces 2
The machinery of obstruction theory 6
Applications of obstruction theory 7
2 CONSTRUCTION OF THE STEENROD SQUARES 12
The complex K(Z2,\) 12
The acyclic carrier theorem 14
Construction of the cup-г products 15
The squaring operations 16
Compatibility with coboundary and suspension 18
3 PROPERTIES OF THE SQUARES 22
Sql and Sq° 23
The Cartan formula and the homomorphism Sq 24
Squares in the и-fold Cartesian product of K(Z2,\) 26
The Adem relations 29
4 APPLICATION: THE HOPF INVARIANT 33
Properties of the Hopf invariant 33
Decomposable operations 36
Non-existence of elements of Hopf invariant one 37
ν

CONTENTS
5 APPLICATION: VECTOR FIELDS ON SPHERES 39
^-fields and V„.k+l 39
A cell decomposition of V„ k 40
The cohomology of P„,k 43
6 THE STEENROD ALGEBRA 45
Graded modules and algebras 45
The Steenrod algebra s/ 46
Decomposable elements 47
The diagonal map of si 47
Hopf algebras 49
The dual of the Steenrod algebra 50
Algebras over a Hopf algebra 52
The diagonal map of si* 53
The Milnor basis for si 56
7 EXACT COUPLES AND SPECTRAL SEQUENCES 59
Exact couples 59
The Bockstein exact couple 60
The spectral sequence of a filtered complex 62
Example: The homology of a cell complex 67
Double complexes 68
Appendix: The homotopy exact couple 70
8 FIBRE SPACES 73
Fibre spaces 73
The homology spectral sequence of a fibre space 76
Example: Я*(П5") 76
Serre's exact sequence 77
The transgression 80
The cohomology spectral sequence of a fibre space 80
9 COHOMOLOGY OF Κ(ττ,η) 83
Two types of fibre spaces 83
Calculation of H*(Z2.2; Z2) 86
H*(Z2,q\ Z2) and Borel's theorem 88
Further special cases of Η*(π,η; G) 89
Appendix: Proof of Borel's theorem 91
0 CLASSES OF ABELIAN GROUPS 93
Elementary properties of classes 93
Topological theorems mod (6 95

CONTENTS
VII
The Hurewicz theorem 96
The relative Hurewicz theorem 98
■tfp satisfies Axiom 3 98
The #р approximation theorem 100
11 MORE ABOUT FIBER SPACES 102
Induced fibre spaces 102
The trangrcssion of the fundamental class 103
Bockstcins and the Bockstcin lemma 104
Principal fibre spaces 108
12 APPLICATIONS: SOME HOMOTOPY GROUPS OF SPHERES 111
The suspension theorem 111
A better approximation to S" 112
Calculation of πη+λ(5"), к < 7 114
The calculation continues 118
Appendix: Some homotopy groups of S3 123
13 n-TYPE AND POSTNIKOV SYSTEMS 128
Cellular approximation 129
л-type 129
Postnikov systems 131
Existence of Postnikov systems 132
Naturality and uniqueness 135
14 MAPPING SEQUENCES AND HOMOTOPY CLASSIFICATION 138
The fibre mapping sequence 138
Mappings of low-dimensional complexes into a sphere 140
The spectral sequence for [A', A"] 142
The skeleton filtration and the inclusion mapping sequence 143
Appendix: Properties of the fibre mapping sequence 145
The map from LIE to LiB 146
15 PROPERTIES OF THE STABLE RANGE 149
Natural group structures 149
Examples: Cohomology and homotopy groups 150
Consequences of Serre's exact sequence 151
16 HIGHER COHOMOLOGY OPERATIONS 156
Functional cohomology operations 156
Another formulation of 6r 159
Secondary cohomology operations 161

CONTENTS
Secondary operations and relations 162
The Peterson-Stein formulas 164
17 COMPOSITIONS IN THE STABLE HOMOTOPY OF SPHERES 170
Secondary compositions 170
The 0-, I-, and 2-stems 173
The 3-stcm 178
The 6- and 7-stems 180
18 THE ADAMS SPECTRAL SEQUENCE 184
Resolutions 185
The Adams filtration 187
Evaluation of F" 188
The stable groups {Y,X) 191
The definition of Ext^(M,N) 192
Properties of the spectral sequence 193
Minimal resolutions 199
Some values of Ext У (Z2,Z2) 201
Applications to homotopy groups 203
BIBLIOGRAPHY 207
INDEX
212

PREFACE
In the past two decades, cohomology operations have been the center of a
major area of activity in algebraic topology. This technique for supplementing
and enriching the algebraic structure of the cohomology ring led to important
progress, both in general homotopy theory and i η specific geometric applications.
For both theoretical and practical reasons, much analysis has been made of the
formal properties of families of operations.
We focus attention on the single most important sort of operations, the
Steenrod squares. We construct these operations, prove their major properties,
and give numerous applications.
In the development of these applications, we treat various techniques of
homotopy theory, notably those useful for computation. In the later chapters,
special emphasis is placed on calculations in the stable range. Here we make
detailed computation in the stable homotopy of spheres and, in Chapter 18,
develop the tool currently found most effective for stable calculations.
This book is intended for advanced graduate students who are well versed in
cohomology theory and have some acquaintance with homotopy groups. It is
based on notes by the second-named author from lectures aimed at such
students and given at Northwestern University by the first-named author. It
attempts to give the student a thorough understanding of the cohomological
methods and their history.
Our choice of treatments has not generally been based on elegance but rather,
we feel, on ease of comprehension; this book is primarily for the student. One
IX

χ
PREFACE
can find in the literature a smoother definition of the squaring operations, a slick
exposition of spectral sequences, or an axiomatic approach to secondary
operations. These methods, important as they are, do not seem entirely appropriate
for the student's first view of the subject. They rarely were the discoverer's first
view either.
The book makes no pretense to being self-contained. To fill in complete
details at every stage would be to compromise the goals of the book, which aims
to give an overview of the theory and practice of the cohomological method and
to attract students into the field. We have collected many results which until now
were scattered through the literature, and some have been proved in detail while
others have only been stated (with reference to sources). We have used various
criteria in deciding when or when not to give details, but of course the final
decision ultimately rests on the personal taste of the authors.
We wish to thank Professors A. H. Copeland, Jr., Daniel S. Kahn, and
R. C. O'Neill for their suggestions and the National Science Foundation for
partial support. And in particular we thank Professor F. P. Peterson, not only
for his suggestions but also for his lectures at the Massachusetts Institute of
Technology, to which the authors acknowledge a substantial debt.
ROUERT E. MOSHER
Martin С Tangora

CHAPTER
INTRODUCTION TO
COHOMOLOGY OPERATIONS
It has long been known that the sphere S"~' is not a retract of the closed
cell E" of which it forms the boundary. Such a retraction/· would, when
composed with the inclusion map ;', give the identity map of S"'1. Passing to the
reduced homology groups, we would have a factorization through H„{E")
of the identity map of /?*(5n-'): R*{Sn~l) Χ Η*{En) A H^S"'1).
This is impossible because the reduced homology of the cell is zero whereas
that of the sphere is not: ^„_,(5"-1) is an infinite cyclic group.
This argument illustrates one of the fundamental methods of algebraic
topology: the non-existence of a topological construct (the retracting map)
is proved by showing that its existence would entail the existence of an
algebraic construct (the induced map of the homology groups) which is
more readily seen to be impossible.
We consider another, more delicate problem. Let P(n) denote real
projective «-space (n > 1). It is known that the fundamental group π,(Ρ(«))
is Z2, that is, the cyclic group of order 2, independent of n. Question: For
η > m > 1, does there exist a map /: P(n) —>P(m) inducing an
isomorphism of the fundamental group ? Again the non-existence of the map can
be demonstrated by considering an induced algebraic map. We consider
the cohomology ring of P(n) with Z2 coefficients. This ring is known to be
a truncated polynomial ring on a one-dimensional cohomology class a„,
truncated by the relation (я„)"+1 — 0. If there existed a map / with the
required property, then the induced map in cohomology would have to
1

2
COHOMOLOGY OPERATIONS
satisfy f*(am) = a„. (This follows from the Hurewicz theorem and the
universal coefficient theorem for H\P{n)\Z2).) Because/* must be a ring
homomorphism, this is impossible, for we would have (an)" ^0 =/*(0) =
/*((aj") when η > т.
This example illustrates the advantage of cohomology over homology
because of the additional algebraic structure given by the cup product. Our
first objective in this book will be to develop a much more extensive
algebraic structure in cohomology; the cup product will be supplemented, or
overwhelmed, by an infinite family of operations. Every topological map
respects not only the homology groups and the cohomology ring but all
the structure afforded by these new operations.
We now make precise what we mean by a cohomology operation.
COHOMOLOGY OPERATIONS AND Κ(π,η) SPACES
Definition
A cohomology operation of type (π,η; G,m) is a family of functions
θχ: Η"(Χ;π) —>Hm(X;G), one for each space X, satisfying the naturality
condition f*0y = 9xf* for any map/: X —> Y.
example: The cup-product square, и —>u2, gives for each η and each
ring π an operation H"(X; π) —> H2"(X; π). Note that this is generally not
a homomorphism.
We will denote by 0(π,η; G,m) the set of cohomology operations of type
(π,η; G,m). In Theorem 2 we will establish a fundamental result on the
classification of such operations in terms of the cohomology of certain
spaces, which we now introduce.
Definition
We denote by Κ(π,η) any " nice " space which has only one non-trivial
homotopy group, namely, πη(Κ(π,η)) = π.
By a " nice" space we mean a space having the homotopy type of a
CW-complex. We will usually assume that the spaces under consideration
are of this type, without making explicit mention.
Recall that the Hurewicz homomorphism h: π;(Χ) —>H,(X) is defined for
any X and any i by choosing a generator и of H,(S') and putting
h: (/1->/·(«) where/: S' ->X.
A space X is said to be η-connected if nt(X) is trivial for all / < n. The
Hurewicz theorem states that, if A" is (n — l)-connected, then the Hurewicz
homomorphism is an isomorphism in dimensions / < η and is still an

INTRODUCTION TO COHOMOLOGY OPERATIONS
3
epimorphism in dimension « + 1. Thus the first non-trivial homotopy
group of X and the first non-trivial homology group occur in the same
dimension and arc isomorphic under h. The statement of the theorem must
be modified for η — 1; in this case the epimorphism h: π{(Χ) —> Ηλ{Χ) has
as kernel the commutator subgroup of πλ{Χ)—and we do not claim that ft
maps n2(X) onto H2(X).
The universal coefficient theorem for со homology gives an exact sequence
0 ->Ext (Я.-ЛЛад -+Η\Χ\π) ->Hom (ΗΠ(Χ),π) ->0
for any space X. If X is (n — l)-conncctcd, this becomes
Η"(Χ;π)^Ηοτη(ΗΠ(Χ),π)
since the Ext term becomes Ext (Ο,π) = 0. Now if π = π„(Χ), then the group
Horn (Η„(Χ),π„(Χ)) contains A-1, the inverse of the Hurewicz homomor-
phism (which is an isomorphism here).
Definition
Let X be (n — l)-connected. The fundamental class of X is the cohomo-
logy class ι ε Η"(Χ;π„(Χ)) which corresponds to A-1 under the above
isomorphism. This class may be denoted ix or /„, to stress its provenance
or its dimension.
In particular, Κ(π,η) has a fundamental class /„ e Η"(Κ(π,η); π).
Later in this chapter we will prove the following theorem.
Theorem 1
There is a one-to-one correspondence [Χ, Κ(π,η)]<-^Η"(Χ;π), given by
[/]<->/*(/„). (Here the square brackets denote homotopy classes, as usual.)
We draw two important corollaries.
Corollary 1
There is a one-to-one correspondence
[Κ(π,η\ Κ(π',η)] <->Hom (π,π')
This follows from Theorem 1, the universal coefficient theorem, and the
Hurewicz theorem:
[Κ(π,η), Κ(π',η)]<+Η"(Κ(π,η);π')
яв Horn (ΗΠ(Κ(π,η)), π') - Horn (π,π')
Corollary 2
The homotopy type of Κ(π,η) is determined by π and n. Moreover, the
identity map of π determines (up to homotopy) a canonical homotopy
equivalence between any two " copies " of Κ(π,η).

4
COHOMOLOGY OPERATIONS
Recall that Κ(π,η) is assumed to have the homotopy type of a CW-
complex. By Corollary 1, any isomorphism π «* π' can be realized by a map
Κ(π,η) —> Κ(π',η). Since all the other homotopy groups are trivial, it
follows by J. H. C. Whitehead's theorem that this map is a homotopy
equivalence. This proves Corollary 2.
Notation
We write Нт(к,п; G) for Hm(K(n.n); G). This is justified by Corollary 2.
We come to the classification theorem. Let 0 be an operation of type
(n,n;G,m). Since /„ e Η"(π,η; π), we have 0(/„) e Hm(n,n; G).
Theorem 2
There is a one-to-one correspondence
0(π,η; G,m) ^ Hm(n,n; G)
given by 0<->0(/„).
proof: We will show that the function 0<->0(/„) has a two-sided
inverse.
Let ψ e Hm(n,n; G). We define an operation of type (π,η; G,m), which
we also denote by φ, as follows: for usH"{X;n), put <р(ы) =/*(<р) е
Hm(X; G) where / is a map X—> Κ(π,η) such that [/] corresponds to и
under Theorem 1.
We now have functions in both directions between 0(π,η; G,m) and
Hm(n,n; G), and it is easy to see that the composition in either sense is the
identity transformation. In fact, if X — Κ(π,η) and u^~ /„, then /must be
homotopic to the identity, and so ψ(ι„) =/*{φ) — ψ. On the other hand, if
φ = 0(/„), then the operation φ is equal to 0, since φ{ιι) -=/*(<p) —f*(0(in)) =
0(/*(/„)) = 0(h). This completes the proof.
Usually we will use the same symbol to denote both an operation and
the corresponding cohomology class, as we have already done for ψ in the
proof above.
Corollary 3
There is a one-to-one correspondence
0(π,η; G,m) ^> [Κ(π,η), K(G,m)]
This follows immediately from Theorems 1 and 2.
Theorem 2 reduces the problem of finding all cohomology operations (of
the type under consideration) to the computation of the cohomology of
Κ(π,η) spaces. Of course Κ(π,η) spaces are not geometrically simple; they
are generally infinite-dimensional, with the striking exception of the circle

INTRODUCTION TO COHOMOLOGY OPERATIONS 5
K{Z,\). Computation of their cohomology will be one of our major
objectives. But first we show that Κ(π,η) spaces exist.
Construction
Let π be an abelian group and n an integer, n > 2. We construct a
С ^-complex Κ(π,η).
To begin, let 0 —> R —> F—> π — > 0 be a free abelian resolution of π, and
let {a,-} and {A,-} form bases from R and F, respectively, where / and У
run through index sets / and J. Let К be the wedge (one-point union) of
«-spheres 5", jsJ. For each ah take an (n -f l)-ccll e,- and attach it to К
by a map /,:ё,—>А^, where [/i] = u/ei„(i)=F and e; denotes the
boundary of et. Then the resulting space X, formed from the disjoint
union of К and the e{ by identifying each point in <?,· with its image under
/;, is of dimension at most η — 1, has π„(Χ) — π, and is (n — l)-connected
(by the Hurewicz theorem).
It remains to kill the higher homotopy groups. This is done by trans-
finite induction, using the following lemma.
Lemma 1
Attach an (m + l)-cell to the С ^-complex У by a map /: Sm —> У, and
call the resulting complex Z. Then the injection j: Y-^Z induces
isomorphisms j# : π,(7) л π,(Ζ), i<m; in dimension m, }# is an cpimor-
phism, which takes the subgroup generated by [/] to zero.
proof: The isomorphism for /' < m follows from the cellular
approximation theorem, since Υ and Ζ have the same w-skeleton. If i,j,k are the
injections, the diagram below is commutative.
Υ -UZ- ru;f"+1
Passing to the induced diagram in homotopy, j'#f# =k#i# =0 (since
i# = 0). If [1] denotes the generator of n,„(Sm) = Z, then/Л/] =j#f#[l] = 0.
Thusy#, being a homomorphism, takes [/] and the subgroup generated
by it to zero. Finally, the assertion that j# is onto nm(Z) follows from
cellular approximation, since any map of Sm into Ζ can be deformed into
the /«-skeleton Zm = Y.
This proves the lemma and completes the construction.
The balance of this chapter is devoted to a survey of obstruction theory
and the proof of Theorem 1.

6
COHOMOLOGY OPERATIONS
THE MACHINERY OF OBSTRUCTION THEORY
We now recall some machinery of obstruction theory which will be
needed to prove Theorem 1.
Let У be a space, and assume for convenience that Υ is η-simple for
every «, that is, the action of π{(Υ) on π„(Ύ) is trivial for every «. Under
this hypothesis we can forget about base points for homotopy groups, and
any map /: S" —> Υ determines an element of π„( Υ).
Let В be a complex and A a subcomplex. Write X" for A u B", where as
usual B" denotes the «-skeleton of B. Let σ be an (« + l)-cell of В which
is not in A. Let g — g„ be the attaching map ά = S" —> X" c B.
Given a map/: Χ"—> Υ, denote by c(f) the cochain in C"+l(B,A; π„(Υ))
given by c(f): σ—>[/ο#σ]. Then it is clear that/may be extended over
Χ" υ9σ σ if and only if /o ga is null-homotopic, that is, if and only if
it/)0) -- 0, and therefore that/can be extended over Γ+'--^υ β"+1 if
and only if the cochain c(f) is the zero cochain. It is a theorem of
obstruction theory that c(f) is a cocycle. It is called the obstruction cocycle or " the
obstruction to extending/over 5"+1."
There are two immediate applications. First, any map of an
«-dimensional complex К into an «-connected space X is null-homotopic.
proof: Take (Β,Α) = (Κ χ /, Κ χ /) and define /: A —>X by the given
map /l —> X on one piece and a constant map on the other piece; then/
can be extended over В because the obstructions lie in the trivial groups
Second, as a particular case, a finite-dimensional complex К is conlractible
if (and only if) π;(Χ) is trivial for all /' < dim K.
Suppose f,g are two maps A"1—> Υ which agree on X"'1. Then for each
«-cell of В which is not in A we get a map S" —> У by taking/and g on the
two hemispheres. The resulting cochain of C(B,A; π„(Υ)) is called the
difference cochain of/and g, denoted d(f,g). The main facts about the
difference cochain are the following (we omit the proofs).
Coboundary formula
dd(f,g) = c(g)-c(f).
Addition formula
d(f,g) + d(g,h) = d(f,h).
Any cochain can be realized as a difference cochain. This is more
precisely stated in Proposition I.

INTRODUCTION TO COHOMOLOGY OPERATIONS
7
Proposition 1
Given /: X" —> Υ as before, and given any cochain d e C"(B,A; π„( Υ)),
there exists g: A"1—> У such that d(f,g) is defined and is equal to d.
The proof is an easy consequence of the homotopy extension property
of complexes.
Denote by [c(f)] the cohomology class of c(f).
Theorem 3
There is a map g: X"+1 —> У which agrees with/on X"'1 if and only if
M/)] = 0> tnat is> c(f) is a coboundary.
proof : Suppose c{f) = <5i/; choose # which agrees with /on I""1 and
such that d{f,g)=-d; then c(f) = Sd= —5d(f,g) = c(f)- c(g). Thus
c(#) — 0, and so g extends over X"+1. The reader can easily supply the
proof of the converse.
We now want to interpret the above results in terms of extension of
homotopies. Let К be a complex. Let (Β,Α) = (Κ χ /, Κ χ /) and let
X" = A u B" as before. Now suppose that/,# are two maps Λ^—> Υ which
agree on /f"'. Then there is an obvious map a: A"1—> Y. Moreover, c(a)
corresponds to d(f,g)—more precisely, to d(f\ K",g \ K")—under the
identification of C"+1(B,A; π„(Υ)) with C(K\nn{Y)). The following
theorem is merely a translation of the previous results into the present
context.
Theorem 4
The restrictions of/and g to K" are homotopic, rel K"~\ if and only if
d(f,g) = 0eC"(K;n„(Y)). They are homotopic, rel/r-2, if and only if
[d{f,g)} = 0sHn{K;n„{Y)).
APPLICATIONS OF OBSTRUCTION THEORY
In this section we will prove Theorem I. But first we present a homotopy
classification theorem. Let Υ have a base point *; we will also denote by *
the corresponding constant map of any space into Y.
Theorem 5 (Hopf-Whitney)
Let К be a complex of dimension n, and let Υ be (n — I )-connected. Then
there is a one-to-one correspondence k: [Κ, Υ] <-> Η"(Κ;π„(Υ)).
proof: Denote by IK, Υ J the homotopy classes, rel/f~2, of maps
(ЛГЛ""1)—>(У». Then there is a natural function 0: {Κ,Υ]->[Κ,Υ]. We
first prove that 0 is 1-1 and onto.

8
COHOMOLOGY OPERATIONS
Given any map/: A"—> У, the restriction/ \K" 'is null-homotopic since
У is (n — l)-connected. By homotopy extension we can find a map
g: (КД""')—>( У,*) such that g~f. This proves that 0 is onto.
To prove that 0 is 1-1, Iel/,# be Lwo maps {Κ,Κ"-λ) —>(У,*) such that
f~g:K^>Y. Let A:A"x/—>У be the homotopy. Write A' for the
restriction of h to K"~2 χ /.
Consider K"'2 xlxl. We have /г': /С""2 χ / χ 0-> Г and
*: К"-2 х/х 1->У
Now /г^"-1 χ /) = *; so /ι':Γ!χ/χΟ->», and thus we have a
partial homotopy between W and *: K"~2 χ /—> У, given by
H: K"-2 x/x/->*
Since У is (я — l)-connected and dim (Α'"-2 χ / χ /) <«, the obstructions
to extending Я over Α""-2 χ / χ / lie in groups with zero coefficients.
Thus we have a map Я: K"~2 χ / χ /—> К, which is a homotopy between
A' and * such that Η: Κ"'2 χ /χ /—>*.
Now consider Kx I x I.We have A:A:x/xO—>У and
Я: А""-2 Х/Х/->У
where Я is a homotopy of h\(K"~2 χ I). We can extend Я to a
homotopy Η: Κ χ Ι χ /—> Υ, using (only) the homotopy extension
theorem.
Now
Я(х,0,0) = Л(х,0) =Дх)
Я(х,0,1)-/'(х)
Я(*. 1,1) = £'(*)
Я(х,1,0)=А(х,1) = #(х)
where /' and g' are defined by the above equations. Moreover, Я = * on
(A"""2 X /x/)u(A""-2 X/X I). Thus, /~/' ~#'~,9: A"-> У, where
all homotopies are rel A"""2. This completes the proof that θ is 1-1.
We next define a transformation k: {K,YJ -^Η"(Κ;π„(Υ)) and prove it
1-1 and onto.
Given/: (Κ,Κ"-1) —>(У,*), let £(/) be the cohomology class of </(/,*)—
more precisely, (/(/lA'",*)—in ЯХА^яДУ)). This makes sense because
dim К = η and hence any «-cochain is a cocycle.
Let f,g be two maps (Κ,Κ"'1) -> (У,*). Then k(f)-k(g) =
[</(/,*) - </(#,*)] = [</(/,$)]. But [</(/,#)] - 0 if and only if/~,9 rel A""2.
This shows that к: [А",У] -^Η"{Κ;πη{Υ)) is well defined and 1-1.

INTRODUCTION TO COHOMOLOGY OPERATIONS
9
Any cocycle и eZ"(K;n„(Y)) may be realized as d(f,*) for some map
/: (K,K"-1) ->(У,*). Then k(f) = [</(/*)] = [и], so that к is onto.
This proves the theorem. The correspondence k0~l,
[K, Y] <-?- [К, Yj -4> H"(K; π„( Υ))
may be described as follows: Every class α e [K,Y] contains a map/such
that f\K"~1 = *. Then i/(/,*) is defined and is a cocycle, and
ИГ1: я->[</(/»]. Г,
We are now ready to prove Theorem 1. If X is a space having the homo-
topy type of а С ^-complex K, we may replace X by К without affecting
the conclusion of the theorem. We therefore consider a C^-compIex К
(with no restriction on its dimension). To simplify notation, write Υ for
Κ(π,η). We must show that [/] —>/*(»„) gives a one-to-one correspondence,
[Κ,Υ]^Η"(Κ;π).
Let / be the inclusion of K" into К and consider the following diagram:
[K,Y]-!%.[K\Y]
Hn(K;n)-^H"(Kn;n)
Now /* is 1-1; moreover, /* is 1-1 since, if fc^g:K" —> У, then also
f^g: К —> У, since the obstructions to extending the homotopy lie in the
zero groups л:т(У), т> п.
We claim that the image of /* corresponds to the image of /'* under the
correspondence к · 0~l. In fact the following statements are easily seen to
be equivalent, where/: (#",#"- ')-> ( Y,*):
1. [/] is in the image of /*
2. /extends over all of К
3. /extends over K"+i
4. c(/) = 0eC+1(K;*)
5. dd(f,*) = 0
6. the cohomology class of d(f,*) is in the image of /*
The equivalence of (2) and (3), for example, follows from the fact that
the higher obstructions to extending/lie in groups with zero coefficients
nm(Y), m>n.
This proves that by going around the above diagram we get a
correspondence <j9 = (/*)-|Ofc-0"IX/#): [AT,Г]->//"(Л», which is 1-1 and
onto.

10
COHOMOLOGY OPERATIONS
It is not immediately obvious that φ is the advertised correspondence
[/] —>/*(»„). One can verify that the correspondence к ■ θ~' is natural and
hence that ψ is natural, i.e., if A is another complex, any map α: Κ^Ά
gives a commutative diagram as shown below.
[K,Y]*^-[A,Y]
Η"(Κ;π)^-Η"(Α;π)
Now read У for A and/for a, and observe that ψ takes the identity map
id, of Υ into /„. Thus cp([f]) = φ(/4[Μγ])) =/>[«/,])) =/*('.)■
This completes the proof of Theorem 1.
DISCUSSION
We have suggested that additional algebraic structure, namely, cohomol-
ogy operations, will be a useful tool in the study of geometric problems.
With the help of obstruction theory, the classification of cohomology
operations has been reduced (Theorem 2) to the calculation of the
cohomology of Κ(π,η) spaces. We will pursue this calculation in some special
cases in Chapter 9.
The Κ(π,η) spaces were first studied in detail by Eilenberg and MacLane,
and they are often referred to as " Eilenberg-MacLane spaces." Later
work includes many calculations by Serre and Cartan.
Another possible approach is the construction of explicit operations. In
the next chapter we construct an extremely important family of operations,
due to Steenrod, who discovered them and proved many of their properties.
The cohomology operations considered in this chapter are sometimes
called primary cohomology operations. In Chapter 16 we will introduce
some more complicated constructions, all of which are known as higher
order cohomology operations but are not cohomology operations in the
sense of the present chapter.
In this chapter we have referred to the Hurewicz theorem, which will be
proved in Chapter 8, and the cellular approximation theorem, which will
be stated in Chapter 13. We have also appealed to the following theorem
of J. H. C. Whitehead: A map of nice spaces inducing an isomorphism on
each homotopy group is a homotopy equivalence.

INTRODUCTION TO COHOMOLOGY OPERATIONS
11
REFERENCES
General
1. D. Husemoller [l].f
2. Ε. Η. Spanier [1].
3. N. E. Steenrod [2].
Κ(π,η)
1. H. Cartan [5].
2. S. Eilenberg and S. MacLane [1,2].
3. J.-P. Serre [2].
CW-complexes and the Whitehead
theorem
1. P. J. Hilton [1].
2. G. W. Whitehead [1].
3. J. H. С Whitehead [1].
Obstruction theory
1. S. Eilenberg [1].
2. S.-T. Hu [1].
3. N. E. Steenrod [1].
4. G. W. Whitehead [1].
t Complete bibliographical material may be found in the Bibliography at the
end of the book, where the authors' names are listed alphabetically.

CHAPTER
CONSTRUCTION OF THE
STEENROD SQUARES
The object of this chapter is to construct the Steenrod squares. These are
cohomology operations of type (Z2, n;Z2, η ~ i).
We remark now, once and for all, that there are analogous operations
for Zp coefficients, where ρ is an odd prime; but they will not be treated in
this book.
THE COMPLEX K(Z2,\)
In Chapter I we constructed a Cff-complex Κ(π,η) for any abelian group
π where n>2. We now have need for an explicit complex A"(Z2,1).
Proposition 1
Let Ρ = P(co) --^ \J„ P(n) denote infinite-dimensional real projective
space, the limit of P{n) under the natural injections P(n) - -»P(n -p 1). Then
Ρ is a A"(Z2,I) space.
proof: The «-sphere S" is a covering space for P{n) with covering
group Z2. From the exact homotopy sequence of this covering, it follows
that 7T,(iD0i)) = Z2 and that п{{Р{п)) is trivial for 1 </'<«. The result
follows.
Let S' denote the infinite-dimensional sphere, i.e., the union of all 5"
under the natural injections 5"—>5,+'. We can easily give a cell structure
for S* as a CH^compIcx. In each dimension />0, we have two cells,
12

CONSTRUCTION OF THE STEENROD SQUARES
13
which will be denoted dt and Tdt. The action of the homology boundary
д is given by <di = dl.x -\- (- I)T</,-,. with iT-= 77, 7T- 1. (A sketch of
S2 will make this plausible.)
We can compute the homology of Sy from these formulas, verifying
that Sr is acyclic. Indeed, in even dimensions the only non-zero cycles are
generated by d2j — Td2j — <d1}¥,; in odd dimensions, only the sign is
changed in the argument.
The homology of Ρ is more interesting. We obtain a cell structure for
Ρ = P(co) by collapsing S' under the action of Z2 considered as generated
by T; in other words, identify i/, with Tdt for every / > 0. Thus Ρ has
exactly one cell in each dimension, denoted by ex\ and the boundary
formula is ?e2j = 2e2J_{, ce2}_x = 0. Therefore Ht{P;Z) must be Z2 for i
odd, 0 for / even.
By the universal coefficient theorems, since H*(P;Z) is Z2 in odd
dimensions, we have also the following: H*(P;Z) is Z2 in positive even
dimensions; H*(P;Z2) and likewise H*(P;Z2) are Z2 in all dimensions.
Thus we know the homology and cohomology groups of A"(Z2,I) in all
dimensions and for any coefficients and in particular for coefficients Ζ
or Z2. We turn to the calculation of the cohomology ring.
Let W denote the chain complex of S'. Then W is a Z2-free acyclic
chain complex with two generators in each dimension / >0.
To compute the ring structure of H*(P), we must give a diagonal map
for W. The action of Ton W7 gives an action of Ton W ® Wby T(u ®v) =
T{u) ® T{v).
Define/·: W^W® Why
Κ</«)--Σο^ι(-1)*'"-4®74 j
r(7V/,.) - T(r(dd)
where T' of course denotes Tor I according to whether у is odd or even.
Then /· is a chain map with respect to the usual boundary in W ®W,
namely, c(u ® v) ~ (cu) ® ν I (-- I)dct:"// ® (cc). The verification is direct
but tedious, and wc omit it.
If/; denotes the diagonal map of Z2, then it is clear from the definitions
that r is h-equfrariaiit, that is, r(qw) h(g)r(w) for gcZ2 and wc W.
Therefore/-induces a chain map 9: W T—> W T® W Γ, where WIT — W\Z2
is the chain complex of Ρ P(x). Explicitly,
Φ,) -Io<;-,(-l)"'~74®<V;
This map s is a chain approximation to the diagonal map Δ of P, and
so we may use it to find cup products in H*(P;Z2). Let a; denote the

14
COHOMOLOGY OPERATIONS
nontrivial element of H'(P;Z2) = Z2. The summation for s(eJ+k) contains
a term e; ® ek with coefficient 1 mod 2. Therefore,
<«; V ak, eJ + ky = <Δ*(α, x ak), e/ + k>
= <«,· x ak,s(e} + k)>
= 1 (mod 2)
so that ocj и ock = ocJ+k. We have indicated the proof of the following result.
Proposition 2
H*(Z2,\ ;Z2) = H*(P;Z2), as a ring, is the polynomial ring Ζ2[αι] on
one generator, the non-zero one-dimensional class a,.
THE ACYCLIC CARRIER THEOREM
To state the fundamental theorem on acyclic carriers, we need some
terminology. Let π and G be groups (not necessarily abelian) and let
Ζ [π] denote the group ring of π. Let К be a π-free chain complex with a
Z[7r]-basis В of homogeneous elements, called "cells." For two cells
σ,τ e B, let [τ:σ] denote the coefficient of σ in δτ; this is an element in
Ζ [π]. Let L be a chain complex on which G acts, and let Λ be a homo-
morphism π —> G .
Definition
An h-equivariant carrier G from К to L is a function С from 5 to the
subcomplexes of L such that:
1. if [t:ff]^0then 6σ с Ст;
2. for χ e π and σ e β, Λ(χ)6σ <= 6σ.
The carrier C is said to be acyclic if the subcomplex 6σ is acyclic for every
cell σ в B. The Α-chain map/: K^L is said to be carried by G if/σ e aG
for every uefi,
Theorem 1
Let С be an acyclic carrier from К to L. Let A" be a subcomplex of К
which is a Z(7r)-free complex on a subset of B. Let /: A" —>L be an
A-equivariant chain map carried by С Then / extends over all of К to
an A-equivariant chain map carried by G. Moreover the extension is unique
up to an A-equivariant chain homotopy carried by С
Note the important special case where K' is empty.

CONSTRUCTION OF THE STEENROD SQUARES
15
The proof proceeds by induction on the dimension; suppose that/has
been extended over all of K4 and consider a (<7 + I)-ceII τ e B. Then
дт = £ α,σ, where a, = [τ: σ,] e Ζ(π). Thus/(2t) = Σ/(α,σι) = Σ Α(α,)/(σ,),
which is in 6τ by properties (1) and (2). Since/is a chain map,/(ct) is a
cycle, but then, since 6τ is acyclic, there must exist χ in 6σ such that
8x =/(8τ). Choose any such x, and put /(τ) = χ. This is the essential step
in the construction ;/is extended over Кя+' by requiring it to be A-invariant.
Uniqueness is proved by applying the construction to the complex Κ χ I
and its subcomplex ί'χ/υίχ/,
CONSTRUCTION OF THE CUP-/ PRODUCTS
Now let К be the chain complex of a simplicial complex, and let W
be the Z2-free complex discussed above. Define the action of Z2 (generated
by T) on W ® К by T(w ®k) = (Tw) ® k, and on К ® К by T{x ®y) =
(- l)d9JCd^( j ® x). Define a carrier С from W ® К to К ® К by
С: (/; ® σ —> С(ст χ σ)
where by С(ст χ σ) we mean the following: К — C(X), the chain complex
of the simplicial complex X. By the Eilcnberg-Zilber theorem, there is
a canonical chain-homotopy equivalence Ψ: C(X xl)-t C(X) ® ^X).
Then for σ a generator of A^, i.e., a simplex of A", by C{a χ σ) we mean the
subcomplex Ψ6"(σ χ σ) of C(T) ® C(A'). Then С is clearly acyclic and
/г-equivariant, where h is the identity map of Z2. Therefore there exists
an A-equivariant chain map
φ: W®K^K®K
carried by G.
We should examine this φ because it plays a principal role in the
definition of the squaring operations. Consider the restriction φ0 = ψ | d0 ® K.
This can be viewed as a map K^>K® K, and as such it is carried by the
diagonal carrier. Thus it is a chain approximation to the diagonal, suitable
for computing cup products in K. The same remarks apply to Τφ0: σ —>
φ{Τά0 ® σ). Since both φ0 and Τψ0 are carried by C, they must be equi-
variantly homotopic. In fact it is not hard to verify that the chain homotopy
is given by φι: K—>K® Κ: σ —><ρ(ί/, ® σ). Further, φχ and Τψχ are equi-
variantly homotopic homotopies; a homotopy is given by φ2; and so
forth.
We now use φ to construct cochain products.

16
COHOMOLOGY OPERATIONS
Definition
For each integer / > 0, define a "cup-/ product"
C(K) ® C"(K)—>Cp+4-!(K): (M~► и u; v
by the formula
(// u,- v)(c) = (u® ν)φ{άί ®c) с е Cp + „_ ,(70
For the definition we must make an explicit choice of ψ, but it will be
seen that this choice is not essential.
Coboundary formula
6{uviv) = {-\)iduviv-., (-l),'+"wu;or-(--l),'wu,._ll'-(-l)'"'uu,._lU
(It is understood that и u_, r = 0.)
The proof is a routine computation from the definitions: let с be a chain
ofC,+„_;+1(/r);then
(<5(и u; v)){c) -(«и, v)(oc) = (u® v)(p{di ® ёс)
By definition, c(d; ® c) = dd-, ® с — ( — 1)'ί/; ® 5c·; so this becomes
(- 1);(и ® i>) <ρ£(ί/; ® c) - (- l)''(w ® iO<p(&/,- ® c)
= (- l)''(w ® υ) οψ{άί ® c) — (— \)\u ® ν)φ(ά;-, ® c)
-(-l)2"'^®^^,.-, ®c)
=■ (- I); <5(н ® t'Mi/; ® c) - (- I )!(u ® ι»)(ρ(έ/,_, ® c)
~{U ® l))7V(i/;_, ® £")
- (-1)"' φ ® i>)<j9(</, ® Γ) - (- 1)'(|/ ® 1>М</;_ , ® C)
-(-1Γ0>®«Μ</,·-. ®0
By definition of 6{u ® r), the first term may be written
(- 1);(<5н ® ν)φ(ά, ® c) τ (- l)''+p(« ® <5i'V(</,· ® c)
and so we finally have, from the definitions,
(5(ии,г))(г) = ( 1)'δ«υ,ι; , ( 1); + phu,<5i> ( lj'wu,.,»
-(-1Г«и(_1И
which is the required formula.
THE SQUARING OPERATIONS
We emphasize that the cup-/ products are defined on integral cochains
and take values in integral cochains. But suppose и с C(K) is a cocycle
mod 2, that is, Su = 2a, a e Cp+l(K). It follows from the coboundary

CONSTRUCTION OF THE STEENROD SQUARES
17
formula that и и, и is also a cocycle mod 2. Wc can therefore define
operations
Sq,: Z"(K;Z2) -> Z2p-'(K;Z2): ι/-> и и, и
in the obvious way. Moreover, wc can compose this with the natural
projection of cocycles onto cohomology classes.
Lemma 1
The resulting function Sqi:Zp{K\Z2)->H2,'-i{K\Z2) is a homo-
morphism.
proof: Let с be a cochain of the appropriate dimension, and write down
Sqiu -'■- v)(c). As expected, one obtains the terms Sqi(u)(c) and Sq,{v)(c)
plus two cross terms, but the sum of the cross terms is a coboundary
mod 2:
(и и; »)(c) -f (ν υ, i/)(c) - <5(н и ,· +, d)(c) (mod 2)
Lemma 2
If и is a coboundary, so also is Sq^u).
proof: If и — δα. Sq;{u) = δ(α υ, δα + a u;_, α) (mod 2)
Proposition 3
The above operation passes to a homomorphism :
Sq,: H"{K;Z2) ->H2p-'(K;Z2)
This follows from the preceding lemmas. Of course " homomorphism "
is understood in the sense of additive groups, not of rings.
Proposition 4
Let/be a continuous map /i—>/.. Then/* commutes with Sqt as in the
diagram below.
HP{L;Z2)^ H2p-!(L;Z2)
fi lft
HP(K;Z2) -?*+ H2p-\K\Z2)
proof: By the simplicial approximation theorem, we may assume/ is
simplicial. Let и be a ^-cochain of L. From the definitions, we have the
formulas
/*(S<7,(h)): с - >(и ® ιήφ,Μ, ®/(г)) - (ι/ ® и)^(1 ®/)(</, ® г)
&/,(/*(«)): г >(/·*// ®f*u)q>M ® г) - (и® ι/)(/"®/)φΛ(</, ® γ)
where г is a (2/? — /)-chain of A". But the two chain maps φ;(1 ®/) and
(/ ® /)<?*: are both carried by the ac>clic carrier С from W ® К to L ® L

18
COHOMOLOGY OPERATIONS
given by C(d; ® σ) = C(fa χ fa). Thus they are equivariantly chain-
homotopic, and hence the two images displayed above are cohomologous.
In fact, if Λ is the homotopy, the difference between the above cochains is
bg where g(e) = {u® u)h{di ® <?).
The definition of the squaring operations is now complete, in the sense
that we can draw the following inference.
Corollary 1
The operation Sq; is independent of the choice of ψ'.
The corollary is proved by putting К — L in Proposition 4, letting φκ, q>L
denote two different choices of φ, and taking for/the identity map.
Proposition 5
If и is a cochain of dimension p, then Squ(u) is the cup-product
square и2.
This follows from the remarks given after the definition of ψ, since
(u u0 u)(c) = (и ® м)<р0(с) and <p0 is suitable for computing cup products.
We have already begun to assemble the basic properties of the squaring
operations, and henceforward it will be more convenient to modify the
notation as follows.
Definition
Denote by Sq' (with upper index) the natural homomorphisms
Sq:: Hn(K;Z2)->Hp+\K;Z2) /-0, 1, ...,p
given by Sq' = Sqp-i. For values of i outside the range 0<i<p, Sq! is
understood to be the zero homomorphism.
Thus Sq1 raises dimension by / in the cohomology of K.
COMPATIBILITY WITH COBOUNDARY AND SUSPENSION
We now wish to define squaring operations in relative cohomology. Let
L be a subcomplex of A"; we have an exact sequence, at the cochain level,
0 -> C*{K,L) -£* C*(K) -L> C*{L) — 0
We may assume that <pL ■=■ φκ \ W ®L, since φκ(ά; ® σ) e (Γ(σ χ σ) с
L ® L for σ г L. This implies thai for u,v e C*(K),j*(u u, v) -- j*u u,/*!>.
Define relative cup-/ products as follows. Let u,v f C*(K,L); then
j*{q*u u, q*v) — 0, so that (</*// u; q*v) is in the image of q*, by exactness;
but q* is one-to-one, and hence we may define и u; υ as the unique cochain

CONSTRUCTION OF THE STEENROD SQUARES
19
in C*(K,L) such that<7*(w U; v)=q*u u;o*i>. It is trivial to verify that the
coboundary formula for cup-/ products carries over into this context.
Therefore we obtain homomorphisms
Sq1: H"(K,L;Z2)-^H^\K,L;Z2)
by the same process as in the absolute case. Jt is obvious from the
definition that q*Sq' — Sq'q*.
In what follows, all coefficients are in Z2 and we drop the Z2 from the
notation.
We recall the definition of the coboundary homomorphism δ*: H*(I.) —>
jH*(K,L). Let a be a cocycle and a its cohomology class, a e a e H"(L).
Then a =j*b for some b e C"(K). Then j*(db) = (δα) = 0, and so Sb =■ q*c
for some с e C+ \K,L). Since o* is one-to-one, с must be a cocycle, and by
definition δ*(a) = c, the cohomology class of c.
Proposition 6
Sq' commutes with δ* as in the diagram.
H"(L) —^-* H"' '(L)
И \ι· (coefficients Z2)
H*+l(K,L)-^ Hp+l+l(K,L)
proof: Using the notation of the last paragraph, Sq'a and Sq'(o*d) are
represented by a up_; a and с up+,_; c, respectively. Now
o*(c up+, _; c) = q*c up+, _; <7*c =-- .56 up+, _, Ob = <56' (mod 2)
where 6' =^ (ό υ,.,, _, δό) + (b u„_, 6). Moreover, j*(b') = 0 + (a up_, a).
Therefore, by definition of о*,
5*(V(fl)) = ^[flup_,fl]=[fup+1_,c]
and the class on the right is Sa'(<5*(a)).
Recall that, given any space X, we may define the cone over X and the
suspension of A' from the product space Χ χ / by collapsing 1x0 or
A' x /, respectively, to a point. In reduced cohomology, we have the
suspension isomorphism S*: fl*(X)—* fl*(SX) defined by the composition
H"(X) -4* tfp+'(CX,X) -г-* Яр+'(SX)
which raises dimension by one. The second isomorphism is proved by an
excision argument and is based on a map. This, together with the naturality
of the squaring operations and Proposition 6, yields the following
fundamental property.

20
COHOMOLOGY OPERATIONS
Proposition 7
The squares commute with suspension:
Sq! ■ 5* - 5* · Sq': H"(X) - >/7p+i+1(5A')
We can apply Proposition 7 to obtain an example of a non-trivial
squaring operation which is not just a cup product. Let К denote the real
projective plane; its cohomology ring with Z2 coefficients is just the
polynomial ring over Z2 generated by the non-trivial one-dimensional
cohomology class α and truncated by the relation a3 — 0. From Proposition
5, Sq\x) -- Sq0(n) — я2, so that S*Sqi(x) is non-zero. Hence SqlS*(a) is
also non-zero, showing that the operation Sq* is non-trivial in H2(SK;Z2).
DISCUSSION
The squaring operations constructed in this chapter are a special case
of the reduced power operations of Steenrod. These operations have been
very important in the development of algebraic topology; most of this book
is devoted to their properties and applications. And there are many more.
The specific construction we have given is neither the simplest possible
nor the most subtle. Steenrod's original definition is more direct; his most
recent definition is far more elegant. The construction we have given is
adopted as a middle ground—one from which the algebraic properties
are easily deduced, and yet the geometric genesis is not totally obliterated.
The reader will observe that we have defined the squaring operations in
the simplicial cohomology theory, yet we will use these operations in
singular theory. We justify this usage as follows. In their book (pp. 123-
124), Steenrod and Epstein show that the squares, if defined for finite
regular cell complexes, have unique extension to both singular and Cech
theory for arbitrary pairs. But a finite regular cell complex has the homo-
topy type of a simplicial complex, on which we have defined the operations.
EXERCISE
1. Suppose the cocycle и e C2p(X;Z) satisfies ёи = 2a for some a.
i. Show that и u0 и + и и, и is a cocycle mod 4.
ii. Define a natural operation, the Pontrjagin square,
P2: H2-( ;Z2)^H*"( ;Z4)

CONSTRUCTION OF THE STEENROD SQUARES
21
iii. Show that pP2(u) = uvu, where p: #*( ;Z4)-*#*( ;Z2) denotes
reduction mod 2.
iv. Show that P2(u + v) = P2(u) + P2(v) + ukj v, where и и ν is computed
with the non-trivial pairing Z2 ® Z2 -+Z4.
REFERENCES
General Other definitions of the squares
1. N. E. Steenrod [2]. 1. N. E. Steenrod [3,5,6].
2. and D. B. A. Epstein [1].

CHAPTER
PROPERTIES OF THE
SQUARES
We assemble the fundamental properties of the squaring operations in
an omnibus theorem.
Theorem 1
The operations Sq', defined (for / >0) in the previous chapter, have the
following properties:
0. Sqi is a natural homomorphism H"(K,L; Z2)^>Hp+i(K,L; Z2)
1. If/ >p, Sq'(x) = 0 for all χ e H"{K,L;Z2)
2. Sq'(x) = x2 for all χ e H!(K,L,Z2)
3. S^0 is the identity homomorphism
4. Sq1 is the Bockstein homomorphism
5. t*Sq! - Sq'd* where δ*: H*(L;Z2)^H*(K,L;Z2)
6. Car tan formula: Sqi(xy) = Yij(SqJ'x)(Sq!~Jy)
7. Adem relations: For a<2b, Sq°Sqb = Yc(b>zu1)Sqa+b~cSqc where the
binomial coefficient is taken mod 2
We remark that the above properties completely characterize the
squaring operations and may be taken as axioms, as is done in the book of
Steenrod and Epstein.
Properties (0), (1), (2) and (5) have been proved in the last chapter. This
chapter will be devoted to the proof of (3), (4), (6), and (7).
22

PROPERTIES OF THE SQUARES
23
Sql AND Sq°
Let β denote the Bockstein homomorphism attached to the exact
coefficient sequence 0—>Z—>Z—>Z2—>0. Then β is a homomorphism
H*(K,L, Z2) —> H*(K,L; Z), which raises dimension by one. Jt is defined
on χ e HP{K,L;Z2) as follows: represent the class .v by a cocycle c; choose
an integral cochain c' which maps to с under reduction mod 2; then be' is
divisible by 2 and ?{Sc') represents βχ.
The composition of β and the reduction homomorphism gives a
homomorphism
S2:HP(K,L;Z1)->HP+X(K,L;Z2)
which we also call "the Bockstein homomorphism"; in fact, it is the
Bockstein of the sequence 0—>Z2—>Z4—>Z2—>0. To show that this is
Sq\ we will use the following lemma, which in the light of (3) and (4) will
be seen to be a special case of the Adem relations (7).
Lemma 1
S2SqJ ■= 0 if7 is odd; <525^J = SqJ + 1 if7 is even.
proof: Given us HP(K,L;Z2), let с be an integral cochain such that
the reduction mod 2 of с is in the class u. Then SqJu is the class of
(c up_j- c). Now Sc — 2a for some integral cochain a e C+1(K,L). Writing
/ for (p —7), we have, by the coboundary formula,
S(c u, c) = (- \)'2a и, с 4- (- l)Jc иг 2α - (- \)'c u,_, с - (- l)"c u,_, с
Thus S2(SqJu) is represented by the mod 2 cocycle
au,c-, fUjfl-^iiXfU,., c)
where the coefficient j is 0 or 1 according to whether 7 is even or odd,
respectively. But the sum of the first two terms is a coboundary, namely,
!>(£■ u/+, a) (mod 2), and the last term represents (j)(5i7J + 1w). This proves
the lemma.
As a special case of the lemma, d2Sq° — Sc/1. This shows that (4) follows
from (3). It remains to prove (3).
Property (3) must be true in the real projective plane P(2), for in that case
t,Sq°{x) - Sql{z) - (cc2) 7^0, and so Sq°(n) J-0, which proves Sq°(x) =■ α
since y. is the only non-zero element of H1(P(2);Z2) = Z2. We can deduce
that (3) is true in the circle 51 by taking a map/: S1- >P(2) such that
f*: ι^>σ, where σdenotes the generator of H*(Sl ;Z2), and applying the

24
COHOMOLOGY OPERATIONS
naturality condition: Sq°a = Sq°f*a =f*Sq°<x=f*a = a. Then (3) holds
in every sphere S" by suspension (Proposition 6 of Chapter 2).
Let К be a complex of dimension n; map К to 5" so that σ pulls back
to a given class in H"(K;Z2) (use the Hopf-Whitney theorem, Theorem 5 of
Chapter 1). Then (3) holds for that class. Now let К be any complex, of
unrestricted dimension; (3) must hold on any «-dimensional cohomology
class in К because the injection j of the «-skeleton K" into К induces a
monomorphism j*: //"(#;Z2)—>//%£";Z2) and the result follows as
before by naturality. Thus (3) has been shown to hold in absolute
cohomology.
Now let K,L be a pair and Kv CL the space obtained by attaching to К
the cone over L, attached at the common subspace L. We then have
isomorphisms, commuting with Sq°,
H*{K,L) % H*(K и CL, CL) л, Н*(К и CL)
The first isomorphism is by excision; the second, by contractibility of CL.
This completes the proof of (3) and hence also of (4).
THE CARTAN FORMULA AND THE HOMOMORPHISM Sq
The Cartan formula (6) has two forms, one where we interpret the
multiplication as the external cross product and another in which it is
interpreted as the cup product. We will prove the first form and deduce the
second as a corollary.
Consider the composition
W ® К ® L -!^U W®W® K®L-t +W®K®W®L
φκ®φ'^- K®K®L®L—+K®L®K®L
where r: W—> W ® W was defined in Chapter 2 and Τ permutes the second
and third factors (we are not concerned with sign changes because we want
conclusions in Z2 coefficients). This composition, which we denote by
<Pk®/.i is easily seen to be suitable for computations of cup-/' products and
hence Sq', in К®L.
Using the same letters to denote (co-)homology classes or their
representatives, and writing p,q,n for dim u, dim r, and ρ -\ q — /', respectively, we
compute as follows:
Sq'(u X v)(a ®b) = ((u ® v) u„ (u ® v))(a ® b)
^(u®v®u® ν)ψκ <g L(d„ ®a®b)
~-(u®u®v® £-)£ φκ((1} ®α)® TJ(pL(d„_j ® b)

PROPERTIES OF THE SQUARES
25
the last step using the definition of r, and the summation being over j,
0<j<n;
Sq'(u X v)(a ® b) =■ Σ (" ^j ч)(а) ® (v u„_, v){b)
= Σ (Sqp~Ju)(a) ® (Sq"-" + jv)(b)
= Σ (Sq'-Ju X 5^"" + yt;)(fl ® 6)
Hence, since 5<y'x is zero for / outside the range 0 < ;' < dim x, we have
Sq'(u Χ ι;) - Σ"-ο Sq"-Ju X Sq^^'v
which completes the proof of the first form of (6).
In order to prove the cup-product form of (6), let Δ denote the diagonal
map of K, if x,y e H*(K;Z2), x u у = A*(x χ у), and so
Sq'(x ν y) = Sq'A*(x X y)
= Δ* Sq'(x X y)
= Δ*χν=05^χ Χ Sq'~Jy
= ^Sqix^J Sq'~'y
This completes the proof of the Cartan formula (6) in both
interpretations.
We remarked before that the Sq' are homomorphisms only in the sense
of groups; the Cartan formula makes it clear that they are not ring
homomorphisms, but they can be combined into a ring homomorphism in the
following sense.
Definition
Define Sq: H*(K;Z2)^H*(K;Z2) by
Sq(u) = Σ; Sq'u
The sum is essentially finite; the image Sq(u) is not in general
homogeneous, i.e., it need not be contained in Hp for some p. (It is understood that
each Sq' is defined on nonhomogeneous elements и е Н* by requiring it
to be additive.)
Proposition 1
Sq is a ring homomorphism.

26
COHOMOLOGY OPERATIONS
PROor: Clearly Sq(u) u Sq(i) - (£,- Sq'u) и (£y SqJr) has Sq'(u u r) as
its (/7 bf/ -i-/)-dimcnsional term, by the Cartan formula. Thus Sq(u) u
5^(1·) -- S</(i/ υ ν).
As an application of this homomorphism, we compute the Sq' on any
power of any one-dimensional cohomology class of any complex, as
follows.
Proposition 2
For и с-И \K;Z2), Sq'(uJ) = (,V '.
proof. St/(u)^ Sqnu - Sqlu — и -f u2 by properties (1) to (3). But
Sq is a ring homomorphism. Therefore Sq(uJ) -— (11 + u2)J — uJ £* (i)uk, and
the proposition follows by comparing coefficients.
In particular this gives us the action of all the Sq' in H*(Z2,\ ;Z2). since
this is generated as a ring by a one-dimensional class.
We pursue this line not only for its intrinsic interest but because it will
serve us in the proof of the Adem relations.
SQUARES IN THE я-FOLD CARTESIAN PRODUCT OF Κ(Ζ2Λ)
Definition
Let K„ be the topological product of η copies of K(Z2,\). Here we may
take for K(ZZ,\) the complex P(x) discussed in Chapter 2.
Since H*(Z2,1; Z2) is the polynomial ring on the one-dimensional class,
it follows by the Kunneth theorem that the ring H*(Kn\Z2) is the
polynomial ring over Z2 on generators x,, ..., л„, where y, is the non-trivial
one-dimensional class of the /lh copy of Κ(Ζ2Λ)- I" this polynomial ring
Z2[v,,. . .,y„], we have the subring 5 of symmetric polynomials, which (by
the fundamental theorem of symmetric algebra) may be written as
Ζ2[σ,,. . . ,σ„] where σ; is the elementary symmetric function of degree j
(for example, σ, — „ν, ι- · · · + γΒ).
Proposition 3
In Η*(ΚΛ;Ζ1),&,'(σ„) = σιρι(\ <ι'<η).
prooi : Sq(a„) = Sq(U x,) - Π Sq( ν,) -- Π (x, + *?) - σΛΠ (· τ ν,)) -
ση Yj = o σ.· The result follows.
Corollary I
In H*(Z2,n; Z2), Sq'i„ ^ 0 for 0 < / < n.
proof: By Theorem 1 of Chapter 1, we can find a map/of K„ into
K(Z2,n) such that /*(!„)= σ„. Then f*Sq\ia) - Sq'f*(K) = VK) -_
σ„σ; ^0, which proves the corollary.

PROPERTIES OF THE SQUARES
27
Wc can show more; we can find a host of linearly independent elements
in H*(Z2,n; Z2) by using compositions of the squares.
Notation
Given a sequence / — {/,,.. .,/',.} of (strictly) positive integers, denote by
Sq' the composition Sq'' ■ · · Sq>r. By convention, Sq' = Sq°, the identity,
when / is the empty sequence (the unique sequence with r — 0).
For example, Sq{2A)(x) = Sq\Sq\x)).
Definitions
A sequence / as above is admissible if /,■ > 2(/, + 1) for every j < r. (This
condition is vacuously satisfied if r < 1.) In this case we may also refer to
Sq' asadmissible. The length of any sequence / is the number of terms, r in
the above notation. The degree d{l) is the sum of the terms, £_,-1). (Thus
Sq1 raises dimension by d(l).) For an admissible sequence /, the excess
e(l) is 2/, -d{l).
For the excess, we have
e(I) ^ 2/, - d(I)
= h - h — · · · — 'r
= (/,-2/2)-(/2-2/3) + ··· +(/,)
The last expression justifies the name, but the first two are more convenient
in practice.
Recall that S denotes the symmetric polynomial subring of the
polynomial ring H*(K„;Z2) — Zi[.y,.. . .,-v„]. We define an ordering on the
monomials of 5 as follows: given any such monomial, write it as
m =σ)\σ)\· ■ ·σ)\ with the У* in decreasing order, jx >j2 > · · ·. Then put
m<iri if У, <j[ or if 7, =--j[ and {m\a}) <{m'lah).
Theorem 2
If d(l) <n, then Sq'(ff„) can be written σ„· Q, where Qt — au ■ ■ ■ σ,ν +
(a sum of monomials of lower order).
We prove this theorem by induction on the length of/; it reduces to the
last proposition if r — 1. Let r be the length of /, and write J for /,, .. ., ir.
We assume the result for sequences of length <r (in particular for J);
ihen, using the Cartan formula,
VOO =~ SqhSqJ(<rn) -~ SqHv.Q,)

28
COHOMOLOGY OPERATIONS
By the induction hypothesis on Qj, Sq'(a„) may be written
<ν*ί,σ/2·' ·σ.ν +σ»σ.', Cower terms of Qj) + ££г£ <VmV~m((?v)
Observe that Sq'{o^) <<ri+J and also that 5^'(σ.) = σϊ <σα· Therefore the
largest possible term after the first term in the above expression is obtained
from the last group of terms by taking m = /, — /2 + 1, so that /, — m =
i2— 1; thus this term is
Now / is admissible, and so 2/'2 — 1 <2/2</,, but this implies that
χ <σ„σ(ισ,2· · -air, and the proof is complete.
As /runs through all admissible sequences of degree <n, the monomials
σ, = σ;ισ,.2· · ·σ,ν are linearly independent in 5 and hence in H*(K„;Z2).
From the above theorem, the Ξη'(σ„) are also linearly independent. We can
draw the following inference.
Corollary 2
As / runs through the admissible sequences of degree < n, the elements
Sq'(i„) are linearly independent in H*(Z2,n; Z2).
Choose a map/: K„^>K(Z2,n) such that/*(/„) = σ„. Then the corollary
follows from the preceding remarks.
Proposition 4
If и e H"(K;Z2) for any space K, and / has excess e(I)>n, then
Sq'u = 0. If e(I) = n, then Sq'u = (SqJu)2, where J denotes the sequence
obtained from / by dropping /,.
This proposition may be considered as a generalization of properties
(1) and (2) of Theorem 1. To prove the first statement, note that if e(l) —
/, — i2 'r>«, then /, > η +/2 +· · · +/r = dim (SqJu), so that
Sq'u = Sqh(SqJu) — 0 by (I). The second statement of the proposition
follows in a similar way from (2).
These results are included in a theorem of Serre, which states that
H*(Z2,n;Z2) is exactly the polynomial ring over Z2 with generators
{Sq'(i„)}, as / runs through all admissible sequences of excess less than n.
We will establish Serre's theorem in Chapter 9, using spectral-sequence
methods.
As a corollary to Serre's theorem, we mention that the map /: K„ —>
K(Z2,n) such that/*(;„) — σ„ clearly has the property that/* is a mono-
morphism through dimension 2n. We will use this fact in the proof of the
Adem relations, and so we call attention to the fact that our proof of
Theorem 1 will not be complete until we have proved Serre's theorem.

PROPERTIES OF THE SQUARES
29
THE ADEM RELATIONS
We now discuss the Adem relations (7).
An Adem relation has the form
R = SqaSqb + Σ^ο] CIS,"') ScT+b-cSqc = 0 (mod 2 )
where a < 2b, and [a/2] denotes the greatest integer < a/2. We usually drop
the limits of summation from the expression, since the lower limit is
implicit in the term Sqc while the upper limit is implicit in the convention
that the binomial coefficient (*) is zero if у <0. We also use the standard
convention that (*) = 0 if χ < у. (As an exercise in the use of these
conventions, the reader may note that the Adem relations give Sq2n~1Sq" = 0 for
every n.)
We will establish the Adem relations through a series of lemmas.
Lemma 2
Let у be a fixed cohomology class such that R(y) = 0 for every Adem
relation R. Then R(xy) = 0 for every one-dimensional cohomology class
ν (and every R).
We will defer the proof of Lemma 2 to the end, since it is elementary
but long and complicated.
Lemma 3
For every R and for every η > 1, R{an) = 0 where σ„ e H"(Kn;Z2) as
before.
proof: Let I denote the unit in the ring H*(Kn;Z2); then /?(1) = 0
for every R, by dimensional arguments. Then R(x{) = R(lxi) = 0 by
Lemma 2; and finally R(&n) — R{x{ · · -.v„) = 0 by induction on η using
Lemma 2.
Lemma 4
Let у be any cohomology class of dimension η of any space K, with
Z2 coefficients, and let R — R(a,b) be the Adem relation for ScfStf where
a+b<n. Then R(y) = 0.
PROOr: By Serre's theorem we have a map /*: H*(Z2,n;Z2)^>
H*(X";Z2) which takes /„ to σ„ and is a monomorphism through
dimension 2n. We have /?(σ„) = 0 by Lemma 2, and so /?(/„) = 0, since these
elements have dimension η + a + b < 2n. The result for у follows by
naturality, using a map g: K^>K(Z2,n) such that g*(in) = y.

30
COHOMOLOGY OPERATIONS
Lemma 5
Let R be an Adem relation. If R{y) = 0 for every class у of dimension
p, then R(z) — 0 for every class ζ of dimension (p — 1).
proof: Let и denote the generator of Hl(Sl;Z2). Clearly Sq'u = 0 for
all /" > 0. Therefore, by the Cartan formula, R{u χ ζ) = и X R(z). But
и χ ζ has dimension ρ; hence R(u χ ζ) ■= 0, and so R(z) = 0.
The Adem relations follow easily from Lemma 4 and Lemma 5 by
induction on dimension.
It remains to prove Lemma 2.
We begin by recalling the formula (£) = (£-!) L(%'), which holds for
all p,q except for the case ρ = q = 0.
Lemma 6
(?) +(,ίι)4-(ί-ιΙ) -L(^i)-=0 (mod 2) except in the cases (j,=9 = 0)
and (p — 0,q — — 1).
This lemma follows from the formula just cited. (The easiest way to see
the sense of these two formulas is to consider Pascal's triangle.)
To prove Lemma 2 is to show R(xy) -— 0 where χ is any one-dimensional
class and у has the property that R(y) — 0 for every R. We begin by
applying the Cartan formula to S(f(xy); since dim .r—1, Sqb(xy) =
xSqby -l-x2Sqb~xy. Again by the Cartan formula,
Sq°Sqb{xy) = Sq°{xSqby ->- x2Sqb~ *y)
-= xSq"Sqby -+ x2Scf~ 1Sqby + x2Sq"Sqh-' ν + 0
ι x*Sq°-2Sqb-\v
the zero coming from Sql(x2), which is zero mod 2. In a similar manner
we find that
Σ (s)Sq°+b-cSf(xy) - -ν Σ (s)Sqa + h-cSqcy + χ2 Σ (s)Sq° + b—lSqcy
H- x2 Σ (.s)Sef + b-cSqc-ly+x* Σ (s)Sqa + b~-2Sqc-1y
where .9 = s(c) — (Jl^1)· ln these two formulas, the first terms match,
since
xSq"Sqby V χ Σ (s)Scf+b-cSqcy -^- xR(y) = 0
Now a <2b implies (a -2)<2(b— 1), and so the fourth terms also
match: since R( y) = 0 for all R, in particular, for R(a 2, b 1),
Sqa-2Sqb-'y - I, C--c2;*2)Sq° + b-c- 'Sq'y
= ^(b,-C2rl)Sq', + b-c'-2Sqi'-ly

PROPERTIES OF THE SQUARES
31
where с = с + 1. Thus it remains to show that
Sq'Sq"-ly + Σ ("„--^l^Sq-^-^'Sqy
= Σ (sW + b-c-xSqey + Σ (s)Sc^b-cSqc-1y
where the second term on the left-hand side (LHS) replaces Scf'^Scfy,
using R(a— 1, b). We consider three cases.
case 1: a = 2b - 2. Then a - 2c = 2{b - с - 1), and so (s) = (2*t) = 0
unless к = 0, that is, unless с = b - 1; so RHS = SqaSqb~1y + Sqa+1Sqb~2y.
Similarly, („I^r-i) = Gt-i) = 0 unless к = 1, that is, unless c = b — 2; so
LHS = S(/,V~V+S<70+1SV'~2.>;, and the two sides are equal.
cash 2: a = 2b — 1. Proved by a similar argument.
case 3: a < 2b - 2. Then, by R(a, b - 1),
ScfScf-V = Ic C=,2;2)S9"+b"""'IS<fi'
Also,
where с' = с — 1, so we are reduced to verifying that
C-Tc1) + C:c2;-i,) = (i:f27') + G:2;-22) (mod 2)
But this follows from Lemma 5, with p = b — c— \, q = a — 2c — 1. The
exceptional cases are excluded automatically, because ρ = 0, q = 0 or — 1
means b = с + 1, a = 2c + 1 or 2c, respectively, contradicting the Case 3
hypothesis.
This completes the proof of Lemma 2.
We attach a short table of representative Adem relations.
Sq'Sq1 = 0, Sq*Sq3 = 0, .. . ; VV + 1 = 0
Sq*Sq2 = Sq3, Sq1 Sq4 =Sq5,...; Sq'Sq2" = Sq2n+1
Sq2Sq2 = Sq3Sq\ Sq2Sqb = Sq'Sq1, . ..; Sq2Sq*n~2 = Sq^'Sq1
Sq2Sq3 = Sqs + Sq*Sq\ ...; Sq2Sq^' = V +' + Sq^Sq1
Sq2Sq* = Sqb + SqsSq\ . . .; Sq2Sq4n = V +2 + Sg^'Sq1
Sq2Sqs = SqbSq\ . ..; Sq2Sq*n+1 = Sq^2Sql
Sq3Sq2 = 0,...;Sq3Sq*n+2 = 0
Sq3Sq3 = Sq5Sql; ...
Sq2n~1Sqn = 0

32
COHOMOLOGY OPERATIONS
DISCUSSION
Theorem 1 lists the major properties of the squaring operations, and, as
we remarked, these properties characterize the squares uniquely. Parts
(0) to (5) are due to Steenrod. The Cartan formula (6) was indeed
discovered by Cartan; the Adem relations (7) were proved independently, and
by very different methods, by Adem and Cartan.
Theorem 2, the surrounding material, and the proof we have given of
the Adem relations include work of Cartan, Serre, and Thorn. From our
point of view this material will be amplified and completed by the
calculations of Serre given in Chapter 9.
REFERENCES
General
1. N. E. Steenrod [2].
2. — and D. B. A. Epstein [1].
The Adem relations
1. J. Adem [1,2].
2. H. Cartan [3].
The Cartan formula
1. H. Cartan [5].
Related properties of the squares
1. H. Cartan [1,2,5].
2. J.-P. Serre [2].
3. N. E. Steenrod and D. B. A.
Epstein [1].
4. R. Thorn [1].
5. С. Т. С. Wall [I].

CHAPTER
APPLICATION:
THE HOPF INVARIANT
Let S" denote an oriented «-sphere, where и > 2. Let there be given a
map/: 52""1—>5". Consider 52""1 as the boundary of an oriented 2«-celI,
and form the cell complex K— S"vfe2n. Precisely, this is the complex
formed from the disjoint union of S" and e2n by identifying each point in
S2""1 = ё2п with its image under/. Then the integral cohomology of К is
zero except for the dimensions 0, n, and 2n, and is Ζ in those three
dimensions. Denote by σ and τ the generators determined by the given orientations
of the cohomology groups in dimensions η and 2n, respectively. Then the
cup-product square σ2 is some integral multiple of τ.
Definition
The Hop/ invariant of/is the integer #(/) such that σ2 = H{f) ■ τ.
PROPERTIES OF THE HOPF INVARIANT
The homotopy type of К depends only on the homotopy class of the
map/used to define K, and thus H(f) also depends only on the homotopy
class of/. We may therefore speak of the Hopf invariant of a homotopy
class; and we have defined a transformation Η: π2„_,(5")^Ζ, namely,
that transformation which takes a homotopy class α into the integer H(f)
where /is any representative of a.
33

34
COHOMOLOGY OPERATIONS
If η is odd, #(/) = 0 for all/ This follows immediately from the anti-
commutativity of the cup product: σ2 = —σ2; hence σ2 = 0.
If η is 2, 4, or 8, there exists a map /: S2n~' —>S" with Hopf invariant
one. For example, in the case и = 2,/may be taken as the natural
projection /: S3—>C7>(I) = S2, viewing 53 as the unit sphere in the complex
plane C2. Such/is the attaching map in the complex projective plane,
CP(2) = S2 ur eA. Similarly, the cases η = 4, η — 8 correspond to the
quaternionic plane QP(2) = S4 ur es and the Cayley plane CayP(2) =
S8 ur e16, respectively. These maps are known as the Hopf maps.
For values of η other than 2, 4, or 8, it is now known that no map of
Hopf invariant one can exist. This result is a very deep theorem and we do
not prove it; Theorem 2 below is a partial result.
Proposition 1
If я is even, there exists a map/: S2"~1 —>S" with Hopf invariant two.
proof: We may consider the product space S" χ S" as the cell complex
formed by attaching a 2«-ceII to the wedge (one-point union) of two spheres
S" ν S", using an attaching map g: S2"-"1—>S" ν S" as suggested by
Figure I. Let F denote the "folding map" F: Sn ν S"—>Sn, and let ψ
S"n
SnvSn
Figure 1
denote the composition Fg: S2""1—>S". (In fact, φ is known as the
Whitehead product [(„,/„] and is a special case of a general construction.)
We claim that φ has Hopf invariant two.
In order to verify this, we must compute cup products in the complex
K— S" uv e2n. Consider the diagram below, where the vertical maps are
e2n _, S2n~ 1 _^ Sn y gn F ^ Sn
S" X S" = (Sn v S")u9e2
J'
5" u„
a:
the inclusions. Now φ ~ iFg: S2"~i—>К is clearly null-homotopic, by
definition of K. The map F is defined on S" ν S" by taking it equal to F
there; but then it can be extended over all of S" χ S" because iFg -= φ is
null-homotopic. Moreover, F is a relative homcomorphism on the pair

APPLICATION: THE HOPF INVARIANT
35
(f2",52"_1). Therefore F* is an isomorphism in dimension 2n. Now the
folding map F has the property F*a = σ, + σ2, where σ is the generator of
H\Sn) and σ,,σ2 are the generators of Hn(Sn ν S") corresponding to the
two pieces. We denote generators of H2n(K) and H2n(S" χ S") by τ and
p, respectively. In the product 5" X S", we have σ2 — σ\ = 0 and σ,σ2 = ρ
where we may choose ρ to make the sign positive. We also choose τ so that
F*{z) ■= p. The Hopf invariant of ψ is defined by σ2 = Η(φ) ■ τ. Then
F*a2 = Η(φ)Ρ*(τ) = Η(ψ) ■ p. But
F*a2 = (F*a)2 - (F*a)2 = (σ, + <т2)2 = σ2 + σ2 + <т,<т2 + σ2σ,
= 0+0+p+(-I)'ap
= 2ρ
so that Η{ψ)ρ = 2ρ and Η{ψ) = 2, as was to be proved.
Proposition 2
The transformation Η: π2„_,(5")^Ζ is a homomorphism.
We first recall the definition of the additive group structure in π2„_,(5").
We denote by Q the "pinching map"
Q:{e2n,S2n'l)^{e2n ν e2n, S2"'1 ν 52""1)
obtained by collapsing an equatorial (2« — I)-ceII to a point. Let F denote
the folding map S" ν S"—> S", as before. Then if/and g represent elements
of π2„_ι(5"), so that/and g are maps of S2"-1 into S", the composition
Sl«-\ 8,52.-1 y S2n-l _fyg>Sn y у f > у
represents, by definition, the sum of the homotopy classes of/and #.
To prove Proposition 2 we consider the following diagram, where ρ
I ! 1
S" < £ 5" ν 5" ξ „ 5" ν S"
is constructed analogously to the map F of Proposition 1, q is based on the
pinching of e2" by (?, and the vertical maps are inclusions. Our notation
for the generators in cohomology will be σε Ηη(Κ),τε H2"(K),pe H2"(L),
τ, and τ2 in H2n{M), and σ„σ2 in Hn(Sn ν 5"). (We also use σ,,σ2 for the
obvious classes in H"(L).)
By definition, σ2 = H{f-rg)z. Thus, ρ*σ2 = H(f + g)p*r. On the other
hand, ρ*σ2 -= (σ, -лга2)2, as, in the proof of Proposition I, for F. Now in
V/, clearly (σ, +σ2)2 = Я(/)т, + Н(д)т2. Pulling this relation back to L,
<σι Ι σι)2 =H{f)q*T\ ; Η(9)4*τι· We can assume that ρ, τ1; and τ2 are

36
COHOMOLOGY OPERATIONS
chosen so that ρ*τ = q*xx = q*x2 = p. Thus, in H*(L),
Щ/ + д)р = Щ/-\-д)р*т=р*а2 = («τ, +<τ2)2 - Я(/)<Л, + H(g)q*r2
~-{H{f) + H{g))p
which implies #(/-(-#) — H(f) + H(g), as was to be proved.
Corollary 1
If я is even, then n2„-l(S") contains Ζ (i.e., an infinite cyclic group) as a
direct summand.
In fact, the cyclic group generated by the homotopy class of a map of
Hopf invariant two must be mapped isomorphically onto the even integers
by the homomorphism H.
DECOMPOSABLE OPERATIONS
We will be able to prove, by means of the squaring operations and the
Adem relations, that there does not exist a map/: S2"-1—>,S" of Hopf
invariant one except possibly when η is a power of 2. The clue to the method
is the simple observation that/has an odd Hopf invariant, that is, σ2 is an
odd multiple of τ in К = S" ur e2n, if and only if Sq"a = τ in the mod 2
cohomology of K. (Recall that Sq"a = σ2.)
We will say that Sq' is decomposable if Sq' = £,<,· a,Sq' where each a, is a
sequence of squaring operations, and we say Sq' is indecomposable if no
such relation exists. As examples, Sq* is obviously indecomposable; Sq2 is
indecomposable, since Sq^Sq' — 0; Sq3 is decomposable, since Sq3 —
SqlSq2 by the Adem relations. Less obviously, Sq6 is decomposable by the
Adem relation Sq2Sq* = Sq6 -J Sq5Sql.
Theorem 1
Sq' is indecomposable if and only if ;' is a power of 2.
proof: We first suppose that;' is a power of 2 and consider the generator
α of Η '(P(co);Z2). Using the ring homomorphism Sq introduced in the
last chapter, we have SV/(a') ■= (Sq a)' — (х-, я2)' == я' ~ a2' (mod 2). Thus
Sq'(x') is zero unless t is either 0 or i, while Sq'ict')^- a2'. Now the fact
that a2' is non-zero shows that Sq1 is indecomposable; for otherwise
a2'- = V(oO = E«,.a<W) = 0.
To prove the converse, let / = a + 2* where 0 < a < 2k. Writing b for
2\ the Adem relations give
ScfSq" = (Ь-')5<ГЬ -i- b>0 (V-7r№+'-'Stf"
Since b = 2k is a power of 2, (b^') - I (mod 2), by the following lemma.
Thus Sq' = Sqa+b is decomposable.

APPLICATION: THE HOPF INVARIANT 37
Lemma 1
Let ρ be a prime, and let a and b have thep-adic expansions a = £™= 0 a-,p\
b = ΣΓ-ο *>.·/>' where 0 < a(, 6; </>. Then
O-nr-C;) (-od.)
proof: In the polynomial ring Zp[.v], (1 -ι- x)p = I — i". Thus (I -1- x)b =
Π (14 x)h'",-n(I тхр')ь'· Now (J) is the coefficient of x" in this
expansion, as is seen from the first expression; but the inspection of the
last expression shows that this coefficient is precisely Γ] (J;).
NON-EXISTENCE OF ELEMENTS OF HOPF INVARIANT ONE
We now state the main result of the chapter.
Theorem 2
If there exists a map/: S2"-1—>S" of Hopf invariant one, then η is a
power of 2.
We have already remarked that the existence of such a map / would
imply that Sq"a — τ in the complex K= S" KJfe2", where σ, τ are the
generators of If*(K;Z2) in dimensions n, 2n, respectively. If η is not a
power of 2, this is impossible, since then Sq" is decomposable, whereas all
Sq'a must be zero in К (for dimensional reasons) if 0 < / < n. This proves
the theorem.
This result is closely related to the question of existence of multiplicative
structures in Euclidean space R", as indicated by the following proposition.
Proposition 3
Suppose there is a map μ: R" χ /?"—>/?" with a two-sided identity e, no
zero-divisors, and the linearity property (tx)y = t(.xy) = x(ty), t s R, where
we have written xy for μ(χ^). Then π2„_ ,(5") contains an element of Hopf
invariant one.
A proof is outlined in the exercises.
DISCUSSION
Our definition of the Hopf invariant is due to Steenrod; Hopf's original
definition is more geometric but less manageable. In view of Proposition 3
and classical results of Hurwitz on the non-existence of norm-preserving

38
COHOMOLOGY OPERATIONS
/?-Iinear multiplications on R", much interest centered on the Hopf
invariant question, that is, for what η is there a map from 52""1 to S" of
Hopf invariant one ?
Hopf's maps showed existence for η = 2, 4, or 8. The case of η odd
was easily excluded. The case η = 2 (mod 4), η > 2, was excluded by
G. W. Whitehead. Theorem 2 is due to Adem. The complete solution,
namely, η must be 2, 4, or 8, is due to Adams.
The Whitehead product, discovered by J. H. C. Whitehead, is a homo-
topy operation in two variables, mapping πρ(Χ) ® π,(Α')—>πρ+ϊ_1(Α').
This operation is closely related to the Hopf invariant, as shown by
G. W. Whitehead.
EXERCISES
1. Let g: S""1 χ S"~l ->S"~\ Choose yeSn~l. Let a denote the degree of g
restricted to S"~1 χ у and β the degree of g restricted to у х S""1. Show
α and β arc independent of the choice of y. Such a map g is said to be of
type (a,/?).
2. Letg: Sa~l χ S""1 ->S"_1. Viev, S2""1 as the join of S"-1 and S"_1, with
coordinates (aj,b), a,b e S"~ \ 1 e I View S" as the suspension of S"~'. with
coordinates (еДсеУле/. Define /г(#): 52""1 -> S". the Hopf
construction on #, by h(g)(a,t,b) = (g(a.b),t). Prove II(h(g)) = a/? if # has
type (a./?).
3. Prove Proposition 3 by using the given map μ: R" χ R" -> /?" to construct a
map^iS""1 χ S"_ ' -► S"" ' of t>pe (1,1).
REFERENCES
Whitehead products
1. P. J. Hilton [I].
2. G. W. Whitehead [2,4].
The Hopf invariant one problem
1. J F. Adams [1,2]
2. J. Adem [I]
3. H. Cartan [6].
The Hopf invariant
1. H Hopf [1,2].
2. N. E. Steenrod [4].
3. — and D. B. A. Epstein [I].

CHAPTER
APPLICATION:
VECTOR FIELDS ON SPHERES
We consider the (/? -- I)-sphere 5""1 imbedded in R" in the usual way.
A tangent vector field, or simply a vector field on S"~ ', is a function
assigning to each point of S"~' a tangent vector at that point such that the vector
thus defined varies continuously with the point. A k-jield is an ordered set
of к pointwise linearly independent tangent vector fields. We will be
concerned with the problem of finding, for each n, the greatest к such that S"~'
admits a &-field; this maximum к will be denoted K(n).
Clearly, for every n, 0 < K(n) <n I. We take as known the fact that
when η is odd K(n) — 0; thus SJ, for example, does not admit a tangent
vector field. (In fact an orientable smooth manifold admits a I-field if and
only if its Euler number vanishes.) For η even, K(n) > I; it is easy to give
a 1-field in this case: assign to the point ν — (.ν,,.. . ,.γ„) the vector
(v2,—.v,,.v4,— v3,. ..,.y„,— v„_,). As in this example, we will identify a
point in R" with the vector from the origin to that point; and tangent
\ectors may be translated to the origin without explicit mention being made.
A-FIELDS AND KnJt+,
Let \ti(x)} denote a &-fie!d on S"~l. By the Gram-Schmidt process we
may assume that at each point χ the vectors {/,-(.\)} form an orthonormal
set. Then the set {t{(x), . .,ik(x),x) is also orthonormal. Such a set is called
a (k + \)-frame.
39

40
COHOMOLOGY OPERATIONS
We denote by K„t+1 the set of (k + I)-frames in R". (The reader is
warned that various authors use various subscripts VitJ for this set.)
Thus a &-fie!d gives rise to a function/: S"1""1—> KBt+1. We also have a
projection ρ: K„it + 1—> S"~l, taking (vu. . .,vk+l) into the last "coordinate"
vk+l considered as a point in S"~\ Clearly the composition pf\s the
identity of S"~'.
We will give the set V„>k+l a topology such that/and ρ are continuous.
Then, if i*1"1 admits a fc-field, there exists a map/: У1—» FBit+1 such that
#/": 5"~' —> Sn~' is the identity. Our line of attack to prove that S"~' does
not admit a &-fie!d will be as follows. We will study the cohomology of the
space FBjt+1 and show that for certain values of (n,k), H"~l(VnJl+l ;Z2) =
Z2, generated by ν -- v„-{ -= Sq'z for some ζ e H*(V„.k+, ;Z2) and some
t>0. Now suppose 5"""1 admits a λτ-field, yielding/: 5"'~1—> K„>ft+1. If
we let σ denote the generator of H"~x(Sn"i ;Z2), we must have ρ*σ = ν,
f*v = σ, since pf— I. Nowct =/*d =f*Sg'z--= Sq'f*z; but/*z is obviously
zero because dim ζ <n— 1; and we thus arrive at a contradiction.
Our first task is therefore to give a topology for Vnk+l. We will actually
give a cell decomposition.
A CELL DECOMPOSITION OF Vnk
Consider R" as the space of column vectors with the usual basis. Let
0(«) denote the orthogonal group, i,e,, the group of orthogonal
transformations of R". We view 0(«) as a subgroup of the group of η χ η matrices
with determinant ± 1, 0(«) acts on R" from the left by matrix multiplication;
it inherits a topology from R"\ We may consider 0(«) as a subspace of
0(« -f- I) by considering a given η χ η matrix as an (n -) l)x(n+ I)
matrix with zeroes in the last row and column except for a 1 on the
main diagonal. Then УПшк may be identified with the left coset space
0(«)/0(« — k), by taking the last к columns of a representative matrix,
This gives V„tk a topology. (In fact V„mk is a smooth manifold of dimension
\k{2n ~ к — I) = Yj7n-k i, called a Stiefel manifold.) We have an obvious
projection map pkJ when 0 <j < k, pkJ: KBjt—> VnJ,
The projective space P(n) may be considered as the space of lines through
the origin in /?"+1. It will be momentarily confusing, but ultimately more
convenient, to refer to this space as Pn+\. We write P„.k = PJPn-k, the
identification space obtained by collapsing the subspace Pn^k to a point
inP„.

APPLICATION: VECTOR FIELDS ON SPHERES
41
We have a map φ: Л,—>0(я) as follows: a point χ in Pn is a line through
the origin of R"; <p(x) will correspond to reflection through the hyperplane
normal to the line x. If χ is considered a point in R" (and hence an «-vector),
ψ(χ) has the formula
for у e R". (Here < , > denotes the scalar product and ,| |; the norm.)
Clearly φ(χ) is an orthogonal transformation. It is also evident that
ψ is 1-1. Actually φ is a homeomorphism into; we omit the verification. We
have, for m <n, the diagram below, where the vertical maps are the stan-
/»„-£* 0(#и)
i I
Pn -** О(Я)
dard inclusions. In particular, taking m = n— I, we obtain a map
φ:Λ/Λ-,->0(/ι)/0(/ι-1)
that is, Λ,.ι—> ΚΒι1. Note that the slash represents, on the left, the
identification of a subspace to a point; on the right, the formation of cosets.
Now VnA is just the sphere S"~ '. On the other hand, Pn has a cell
structure P» - e° и е1 и ··· и е"~2 и e""1, so that />„ = />„_, и e""1; thusP„,,
is also S""1.
Proposition 1
ψ: Pn_,—> F„_, is a homeomorphism.
Rather than give a detailed proof, we give a geometric description which
should suggest the proof. Think of the case η — 2 so that F„ , = 5""' is a
circle, and we will consider P2 — P{\) as the lower semicircle, with its
endpoints identified to a single point. Our transformation is based on the
map ψ: />„—>0(я) which takes .v to reflection in the hyperplane (line, when
η — 2) normal to .v, followed by projection onto 0(«)/0(« — I), i.e.,
considering the effect on the last coordinate. Using the language of a clock face
for the case η — 2, we must take the unit vector u2, that is, the unit vector in
the 12 o'clock sense; choose χ in P2—a point between 3 and 9 o'clock;
reflect u2 in the line normal to the vector to x; and consider the image as
a point on the circle V2A. The reader may verify that as χ runs from 3 to
9o'cIock, the image of «„runsfrom 12to 12. For example, if .vis at 4 o'clock,
the normal line runs from 1 to 7 and u2 reflects over to 2 o'clock.

42
COHOMOLOGY OPERATIONS
Proposition 2
The matrix multiplication
μ:(Ρ„χ Vn-uk-„Pn-xx K,-,.*-,)->(l/„.*,*'„-,.*-,)
is onto V„k and is a relative homeomorphism.
This map is defined by identifying P„ as a subspace of 0(«), using
the map φ which is a homeomorphism into—and recall that УпЛ —
0(/?)/0(/? - k). etc.
Wc first prove that the inverse image of K„_lt_, is precisely
/>„_, x K„_lt_,. Suppose A,Be Pn χ 0(/7 — 1) are such that ΑΒΌ{η ~ к)
is contained in 0(« — 1); we must show that А е P„_,. Now the last vector
is left fixed by β, by 0(/i —/c), and by the product; hence certainly
AeO(n~ 1). But P„nO(/i- 1) = />„_,, so that Л е />„_,.
We prove next that μ is one-to-one on (P„ — P„_,) χ F„ _,.*_,. Suppose
A,CePn -?„.„β,ΰεΟ(/ι- I) are such that Л ЯО(я - /V) = CDO(n- к).
Then, since /?,£> e 0(/i — I), we have ЛО(/? — 1) — CO(n — I). Therefore
<M --- q>C in K„,; and so A — C, by Proposition 1. Hence BO(n - k) —
DO(n — k). But then Sand D have the same image in K„_1Jt_,.
Finally we prove that μ is onto Vnk. Suppose A e 0(/;). By Proposition 1,
AO(n — 1) = CO(n — I) for some С e Pn. This means that for some D <=
0(n —\),A = CD. But then //(CD) represents A.
We can now give the cell structure of Vnk. Recall that P„ = e° u
?'υ···υ?""'. By a normal cell of Vnk we mean a cell of the form
eh-\ χ ... χ ew- 1 _^p^ χ ... χ />iV_> ynk n >(·,>···> /r > Я — /С
Theorem J
The cells of KnJt are exactly the normal cells and the 0-ceII corresponding
to the identity matrix.
Consider the case к ~ I; then we must have ;', — η and /' — I. Thus the
only normal cell is e"~'- >P„—> V„.,, and the theorem holds. We can
tlierefore prove the theorem by induction on η and /V,for (n — k) fixed. Thus
suppose the theorem is true for Vn_ik_ ,; we are to prove it for Vnk.
By Proposition 2, νηΛ^νη.ΧΛ.^μ{{Ρη~Ρη.χ) χ Κ„_,.»_,). By the
induction hypothesis, then,
К„л-(£>°и {С'''-' Х ··· Xf^'jlU^f""1 X (f°U {ε-'''"1 Χ ··· ΧΛ"1}))
where the cells in braces are subject to the restrictions (//—!)>(',>■■·>
ir > (n - k). But this agrees exactly with the assertion of the theorem for
Vak. Thus the theorem is proved.

APPLICATION: VECTOR FIELDS ON SPHERES
43
Corollary 1
\f2k <//, then Pnk is the /ί-skeleton of V„k.
frook: When 2k <n, the lowest dimension in which a normal cell may
occur that has at least two factors is (w к) — (// — к — 1), since it
corresponds to the case ι",.--·η — к ■ I and thus the two cells have dimension
ir — \ = η — к and ir-\- \>n — k — \. But we have (n — k) \-
(n — к + 1)^2/ί — 2k \>n- I. Thus the theorem above implies that
the /ί-skeleton (V„mk)n consists of the 0-ceII and of normal cells with only
one factor:
(Kj"-e°u еп-к^еп-к + \ u ...U£,"-i
(As it happens, there is no //-cell.) Now Pnk is also a union of cells, of the
same dimensions as in (Vn-k)n. Moreover, we have a map φ: P„.k^VnJi,
just as in the case к = I discussed in Proposition I; this map is a homeo-
morphism into, and its image is evidently (V„>k)". This proves the corollary.
THE COHOMOLOGY OF Pnk
We turn now to consideration of the cohomology of these spaces. Recall
that the ring H*{Pn\Z2) is the truncated polynomial ringZ2[a]/a". We have
the sequence Pn^k-^Pn ->-+ РпЛ =■ Pn/Pn.k, where ;' is the injection andy the
identification map. Now H*(P„-k;Z2) = Ζ2[β]/β"~" and /? = /'*я. By the
exact cohomology sequence of this pair, H"(P„ k;Z2) = Z2 in the range
(/; — k) < q < η and is zero otherwise.
Let v„ denote the generator ofH"(P„,k; Z2); then/*(t>,) = a". But Stf'OO =
(?)οί,+', and so Sq'rq = (3)pq, „ provided (n — k)<q, (q | t)<n. In pax-
ticular, Sqk-'vn-k-C--\)vn-v
Now if 2k <n, we have (УПшк)" — Р„.к, and thus the cohomology of
V„ k agrees with that of PnJi through dimension η — 1. Therefore in this
case H"~\Vn.k\Z2) is generated by i>„_, and Sqk~lvn-k - (ZI')»-·„- .·
We now prove a non-existence theorem for tangent vector fields. Write
the integer /? in the form 2m(2s ■ I); thus 2m is the highest power of 2
dividing n.
Theorem 2
S""1 does not admit a 2m-field.
If s — 0, then 2m — η > η - I and the result is trivial; if m = 0, then η
is odd and the result is known. We therefore suppose m,s > 1.
Recall the argument, given early in the chapter, that if S""1 admits a
2m-field, then there is a map/: S""'-> K„>2,„+1 such that/followed by the

44
COHOMOLOGY OPERATIONS
projection ρ: F„,2m+1—>S"'~1 is the identity of S"~\ Now η = 2m(2s + 1) =
2(2m)j + 2(2m~')>2(2m + I) since ra,j>l. Therefore, if we take k =
2m 4-1, we have 2k <n, and the above results on the cohomology of
V„ik are applicable: H"~\Vnik) = Z2 generated by vn-u and, since pf is
the identity, ρ*(σ) = ι>„_, where σ is the generator of #" ~' (S" ~' ;Z2).
We have S^-lu11_it = G-iK-i; but η - к = п - (2m + l)-2m+1j-l
and к — 1 — 2m; the binomial coefficient is seen to be non-zero by Lemma 2
of Chapter 4.
Thus we are led, from the assumption that a 2m-field exists, to the
contradiction
a =/4-, =/W4.-2-,) = W4-2-.) = V"(0) = 0
This completes the proof.
DISCUSSION
Write η in the form η = 24o+ b(2s + I) with 0 < b <4. Classical algebraic
results of Hurwitz and Radon, treated recently from a modern viewpoint
by Atiyah, Bott, and Shapiro, show that K(n) > 2b 4- 8a - 1; that is, S"-1
admits a A:-field with к = 2b \ 8a — 1.
On the other hand, Theorem 2, due to Steenrod and J. H. C. Whitehead,
gives an upper bound for K(n). For η < 16 and many other values, this
upper bound coincides with the Hurwitz-Radon lower bound and
completely determines K(n).
The complete solution of this problem was given by Adams, who proved
that for all n, the Hurwitz-Radon number is an upper bound as well; thus
K(n) = 2" + Sa - I for all n.
General
1. J. Milnor [I].
2. N. E. Steenrod [1].
3. —and D. B. A. Epstein [I].
Cell structure of Stiefel manifolds
1. С Ε. Miller [I].
REFERENCES
The Hurwitz-Radon theorem
1. M. F. Atiyah, R. Bott, and
A. Shapiro [I].
2. B. Eckmann [1].
Vector fields on spheres
1. J. F. Adams [3].
2. N. E. Steenrod and
J. H. С Whitehead [1].

CHAPTER
THE STEENROD ALGEBRA
In this chapter we describe the algebraic system formed by the Steenrod
squaring operations and derive some of its properties.
GRADED MODULES AND ALGEBRAS
Let R be a commutative ring with unit.
By a graded R-module Μ we mean a sequence M; (/' > 0) of unitary
/^-modules. A homomorphism/: M—> N of graded /?-modu!es is a sequence
f, of /Miomomorphisms,/(: Л/,—>#(. The tensor product Μ ® Ν of two
graded Λ-modules is defined by setting {M ® N), = £,-Л/; ® #,_,.
By a graded R-algebra A we mean a graded /?-modu!e with a
multiplication φ: A ®Λ—>Λ. where φ is a homomorphism of graded /?-modu!es
and has a two sided unit. By a homomorphism of graded /?-a!gebras we
mean a homomorphism of graded /?-moduIes respecting the multiplications
and units.
A graded /?-algebra A is associative if ψ ο {ψ ® 1) — φ ο (I ® <р), where
1 denotes the identity homomorphism A—>A;\n other words, if the diagram
below commutes.
A ®A®A φ.®'> Λ®Λ
1 ® ψ\ \ψ
A® A—2-» Λ
45

46
COHOMOLOGY OPERATIONS
We say that A is commutative \ΐψοΤ— φ: Α ® Α^>Α, where Τ: Μ® ./V—>
N ® Μ is defined by T(m ®n) = (— 1)'J(« ® m), /' = deg n, j = deg m.
If we are given an algebra homomorphism e: /!—>/?, where Л is
considered as a graded Λ-aIgcbra by the convention R0 = R, Rt = 0 (/ Φ 0),
then A is said to be augmented. An augmented /?-a!gebra is connected if
e: Λ0—> /? is isomorphic.
If A and В are graded /?-a!gebras, then A ® В (the tensor product as
graded Λ-moduIcs) is given an algebra structure by defining φΑ®Β =
{φ α ® φ B) о (1 ® Τ® 1). In other words, if we write products by
juxtaposition, (a, ® A,)(a2 ® 62) = (- l)*(fl|fl2) ® ФхЬ2), к - (deg a2)(deg bx).
For example, let Μ be an /?-moduIe. We define the tensor algebra Y(M)
as follows. Write M° for R, Λ/1 for Λ/, M2 for Μ ® Μ, and in general Mr
for the r-fold tensor product Μ ® · · · ® M. Then T(M) is the graded
Λ-aIgebra defined by T{M)r= Mr where the product is given by the usual
isomorphism Ms ® M' я* Ms+I. The tensor algebra Γ(Μ) is clearly
associative, but not commutative.
THE STEENROD ALGEBRA Л
Now take R = Z2, and let Μ be the graded Z2-moduIe such that M; = Z2
generated by the symbol Sq' for each i > 0. Γ(Λ/) is thus bigraded. For each
pair of integers (a,b) such that 0 < a < 2b, let
R{a,b) = Sq" ® Sqb + Isfr-VW" ® Sqc
Let Q denote the ideal of Г(М) generated by all such R(a,b) and 1 + Sq".
Definition
The Steenrod algebra Л is the quotient algebra Γ(Μ)/(λ
Note that Λ inherits a grading from the gradation on M. The elements
of the Steenrod algebra are the polynomials in the Sq', i >0 (Sq° = 1),
coefficients in Z2, and subject to the Adem relations.
Theorem 1
The monomials Sq', as / runs through all admissible sequences, form a
basis for Λ as a Z2-moduIe.
The fact that the Sq' are linearly independent (for / admissible) follows
from the linear independence of the elements Sq'(i„)e H*(Z2,n; Z2) for
all admissible / of degree <n, which is the assertion of Corollary 2 of
Chapter 3.

THE STEENROD ALGEBRA
47
The other part of the proof uses the Adem relations and a reduction
argument. For any sequence /= {/',,.. .,ir} (not necessarily admissible),
define the moment m(I) by the formula m(I) = £s. iss. It is not hard to see
that if / is not admissible, starting at the right and applying the Adem
relation to the first pair /s,/s+, with /, < 2/s+, leads to a sum of monomials
of strictly lower moment than /. Since the moment function is bounded
below, the process terminates and the admissible Sq' actually span Λ.
As an example, Λ-, has as basis Sq1, Sq6Sq\ Sq5Sq2, Sq4Sq2Sq\
The basis given by Theorem 1 is known as the Serre-Cartan basis for Λ.
We will give a quite different basis later in this chapter.
DECOMPOSABLE ELEMENTS
Let A be a connected graded /?-a!gebra, and let A denote the kernel of
the augmentation r; thus A is the ideal of elements of strictly positive
degree. It is called the "augmentation ideal."
Definition
The ideal of decomposable elements of A is the ideal ψ{Α®Α) c A.
The "set of indecomposable elements" of A is the Λ-moduIe Q(A) —
Α'/φ(Α ® A).
The language is somewhat misleading, since elements of Q(A) are not
elements of A but rather cosets. Note that Q(A) is not an algebra, since
it fails to have a unit.
The present definition of decomposable elements is consistent with the
ad hoc definition given in Chapter 4 for Sq'. There it was proved that Sq'
is decomposable if and only if / is not a power of 2. The following result
is now obvious.
Corollary 1
{Sq2'}, /' > 0, generate Λ as an algebra.
We remark that these elements do not generate Λ freely; for example,
Sq2Sq2 = Sq3Sql - - SqlSq2Sq' and Sg'V = 0.
THE DIAGONAL MAP OF Λ
Our next objective is to show that the algebra Λ possesses additional
structure, namely, an algebra homomorphism Λ—> Λ ® Λ. This mapping
will prove to be a powerful tool in the study of Λ.

48
COHOMOLOGY OPERATIONS
As before, let Μ be the graded Z2-module generated in each / > 0 by
Sq'. Define φ: Γ(Μ) —> Г(М) ® Г(М) by the formula
and the requirement that φ be an algebra homomorphism. Thus
φ(ΞςΓ ® Sqs) = φ{$?) ® φ(Ξς5) = (J. S?° ® S?'-) ® (£„ S/ ® S^~*).
Theorem 2
The above ι/' induces an algebra homomorphism φ: A^>A®A.
proof: Let ρ: Γ(Μ)—>./fc be the natural projection. We must show that
the kernel of ρ is contained in the kernel of the above φ. As before, denote
by K„ the «-fold Cartesian product of the space P(oo) = K{Z2,\)· Define
a mapping w: Λ—> H*(Kn;Z2), raising degree by n, by w(0) = θ(ση); that is,
w is evaluation on σ„. By Theorem 2 of Chapter 3 and Theorem 1 above, we
know that w is monomorphic through degree η (degree in A). Let w' be
defined similarly by evaluation on σ2„, and consider the following diagram,
Γ(Μ) p- > A -~^_^^^
A ® A -^®^> H*(Kn) ® H*(Kn) -£* H*(K^7~KS^H*(K2„)
where cohomology is understood to be with Z2 coefficients, and the
isomorphism α follows from the Runneth theorem (Z2 is a field). We wish to
show that the quadrilateral in this diagram is commutative.
Using the Cartan formula, it is clear that the diagram commutes on the
generators Sq\ as in the following computation:
w'p{Sq') = w'(Sq') = Sq'{a2n) = Sq'(an χ <τ„)
= {w® w^{Sq')
Now/?, φ, and α are algebra homomorphisms, but w' is not; so commuta-
tivity in general does not follow immediately from commutativity on Sq'.
However, we may argue as follows. A basis for Mr= Μ ® · · · ® Μ is
given by {Sq'1 ® · · · ® Sqlr}, where /= {/,,.. .,/r} need not be admissible.
Abbreviating such elements by Sq'®, we have
w'piSq1®) = w\Sql) = Sq'(<r2n) = Sq" · · -Sq'-' £s Sq\an) χ Sq^~\an)
Inspection shows that cc(w ® \ν)φ^1®) has the same formula, and this
completes the proof that the diagram is commutative.

THE STEENROD ALGEBRA
49
Now suppose p(z) = 0, ζ e T(M). Then 0 = w'p(z) = a(w ® νι>)φ(ζ). But
α is isomorphic, and (w ® w) is monomorphic in a certain range; thus, for
any z, we choose η sufficiently large (with respect to deg z) so that (w ® w)
is monomorphic when restricted to elements of degree = deg z. Thus
φ(ζ) = 0 and (kerp) с (ker ι/*). This completes the proof of the theorem.
HOPF ALGEBRAS
Let A be a connected graded Λ-aIgebra (with unit); the existence of the
unit can be expressed by the fact that, in the following diagram, both
compositions A^>A are the identity map, where η, the "co-augmentation," is
the inverse of the isomorphism ε\Α0: /ί0—> Λ.
A®R
A®A-^ A
/n® ι
R®A
Now let A be a graded /?-module with a given /?-homomorphism
ε: A—> R. We say that A is a co-algebra (with co-unit) if we are given an
/?-homomorphism φ: Α^Α ® A such that both compositions are the
identity in the dual diagram:
A ® R
A ®A
Α. /ε® 1
R®A
φ is called the "co-multiplication" or "diagonal map."
Now let A be a connected graded /?-a!gebra with augmentation ε. Suppose
also that A has a co-algebra structure with co-unit ε, and that the diagonal
map φ: A—>A ® A is a homomorphism of /?-a!gebras. Then we say Λ is
a (connected) Hopf algebra.
As an example, let Xbe a connected topological group, with m: Χ χ Χ—>
A'the group multiplication map and Δ: .Υ—> Χ χ Xthe diagonal map. Let
Fbe any field. Then II*(X;F) is a Hopf algebra with multiplication m* and
diagonal map Δ*. Also, H*(X\F) is a Hopf algebra, with multiplication Δ*
and diagonal map m*. Since F is a field, H*(X;F) is the vector-space dual

50
COHOMOLOGY OPERATIONS
to H*(X,F), and in the duality, m* corresponds to m* and Δ* to Δ*. In a
moment we will discuss such duality in the general setting.
At the beginning of this chapter we gave the conditions for the
multiplication φ to be associative or commutative. Dualizing, ψ is said to be
associative if the following diagram is commutative:
A * >■ A ®A
ψ\ U ® ι
A ®A -11** A ®A ®A
while ф is commutative \f Τ с ψ = ψ: Α^> Α ® A, that is, if the following
diagram is commutative:
A ®A
V
A\ Γ
φ ч. ψ
А ®А
We may interpret Theorem 2 as asserting that the Steenrod algebra A is a
Hopf algebra, since the diagonal map ψ given in Theorem 2 has the
required properties. The multiplication of A is associative but not
commutative. However, its co-multiplication is both associative and commutative;
we leave the easy verification to the reader. (Since ψ is an algebra homo-
morphism, it suffices to check on the generators.)
THE DUAL OF THE STEENROD ALGEBRA
Now assume that R is a field, and let A be a (connected) Hopf algebra
over R, of finite type (that is, each A{ is finite-dimensional over R). We
define the dual Hopf algebra to A, A*, by setting (A*) ^~ (A;)*, that is, the
dual to Α ι as a vector space over R. The multiplication φ in A gives the
diagonal map <p* of A*, and the diagonal map ψ of A gives the
multiplication ψ* of A*. It is easy to verify that A* is a Hopf algebra. Moreover, φ*
is associative if and only if φ is associative; and similarly for φ*
commutative, and for ф* associative or commutative. We note that A and A* are
isomorphic as /?-moduIes, but certainly not as algebras in general.
Let A* denote the dual to the Steenrod algebra. Then A* is a Hopf
algebra with associative diagonal map φ* and with associative and
commutative multiplication ψ*. It will turn out that ψ* is particularly simple: we
will prove that as an algebra A* is a polynomial ring.
For convenience we now consider infinite sequences of integers. Let Λ
denote the set of all infinite sequences of non-negative integers with only

THE STEENROD ALGEBRA
51
finitely many non-zero entries. Such a sequence will be called admissible
if it consists of an admissible sequence in our previous sense, followed
by all zeroes. Precisely, /~ {/,,/2. ..,/„...} is admissible if, for some
r > 0, ir > 0, /', > 2/„_, for 1 <q <r, and /s --= 0 for s > r. Let 3 denote the
subset of Л consisting of all admissible sequences.
For each integer к > 0, let Jk be the admissible sequence
{2*-1,.. .,2,1,0,. · ·} (/° denotes the zero sequence).
Lemma 1
Let л- denote the generator of Hl(Z2,\; Z2). Then Sq'(x) = x2" if / = /*,
and Sq'(x) ^- 0 for all other non-zero admissible sequences /. Further, for
any non-zero / (admissible or not), Sq'(x) = 0 unless / is obtained from
some /* by interspersion of zeros.
This lemma follows immediately from the observation that Sq(x2J) =
(Sq X)2J = (x + x2)2i = x2i + x2J + ' = Sq°x2J + Sq21x2i.
Definition
For each />0, let ξ, be the element of --t^-, characterized by
Ϊί0)(χ)2' = 0(x) e H2'(Z2,l; Z2) for all 0 e Λ2,_, (ξ0 is the unit of Λ*).
It will sometimes be convenient to write <£,,0> for £,((?).
Proposition Ί
Let / be admissible. For k>\, <^,S^'>=1 if / = /*. Otherwise,
^k,Sq'} = 0. Further, for arbitrary /, <^,S^'> = 0 unless / is obtained
from /* by interspersion of zeros.
This follows immediately from Lemma 1. (We remark that this property
could be taken as the definition of cA.)
Now define a set isomorphism y:3—>Л by y({au.. .,ak,0,.. .}) =
[a, -2a2, a2~2a3,. ..,ak, 0,...}.
For each R e A, we define ξ* e Л* by ξ*--= Y\?= t (й,)'' where /? =
[rx,r2,...}. Note that for / ε 3, the degree of Sq' is exactly the degree of
We order the sequences of 3 lexicographically from the right. Thus, for
example, {8,4,2,0,. .. } > {9,4,1,0,. . . } > {9,4,0,. . .} > {17,3,0,.. . }.
Theorem 3
For/,7e3, <£',7,,V>~0, /<J; if / — У, <<rl7l,V> = 1.
We will prove this by a downward induction. It is trivial for J ^ {0,...}.
let J= {a,,. . .,ak,0. ..} and /={&,, bk,0,...} where, assuming
J > Λ we have ak > bk ^ 0, ak > 1. Put
J'={al-2k-\a2 -?-2,...,ак-\,0,...}

52
COHOMOLOGY OPERATIONS
Then ξ"υ) = ξ,ΗΓ)ξ* since y{J') = y{J) except in the kth place. Calculating,
we have
<i»(",s?'> = <ίϊ(/''ί»,ν> = <Ψ*(£ν(/Ί ® W, V>
Using the definition of the diagonal map φ in Л,
<iy</), V> = <f'"') ® ί». Σ V ® s</'2>
= £<iy('W><i»,S7,i>
where the summation is over sequences /,,/2 (not necessarily admissible)
such that /, -\-[2 = I (in the sense of termwise addition).
Now if bk = 0, 12 has 0 in the k\\\ place and thereafter; hence, by the
above proposition, ^k,Sqll)> = 0. If bk 7^0, we see that the only nonzero
term in the above summation occurs for /2 = /*. Thus (,uy(J\Sq'} =
(ξη1Ί,Sqv)> where /' = /—/*; for this is the only non-zero term in the
above summation.
Descent on bk and к completes the proof.
Corollary 2
As an algebra, Λ* is the polynomial ring over Z2 generated by the
{ί,}(/>1).
proof: The admissible sequences give a vector-space basis for Л. As
J runs through 3, ξνυ) runs through all the monomials in the ξ;. But the
above theorem shows that the ξ'/υ) (J admissible) form a vector-space
basis for Л*. However, a polynomial ring is characterized by the fact that
the monomials in the generators form a vector-space basis. This proves the
corollary.
ALGEBRAS OVER A HOPF ALGEBRA
Let A and Μ be a graded /?-algebra and a graded Λ-moduIe, respectively,
where R is a commutative ring with unit. We say that Μ is a graded
Α-module if we are given an /?-module homomorphism μ: Α® Μ-> Μ
such that μ(1 ®m) = m and μ °(φΛ® Ι) --μ ° (I ® μ): A ® A ® M—+M,
where φΛ is the multiplication in A.
Now suppose that A is a Hopf algebra and that Μ is an Λ-moduIe (as
above) which is also an /?-a!gebra. Then Μ ® Μ has a natural structure
as an Λ-moduIe under the composition
μ': A ®M®M »«"«"„ A®A®M®M-L»
A®M®A ®M -ii^i> Λ/®Μ

THE STEENROD ALGEBRA
53
(where φ is the diagonal map of A). We say that Μ is an algebra over the
Hopfalgebra A if the multiplication <pM: Μ ® M—>A/is a homomorphism
of Λ-moduIes, that is,
A ®M ®M -ul* M®M
1 ® (/>M
Λ ®Λ/
Л/
is a commutative diagram.
As an example, take Л = Z2, Л = yfc, and Μ = H*(X;Z2) where A"" is
any space. Because of the formal relationship between the Cartan formula
and the diagonal map φ of A, we have, for 0 e A and x,y e H*{X;Z2),
0(φΜ(χ ® j>)) = 0(xy) = φΜ(Φ(0Χχ ® v))· Thus the cohomology (with Z2
coefficients) of any space is an algebra over the Hopf algebra A.
As another example, let A be any Hopf algebra, and let β be a graded
Λ-moduIe. Then the tensor algebra T(B) has a natural structure as an
algebra over the Hopf algebra A; we leave the verification to the reader.
THE DIAGONAL MAP OF A*
In what follows we will write Я* and Я* for Η*{Χ;Ζ2) and H*(X;Z2),
respectively. The space X will be assumed to be a complex of finite type.
Thus we have duality: (Я*)* = #*, etc. We have a diagonal map
Δ: Χ—> Χ χ X, and the induced maps Δ*: Я* —>Я* ® Я* and Δ* = (Δ*)*:
Я* ® H*^>H*. Recall that Я* is an algebra over the Hopf algebra A, by
the diagram (1), where the composition along the top row is μ', which
A®H*®H*-!UA®A®H*®H*-U
1 ®Δ*
A ® Я* i! > Я*
A®H*®A®H*
-^-s- Я* ® Я*
0)
makes Я* ® Я* an ^-module.
Define ;.: H*®A->H* by </(.v,fl), j-> = <x, μ(Μ> for ^сЯ*,
χ e Я* - (Я*)*, and Oei.
Proposition 2
λ is a module operation; i.e., the following diagram commutes.
Я* ® A ® Λ AS"> Я* ® Л
ι ® J u
Я* ® .4 -^i-» я*

54
COHOMOLOGY OPERATIONS
proof: The fact that μ is a module operation, μ ο (Ι ® μ) = μ 0 (ψ ® 1),
which can be expressed by a diagram closely related to the above, is the
essential step in the following calculation:
<A о (1 ® <p)(x,0,0'), y> = <λ(χ,φ(0,0')), у>
= (χ, μ(ψ(θ,θ'), у)}
= (Χ, μ(0,μ(θ^))>
= (λ(χ,θ), μ(θ',γ)}
- <λ(λ(χ,θ),0'), у}
= <Αο(Α® 1)(χ,0,θ'), у>
Proposition 3
The following diagram commutes:
Я* ® #* ® Л ® Л -U Я* ® Л ® Я* ® Л Л0"> Я* ® Я*
Δ. ® ψ] |Δ.
Я* ® Л 4 ► Я*
proof: This diagram is related to diagram (1) as the diagram for λ is to
that for μ (in Proposition 2), and diagram (I) plays a similar essential role
in the proof, which proceeds by the following calculation, where we write
Δ*χ = Σ x\ ® x" and φ(θ) = ^0)® θ]:
<Δ* ο λ(χ,0), y®y'}-~ (λ(χ,θ), A*(y ® /)>
=-<.Υ,μ(0,Δ*Ο'®/))>
= (χ, Α* ο (μ ® μ) ν Τ α ф(9,у,у')} by diagram (1)
= <Δ*χ, (μ ®μ)(Σ 0'i®y® 0} ®/)>
- <Χ Χ! ® Χ}, Χ μ(θ'^) ® /i(0},/)>
= ΣΣ <-νί, K0'j,y)Xxi, μ(0'},/)>
= H<Kx't,Oj),yXKx'i,(rj),y'>
-= <(;. ®;.) ΣΣ (*;· ®°)® *.- ® °'}1 у® У У
-= <(;. ®;.) ο τ ο (δ* ® ^)Cv,y), ^ ® у >
Definition
Let Я* be the dual of Л; ;.*: H*^>(H* ® Л)* = Я* ® ./t* where we write
® as a reminder that infinite sums are allowed.
Proposition 4
λ* is an algebra homomorphism.
The proof consists of dualizing the diagram of Proposition 3.

THE STEENROD ALGEBRA
55
Proposition 5
The following diagram is commutative.
#* ® A* ® Λ* <л*®' Я* ® -4*
ι ® v>*| ΐι*
я* ® л* < ϋ h*
This is the dual of Proposition 2.
Proposition 6
For j e #*, the following formulas are equivalent:
1. /*{y) = Σ У; ® Wr 0>,- e #*, wf e Л*; sum possibly infinite)
2. /i(0,j;) = X <0,iv,>j>, for all Ов Л
I'roof: Assume (1). For χ e Я*,
<x, ^(o,^)> = <я(х,о), yy = <x ® o, ;.*^>
= <χ ® ο, Σ у, ® w,>
= Σ (здХ^г)
= <x, Σ <0,и>,->^>
which implies (2) by the duality of #* and #*. Conversely, assuming (2),
<x ® Θ, Х*уУ = <x, μ(0,γ)>
= <x, Σ <0,и>(>^>
= Σ <x,y;Xe,W;y
= <х®0,5>.®н>;>
which implies (1).
Proposition 7
Let χ denote the generator of #'(Z2,1: Z2). Then Я*(х) = £lS, 0x2' ® £,.
proof: By Proposition 6, we must show μ(Ξη',χ) = Σ iiuSq'^x1', and
it is enough to check for admissible I. Now (ξ;,Ξη'} = 1 when 7 = 7' =
{2'~1,2,'~2,.. .,2,1,0,. ..} and has value 0 otherwise. This means that
Σι (ZhSc/'yx2' = 0 except for the particular /above, for which the
summation becomes just x1'. But on the other hand μ(Ξη',χ) = Sq'x, and the
theorem follows immediately, by Lemma 1.
Corollary 3
proof: From Propositions 4 and 7,
λ*(χ>) = (λ*χ)2' = (Σ x2J ® ξι)2' = £/*2' ® ξ})2' = Σ; x2'*' ® (£;)2'

56
COHOMOLOGY OPERATIONS
Theorem 4
The diagonal map ψ* of A* is given by the formula
Ρ*(ί») = ΣΪ=ο(ί»-ι)2'®ίι
proof: Using the above formulas, we have
(i ® φ*)λ*{χ) = (i ® <р*)&кх1к ® ξ,) = Σ*χ2" ® φ*(ξ>)
and also
(я* ® i)A*w = α* ® ΐ)(Σ.·*2' ® ίι) = ΣΣ χ2'*' ® f2< ® ί.·
By Proposition 5, these expressions are equal. The theorem follows.
THE MILNOR BASIS FOR A
Recall that {ξκ}, R e Л, forms a basis for A*. The dual basis, whose
elements are denoted {SqR}, is called the Milnor basis for the Steenrod
algebra A. By definition, <£R,V>=1 if /? = /?' and <£R,V> = 0
otherwise.
This is a different basis entirely from the Serre-Cartan basis of
admissible sequences; some caution is necessary to distinguish, for example, the
element Sq{2A'°" ' of the Milnor basis from the element Sq2Sql of the
Serre-Cartan basis, especially since one tends in the course of long
calculations to abbreviate both of these elements (which do not even have the same
dimension) to Sq1'1. However, we do have the following compatibility.
Proposition 8
Sqil·0· » = Sq'.
proof: By Theorem 3, for I,J admissible, (£'u>,Sq'} has value 0 for
/ < J and is equal to 1 for / = J. Observe that for R = {/,0,. ..}, y(R) = R.
Then ξηΚ) = ξκ, and so clearly ^R,Sq") = 1. If, on the other hand, J is
another sequence of the same degree as R, then J > R, so that (ξr(J\Sqi} =
0. Thus Sq( is dual to ξκ and thus Sq' = SqR, R = {/,0,...}, as was to be
proved.
We illustrate with some further remarks showing the usefulness of the
dual A* in the analysis of the Steenrod algebra A. For / > 0, let J, denote
the two-sided ideal of A* generated by
{(ί.)2',(ί2)2'",· ··,(£,·)2'"'*',· · -,(ξ,)2; ί,+„ί,+2,. · ·}
Define С, as the quotient A*/J,. As an algebra, C, is the polynomial ring
in ξι,.,.,ξ, truncated by the relations (ξ-У"' + ' = 0 (!</</)· C, is

THE STEENROD ALGEBRA
57
clearly finite-dimensional. Now <р*(Уг) c J, ® A*, since
^(ω2,"*+, = Σ«(ί»-«)2'"*+,+,®(ί«)2'"*+'
which lies in J, ® A*. Therefore ψ* defines a map C,—> C, ® C(. Thus C, is
actually a quotient Hopf algebra of A*.
Note also that J, is maximal among ideals У of Λ* having the following
properties:
1. ξ\ e У if and only if i> 2'
2. <р*(У) су® л*, and hence A*/J is a quotient Hopf algebra of A*
Let Л, be the dual of C,. Then A, is a finite-dimensional subalgebra of A.
Clearly Sq' e A, if and only if (£,)' is not in У,. Thus every Sq' is contained
in a finite-dimensional subalgebra of A It follows that every Sq' is nil-
potent. In fact A, is exactly the subalgebra of A generated by {Sq1: i < 2'}.
Indeed, let B, be the subalgebra of A generated by the Sq', i < 2'. From
the formula for the diagonal map of A, B, is clearly a sub Hopf algebra of
A. Surely β, c A,. But by duality sub Hopf algebras of A correspond to
quotient Hopf algebras of A*, which in turn are quotients of A* by ideals
satisfying property (2) above. The result follows by maximality of У, with
respect to properties (1) and (2).
By refinement of the proof of Theorem 1 of Chapter 4, we can easily
verify that A, can also be described as the subalgebra of A generated by
{Sq2': i <t— 1}. In recent literature the notation has shifted and this
subalgebra is often denoted A,_x.
Note finally that SqR e A, if and only if ξκ is not тУ„ that is, if and only
if we have simultaneously /·,< 2', r2 < 2'~ ',...,/·, < 2, and rk = 0 for all
k> t. Thus for every R there exists some / sufficiently large such that SqR
is in A,. Thus every element of A is nilpotent (except of course 5^° = 1).
DISCUSSION
What we have done in this chapter is to construct an algebraic system
with the properties of the Steenrod squaring operations built in. The Adem
relations are imposed in the definition of the Steenrod algebra A. The Cartan
formula appears in more subtle fashion. Formally, the Cartan formula
suggests the diagonal map which makes A into a Hopf algebra; however,
the force of the Cartan formula is that the cohomology ring, under cup
product, of a space is then an algebra over the Hopf algebra A.

58
COHOMOLOGY OPERATIONS
The algebra Λ is of course very complicated. The calculation of the
Serre-Cartan basis was an important step but left many questions about Λ
open. Milnor's study of Λ, based on the dual algebra, yielded a new basis
and many new results. However, work in homotopy theory, especially by
Adams, has shown that much more deep algebraic information about Λ
will have geometrical application. In Chapter IS especially we will
illustrate this; let it suffice now to say that Λ remains an important area of
study.
EXERCISES
1. Using the Serre-Cartan basis and the Adem relations, calculate the sub-
algebra of Λ generated (as an algebra) by Sq°, Sq1, and Sq2.
2. Write the elements in Exercise 1 in terms of the Milnor basis.
REFERENCES
General
1. J. Milnor and J. С Moore [1].
2. N. E. Steenrod and
D. B. A. Epstein [1].
The structure of the Steenrod algebra
and the dual algebra
1. J. F. Adams [1].
2. H. Cartan [6].
3. J. Milnor [2].

CHAPTER /
EXACT COUPLES AND
SPECTRAL SEQUENCES
In Chapters 2 through 6 we have considered an explicit family of coho-
mology operations, namely, the Steenrod squares. As we indicated in
Chapter 1, we will also wish to make some explicit computations of
Η*(π,η; G). For this purpose, and others, we stop to introduce some
algebraic machinery—that of the spectral sequence. Our exposition is based
on the notion of exact couple.
EXACT COUPLES
An exact couple consists of a pair of modules D,E together with homo-
morphisms i,j,k such that the following triangular diagram is exact at
D-UD
\ /
Ε
each vertex. If we write d = jk:E^E, then d is a differential on E, since
dd=j(kj)k-0.
Given an exact couple as above, we define the derived couple as follows:
D'-^D'
E'
59

60
COHOMOLOGY OPERATIONS
D' = iD^ D; E'= H(E,d); i' = i\D'\ j' = j° /'"'; that is, / takes i(x)
into the i/-homology class [j(x)], χ e D, and &' takes [y] into &(j>), ;e£
In order that this make sense, certain verifications are necessary: that7'(x)
is a cycle and that 7 "(ζ), ζ e D', depends only on z; that k(y) depends only
on [y] and is in the submodule D' of D. We leave these verifications to the
reader, along with the proof of the following proposition, which consists
of six easy exercises in diagram-chasing.
Proposition 1
The derived couple is exact.
We may therefore repeat the process of derivation, obtaining an infinite
sequence of exact couples. The nth derived couple is
D"J1+ D"
E"
where D" is clearly a submodule of the original D and /'" = /1 D". The
description of E", j", and k" is more complicated. E" is a subquotient of
E"~\ namely, H(E" ~ V ~') where dr = j'k'. Let us consider d". First of all,
dx = d' — j'k': E'^>E' takes the i/-homology class [y] into the class [j(x)]
where χ satisfies i(x) = k(y). (Such χ exists because у is a cycle, i/(j) = 0;
hence k(y) e kery= im /.) Now d2 is defined on E1 = H{E',d')\ if d'[y] = 0,
[j] e £' = £", then i/2 =/A:2 takes {y] = [[y]] e E2 into [/(x)] e £2 where
χ e D' satisfies i'{x) = k'[y] (such χ exists because d'[y] = 0). Thus i/2[j]j =
[j'(i')~lk'[y]] where it is defined. If we pursue this analysis, we find that
d3 is defined when d1 and d2 are zero and is equal to/(/")"'('")" lk'; and in
general d" is defined when d\...,d"~l are all zero and is equal to
j'{i'Y("~l)k'. (In the last expression, the superscript denotes an iteration,
and should not be confused with the indexing superscripts, as in /".)
When we are given an exact couple (D,E; i,j,k) and form the sequence
of derived couples (D",En; i",j",kn), each E" is a differential group with
differential d"=j"k" and £"+1 = H{En,d"). The sequence (E",dn) is called
the spectral sequence of the exact couple; in general, the term "spectral
sequence" denotes a sequence of differential groups with the property
E"+i = H(En,d").
THE BOCKSTEIN EXACT COUPLE
An important example is the Bockstein exact couple, where /' is induced
by multiplication by 2 inZ;7' is induced by the reduction homomorphism

EXACT COUPLES AND SPECTRAL SEQUENCES
61
£>' = Я*( ;Z)^UH*{ ;Z)
£' = Я*( ;Z2)
ρ: Ζ —>Z2; and kl is the Bockstein homomorphism β (see Chapter 3). The
differential d=jxkx of this exact couple is just the Bockstein
homomorphism δ2 (see Chapter 3). We will take the above exact couple as the first in
a sequence and consider the derived couple E2, etc. For later convenience
we will use subscripts (rather than superscripts) to index the successive
Bockstein differentials dr; thus dx ~ d as above; d2 =j2k2: £2—>£2; etc.
The operation dr acts essentially as follows: take a cocycle in Z2
coefficients; represent it by an integral cocycle; take its coboundary; divide by
2r (this is possible because dr is defined only on the kernel of i/r_,); and
reduce the coefficients mod 2. Notice that every dr raises dimension by
1 in Я*( ;Z2).
The Bockstein differentials act on cohomology with Z2 coefficients, but
they can be made to give information about integral cohomology. To see
this, let X be a space and recall that
H"{X;G) = Horn (HP(X),G) ® Ext {HP.X{X),G)
by the universal coefficient theorem. Therefore a direct summand Ζ in
HP(X;Z) gives rise to a summand Ζ in HP(X,Z) and to a summand Z2
in H"(X;Z2). A summand Z2„ in H„(X;Z) gives rise to a Z2„ in Яр+ l(X;Z)
and to summands Z2 in HP(X;Z2) and Hp+i(X;Z2). The following
proposition is now self-evident.
Proposition 2
Elements of H*(X;Z2) which come from free integral classes lie in
ker dr for every r and not in im dr for any r. If ζ generates a cyclic direct
summand of order 2r in Ял+ l(X;Z), then there exist cyclic direct summands
of order 2 in H"(X;Z2) and H" + 1(X;Z2), generated, say, by z' and z",
respectively; d,(z') and d^z") are zero for / < r, and dr(z') = z". (Implicitly,
z' and z" are also not in im dt for / < /·.)
We say that the image by ρ of the free subgroup of H*(X;Z) " persists
to £°°" and that z' and z" "persist to E' but not to £r + 1."
The following application will be important in our computations of the
homotopy groups of spheres.
Corollary I
Suppose that for some X, H'(X;Z2) = 0, i<n, and H"(X;Z2) = Z2,
generated, say, by z. Then we can infer H"(X;Z), except for torsion in

62
COHOMOLOGY OPERATIONS
odd primes, as follows: H"(X;Z) = Ζ if drz = 0 for all r; and H"(X;Z) =
Z2n if dtz = 0, / < n, and i/„z -£ 0.
THE SPECTRAL SEQUENCE OF A FILTERED COMPLEX
We turn now to a detailed study of another example: the exact couple
of a filtered complex, and the associated spectral sequence.
Let (K,d) be a chain complex. Let {Kp} be a filtration of K, that is, a
family of subcomplexes of К such that Kp с Kp+l for every p. Later we
will assume that K" = 0 for ρ < 0 and that U„ ^" = #> but for the present
these hypotheses are not necessary.
From the short exact sequence
0 -> Kp-' -»л:р-> л^/л:"- ' ->o
we obtain a long exact sequence in homology,
• · --UH^K'-^-U Нри{Кр)-иНри{К-1Кр->)-+
#,+,_,(*'-)-*··· (1)
where the coefficient group, understood to be fixed, is suppressed from the
notation. We thus obtain a bigraded exact couple by taking
D„,q = Hp+q(K') Ep,q = Яр+,(*'/*'-)
DP.,->Z?P+,,,_,: Hp+q(K^Hp+q(K'+l)
Dp,q^Ep,q: Η,^Κ^Η,^ΚηΚ'-*)
Jp.q
k„„
so that the portion of the long exact sequence (1) displayed above becomes,
in the notation of the exact couple,
. . . *Р.Ч+ ' П "Ρ" ! Ч* ' . Г) Jp.q . /Г *Р.Ч , Г) 'р~ 1,0 . . . .
■*■ υρ- 1 .<)+ 1 * Up,q * ^p.q * Up-\.q *
p,q —p.q
Finally, the differential of the exact couple is
d=jk:Ep,q->E,,-l,q:Hp+q(K'IK'-i)^Hp+q-l(K'-i/K>-2)
In this exact couple, the groups D and Ε are bigraded; the bidegrees of
the maps i,j,k,dare implicit in the representations given above. In Figure 1
we display a portion of the diagram in which all these groups and maps
appear. The reader should trace the long exact sequence (1) through this
diagram and should also see how the differential d appears there.

I
s-
51
+
)
Ct.
δ!
* tq
Ct.
+
Ct.
5;
ι Q
ί Τ
il
-^
Ct.
к
о-
+
ь,
*
•ί
^-~-
ι
ft.
^
+ -
a.
1
^
<ϊ"~
^
*
+
. о.
5;
ι
О-
* —·
+ s~^
* d.
: s<
.?■ "^
—»■ +
o,
tq
+
+
1
^
a.
+
&
+
Ct.
a:
t
^
*Λ
wty
t II
+ s_^
Ct.
a:
t
^
f4
+
Si
^^
+
o·
+
Ct,
a:
ί
во
ε
5:
t

64
COHOMOLOGY OPERATIONS
Now attach a superscript 1 to D,E,i,j,k,d in the above, and consider the
derived couple. We have
Z?^ = im ι:Z?'_li<+!->/?;,,
ker dx · Fl > F1
^P.q ПУЪрл," ) — ■ ,| . pl pi
1111 Up+ \,q ■ ^p+ \.q * t-'p.q
As before, the maps /2, etc., are given subscripts to agree with those of the
domain; thus
ι2 · Л2 __> D2
'p.q- UPA * Up+\,q~\
and so forth. The reader should verify that in the «th derived couple the
bidegrees of the maps are as follows:
MAP
i"
r
k"
d"=j"k"
BIDEGREE
O.-i)
(-Я + 1.Я-1)
(-1,0)
{—n,n— 1)
Observe how d' appears in Figure 1. In the first place, dp\q goes from the
group EPt4 (in the center of Figure 1) two steps to the right, by к and then j.
If an element is in the kernel of d\ d\x) = 0(xeEp\q), then k(x) is in the
image of the appropriate /, by exactness of (1); and d2(x) is essentially the
class obtained by taking χ " right one, up one, right one." We have remarked
before that d2 is essentially j1(il)~lkl. In general dr+i is essentially j(i)~rk,
which on the diagram means "right one, up r, right one"—this making
sense if d' vanishes.
We have the following formula for E'p.r
Proposition 3
Erpq is isomorphic to the quotient group
im :Ηρ,4(κηκ·-^^ΗιΗ4(κηκ·'-ί)
im д: Hp+q + l(Kp+r- '/*>)-> Ηρ,4{ΚηΚ'-<)
For convenience we shall abbreviate this quotient to "/ί/β." The map д
which is used in В is the boundary homomorphism of the exact homology
sequence of the triple {Kp+r~\Kp,Kp~l), while the map of which A is the
image is one of the maps in the exact sequence of the triple (KP,KP~ \KP~').
Note that, by exactness of this last sequence, A = ker c: Hp+q{KpIKp~l)^>
Hp + q-l(Kp~1/Kp~r), where д is the boundary homomorphism of the triple
in question.

EXACT COUPLES AND SPECTRAL SEQUENCES
65
proof: If r— 1, the proposition reduces to a triviality. If r = 2, we have
A = ker 8: HpJr„{KpIKp-l)^HpJr4.l{Kp-'IKp-2), but this map is exactly
the map dp\q; moreover, В is just im dp+,.,; thus the proposition is obvious
in this case also.
Now suppose r>2. We seek an isomorphism /: A/5—> ££.,. We first
note that there is a natural homomorphism /:Л—>£,"'. To see this,
suppose χ e A (= ker d as above). This means that χ is an element of
Hp+q(Kp/Kp-l) = Ep,q such that ад = ОеЯ,+1.,(Г'^г'). Thus
*„,,(*) is in the image of /"-": Hp + q.l(Kp-r)^Hp+q.l(Kp-1) (use
Figure 1, where these groups and maps are displayed). Then i~(r~2)k(.x)
is in im /'= ker j, so that d'~\x) — 0. Thus we have shown not only that
χ projects to E'p~q' but that it projects into ker dp~'. We therefore have a
homomorphism/: Λ —>Zrp~' where
71·-1 _ bpr jr-l . c-r-l EV-1
In order to show that/induces a map from /1/5 to E'PA, it is now sufficient
to show that/(5) <= я;-' where
Dr-I :m Jr-l . rr - 1 ^ pr - 1
Cp.4 im Up+r-l,q-r+ 2- I-'p + r-l,q-r+ 2 * ^p,q
But this is clear; for, if χ e 5, then χ = £(j>) for some
ysHp+q+i(Kp+r-l/Kp)
Let / be the image of у in Hp+q+i(Kp+r-1/Kp+r-2)—the latter group
appears in the lower left corner of Figure 1; then/(x) = d'~\y').
Thus we have a natural homomorphism
/:AjB^E;,q = z;-ljBp-1
To see that/is onto, let χ be an arbitrary element of Zp~l. Choose
χ e Ep\q such that x' projects onto x. Then k(x') = i(r~ 2)(y) for some
ye Hp+q_i(Kp~r + i) and we can assume j(y) = 0. Buty(j>) = 0 implies
у e im /, or k(x') e im(/Cr_ 1J), so that χ'ε A = ker 3: Hp+q(K"/Kp-l)^>
Hp+q-l(Kp-iJK"-r) and/(x') = x. This proves that/is onto.
To see that / is 1-1, suppose f(x) e Brp~ql; we must show χ e B. Now
x = d'~ \y) for some у e Hp+q + 1(X',+r" l/Kp+r-2). Since A:(j) is in the image
of /(r_2), if we consider the triple (Kp+r~l,Kp+r-2,Kp), we have d(» = 0 in
Hp+q(Kp+r-2/Kp), and so there exists ζ e Hp+q+l(Kp+r-l/Kp), which goes
to j in the exact sequence of that triple. Denoting by w the image of ζ in
HpXq{Kp\ we have /(r"2>(iv) = %); hence x = d'-\y)=j{w) = o(z), so
that χ e 5.
This completes the proof of Proposition 3.

66
COHOMOLOGY OPERATIONS
Definition
Fp,q = im:Hp+q(Kp)^Hp+q(K).
Definition
^Р,Ч = * P,ql* p- 1,4+1-
The second definition makes sense because Fp_Uq+l = im: Hp+q(Kp~1)^>
Hp+q(K) and this latter map can be factored through HP+Q(KP).
The Fp-q form a filtration of Н*{К) and the Ep\q form the associated
graded group. There is no a priori connection between E' and £x, but the
notation is suggestive, and the following formula, analogous to
Proposition 3, reinforces the suggestion.
Proposition 4
E£q is isomorphic to the quotient group
im:Hp+q(Kp)^Hp+q(Kp/Kp~l)
im d: Ηρ + η + ι(Κ/Κρ)^Ηρ+η(Κ^Κ"-1)
where the maps in question come from the homology exact sequences of
the pair (Kp,Kp~l) and the triple (K,KP',KP~"'), respectively.
We shall abbreviate this quotient as "C/D".
The proof follows immediately from the diagram below and a standard
argument (see Exercise 1).
/WO , '^Hp+q+i(KjKp)
:Hp+q(Kp):
Hp+q(K)^ ^Hp+q{KpIKp-')
Our intention is to use the spectral sequence to get information about
H*(K) from knowledge of the subcomplexes {Kp}. In order to press this
program, we now consider the following convergence conditions on the
subcomplexes {Kp}:
1. K" = 0 for all ρ < 0
2. El4 - Hp+4{KpIKp-') = 0 for all 9 < 0
3. K=\JPK>
It follows from (1) that Ep\q = 0 for all ρ <0, and thus it follows from
(1) and (2) that E'p-q = 0 for all r whenever ρ < 0 or q < 0. The next two
results show how much more we can deduce from conditions (1) to (3).
Proposition 5
If (I) and (2) hold, then £; = E'p*„' for any r > max {p, q~\).

EXACT COUPLES AND SPECTRAL SEQUENCES
67
The proof is easy: by definition, E'p*q' is the homology of Е'рл with respect
to d' as follows:
f-'p* r,q-r+ 1 * ^p.q * ^p- r.qXr- 1
but when r > max (p, q + 1), the two groups at the ends of this sequence
are both zero, as a consequence of (1) and (2).
Theorem 1
If convergence conditions (1) to (3) hold and r > max (p, q + 1), then
£p.4 «a £*,. Thus the convergence conditions imply that, for each (p,q),
the groups £p„ become stable with sufficiently large r and the stable group
is exactly E^.
proof : Recall from Propositions 3 and 4 that Е"м «а А/В and E*q «a C/D.
(For the definitions of А, В, С, D, refer to Propositions 3 and 4.)
We first show that В — D, provided r > q + 1. In the diagram below,
the row is exact, and the last group in the row is £j+(J„_r+1 = 0 by
condition (2), so that φ is onto and thus im о = im o'. Under hypothesis (3), it
follows that В = D.
Ur>p, then (1) implies that Hp^q{KpIKp-') = HpU{K"). Thus we have
A = C. Therefore when r > ρ and r >q + 1, we have A/B= C/D, and the
theorem is proved.
Recall that Ep\= FpJFp.Kll+l where Fp,,=--im: Hp+q(K>)^Hp+q(K).
Assuming the convergence conditions (1) to (3), we have Ep;q =- 0 for ρ < 0
or q < 0. Therefore the following series is finite:
H„{K) = Fn.0 = F,.,, = - = F1>n_, = F0,„
and the successive quotients are the groups £,>-,·. This composition series
gives us considerable information on H„(K); we say " the spectral sequence
converges to Η^(Κ)."" Note for fixed n, only finitely many computations
are needed to obtain a composition series for H„(K).
KXAMPLE: THE HOMOLOGY OF A CELL COMPLEX
As an application, let К be the singular chain complex of a cell complex
(also denoted by K) and Kp be the singular chain complex of the ^-skeleton;
W* will denote singular homology (with some fixed coefficient group).

68
COHOMOLOGY OPERATIONS
The convergence conditions (1) to (3) are clearly satisfied; moreover we
have a strong form of condition (2), since Epq = Hp+q(KpJKp~1) = 0 for
all q Φ 0. It follows from this last observation that Е'рл = 0 for q Φ 0,
and Ерл likewise. The composition series collapses, and we have
H„(K) — F„i0 = £„_„
Also, dr must be identically zero for r > I, since it changes the second
grading; thus H„(K) = E„>0 = E„\0. From the definitions we easily see that
Hn(K) is nothing but the homology at Hp{KPIKp~l) in the following long
sequence (not exact),
>нр+1(кр+1/кр)-^н„(кр/кр-*)-!инр-1(кр-*/кр-2)^...
— pi — F1 — Fl
where each dl is the boundary homomorphism of a certain triple of the
form {KP,KP'\KP~2). Thus we have proved the familiar result that the
singular homology of the cell complex A^may be computed as the homology
of a complex C{K), in which CP{K) = Ηρ{Κρ\Κρ-χ\
DOUBLE COMPLEXES
An important source of filtered chain complexes is the double complex.
A double complex is a doubly indexed family {Cp,} of abelian groups, with
two differentials,
· *~p,q >^p-l,«s " · ^P,q >^P,«-1
such that </'</'= 0, d"d" = 0, and d'd" ^d"d' = 0. We also assume that
Срл = 0 for ρ < 0 or q < 0.
An example of a double complex is given by setting Cp,q = Ap® Bq,
where (A,dA) and (B,dB) are given chain complexes; we put d' = dA® \B,
d" = (-\)p\A®dB:Ap®Bq^Ap®Bll.l.
A double complex (Cpq,d\d") gives rise to a chain complex (C„,d) if we
set C„ = ΣΡ+,=η Cpq and d = d' + d". There are two obvious filtrations of
the complex (C„,d):
1 TP = V- Γ
JSP
Τ "(~q _ V- /~
z·· *"n — L*p + k=n *~p,k
k <q
Each of these filtrations gives rise to a spectral sequence. For instance,
in (1), one has '£„',, = tfp+„('С/'С-')· Here {'C'/'C'-\ = CP,K-P or,

EXACT COUPLES AND SPECTRAL SEQUENCES
69
equivalently, ('Cp/'C~1)p+q = Срл. One verifies that the above homology
group Έ1 is computed by means of d" and that the differential dl is induced
by d'. Thus, in a rough but natural notation, we may write ΈΡΛ = HpHq'(C).
In (2) the situation is similar but the roles of the two indices are
interchanged. We write "Егрл = Hq'H'p(Q.
Each filtration obviously satisfies the convergence conditions, and
therefore each spectral sequence converges to the same thing, namely, H*(C).
We illustrate these ideas with a theorem in homological algebra. We
recall the definition of Tor (A,B) for abelian groups A,B. In the first place,
we may define 'Tor (A,B) by taking a free abelian resolution
0->Л'->Г'->Л->0
of A; 'Tor (A,B) is characterized by the exact sequence
0—>'Tor (/1,Я)—>/?'®Я—>F'® 5—>Л®Я—>0
In the second place, we may define "Tor (A,B) by a resolution of В
0->Л"->Г'->Я->0
and the requirement that the following be exact:
0 -> "Tor (A,B) ->Л ® /?"->Л ® Г"->Л ® В -> 0
From either definition, it follows that Tor (A,B) = 0 if either A or B is
free.
Proposition 6
'Tor = "Tor.
proof: Let X be the chain complex with X{ — R', X0 = F, Xt = 0
otherwise. Let У be the complex У, = R", Y0 = F", Y, = 0 otherwise. Let
С be the double complex X ® Y. Then
[C„.q]
F'®R" R'®R"
F' ® F" R' ® F"
We first compute //*(Q from the spectral sequence of the filtration (1).
First,
and finally,
"Tor (F',B) "Tor (R',B)
F'®B R'®B
0 0
F'®B R'®B
MP(Hq'(Q)
0 0
Α® Β 'Ίοτ(Α,Β)

70
COHOMOLOGY OPERATIONS
This is E2 and hence also E". Thus H0(C) = A® 5 and Я,(С) = 'Tor (A,B).
On the other hand, computing from the spectral sequence of (2),
HP{C)
A®R" 'Тог(Л,/?") = 0
A®F" 'Jot{A,F") = Q
"Ίοτ{Α,Β) 0
A® B 0
By this route we obtain H0(C) = A® B, but Hl(C)= "Tor (A,B). Thus
'Tor (A,B) = HAQ = "Tor (A,B).
We have thus also proved that Tor (A,B) is independent of the choice
of free abelian resolution.
DISCUSSION
We have chosen to emphasize the exact couple as the starting point for
the theory of spectral sequences. This point of view, due to Massey, seems
conceptually simpler than earlier methods, which were based directly on
differential filtered groups, or than the more abstract methods of Cartan
and Eilenberg.
On the other hand, while we have developed the spectral sequence of a
filtered complex as an example of a bigraded exact couple, we now note
(with Cartan-Eilenberg) that the results obtained do not really depend on
the existence of the complex K. To construct the exact couple, we need
only a family of groups fitting into the diagram of Figure 1, that is,
playing the roles of Н*{КР) and Η*(ΚΡ+1/ΚΡ). If we have as well groups playing
the roles of #*(#), H*(K/KP), and H*(Kp+r/Kp)—that is, fitting into the
relevant exact sequences—and satisfying suitable convergence conditions,
an analogue of Theorem I will apply. The reader will find examples in the
Appendix below and in Chapter 14.
APPENDIX: THE HOMOTOPY EXACT COUPLE
As another example we describe the homotopy exact couple. The results
include a derivation of a certain exact sequence of J. H. C. Whitehead, who
discovered it by other methods. For this example we assume the relative
Hurewicz theorem.

EXACT COUPLES AND SPECTRAL SEQUENCES
71
Let К be а С ^-complex; consider the exact homotopy sequence of the
pair (#",#"-')· By setting
D\A = np+q(Kp) Elq = np+q(Kp,Kp~l)
we obtain the homotopy exact couple, with a diagram analogous to Figure I
of this chapter. (Wc remark first that Ε' is not a homotopy invariant of К and
second that we leave to the reader the definitions of Dpq and Ep\q for small
ρ and q.) The convergence conditions will clearly be satisfied if we have
the relative Hurewicz theorem for A^(so that Ep\q = 0 forg <0); hence we
assume К simply connected. Then this exact couple gives a spectral sequence
which converges to nJJC). Consider the derived couple (which is a homotopy
invariant, but we leave the proof to the reader); a typical portion of this
couple appears as follows:
· · —* c'p+ 1,0 *■ ^p.O * 1Jp+ 1,-1 * ^ρ,Ο —* · · · \l)
It can be shown that £p2,0 «a HP(K). In fact it is clear that £„',0 ^ CP(K);
and one verifies that dl is the same as the homology boundary. Thus the
above sequence (I) contains the homology of K. Moreover, Dp+1_, =
imy: πρ(Κρ)—>πρ(Κρ+1)= πρ{Κ), but j is onto, so that the sequence (1)
also contains the homotopy of K. If we denote by ГД/Q the group
D2pQ = \mj: πρ(Κρ~ι)^>πρ(Κρ), then (I) becomes the exact sequence of
J. H. С Whitehead:
• · · ^Ηρ+ι(Κ)^Γρ(Κ)^np(K)^Hp(K)^ ■ ■ ■
As a corollary, suppose К is (n — I)-connected. Then the inclusion
K"~l с К" is null-homotopic, by an easy obstruction-theory argument;
hence ГДА^) = 0 for ρ <n, and hence, if К is (n — I)-connected, then the
Hurewicz homomorphism 7r„ + 1(/Q ->W„ + 1(/Q is onto.
EXERCISES
1. Prove the following "butterfly lemma": Given a commutative diagram

72
COHOMOLOGY OPERATIONS
of abelian groups and homomorphisms, in which the diagonals pi and qj are
exact at C, there is an isomorphism
im<7 ^ imp
imf img
(Proposition 4 is a special case.)
2. Prove that the homotopy exact couple is not a homotopy invariant but that
the first derived couple is (see Appendix).
REFERENCES
Expositions of spectral sequences Appendix
1. H. Cartan [6]. 1. P. J. Hilton [1].
2. — and S. Eilenberg [1]. 2. J. H. С Whitehead [2].
3. R. Godement [1].
4. S.-T. Hu [1].
5. W. S. Massey [1,2].
6. J.-P. Serre [1].

CHAPTER
FIBRE SPACES
In this chapter we recall elementary properties of fibre spaces and state
properties of Serre's spectral sequence for the homology of a fibre space.
FIBRE SPACES
Let Ε and В be topological spaces, and let ρ be a map of Ε onto B. The
map ρ is a fibre map in the sense of Serre if it satisfies the covering homo-
topy condition for finite complexes; that is, given any finite complex K,
a map g: A^—> E, and a map F: Κ χ /—> В such that F(x,0) = p{g{x)) for all
xe К (thus F is a homotopy of pg), then there exists a map G: Κ χ /—>£
such that G(x,0) = g(x) and such that p(G(x,t)) = F(x,t). (Thus С is a
homotopy of # which covers the homotopy F.)
In the same context, ρ is a fibre map in the sense of Hurewicz if it
satisfies the covering homotopy condition for all topological spaces; i.e.,
if in the detailed definition above, К is an arbitrary topological space.
Clearly every Hurewicz fibre map is also a Serre fibre map; the converse is
false.
We say that (Ε,ρ,Β) is a fibre space if ρ is a fibre map.
We will always assume that B, the " base space," is arc-connected.
(This makes redundant the assumption that ρ be onto B.) And when the
sense is not specified, "fibre map" will be understood to mean: in the
sense of Serre.
73

74
COHOMOLOGY OPERATIONS
Choose a base point * in B. Then p~\*) is a subspace of E, called the
fibre F of the fibre space. For any b e B, the subspace ρ~' (b) of Ε is called
the fibre over b. In a Hurewicz fibre space, all the fibres have the same
homotopy type: in a Serre fibre space, any two fibres which arc both finite
complexes must have the same homotopy type.
A fibre map ρ induces, in homotopy, an isomorphism
/>*: n*(E,F)*s π*(Β)
and, using this isomorphism, the exact homotopy sequence of the pair
(E,F) becomes the exact homotopy sequence of the fibre space:
■ ■ ■ -> n„(F)-> n„(E) — n„(B) -> π„_,(/=-)-> · · ·
We assume that the reader is familiar with the general theory of fibre
spaces, including the notions set forth above, as well as certain others, in
particular the action of π,(β) on the fibre and the coefficient bundle 3L4(F)
over B. (Although results will often be stated in terms of (co)homology
with local coefficients, in practice we will assume that the local coefficients
are simple.) However, we will further discuss the general theory of fibre
spaces later, especially in Chapters 11 and 14.
We assemble several important examples.
Example 1
(R,p,Sl) where R is the real line and S1 may be considered as the reals
modulo 1, the interval /= [0,1] with the endpoints identified;/? is the
natural projection. The fibre is Z, that is, an infinite discrete group which we
identify with the integers. From the exact homotopy sequence of this
fibre space we deduce the homotopy groups of S1 :
π,(5')^Ζ 7t„(S') = 0 for η > 1
Example 2
(S3m,S2) where η is the Hopf map. The fibre is S1. From the previous
example, n„(Sl) = 0 for n > 1, and therefore the exact homotopy sequence
of the Hopf fibration takes the form
>0^n3(S3)^n3(S2)^0^n2(S3)^n2(S2)^Z^0
which confirms that n2(S2) ^Z and shows that, for all n >2, 7r„(S3) «w
7r„(S2). Thus in particular n3(S2) — Ζ generated by the class of η.
Example 3
(Ε,ρ,Β) where В is an arc-connected space with base point *, Ε is the
space of paths in В beginning at *, and ρ projects a path onto its terminal

FIBRE SPACES
75
point. The path space Ε is given the compact-open topology. It is easy to
show that Ε is contractible; the usual contracting map pulls each path
back along itself to the base point. The fibre is ΩΒ, the space of loops in
В at *; since Ε is contractible, the exact homotopy sequence shows that,
for any space B,
π„(Ωβ)^πη+1(β)
Example 4
(C,p,B) where С is a regular covering space of В and ρ is the covering
map, which is a local homeomorphism. The fibre С is a discrete group, the
group of covering translations. Since G is discrete, we have, from the exact
sequence, n„(C) «* π„(Β) for η > 1, whereas the exact sequence terminates
as follows:
я,(С)^ я,(С)^> π,(β)^ G^ я0(С)
and the groups on the ends are trivial, so that G «а л,(В)/'^л,(С). (Since
С is a regular covering, π,((Γ) is a normal subgroup.) If С is simply
connected (the universal covering), then G ъ π,(β).
Example 5
(S2"~l, p, CP(n— 1)), the base space being complex projective space;
the fibre is 5"; it is understood that η > 1. From the exact sequence, we
have n2(CP(n - 1)) ^ n^S1) % Z, and n{CP{n - I)) <* n{S2n~') for i > 2.
Thus the first non-trivial homotopy groups of CP(n — I) are infinite cyclic
groups in dimensions 2 and 2n — I. It follows that CP(cc) is a K(Z,2)
space.
Example 6
(0(/i + 1), p, S") where the total space 0(/i — 1) is the orthogonal group
in /i + 1 variables and ρ is evaluation at a fixed point of S". The fibre is
0(/i). From the exact sequence
π;+ ,(S") -> π,(Ο(/0) -> π,(0(/; + 1)) -> 7t,(S")
we have, if/i > / ! 1, π,(0(/ί)) »» π,(0(/ί + 1)), so that π,(0(/ί)) is
independent of/; for n sufficiently large (n > / 4 1). This limiting group is called
the stable group π,(Ο). These groups are known; by the Bott periodicity
theorem, π,-(Ο) ^ πί+8(0), / >0. The values are
/ mod 8
π,.(Ο)
0 1
z2 z2
2
0
3
ζ
4
0
5
0
6
0
7
Ζ

76
COHOMOLOGY OPERATIONS
THE HOMOLOGY SPECTRAL SEQUENCE OF A FIBRE SPACE
Fibre spaces may be thought of as generalized product spaces. In homo-
topy, the analogy with product spaces is expressed through the exact
homotopy sequence of the fibre space. In homology the situation is more
complicated. The main results express the analogy between the total space
Ε and the product space fixfby means of a spectral sequence which,
under certain hypotheses, starts from the product of the homology of
В and of F and converges to the homology of E.
These results are obtained by means of cubical homology theory, or,
to be precise, based normalized cubical singular homology theory. We will
not derive them. The main theorem is as follows.
Theorem 1 (Serre)
Let (Ε,ρ,Β) be a fibre space with fibre F, and suppose that В and F are
arcwise connected.
1. The chain complex C(E) admits a certain filtration, giving rise to a
spectral sequence.
2. In the resulting spectral sequence {Er,dr}, the bidegree of if is (—r, r — 1).
3. Elq=Cp{B)®Hq{F).
4. Epq = Hp(B,3Cq(F))—local coefficients. If В is simply connected, then
Eiq = H„(B;Hq(F)).
5. The spectral sequence converges to Я*(£).
6. /'*: #„(.F) —> #„(£) may be computed as follows:
tf„(F) = El,n^E'Q,n = Е$л = F0;„ с Нп{Е)
r being taken sufficiently large so that £o,„ = E%„. Note that every dr
vanishes on ££„; he nee the map E^„—*ЕГ0„ is defined.
7. p„: Hn{E)^Hn{B) may be computed as follows:
Hn(E) = F„,Q^£»0 = Ern,0 с El,o = H„(B)
where again r is taken sufficiently large. The existence of an inclusion
Ern0 c £^,о follows from the obvious fact that no element of £„r;0 can be
a boundary for any r.
EXAMPLE: H*(nS")
To illustrate the application of Serre's spectral sequence, we compute
Я*(П5'"), the integral homology groups of the loop space of the «-sphere.
(We assume n>2, so that B=S" is simply connected.) Recall, from

FIBRE SPACES
77
Example 3 above, that HS" is the fibre of a contractible fibre space over
S". Hence, in the spectral sequence of this fibre space, E*q = 0 except for
Eq>0 «w Z. We know Hi{Sn) % Ζ for / = 0 or n, 0 otherwise. It follows from
item (4) of the theorem that E2pq = 0 unless/? = 0 or n. Thus the only
differential in the spectral sequence which can possibly be non-zero is d":
£;,,—>£S.4+n-i· Consider Figure 1, which displays E". We have Ε"πΛ) «aZ.
я
2(и-1)
(и-I)
О
О η
Figure 1
Since Ε"π V = £"o = 0, the generator of £„" 0 cannot be a cycle under d";
moreover, since E"0^nll = ££„_, = 0, this generator must map Е"пЛ
isomorphically onto £ο.η-ι· This tells us that Я„_,(П5") ^ £o,n-i =
£ο.η-ι *ΐ— £ϋ.ο ^ 2. Of course we also see that £o., must be zero for
0<<7<«— I, and hence Hq(ClS") likewise. In informal language, the
groups £02,<, for 0 < q < η — I must be zero groups because there is nothing
that can " kill" them, and in order to "get rid of" the Ζ at (/7,0) we must
have a Z at (0, η — I).
We now apply similar reasoning to the next (n — 1) rows. From item (4),
£*,,-, - Hn{Sn;Hn.i{ClSn)) = Hn{Sn;Z) = Z. To get rid of this Z, we must
have an isomorphism d": £„",„-, —■+ ЕоЛ(„_1}, and the inference is that
Hq(ClS") = 0 for (n - \)<q<2(n~- I), with Я2(п_п ^Z.
Once the pattern is established, it is obvious that an inductive proof
gives Hq(ClS") *Z for q an integral multiple of (n — I), and zero otherwise.
This completes the calculation.
SERRE'S EXACT SEQUENCE
Although in general a fibre space does not yield a long exact sequence
in homology as it does in homotopy, the Serre spectral sequence gives the
following result in that direction.

78
COHOMOLOGY OPERATIONS
Theorem 2
Let (Ε,ρ,Β; F) be a fibre space, with В simply connected. Suppose that
Η{B) = 0 for 0 < / <p, and that Hj(F) = 0 for 0 <j <q. Then there is
an exact sequence (of finite length),
Hp+q-\{F)-^+ Яр+,_1(£)-^ #,,+,_,(/?)-^· Hp+q-i(F)
-■··-//,(£)-<)
proof: It follows from the hypotheses that Ejj = 0 when either
0 < / <p or 0 <j <g. Thus the normal series for H„{E), which contains
the Efi} terms for which / -\-j = n, collapses to the exact sequence
0 ->££„-> Hn(E) ->£&->0
Now in the general case we have the exact sequence
«—> ^-Ίι.Ο^ Ln,0 * ^"O.n-l -* ^Ο,η-Ι —> «
When n<p+q, we have Enn0 ъ H„(B) and £S.n-i ^#η-ι(0· Making
these substitutions in the last above sequence and splicing together the
two types of sequences displayed above (for all n<p^-q) yields the
sequence of the theorem.
We refer to the sequence of the theorem as the exact homology sequence
of the fibre space, or Serre's exact sequence (in homology).
The map τ: #„(5)—>#„_,(F), which corresponds to d^0, is called the
transgression. Thus far it is defined only when η <p-\-q where p,q are
as in the hypotheses of the above theorem; later we will study the
transgression in greater generality. That the other maps in the exact sequence
are actually /'* and p„ is stated in terms (6) and (7) of the main theorem.
Our theory casts new light on the Hurewicz theorem.
Theorem 3 (Hurewicz)
Let A' be a connected space with π;(Χ) = 0 for all./ <n (n > 2). Then
Ht(X) -= 0 for all / < η and πΠ(Χ) % Hn{X).
We take as known the corresponding theorem in which η —- I and the
last statement is replaced by n^xyc «* H^X) where С is the commutator
subgroup of π,(Λ'). We can then prove the present theorem by induction
on n, using Serre's exact sequence. Consider the contractible fibre space
(Ε,ρ,Χ; ΩΧ). By the induction hypothesis, H{X)=0 for all / <n— I,
and H„-i(X) ъ π„_ {(X) = 0. It remains to show that Hn{X) «w π,,(Λ'). Now
π/ΩΑ') «a nj + i(X); so we can apply the induction hypothesis to the space
nA'andweobtaintfy(nA') = 0foraM7 <n- I and#„_,(nA') «a π„_,(ΩΛ')^
π„(Χ). We apply Serre's exact sequence, with p=n and q = n — \;

FIBRE SPACES
79
p+q- \=2n-2>n, so that Hn{X) ъ Нп.х(С1Х). But Нп-х(С1Х)ъ
π„(Χ). This completes the proof.
(We omit to show that the isomorphism of the present theorem is in
fact the usual Hurewicz homomorphism,)
In somewhat similar fashion one can give a proof of the relative Hure-
wicz theorem. We omit it here, however, because we will state and prove
this theorem in greater generality in Chapter 10.
It is well known that any map/: X^> Υ is homotopically equivalent to
an inclusion map/': X^Y' where Y' has the homotopy type of Y. (The
standard procedure is to take for Y' the mapping cylinder of f.) This lends
weight to the following theorem of J. H. C. Whitehead.
Theorem 4
Suppose/ X с у, where X is arc-connected, Υ simply connected, and
π2(Υ,Χ) abelian. Then, for η > I, the statement that/, (the induced map
in homology) is isomorphic for dimensions j <n and epimorphic in
dimension // is exactly equivalent to the corresponding statement for/#
(the induced map in homotopy).
The proof is easy by our present methods. Write the homotopy-homology
ladder of the pair (Y,X):
πη(Χ)-1^πη(Υ)->πη(Υ,Χ)-*πη_,(Χ)->---
Hn(X) -j? h\ Y) -> tf„( Y,X) - H?- ,(*)->·■■
Now consider the following statements:
1. f^ is isomorphic, j < n, and epimorphic, j = n
2. The pair ( Y,X) is «-connected
3. Ηj(Y,X) = 0 for j<n
4. /, is isomorphic, у <n, and epimorphic, j = η
Statements (I) and (2) are equivalent, by the exactness of the homotopy
sequence; (3) and (4) are equivalent by the exactness of the homology
sequence. Finally, (2) and (3) are equivalent by the relative Hurcwicz
theorem. Thus (I) and (4) are equivalent, which establishes the theorem.
Corollary 1
If X and Υ are simply connected and/: X > Y, then/, is isomorphic
for all /i if and only if/* is isomorphic for all n.
This follows from the theorem upon observation that if π,(A') is abelian,
then 7t2( Y,X) is abelian (as required in the theorem). Of course Υ is replaced
by an equivalent space such that/becomes an inclusion; this does not affect
the desired conclusion.

80
COHOMOLOGY OPERATIONS
THE TRANSGRESSION
We return to the study of the transgression. The transgression has
appeared, in the case where the fibre F is {q— I)-connected and the base
space В \s(p — I)-connected, as a map τ: H„(B)^H„-1(F), defined when
η <p -\-q and given by i/„"0.
We now define the transgression more generally as d^O, without
supposing anything about connectivity of F and B. Then the transgression τ
in general has a subgroup of H„{B) as its domain and takes values in a
quotient group of //„_,(.F).
There is an entirely different description of the transgression which will
be important in what follows. In the fibre space (Ε,ρ,Β; F) let p0 denote
the map of pairs induced by p,p0: (E,F) —>(/?,*). We define (for any ri) a
transformation f from im {p0)*, which is a subgroup of H„(B,*), to a
quotient group of //„_,(F) as follows: if χ e im (p0)* c Hn{B,*), choose
у e H„(E,F) such that (p0)*(y) = χ and take cy as a representative of τ(χ).
The value of f(x) lies in a quotient group because of the indeterminacy
arising from the choice of y.
We say that χ e H„(B) is transgressive if τ(χ) is defined. Identifying
H„(B) with £^0, this is equivalent to the condition that d'(x) = 0 for all
/ <n. We assert that this is in turn equivalent to the condition that χ lie
in im (Po)* and that, moreover, if χ = (p0)*(y), then τ(χ) is the homology
class of dy. Thus τ = f. We omit the proof; the reader is referred to Serre's
paper [I].
THE COHOMOLOGY SPECTRAL SEQUENCE OF A FIBRE SPACE
We now outline the situation for cohomology, which is entirely
analogous to the situation for homology as described above but is enriched by
the ring structure and the squaring operations.
As in Theorem I, we have a spectral sequence (Er,dr) of which Е{л is
given by H"(B;3Zq(F)) or by H"(B; Hq(F)) if В is simply connected. We have
dpr'q: Ергл^>Ерг+гл~г+\ so that the bidegree of dr is (r, -r + I), just the
opposite of that of the homology differential d'. The spectral sequence
converges to H*{E). The reader can translate most of Theorem I and the
related discussion into the cohomological case.
As for the ring structure, if R is a ring, then we have the following
results (we omit the proof).

FIBRE SPACES
81
Theorem 5
There is a spectral sequence with EP,q = HP(B; 3tq(F;R)), converging to
H*(E;R); moreover:
1. For each r, Er is a bigraded ring, i.e., the ring multiplication maps
Ep'q®EP'-q'\ntoEp+p'-q+q'
2. In Er, dr is an anti-derivation (with respect to total degree), i.e., the
following is true whenever it makes sense:
dr(ab) = dr(a) -Ь + (-\)ка- dr(b) к = total degree of a
3. The product in the ring Er+1 is induced by the product in Er, and the
product in the ring Ex is induced by the cup product in H*{E;R).
If R is a field (and in fact we will usually work with R = ZP for ρ prime),
then the Kunneth theorem implies that
E2 = H*(B;R) ® H*(F;R) (tensor product of rings)
Serre's exact sequence in cohomology is exactly analogous to the
sequence in homology; if В and F are (p— I)- and (q— I)-connected,
respectively, the sequence terminates as follows:
> Hp+q~2(F)-^ Hp+q-\B)-?UHp+q-\E)-£+ Hp+q~\F)
The (cohomology) transgression is given by
τ = ί/η0·"-1:£η°·"-1^£;·0
We say that χ e H"~\F) is transgressive if τ(χ) is defined or, equivalently,
if d;(x) = 0 for all i <n; as before, this turns out to be equivalent to the
condition that δχ lies in \m p* с H"(E,F); and, moreover, if dx = p*(y),
then τ(χ) contains y.
Proposition 1
If χ is transgressive, then so also is Sq'(x); moreover, if ует(х), then
Sq!(y) s r(Sql(x)).
The proof is an immediate consequence of the preceding remarks; in
fact, if у ε τ(χ), then p*y = δχ, so that Sq!(p*y) = Sq'(dx). By naturality,
p*(Sq'(y)) = d(Sq!(x)); but this is equivalent to Sq'(y) = z(Sq'(x)).
DISCUSSION
The spectral sequence of a fibre space has been a calculational tool of
the utmost importance in the development of homotopy theory. The main
results (Theorems I, 2, and 5) of this chapter were proved by Serre; at the

82 COHOMOLOGY OPERATIONS
time they represented a major breakthrough in technique. While we have
not proved these theorems here, we urge the reader to consult Serre's
paper [I ] both for the proofs of these theorems and for many other
examples and applications.
We call the reader's attention to Figure 1 and suggest that he use such a
graph of E1 (and higher Er) to facilitate his own computations.
Theorem 3 (the Hurewicz theorem) and Theorem 4 were known before
Serre's work; however, use of his results shortens the proofs.
The periodicity theorem of Bott, mentioned in Example 6, has generated
much recent interest in several areas of topology and geometry; we do not
pursue any of these directions here.
EXERCISE
1. Compute the ring H*{CiS";Z). Your result should depend on the parity of n.
Fibre spaces
1. E. Fadell [I].
2. P.J. Hilton [I].
3. S.-T. Hu [I].
4. J.-P. Serre [1].
5. N. E. Steenrod [I].
REFERENCES
The spectral sequence of a fibration
1. S.-T. Hu [I].
2. J.-P. Serre [I].
The periodicity theorem
1. R. Bott [1].
2. D. Husemoller [1].

CHAPTER
COHOMOLOGY OF Κ(ιτ,η)
Since K{Z,\) may be taken to be the circle S\ its cohomology is
completely known. Also, in Proposition 2 of Chapter 2 we showed that the
cohomology ring H*(Z2,\ ;Z2) is just the polynomial ring Z2[a] where α
is the one-dimensional generator. These two examples provide a starting
point in the computations of Hm(n,n; G) in general, which is equivalent
to the classification of cohomology operations of type (π,η; G,m) (Chapter 1,
Theorem 2). We will here compute Η*(π,η; G) in a number of cases.
Our method will be to use the Serre spectral sequence of fibre spaces in
which, of the base space, total space, and fibre, two have known
cohomology, thus permitting the calculation of the cohomology of the third.
TWO TYPES OF FIBRE SPACES
There are two types of fibre spaces which will be important in this attack.
One is the type of Example 3 of Chapter 8: starting from the base space B,
we construct (Ε,ρ,Β; ΩΒ) where the total space is the contractible space of
paths in β initiating at the base point of Sand the fibre is the space of loops
in B. Since Ε is contractible, it follows from the exact homotopy sequence
that if Β = Κ(π,η), then F =- ΩΒ = Κ(π, η - 1).
The other type originates from a given short exact sequence of abelian
groups,
О—>Л—>Я-»С—>0
83

84
COHOMOLOGY OPERATIONS
Such an exact sequence gives rise to a fibre space for each n. To see this,
K{A,n)^K{B,n)
t
K(C,n)
we use the following fact.
Proposition 1
Every map/: X —> Υ is homotopically equivalent to a map/': Л"—> К
where (A",/', K) is a fibre space in the sense of Hurewicz. Moreover, X
may be taken to be a deformation retract of X'.
proof: Take X' to be the subspace {(χ,λ):/(χ) = λ(0)} of Χ χ У" and
put/'(χ, A) = /.(1). A homotopy equivalence between X and A"isgiven by
the map φ: X —> Χ': χ —> (χ,/(χ)*), where/(χ)* denotes vthe constant path
at/(x), and by the map ψ: A"—>A': (χ,λ) —> χ. Then φψ is homotopic to
the identity map of X' under a homotopy which retracts all paths back to
their initial points. Since φφ is itself the identity map of X, this shows that
AMs a deformation retract of A". Clearly/V =/ so that/and/' are
homotopically equivalent.
It is not hard to see that X' is a fibre space in the sense of Hurewicz. Let
g0 be a map of a space A into X' and let {h,} be a homotopy of the map
h0 —f'gQ. For each a e A,g0(a) = (χ(α),λ(α)) gives a path A(o) starting at
x(a) and ending at h0(a). It is required to give a homotopy {gt} of the map
g0: A —> A" such that#,(a) = (х,(й),;.,(я)) projects to Л,(й), which means that
the path λ,(a) must end at the point h,(a). This is easy; we let x,(a) = x(a)
for all / and obtain λ,(α) by attaching to λ(α) the path from h0{a) to h,(a)
given by the homotopy {//,}, with a suitable reparametrization. We leave
the explicit details to the reader.
Now suppose we are given an exact sequence of groups, as above. We
may realize the homomorphism B^C by a map K{B,ri) —> K(C,n), using
Corollary 1 of Chapter 1. We convert this map into a fibre map, and it
is obvious from the exact homotopy sequence of the fibre space that
the fibre has the homotopy groups of K(A,n). What is not obvious is that the
fibre has the homotopy type of а СИ''complex. We omit to prove this fact.
To illustrate the use of the first type offibration, we calculate H*(Z,2;Z).
Proposition 2
H*(Z,2; Z) is the polynomial ring Z[i2] where i2 is of degree 2, that is,
(i2)" generates Η2η(Ζ,2;Ζ).
We use the fibre space
S' = A:(Z,l)->£
1
K(Z,2)

COHOMOLOGY OF Κ(π,η)
85
where Ε is contractible, and consider the cohomology spectral sequence
with integral coefficients. Here E2·4 — 0 for q > 1, so that nonzero groups
can appear only in rows 0 and 1 of E2 (and consequently Er). Thus only d2
can be non-zero; all higher dr are automatically zero for dimensional
reasons. Moreover E™ — 0 if (p,q) Φ (0,0), since Ε is contractible. It
follows immediately that Η"(Ζ,2; Ζ) ъ> Ερ2·° is Z in even dimensions and
zero in odd dimensions; see Figure 1. It remains to obtain the ring structure;
я
(Er,dr) —*■ H*(E), £ contractible
I · · · ρ
0 1 2 3 4 5 6
Η* (Ζ, 2; Ζ)
Figure 1 Computing H*(Z,2; Ζ)
for this, we use the fact that d2 is an anti-derivation and the observation
that άρ2Λ is an isomorphism for each ρ and hence carries generators to
generators. Thus, denoting the generators of f^'1 and E\·0 by /, and /2,
respectively, we have
dl(ll® ll) = i/2('2)® Ί L»2®^2('l)= -1*2® ! 2
which shows that /2 ® г2 is a generator of E2'°, and hence (z2)2 generates
H\Z,2; Z); similarly,
i/2(/2 ® /,) = /2 ® i/2(/,) = !2 ® *2
so that (z2)3 generates Я6; and so forth. This proves the proposition.
Observe that the cohomology rings of the base space and the fibre are a
polynomial algebra and an exterior algebra, respectively.
To illustrate the use of the second type of fibre space, we calculate
H*{Z2,\; Z2) directly from #*(Z,1) without recourse to the construction
given in Chapter 2. From the exact sequence
0 >Z—>Z->Z2—>0
we construct a fibre space in which both the total space and the fibre have
the homotopy type of S\ while the base space is A"(Z2,I). Although in this
case π,(β) f 0, it can be shown that 3C*(F) is simple. We can then calculate
H*{Z2,\; Z2), using the spectral sequence of this fibre space (coefficients
/Γ(5Ί)
1

86
COHOMOLOGY OPERATIONS
in Z2)—see Figure 2. Only two rows can contain non-zero groups, since
F= Sl. Since Ε — Sl, Ery-q = 0 except when (p + q) = 0 or 1; moreover,
since d2 is the only non-zero differential in the spectral sequence, it is
readily seen that E°/° =--E},;° = Z2 while ££·«--= 0 otherwise. One then
deduces Εξ'" (q = 0,\) inductively with increasing ρ and obtains the desired
groups. As for the ring structure, the only point which is not obvious is
whether a2 -f-O where a generates H\Z2,\; Z2). Since a2 = Sqlx and Sql
is the Bockstein homomorphism, it is enough to show that the Bockstein
is non-zero on a. We leave this to the reader.
H*(S4Z2)
(Er,dr) -»- H*{SKZ2)
(и) . («®t,)
2 3 4
H*(Z2,\;Z2)
Figure 2 Computing H*(Z2,\; Z2)
These examples should convince the reader that skillful use of the
spectral-sequence diagram can (in some cases) reduce the calculation to
routine. Henceforward we will depend on the reader's ability to work with
these diagrams. One generally draws a single diagram and considers it to
represent all the Er superimposed upon one another; only in very
complicated cases is it necessary to draw separate diagrams for each Er.
CALCULATION OF tf *(Z2,2; Z2)
We now consider a more complicated example: we calculate H*(Z2,2; Z2),
at least in low dimensions. For this purpose we use the fibre space over
B^K(Z2,2) with contractible total space and with fibre F=A"(Z2,1).
The spectral sequence (with Z2 coefficients) converges to the cohomology
of the total space; hence E\-q — 0 except at (0,0). The cohomology of the
fibre we know to be the polynomial ring on a one-dimensional generator a.
We proceed as follows with the calculation of the cohomology of the base
space—see Figure 3.

COHOMOLOGY OF Κ(π,η)
87
By the Hurewicz theorem HL(B;Z2) = 0; hence Er1·" = 0 for all q and all
r. Then α is transgressive and τ(α) = d2oc — i2, which is the generator of a
Z2 at £220, indicating that H2{B;Z2) = Z2. Now H2(F;Z2) is generated by
a2, and since i/2 is a derivation, ί/2(α2) = 0; more generally d2(a2k) = 0 for
all k. On the other hand, d2(oc2k+l) = d2{a) ® a2k = i2 ® a2*. Since £2 is
the tensor product of H*{B;Z2) and H*{F;Z2), Ef = Z2 generated by
/2®α". The groups in this column with q even are killed by d2,Q + 1.
The others are not cycles; £22·2k +1 = Z2 generated by ι2®α2*+1, and
ί/2ί'2®«"+,):
ι (j2 ® a2*) = (/2)2 ® a2*. In particular this shows that
E2·0 contains a Z2 generated by (/2)2. Note that, since /2 is of dimension 2,
02)2 = ^2(/2).
α2(=5'91α) 2"
(£r,i/r) —»■ H*(E;Z2), Econtractibie
2 3
(ι;) (5?ΐι2) d'.2=59212)
Figure 3 Computing H*(Z2,2; Z2)
Thus the column ρ — 2 contains groups that alternately kill by or are
killed by d2. We have not yet completed the study of the column /7 = 0,
where d2(x2k) = 0. Since α is of dimension 1, α2 = 5^'α. But α is
transgressive; hence, so is a2 and
d,{*2) = τ(α2) = riSqU) = Sq\rx) = S7O2)
This shows that E2·0 contains a Z2 generated by Sql(i2). If we consider a4,
then, since a4 = (a2)2 = Sq2Sq1cc, this element also is transgressive and there
must be a Z2 at (5,0) generated by Sq2Sq\i2).

88
COHOMOLOGY OPERATIONS
The reader may pursue this calculation further if he wishes. We remark
that H5(B;Z2) contains not only the Z2 already mentioned but also a Z2
generated by the product (S<7'/2)(/2) which arises because dlA must be
non-zero. Also, it is not hard to verify that a" is transgressive if and only
if η is a power of 2.
In fact, H*(Z2,2;Z2) is the polynomial ring over Z2 generated by all
Sq'(h) where / is an admissible sequence of excess less than 2. We can
verify this in low dimensions by computations such as those outlined
above; but to prove it in general we need better tools. A famous theorem
of A. Borel is such a tool.
H*(Z2,q;Z2) AND BOREL'S THEOREM
A graded ring R over Z2 is said to have the ordered set X\_,x2,-.. as a
simple system of generators if the monomials
{x;,.Yi2· · ·*,„: /, < /2 < · · · < /r}
form a Z2-basis for R and if for each η only finitely many x, have
gradation n.
Examples of rings with a simple system of generators include exterior
algebras and the locally finite graded polynomial ring Z2[x{,x2,...]; in
the latter case, the {xf} form a simple system of generators.
Theorem 1 (A. Borel)
Let (Ε,ρ,Β; F) be a fibre space with Ε acyclic, and suppose ff*(F;Z2)
has a simple system {xa} of transgressive generators. Then H*(B;Z2) is
the polynomial ring in the {τ(χ,)}.
A proof of this theorem is outlined in an appendix to this chapter.
Borel's theorem is precisely what we need in order to calculate
H*(Z2,2; Z2); in fact, we have the following more general result.
Theorem 2
H*(Z2,q;Z2) is the polynomial ring over Z2 with generators {Sq'(iQ)}
where / runs through all admissible sequences of excess less than q.
The following proof of this theorem consists of an application of Borel's
theorem and a lengthy exercise in admissible sequences.
We introduce the notation L(p,r) for the sequence 2r~lp,2r~ 2p,...,
4p,2p,p where ρ > 0 and r > 0. (If r — 0, wc write L{ ,0).) Then the excess
of Ц/у) is/) and the length of L{p,r) is/·; the degree is seen to be/7(2r— I).

COHOMOLOGY OF Κ(π,η)
89
The theorem is known for q = 1 (Proposition 2 of Chapter 2). We prove
it by induction on q. We suppose it true for q and consider the fibre space
F=K(Z2,q)^L
I
B = K(Z2,q + \)
By hypothesis H*(F;Z2) is the polynomial ring over Z2 with generators
{Sq'(iq): /admissible, e(I) < q). To keep the notation intelligible, we write
H*(F;Z2) = P[{ij}] where {z}} are the Sql(iq) suitably indexed. Let pj
denote the dimension of z} (which is q plus the degree of the corresponding
/). Then (Zj)2"= SqL{pj-r,(Zj). And the {(zj)2r} form a simple system of
generators for H*(F;Z2). Since /„ is obviously transgressive, τ(ιη) =;, + ,,
all these generators are transgressive and Borel's theorem applies:
H*(B;Z2) = P[{z(z2')}] = P[{TSqL<"'%}]
= P[{SqL^-r4^)}) ■■= P[{SqL^^Sq'iq + l}] (I from 2j)
Thus the theorem will be proved when we have shown that, as / runs
through all the admissible sequences with e(f) < q, L{q-rd{I), r)-f runs
exactly once through all the admissible sequences with e(I) < q -j- 1.
It is clear that every L(- ■ ■)■/ is admissible. We will now construct an
inverse function /—>L/ where J runs through the admissible sequences
with e{J) < q -f 1; this will complete the proof.
Any admissible sequence J = {j\,j2,.. .,j\} may be written (in at least
oneway) as J = {/,,... J,} · {;',+ „... ,js} where y, = 2(j)+,) for all / < / -- 1.
(Here t may be zero.) Then J=L{jt,t)-f and e(J)=j, \ e{I) — 2{ji+i) if
t>\;e(J) = e(f) if / = 0. Recall the formula e(J) = 2/, -d{I).
Substituting this gives <?(/) =j, — d{I) for i > 1.
Now, if / = 0, each J with e(J)< q has a unique expression L( ,0)·/
with f(/) < q; if e(J) = q, then У obviously has a unique expression of
the form Li.
If f > 1, we have e{J) = /', — i/(/); hence e(J) -=qif and only if7, = q \
d{I). Thus, if <?(/) < q, J has a unique expression Ц ,0)·/ with e{l) < q\
if e{J) = q, we choose f the maximum of all allowable / such that
J = L{q \-d{I), t)-I (with e{I) < q), namely, the t for which j, >2(v', + 1).
This completes the proof of the theorem.
FURTHER SPECIAL CASES OF B*(n,n;G)
The above results and methods enable us to assemble the following
results about the cohomology of Κ{π,η) with Z2 coefficients.

90
COHOMOLOGY OPERATIONS
Proposition 3
#*(Z,2;Z2) is the polynomial ring over Z2~ generated by a two-
dimensional class i2.
proof: Essentially the same as for #*(Z,2;Z) given earlier
(Proposition 2).
Theorem 3
H*(Z,q;Z2) is the polynomial ring with generators {Sq'ii^)} where /
runs through admissible sequences of excess e(f) <q and where /,, the last
entry in /, is different from 1.
proof : By the same methods as the proof of Theorem 2. The proviso
ir > 1 arises because Sq\iq) = 0.
Proposition 4
H*(Z2„„\ ;Z2)=-= P[dm{ix)) ® £(!,) for m >2. Here dm denotes the mth
Bockstein homomorphism of Chapter 7; P[dm(ii)] denotes the polynomial
ring over Z2 on one generator dm(i{); and £Όι), the exterior algebra on
one generator /,.
proof: Compare the calculation for m — 1 given earlier in this chapter.
Theorem 4
H*{Z2n,,q\Z2) is the polynomial ring with generators {Sq'"'(lq)} where
we define Sq'" = Sq' if / terminates in /', > 1 and Sq'" = Sqh · · •Sqi'-'dm
(i.e., Sq1 with d,„ replacing Sqir) if/, — I; and where / runs through
admissible sequences of excess e(f) < q.
Proof by the same methods as before.
DISCUSSION
The main results (Theorems 2, 3, and 4) of this chapter were proved by
Serre. Theorem 2 completes at last the proof of Theorem 1 of Chapter 3.
In Chapter I we commented that we would study cohomology
operations both by construction of explicit operations and by general study of
Η*(π,η; G). The reader should note that in the cases covered by Theorems
2, 3, and 4 the two approaches have merged. We have both classified the
operations and specifically identified them, as well as having made in
Chapters 3 and 6 detailed analysis of many of their properties.
Theorem ΙΛ of the Appendix, a spectral-sequence comparison theorem,
is by no means stated in full generality and is in fact a special case of a
theorem of Zeeman.

COHOMOLOGY OF Κ(π,η)
91
We have used (and will use again) the following consequence of a result
of Milnor. Let /: .Y —> У a map of spaces having the homotopy type of
CW'-complexes with a finite number of cells in each dimension. Convert
/into a fibre map (by Proposition I). Then the fibre is again such a space.
APPENDIX: PROOF OF BOREL'S THEOREM
We outline a proof of Borel's theorem based on a simplified version of
the spectral-sequence comparison theorem.
In this appendix we consider only spectral sequences satisfying the
following conditions:
1. ErM = 0if/><0 or q<0
2. Ε?·" = Ef° ® E?·"
3. drpq has bidegree (r, —r + 1)
Note that the Serre spectral sequence for the mod 2 cohomology of a
fibre space has this form.
By a homomorphism/: Ε — >E of spectral sequences we mean a system
of maps {/Г},/Г--ЕГ^£Г such that djr=frdr,fr+i is induced by
fr and/Γ" — βΛ® fi'"· We now state a comparison theorem.
Theorem 1A
Let /: E—>E be a homomorphism of spectral sequences satisfying
conditions (1) to (3). Suppose f^4: Е%* —>£0''' is an isomorphism for each
q. Suppose ££« = EPJ> = 0 except for (p,q) = (0,0). Then/™ is an
isomorphism for all (p,q) and all r > 2.
remark: Under the hypotheses, the conclusion is equivalent to the
assertion that/2P'0 is an isomorphism for each p.
The proof is by induction on the column p. By hypothesis,/2υ* is an
isomorphism. We indicate the induction step. Suppose we have proved
/2S'° an isomorphism for s<p. For r<p, E?·0 = ££° = 0 = ££0 = £,"·0
and frp-° is an isomorphism. Now proceed by descent on r—i.e., assume
/A° is an isomorphism. Using the fact that Ε^~'-'~λ = E"~r-r~' = 0 and
the induction hypothesis on p, it is easy to see*that//'0 maps im d^~r-'~l
isomorphically onto im d?~r-r~\ The five lemma and the diagram below
0— im dpr ' — Ef-° — £Л0, — О
/,'
p.o
fi
p.o
1
p.o
№
0-> im #-'■'-' — E?-° — Etfi — 0

92
COHOMOLOGY OPERATIONS
'show that/p0 is an isomorphism. The downward induction on r then shows
that//0 is an isomorphism, which completes the induction on ρ and hence
the proof.
We now apply Theorem 1A to prove Theorem 1. Take for Ё the spectral
sequence in question, namely, the Serre sequence for an acyclic total space
with H*(F;Z2) having the simple system of transgressive generators {хг}.
For case, assume deg хг > 1. We will construct a spectral sequence Ε and
a map/: Ε —>£ in such a way that the comparison theorem applies.
Let Ρ be the graded polynomial ring over Z2 with one generator y\ for
each x% and with deg ya — deg x% 4 1. Let A be the graded exterior algebra
over Z2 with one generator za for each xx and with deg za = deg xx. Let
R = P® A. Define a differential δ of total degree 1 on R by the formula
6{y ® zx) = yy\ for у e Ρ and the requirement that <5 be a derivation.
Filter R by setting F"(R) =Σι>ρΡι® Λ and consider the resulting
spectral sequence Ε for Η *(/?). It is easy to check that Εξ·" = Rp\ all
(p,q). Thus £ satisfies properties (1) to (3). The acyclicity is easily verified
if {x,,} has only one element; the general case follows by the Kunneth
formula.
Define a ring homomorphism f*-°: P--> H*(B;Z2) by j-e—>фгв) and an
additive (not, in general, multiplicative) homomorphism /20'*: A —>
H*(F;Z2) by zfl · · -z,,-**,-, ■ · -.vlr, /,<···< /r. Define //■« =/2"·0 ®j^.
Once/r is defined, define fr+l by requiring that/r+1 be induced by/r. This
is possible since/X — drfn which is readily checked (and uses the
hypothesis that each xx is transgressive).
We now may apply Theorem 1A. It follows that/*0: Ρ -^>Η*(Β;Ζ2)
is a ring isomorphism.
General
1. H. Cartan [2,5].
2. A. Dold [I].
3. J.-P. Serre [2].
4. N. E. Steenrod [2].
REFERENCES
Milnor's theorem
I. J. Milnor [3].
The spectral-sequence comparison
theorem
I. E. С Zeeman [I].

CHAPTER
CLASSES OF ABELIAN GROUPS
In subsequent chapters we will use the Steenrod operations to make many
calculations. The material of this chapter will be an important tool in the
calculations and is of interest in its own right.
Let A be an abelian group. By the p-component of A, where ρ is a prime
integer, we mean the quotient group obtained from A'by factoring out the
subgroup of all torsion elements of order prime to p. For example, if
Λ = Ζ φ Ζ2 Θ Z3, then the 2-component of A is A/Z3 mZ®Z2.
In this chapter we will obtain the following result: Suppose Xis a simply
connected space such that Hk(X) is finitely generated for all &. Suppose У is
another such space and that we have a map/: X—> Υ such that/induces an
isomorphism on cohomology with Zp coefficients. Then the p-components
of the corresponding homotopy groups are isomorphic.
This result will be a fundamental tool in our calculations of homotopy
groups. Its proof is based on Serre's theory of classes of abelian groups.
ELEMENTARY PROPERTIES OF CLASSES
Definition
A class of abelian groups is a collection С of abelian groups satisfying the
following axiom:
I. If О—>Л'—>Л—>Л"—>0 is a short exact sequence, then A is in С if and
only if both A' and A" are in С
93

94
COHOMOLOGY OPERATIONS
Thus a class of abelian groups is closed under the formation of
subgroups, quotient groups, and group extensions.
When we use the letter C, it will be implicit that С is a non-empty class of
abelian groups.
Definition
A homomorphism/: Л—>5is said to be a C-monomorphism ifker/e C,
a C-epimorphism if coker/e C, and a C-isomorphism if both ker/ and
coker/are in С
warning: a C-isomorphism may fail to have an inverse.
Consider the relation ~ defined by A ~ В if and only if there exists a
e-isomorphism of A to B. This relation is reflexive but need not be
symmetric. We say that two groups A,B are C-isomorphic if they are equivalent
by the smallest reflexive, symmetric, and transitive relation containing the
above relation. Thus A and Bare C-isomorphic if and only if there exists a
finite sequence {A = A0,A{,A2,.. .,A„ = B] and, for each / (0 < / <«), a
C-isomorphism between A; and Ai+l in one direction or the other.
We will have need of the following further axioms at various times.
2A. If А,В e C, then A ® В е С and Tor (A,B) e С
2B. If A e C, then Л ® В е С for every abelian group В
3. If As С, then Я„(Л, 1 ,· Ζ) e С for every η > 0
Note that (2B) implies (2A), for Tor (A,B) is a subgroup of Λ ® /? for a
certain group /?.
The most important examples in our work are the following.
Example 1
C0, the trivial class, containing only one group, the trivial group. This
class clearly satisfies Axioms (1), (2B), and (3).
Example 2
GFG, the class of finitely generated (abelian) groups. This class satisfies
Axioms (1), (2A), and (3). A counterexample for Axiom (2B) is given by
A = the integers, B= the rationals. As for the proof of (3), it is enough to
verify (3) for Ζ and for Zp, ρ prime. It is clear for Z. For Z2, we have a
cell structure of finite type, from which the result is clear. An analogous cell
structure can be given for ρ > 2. We omit the details.
Example 3
(?„, where ρ is a prime: the class of abelian torsion groups of finite
exponent such that the order of every element is prime to p. This class
satisfies Axioms (1), (2B), and (3). The proof of (3) is not obvious and we
will give it later.

CLASSES OF ABELIAN GROUPS
95
If the class G in question is taken to be C0 (Example 1), it is readily seen
that all the C-notions, such as e-isomorphism, reduce to the usual classical
notions.
We remark that many standard results of homological algebra have
analogues in C-theory; for example, there is a "five-lemma mod C."
TOPOLOGICAL THEOREMS MOD С
We are interested in the following C-theory generalizations of three
classical topological theorems which have been discussed in earlier chapters.
Theorem 1 (Hurewicz theorem mod C)
Let X be a I-connected space. Let С be a class satisfying Axioms (1),
(2A), and (3). If π,-(Α') e С for all /' < n, then #,(*) e С for all / < η and the
Hurewicz homomorphism h: π,,(Λ')—> Hn(X) is a e-isomorphism.
Theorem 2 (Relative Hurewicz theorem mod C)
Let A be a subspace of X; let X and A be 1-connected, and suppose also
that i#: π2(Λ)—> π2(Χ) is an epimorphism. Let С be a class satisfying
Axioms (I), (2B), and (3). Then if π^Χ,Α) e С for all / < n, it follows that
Hi(X,A)e С for all \<n and h: πη(Χ,Α)^>Ηη{Χ,Α) is a e-isomorphism.
Theorem 3 (Whitehead's theorem mod C)
Let/be a mapping A —> X, where A and X are I-connected, and suppose
f# is an isomorphism on π2. Let С satisfy (I), (2B), and (3). Then/* is a C-
isomorphism on π, for all i <n and a C-epimorphism on π„ if and only if
the corresponding statements hold for/* on #*.
From the Hurewicz theorem mod CfC (see Example 2) we can
immediately draw a useful consequence.
Proposition 1
All homotopy groups of a finite I-connected complex are finitely
generated.
The proof is immediate from the Hurewicz theorem, in the light of the
following obvious property of classes.
Proposition 2
If the groups А, В are C-isomorphic and A e C, then Be C. (Here С need
only satisfy Axiom (I).)
We will indicate the proofs of the above theorems. First we make some
remarks concerning spectral sequences.

96
COHOMOLOGY OPERATIONS
Remark 1
If E\.q e C, then £'м е С for all r > 2.
This is clear from Axiom (1).
Remark 2 A
If С satisfies (2A), then, in the spectral sequence of a fibre space, if
£loeC,0<p< m, and El,„ eC,0<q<n,\t follows that E2Me<Z for all
p,q in the given range.
Remark 2B
If С satisfies (2B) and H„(B) e C, where Sis the base space of a fibre space,
then E^q e C, for all q, in the spectral sequence.
The proofs are easy.
THE HUREWICZ THEOREM
We now indicate the proof of the Hurewicz theorem.
Recall (Theorem 3 of Chapter 8) that we proved this theorem for the
case С — CJ0 by spectral-sequence methods, using induction on n. The
induction began by using classical methods to prove the anomalous case η = I.
The essential step in the proof is the sequence of isomorphisms
πη(Χ)^πη.ι(ΩΧ)^Ηη^(ΩΧ) *Hn{X)
where the first isomorphism is well known, the second comes out of the
induction hypothesis, and the third is proved by means of the spectral
sequence of the fibre space (Ε,ρ,Χ; ΩΧ) where Ε is contractible.
In the present case, we assume X is I-connected, and so the induction
begins without trouble, but we cannot apply the induction hypothesis to
ΩΧ because it may fail to be simply connected. We are therefore obliged to
introduce T, the universal cover of ΩΧ, and use the following detour to
obtain what corresponds to the second isomorphism in the above sequence:
where we write A p> В or, sometimes, А «в В (mod С) to indicate that A and
В are C-isomorphic. Since X is implicitly assumed " nice," the theorem of
Milnor (see Chapter 9) permits us to assume that ΩΧ (up to homotopy
type) has a universal cover T.
Thus our proof consists of establishing five (e-)isomorphisms. The first
of these, π„(Χ) <*;ππ_)(ΩΛ'), is well known (Chapter 8, Example 3).

CLASSES OF ABELIAN GROUPS
97
It is also well known that π„_,(ΩΙ)*π„_,(Γ). Indeed, the universal
cover of a space has the same homotopy groups as the space except that it
is simply connected; but, as our theorem is obvious for η = 2, we may
assume in our proof that η > 2; hence the result.
The next step is the C-isomorphism of π„_,(Γ) and Я„_,(Г). But
π,(Γ) ζαπ^ΩΧ) &ni + l(X) for / > I; thus Γ satisfies the hypotheses of the
theorem for dimension η — I, and thus the result is a consequence of the
induction hypothesis.
The next, and crucial, step is to establish that ЯДТ) ъ Η;(ΩΧ) (mod С)
for all /</;. This depends on arguments in the spectral sequence of the
fibre space (ΩΛ',^,λ'(π,Ι)), with fibre T, where we write π for π,(ΩΑ') «s
π2(Χ). Since the base space is not simply connected, it is not immediately
clear that the spectral sequence of Chapter 8 is applicable; in fact, it does
apply. We omit the proof. The construction of the fibre space itself may be
carried out as follows: Let И^Ье a π-free acyclic complex. Since π acts on T,
we may define W x „ Tin the customary way as the quotient of W χ Γ by
the relation (wg,t) ~ (w,gt) (gen). This is the total space; since W is
acyclic, W χπ T~(pt.) χπ Γ~7/π^ΩΧ. The total space maps naturally
to W/π, that is, to λ"(π,I), which is the base space. (Compare the
construction of K(Z2,\) in Chapter 2.) The fibre is clearly T.
Consider, then, the spectral sequence of this fibre space. Since π = π2(Χ)
is in C, ££„ ъ Ηρ(π,\; Ζ) is in C, for all p, by Axiom (3). By the induction
hypothesis, Egq ъ Hq(T) e С for all q < η — I. Using Remark 2A, we have
Егрл е С for all p,\fq<n — I. Thus clearly Η,{ΩΧ) e С for / <η - I. We
now must show that Η„^{(ΩΧ) and Я„_,(Г) are C-isomorphic. Clearly
Ηη_\(ΩΧ) is C-isomorphic to £O'.n-i and Я„_,(Г) ^ fo.n-i· But ^o.n-i and
£ΌΧ„_, are C-isomorphic, for £o,n-i 's the quotient of Eq,„-i by a group in
C; similarly for £o,„_,, etc.
This establishes the isomorphism Η,{Τ) ^ Η^ΩΧ) (mod C) for all /' < n.
Finally we must show Ηη^χ{ΩΧ) «a Hn(X) (mod C). Here the argument
is nothing but the C-theory form of the corresponding argument in the case
C = C0. We use the spectral sequence of the contractible fibre space
(Ε,ρ,Χ) with fibre ΩΑ'. We have shown above that E^q ъ Я/ΩJ) ε С
forq </ί. Also, by the induction hypothesis, EpA) ^ HP(X) ъ πρ(Χ) e С for
ρ <n. Now H„(X) <*> El0. As argued above, E^0 ъ £„"„ (mod C). But Ε is
contractible, and so d": E^0^> £,"ιΠ_, is a C-isomorphism. Again, Εΰ,„- , «в
££„_, (mod C), but this is just Я„-1(ΩA'). Thus H„(X) л» Η„.\(ΩΧ)
(mod С).
This gives us the five isomorphisms which yield π,,(Α') л» Н„(Х) (mod С)
and hence the Hurewicz theorem.

98
COHOMOLOGY OPERATIONS
THE RELATIVE HUREWICZ THEOREM
In order to prove the relative Hurewicz theorem, we introduce the
relative fibre space
ΩΧ^(Ε,Υ)
i
(X,A)
where (Ε,ρ,Χ; ΩΧ) is the usual contractible fibre space over X and Υ =
ρ'1 (A). It gives rise to a spectral sequence, entirely analogous to that for a
fibre space, converging to //*(£, Y) and with E]q = HP{X,A; Ηη(ΩΧ)).
Recall that A'and A are assumed I-connected and that the hypotheses of
the theorem also imply π2(Χ,Α) — 0. We operate by induction on n; the
theorem is obvious for η = 2; we assume it for η — I and deduce it for n. It
is immediate from the induction hypothesis that Η^Χ,Α) e С for /' < n. We
must show H„(X,A) ^πη(Χ,Α) (mod C).
In the spectral sequence of the relative fibre space, since Η^Χ,Α) e С for
/ < n, we have £/„ e С for ρ < η (for all a), using Remark 2B. By
arguments similar to those of the proof of the Hurewicz theorem,
Hn(E, Υ) ψ £«o % Elo ~ H„(X,A)
On the other hand, π^Χ,Α) ъ π;(£, Υ) by the covering homotopy property
and nj(E,Υ) ^π;_,( Υ) because £is contractible. Puttingy'= 2 shows that
Υ is simply connected and the Hurewicz theorem applies to Y, giving
π„_,( Y) % tf„_,( Y) (mod C). Thus
πη{Χ,Α) *πη{Ε, Υ) ***„_,(/) ψ Я„_,( Υ)*Ηη(Ε, Υ) ψ Ηη(Χ,Α)
which proves the theorem.
The reader should now be able to deduce the Whitehead theorem from
the relative Hurewicz theorem by essentially the method given in Chapter 8.
e„ satisfies axiom 3
We now focus our attention on the class Cp. Recall that a group G
belongs to Cp if nG = 0 for some integer η and if every g с G has finite order
prime to p.

CLASSES OF ABELIAN GROUPS
99
It is clear that Cp satisfies Axioms (I) and (2B). We now prove that
Axiom (3) also holds. Thus let A e Cp; we show that Hn{A,\; Z) e Cp for
all«>l.
Indeed, let A have exponent q prime to p. We prove by induction on η
that there exist non-negative integers e„ such that H„(A,\) has exponent
dividing qe", n>\. This assertion is trivial for η = 1.
Let j be a positive integer. Multiplication by s in A induces on K{A,\) the
map s: K(A,\)^>K(A,\) given by
K(At\)±*Kt(A,\)x ■■■ xK,(A,\)JUK(A,\)
where Δ is the diagonal and μ the iterated multiplication. Note that
q: K(A,\)^>K(A,\) is null-homotopic.
Consider Hn([\*=iK<(A, I)) as given by the Kunneth formula. If we use the
induction hypothesis and Axiom (2A) on (%, it follows that this group may
be written as £-= , H„(K,{A,\)) Θ G where G has exponent dividing qf for
some/.
Let ae H„(K(A,\)). Then we may write Δ*(α) = Σϊ=\ я,ф# for some
gsG. Thus J*(a) = л? + μ *(.#)· Taking s = q, recalling q^{a) = 0, we have
qa +μ*(#) = 0 for some g e G. Thus<7/+1a = Oand we may take e„=f+ I.
This completes the proof.
We now state for future use another result about Cp.
Lemma 1
Let f: A, —> A 2 be a homomorphism of finitely generated abelian groups.
Suppose/is a ep-isomorphism. Then Ax and A2 have isomorphic
^-components.
proof: Let p{At) denote the subgroup of A; consisting of elements
of finite order prime to p. We must show Ax\p{Ax) «= A2/p(A2). Clearly
f[p{A\)] c p(A2). Since Λ,- e efC, p(A ,·) g Ср. Thus we have the diagram
0->/>(Λ.)-»4.->Λι//>(Λι)->0
1 к I'
0->^2)^^2->^2/^2)->0
By the 5-Iemma mod Gp, f is a Cp-isomorphism. Using this fact, the
structure theorem for finitely generated abelian groups, and the observation
that AJp(Ai) has no torsion prime to p, the reader may easily deduce the
following:/is a monomorphism, induces an isomorphism of torsion
subgroups, and has for its image a subgroup of maximum rank. The result
follows.
We remark that/need not be an isomorphism.

100
COHOMOLOGY OPERATIONS
THE Cp APPROXIMATION THEOREM
We can now give the theorem announced in the introduction to this
chapter.
Theorem 4 (Cp Approximation Theorem)
Let X and A be simply connected (nice) spaces such that H:(A) and
Hi(X) are finitely generated for every /'. Let/be a map A —> X such that
f# : π2(Α) —>π2(.Υ) is epimorphic. (We may assume, without loss of
generality, that/is an inclusion.) Then conditions (1) to (6) below are equivalent
and imply condition (7).
1. /*: Η'{Χ\ΖΡ)^Ή\Α;ΖΡ) is isomorphic for i <n and monomorphic
for / = η
2. /*: Η:(Α;Ζρ)^>Ηί(Χ;Ζρ) is isomorphic for i <n and epimorphic for
i=n
3. H{X,A; Zp) = 0 for / < η
4. ΗΧΧ,Α; Ζ) eCp for i <n
5. πχΧ,Α) e C% for / < η
6. /#: π;(Λ)—> π,(Χ) is Cp-isomorphic for /'<« and (Jp-epimorphic for
\ = n
7. π,(/ί) and π,(Α') have isomorphic ^-components for i <n
Thus this theorem reduces the problem of computing the ^-component of
П;{Х) to that of finding a space A with the same cohomology in Zp
coefficients, together with a map of A into X inducing isomorphisms in Zp
cohomology.
The proof is not difficult with the tools we now have at hand.
Conditions (I) and (2) are equivalent by vector-space duality, since Zp is a
field and all //,· are finitely generated.
Conditions (2) and (3) are equivalent by the exact homology sequence of
the pair (X,A).
To see that (3) implies (4), recall that we have, from the universal
coefficient theorems, an exact sequence
0 -> H,{X,A)® Zp -> Н{Х,А; Zp) --» Tor (//,_ ,(X,A\ZP) -> 0
From (3), the group in the middle is zero, and so the group on the left is also
zero. But H;(X,A) is finitely generated, so that Η,(Χ,Α)®ΖΡ — 0 implies
that Hi(X,A) must be the direct sum of finite groups of order prime to p;
thus Hi(X,A) e Cp, which is condition (4).

CLASSES OF ABELIAN GROUPS
101
Conversely, from (4) (applied both to #,· and #,_,) and the above exact
sequence, we have (3), by Axiom (2B).
Conditions (4) and (5) are equivalent by the relative Hurewicz theorem.
Conditions (5) and (6) are equivalent by the exact homotopy sequence,
mod ep, of the pair (X,A).
Now by the Hurewicz theorem mod (iFU, the groups π,(Λ) and n,{X)
are finitely generated. Thus by Lemma I, (6) implies (7).
This completes the proof.
DISCUSSION
Of the results of this chapter, we shall make the most use of Theorem 4,
the Cp approximation theorem. It is this theorem which will justify our
calculations of 2-components of some homotopy groups of spheres (and
which justifies similar computations in the ^-component for odd primes^).
The material presented here was developed by Serre. The reader may
refer to his paper for a fuller discussion of relative fibre spaces.
Rather than show directly the triviality of the coefficient system in the
spectral sequence for the fibre space (ΩΛ^,Λ^π, 1)) with fibre T, we remark
that in this particular case the existence of the spectral sequence was known
before Serre's work.
EXERCISE
1. Prove Theorem 3.
REFERENCES
General The spectral sequence of a covering
1. S.-T. Hu [I] I. H. Cartan [4, Exp. XI-XIII].
2. J.-P. Serre [1,3]. 2. — and S. Eilenberg [I].
3. Ε. Η. Spanier [I]

CHAPTER
MORE ABOUT FIBRE SPACES
In the chapter following this we will launch our calculations of homo-
topy groups. The present chapter consists of several lemmas about fibre
spaces which will be needed in the calculations and in the later
development.
INDUCED FIBRE SPACES
Let ξ denote a fibre space (Ε,ρ,Β; F) and let/ be a map of a space X into
the base space B. The induced fibre space /*(ξ) is the fibre space with X as
base space, ^as fibre, and as total space the space £" = {(x,e): f(x) = p(e)}
topologized as a subspace of Χ χ Ε. The projection map of /*(£) is the
natural projection π,: £" —> X : (x,e) —>.v. Note that there is also a natural
map π2: E' —> £; {x,e) - >e. It is easy to verify that if ξ is a fibre space in
the sense of Serre or of Hurewicz. then/*(<!;) is likewise (see Exercise I).
Observe that if A' is a subspace of В and/is the inclusion, then Ε' may be
identified in a natural way with a subspace of E, namely, p~\X), and
/*(£) in this case is just the restriction of ξ to the smaller base space X.
Two fibre spaces, ξ — (Ε,ρ,Β; F) and ξ' = (E',p,B; F'), are said to be
fibre homotopy equivalent if there is a homotopy equivalence between the
total spaces which is compatible with the projections. Precisely stated, this
means that there exists a homotopy equivalence {φ,φ), φ: £ —> Ε', φ: Ε' —> £,
102

MORE ABOUT FIBRE SPACES
103
where the homotopies H: Ex I^>E and Η': Ε' χ /—>£", which deform
the compositions φφ and φφ into the respective identity maps, are required
to satisfy p(H(t,e)) =p(e) and p'(H'(f,e')) =p\e') for all points e,e' and all
t e /. It is obvious that this gives an equivalence relation among fibre spaces
over a fixed base space В and that the restriction of (ψ,φ) gives a homotopy
equivalence of the fibres. In Exercise 2 the reader is asked to verify that
homotopic maps induce fibre homotopy equivalent fibre spaces.
We will make extensive use of the next result, which shows the
importance of the idea of induced fibre space.
Proposition 1
Suppose we have the situation shown in the diagram, where ξ = (Ε,ρ,Β)
Ε'-21+Ε
y'Y \
is a Hurewicz fibre space and £" is the induced fibre space/*(^). Suppose
the composition fg: Κ—> Β is null-homotopic. Then there exists a lifting of
g, that is, a map A: K—>£' such that nxh = g. The conclusion also holds if^
is only a fibre space in the sense of Serre, provided that Υ is a finite complex.
prooi·: Using the covering homotopy property, we lift a null-homotopy
of/# to obtain a map hb: Κ—> Ε such that phE —fg. Then the maps g and
hE combine to give a map h =g χ hE: Y—> Χ χ £ which has its image in E'
and which obviously satisfies nji = g.
It may seem trivial to remark that if the space £in the above proposition
is a contractible space, then the converse holds—that isbfg~0 if and only
if g may be lifted to £". (This is obvious, since if g has a lifting h, thenar
factors through £, using π2.) However, this remark will have wide
application, since an important case of induced fibre space is that in which ξ is the
standard fibre space of paths over В (Chapter 8, Example 3).
THE TRANSGRESSION OF THE FUNDAMENTAL CLASS
Let ξ^(Ε,ρ,Β; F) be a fibre space such that the fibre F is (n — ])-
connected. Then there is a fundamental class ih- e H"{F\nn(F)). Moreover,
the transgression of ih- is defined, in the narrow sense. By definition, the
characteristic class χ(ξ) of the fibre space ξ is this cohomology class
t(/f)eW"+ '(β; *„(/="))·
Let Ε be the standard contractible fibre space over Κ(π, η + I) (where
π is an abelian group), that is, the fibre space of paths. Then the fibre is

104
COHOMOLOGY OPERATIONS
ΩΑ^(π, η -j- I) = Κ(π,η). From the cohomology spectral sequence of this
fibre space, it is clear that the transgression
τ: Η"(Κ(π,η); π)^Η""(Κ(π, η + Ι); π)
is an isomorphism. Denote the fundamental classes of the base space and
fibre by i„+1 and i„, respectively. Wc claim that the above isomorphism
mapsi„ to i„+1. This is certainly plausible; we urge the reader to verify it for
himself, using the definition of fundamental class (Chapter I) and the
formulation of τ in terms of p* and δ (see the discussion of transgression in
Chapter 8).
Suppose now that / is a map of a space X into K{n,n-r 1) and let
(E',qbX; Κ(π,η)) be the induced fibre space. We then obtain a map of the
Serre exact sequences in cohomology (the coefficients, which lie in π, are
>Η"(Κ(π,η))-^ Η"+ι(Κ(π, η + Ι))-ϋ-% #"+1(£)
l· I" !■■·
>Η"(Κ(π,η))^Η"+ί(Χ) ? >Η" + ι(Ε')
suppressed for clarity), where commutativity for the square containing the
transgressions may be verified using the formulation in terms of p* and «5. We
therefore have the following relation,
Formula 1 /*('„+1) =- t,(0
which is fundamental in what follows. Since/*(i„+,) is just the cohomology
class which represents/, in the sense of Chapter I (Theorem 1), the above
relation may be stated thus: in the fibre space induced by /from the
standard fibre space of paths, the fundamental class transgresses to the
cohomology class which corresponds to/. Observe that q*(f*(i„+,)) = 0 in
Η" + ι(Ε';π), by exactness.
BOCKSTEINS AND THE BOCKSTEIN LEMMA
Recall (from Chapter 7) the Bockstein exact couple,
#*( ;Z)-^U//*( ;Z)
\ s^
H*{ ;Z2)
where β denotes the Bockstein homomorphism of the exact sequence
0—> Z—> Z—>Z2 —> 0 and ρ denotes reduction mod 2. In the resulting
spectral sequence, dr is the 'Vth Bockstein operator"; dx in particular is the
homomorphism <52 = ρβ of Chapter 3.

MORE ABOUT FIBRE SPACES
105
The operator dr has been defined in terms of the spectral sequence, so
that its domain and range appear, for large values of r, as complicated
"subquotients" of H*( ;Z2). However, it is clear from the definition that
the domain is a quotient group of the subgroup of H*( ;Z2) of elements for
which d, = 0, i < r, or in other words a quotient group of Π '=! ker (*/,-)·
Moreover, it is convenient to consider dr as defined on f] r;=\ ker (d;) itself,
rather than on a quotient of this group. In this way we can make sense of a
composition of Bockstein operators drds even when r φ s; and it is clear
from the definitions that such a composition is always zero, modulo the
indeterminacy of dr. (When a homomorphism, such as dr, takes values in
a quotient group A/B, we say that dr has indeterminacy B.)
Recall also from Chapter 7 that d, vanishes on any cohomology class
which is the image, under p, of a class in coefficients 2, + 1. Therefore the
composite dtp gives a well-defined map #"( ;Z2i)—>Я" + 1( ;Z2), and d,p
may be identified with a map (or rather a homotopy class of maps) from
K(Z2,,n) into K(Z2, л + 1).
In Chapter 9 we indicated a procedure for constructing a fibre space from
a short exact sequence of abelian groups. Another technique which will be
useful in the present context is the following.
Lemma 1
The map d;p\ K(Zv,n) -^>K(Z2, η \- 1) defined above induces, from the
standard fibre space of paths over K(Z2bn -j I), a fibre space with fibre
K(Z2,n) and total space A^(Z2,+ ,,/?). Moreover, the injection and projection
of the induced fibre space correspond to the maps in the exact sequence
0—>Z2—>Z2i.,—>Z2,—>0
(under the correspondence of Chapter I, Corollary I).
The proof is easy, with the machinery available to us. The fibre is clearly
ΩΚ(Ζ2, η — 1)^ K(Z2.n). From the homotopy exact sequence of the
induced fibre space, it is clear that the total space must be K(G,n) where
G/Z2i <*zZ2. To see that the group extension is the non-trivial one, write the
spectral sequence (in Z2 cohomology) of the induced fibre space and
observe that z(it)- = (d;p)*(iB) by Formula 1, where it and iB denote the
fundamental classes in the fibre and base of the induced fibre space. Thus
we see from this spectral sequence that H"(E';Z2) ^Z2, so that G is
cyclic; hence G *a Z2,. ι, as claimed. Finally, since the injection and
projection are clearly not null-homotopic, they must represent the homomor-
phisms stated, since those are the only non-trivial ones between these
groups.

106
COHOMOLOGY OPERATIONS
Thus the above technique using the induced fibre space produces the
same fibre space as the technique of Chapter 9. This is actually a special
case of a much more general result about the classification of fibre spaces
with Κ(π,ιή as fibre. The more general result will be obtained in Chapter 13.
The next lemma needed for our calculations is the following, known as
the " Bockstein lemma."
Theorem 1
Let (Ε,ρ,Β; F) be a fibre space. Let the class и е H"(F; Z2) be transgres-
sive, and suppose that, for some integer / (/> 1) and for some class
ν e H"(B;Z2)b dp = z(u). Then di+ip*o is defined, and moreover
j*di+lp*(v) = dl(u)
where 7 is the inclusion F <= E. Here the members of the formula d;v = τ{ύ)
and of the formula of the conclusion lie in appropriate quotient groups of
Hn + 1(F;Z2). The equalities should be suitably interpreted.
This theorem is proved by the "method of universal example": we first
prove it in a special case, or " universal example," and then show that the
general case can be mapped into the special case in such a way that the
general conclusion follows by naturality.
For the universal example, take the fibre space
F= K(Z2,n) -L>E=K(Z2, + i,n)
I
B=K(Z2„n)
which was discussed in Lemma I. For u, take the fundamental class
iF e H"(F,Z2), and for v, take p(iu) e H"(B;Z2), the reduction mod 2 of the
fundamental class of B. It is immediate that i/, is zero on Hn+l(E;Z2)
(Chapter 7), and so d!+ip*(v) is defined. Now consider the Serre exact
sequence of this fibre space, with coefficients in Z2 (the coefficients are
suppressed for clarity):
O^HXB)-£+II"(E)^+ir(F)-UH" + l(B)-!^ir" + l(E)^+Hn + l{F)
ν 1И и d{v) di+l(ih) dx{u)
Each group written here is isomorphic to Z2 and is generated by the
element written beneath it, according to the results of Chapter 9. By exactness,
the first p* is monomorphic and hence isomorphic, so that p*(r) = iE. Then
the first j* is zero, by exactness; so τ is isomorphic and x(u) = dj(v) in
H"+i(B;Z2). Thus the example meets the hypotheses of the theorem. Since
τ is isomorphic, the second p* is zero and the second j* is isomorphic, so
that
j*di+ip*(v) =j*di + 1(iE) = d,{u)

MORE ABOUT FIBRE SPACES
107
which is the assertion of the theorem for this case. In this special case there
is zero indcterminancy, that is, the subgroup to be factored out is trivial,
since i/r-0on H"ь \E;Z2) for r < i.
Before proceeding with the proof of the general case, we alert the reader
to the kind of language which will be used when dealing with maps into
Κ(π,η). Consider the following situation:
According to Theorem I of Chapter 1, there is a certain natural
correspondence between homotopy classes of maps К—>Л^ and elements of
Η"(Υ;π). By virtue of the identifications
Ы<->#*(0
and also because of the tendency to speak ofg when we really should speak
of the homotopy class [g], we will be led to make such statements as
In the long run, this abuse of language will serve us well and we find it no
more confusing than the cumbersome formulas which result if one
meticulously preserves the distinctions between g, [g], and g*{i„).
Consider now the general case of the theorem. We will now use E, B, F, p,
j, u, v, etc., to denote the objects and maps in the general case, and we
distinguish the corresponding objects and maps in the universal example by
a subscript zero. We work with the following diagram:
F J—
*!
F0 = K(Z2,n)—^E0
It is implicit in the hypotheses, since dt{v) is defined, that d}(v) = 0 for all
j<i, which means that vs H"(B,Z2) can be considered as the reduction
mod 2 of a class w e H"(B;Z2i.). This class is represented in the diagram as
a map В —> B0. The resulting triangle is commutative, vaw~v, since v0 is
simply reduction mod 2. It is also clear that the other "triangle" is
commutative: (djV0)w~diV·, for these two maps obviously have the same effect
on the fundamental class of K(Z2, η — 1).
Now (djV0) , w p~(djV) ο ρ ■= p*(d,v) and d,v =- τ(ι/), so that p*(div) =
р*(т(и)) ^- 0. But the universal example is induced by the map i/fi;0, and so
(i/;U0) ^ и· о ρ ~0 implies that the composite w c ρ can be lifted to a map

108
COHOMOLOGY OPERATIONS
g: E—>E0 as shown in the diagram, and the square is strictly commutative:
wp = p0g. Since pj is a constant map, the restriction of g to F, that is, gj,
is a map into F0, which we denote by h, making the left square strictly
commutative also.
It would be nice if/; were exactly u. We do not know this, but we have
τ(Λ) = »ν*(τ0(/Κο)) by naturality and we know t0(iFu) = dsv0 from Formula I
• and Lemma I; and therefore
t(/j) = w*(d,v0) = (div0)w ~ dp = r(w)
so that at least τ(Λ) = т(и); this will be enough.
From the universal example, j0*di+ip0*v0 — dxu0. Applying h* to this
relation, using commutativity of the diagram and naturality of the Bock-
stein operators, we deduce that j*d^xp*v = ί/,Λ; but d^h—-dxu modulo
i/,(ker τ) = i/,(imv'*). This proves the theorem.
PRINCIPAL FIBRE SPACES
We now consider fibre spaces having certain additional structure. This
sort of fibre space will be useful in Chapter 13.
Let (Ε,ρ,Β; F) be a fibre space (in the sense of Serre, as usual) and denote
by E* the space {(e^ej): pex = pe2}, topologized as a subspace of Ε χ Ε.
We say that (Ε,ρ,Β; F) is a principal fibre space if there are maps
φ: Ex F—>.Eand h: £*—уFsuch that:
Ι. φ acts fibrewise, i.e., the following diagram is commutative:
Ex F-^E
(p'id)| |"
BXF^-B
2. The restriction of φ to F χ F, where F= p~\b0) is a fixed fibre (b0 the
base point of B), is a map F χ F-* F which, considered as a
multiplication map, has a two-sided unit and a two-sided homotopy inverse
3. The composition
E* ""■'") EX F-^E
is homotopic to the projection π.
For example, let Ε be the space of Moore paths over B, i.e., the space
of paths (/,/■) where r is a non-negative real number and/ is a map of the
closed interval [0,r] into B, with/(0)=^ b0. We have a natural projection

MORE ABOUT FIBRE SPACES
109
ρ: Ε —> β given by p{f,r) =/(/■)■ Then the fibre F = p~\b0) is the space of
Moore loops in B. (All spaces of functions are given the compact-open
topology.) We obtain a map φ: Ε χ F—>£ by letting <p(e,/) be the path
obtained by first going around the loop/and then along the path e. We
obtain a map h: E*—>F by letting h{eue2) be the loop which goes out
along e2 and returns along e\'. It is easy to verify that the three conditions
are satisfied so that {Ε,ρ,Β; F) is a principal fibre space. For instance,
condition (3) becomes the statement that {e2e[1)el is homotopic to e2 (where
ex, e2 are paths in B).
The reader will note that any principal fibre bundle in the sense of
Steenrod is a principal fibre space in the sense above.
If {Ε,ρ,Β; F) is a principal fibre space, then the fibre space induced by a
map of another space X into β is a principal fibre space. The proof is
straightforward and is left to the reader (see Exercise 4).
The importance of principal fibre spaces lies in the following result.
Proposition 2
Let {Ε,ρ,Β; F) be a principal fibre space, and let v, v' be two maps of a
space X into E. Then the composite maps pv, pv' are homotopic if and only
if there exists a map w: X^F such that the composite φ ο {v,w) is homo-
topic to v'.
proof: Given w as above, since ρφ = n{{p,id) by condition (I), we have
pv' ^ρφ{υ,\ν) c^nx{p,id){v,w) r~^pv
Conversely, if it is given that pv ~/n>', we can use a covering homotopy to
replace v' by another map in the same homotopy class such that pv = pv'.
Then {v,v')\ X—>Ε χ Ε is a map of X into E* and the composite
w = h{v,v')\ X—>F has the required property.
DISCUSSION
The characteristic class of a fibre space has other interpretations. In
particular, the reader is advised to find out the meaning (and proof) of the
statement that" the characteristic class χ(ς) measures the first obstruction to
a cross section of ξ, assuming π,(β) — 0."
The innocuous-looking Bockstein lemma (Theorem 1) will be crucial in
the forthcoming calculations. We will prove a generalization of this lemma
in Chapter 16 (Theorem 3). However, even in its ungcneralized form, this

110
COHOMOLOGY OPERATIONS
lemma played an important role in the history of calculation of homotopy
groups; more accurately, failure to understand this lemma (and its mod;»
analogue) led for a time to some serious confusion in the field.
With this chapter we have at last completed our preparations; in Chapter
12 we compute.
EXERCISES
1. Verify that if ξ is a fibre space (in the sense of Serre or Hurewicz), then the
"induced fibre space"/*(£) 's actually a fibre space (in the same sense).
2. Show that /*(ζ) depends essentially only on the homotopy class of /, not
on the choice off within that class; in other words, by varying / within its
homotopy class, we obtain fibre spaces which are all in the same fibre
homotopy equivalence class.
3. Show that the characteristic class of a fibre space is a fibre homotopy
invariant.
4. Verify that if (Ε,ρ,Β) is a principal fibre space and if/is a map into B, then
the fibre space induced by/is a principal fibre space.
REFERENCES
General properties of fibre spaces Principal fibre spaces
1. E. Fadell [I]. 1. J.-P. Meyer [I].
2. P. J. Hilton [1]. 2. F. P. Peterson and N. Stein [I].
3. S.-T. Hu [I].
Cross-sections of fibre bundles
1. N. E. Steenrod [I].

CHAPTER
APPLICATIONS:
SOME HOMOTOPY GROUPS
OF SPHERES
Using the machinery which has been developed in the last few chapters,
we can now make some computations in the homotopy groups of spheres.
The idea behind the technique is quite simple. To obtain the 2-com-
ponents of the homotopy groups of 5*", we begin with K(Z,n), which has the
same cohomology and homotopy groups as S" up through dimension n.
However, K(Z,n) has non-zero cohomology in higher dimensions, whereas
S" has non-zero cohomology only in dimension n. We will modify K(Z,n)
so as to eliminate or "kill" the non-zero mod 2 cohomology classes in
dimensions η + l,и +2,. .., and then the 2-components of the homotopy
groups of the resulting space will be the same as those of the sphere in those
dimensions, by the Cp approximation theorem (Theorem 4 of Chapter 10).
THE SUSPENSION THEOREM
It makes sense to talk about " the homotopy groups of S"" because to a
great extent these groups are independent of n. To be precise, n„+k(S") is
independent of η provided only that n>k + 2. This follows from the
Freudenthal suspension theorem. We will indicate the proof of this
standard result in homotopy. For convenience we suppose that each space X
comes provided with a distinguished " base point," denoted by an asterisk
(*). The reduced suspension of X, SX, is obtained from Χ χ I (where / as
111

112 COHOMOLOGY OPERATIONS
usual is the unit interval) by collapsing to a single point the subspace
(JIT x 0) и (JIT χ 1) υ (* χ /)
Then it is well known that fl^X) *» Hi+i(SX) (where /7 denotes the
reduced homology). There is a natural map φ: .Υ—> QSX, given by defining
φ(χ) as the path /—>(x,f), in a natural notation. Then the suspension
homomorphism Ε i^ox Einhangung) is defined by the diagram below. In fact Ε can
be defined more generally as a homomorphism of [Y,X] into [5Ύ,5Ύ]
as a homomorphism, that is, whenever the set of homotopy classes [Y,X]
has a natural group structure).
ni(X)-^^ni(aSX)
X i-
Now suppose that X is (n — l)-connected; then it follows that SX is
«-connected and ΩΞΧ is (n — I)-connected. If LX is the standard contrac-
tible fibre space of paths \nSX, then the Serre homology exact sequence of
this fibre space begins as follows:
H2„(LX) _> H2n(SX) _U Я2„_ t(ClSX) — H2n.t(LX)
Я2„_,(Л
where the groups on the ends are zero since LX is contractible. It can be
shown that τ and φ are compatible, i.e., that the triangle in the diagram is
commutative, and thus φ induces isomorphisms on Hk for all к < 2n — 1
and an epimorphism for H2n-1. But then the corresponding assertions hold
for the induced maps in homotopy, namely, for the suspension
homomorphism E. Thus we have indicated the proof of the following important
result.
Theorem 1 (suspension theorem)
If X is (n — l)-connected, then the suspension homomorphism Ε: π:(Χ) —>
n, + l(SX) is isomorphic for i<2n— 1 and epimorphic for i<2n— 1.
In particular, taking X = S" and observing that S(Sk) = S* + 1,we have the
previously stated corollary that n„+k(S") is independent of η for η > к +2.
A BETTER APPROXIMATION TO S"
Throughout the remainder of this chapter, when we say " homotopy
group," it is understood that we mean the 2-component of the homotopy

APPLICATIONS: SOME HOMOTOPY GROUPS OF SPHERES 113
group; the symbol πη(Χ) is to be interpreted in the same way. Cohomology
will be with Z2 coefficients.
Recall that H"+\Z,n;Z2) = 0 and that Hn + 2{Z,n; Z2) = Z2 generated
by Sq2(in) where /„ is the fundamental class. This cohomology class defines a
map (or rather a homotopy class)
Sq2:K(Z,n)^K(Z2,n + 2)
by Theorem I of Chapter 1. From the standard contractible fibre space over
K(Z2, η -r 2), this map induces a fibre space Xx over K(Z,n) with fibre
ΩΚ(Ζ2, η +2) = K{Z2, η -\ ]). We view Xt as a better approximation to
S" than K(Z,n). We say this because Hn + \XX,Z2) is zero, the class
Sq2 e Я" + 2(Z,«; Z2) having been killed. This is verified by considering the
cohomology exact sequence of the fibre space F, —>.Y, —>β, where
Ft = K(Z2, η + I) and B= K{Z,n). By construction, the fundamental class
!B+1 of F, transgresses to Sq2(i„). Then Sq\i„+l), which generates #" + 2(F,),
transgresses to SqiSq2(i„) = Sqi(i„). Thus, in the Serre cohomology
sequence (coefficients in Z2),
■-Я"+'(^)ЛЯ"+2(Й)ЛЯ"^,)ЛЯ"+2(£,)ЛЯ"+3(Й)-·
the transgression τ is onto Hn + 2(B) and monomorphic on #" + 2(F,), so
that, by exactness, ;>* and i* are both zero in the dimension shown and thus
H" + 2(Xl) = 0. Note that the method requires not only that the
fundamental class of the fibre transgress to the appropriate class in the base but also
that the transgression be monomorphic on the next dimension of the fibre.
Now \etf:S"->K(Z,n) represent the homotopy class of a generator of
πη{Κ{Ζ,η)) — Ζ. Then, in cohomology,/* is isomorphic through dimension
η and monomorphic in dimension η -τ Ι—the last is obvious because
H"'l(Z,n; Z2) = 0. Therefore, in homotopy, f# is isomorphic through π„
and epimorphic on π„+1 (mod C2), by the Cp approximation theorem. This
much is not new. But the composition
S" -U K(Z,n) -^ K(Z2, η + 2)
is null-homotopic, since π„(Κ(Ζ2, η + 2)) = 0. Therefore /"may be lifted to
a map/,: S"—> Xx. Now/, meets the hypotheses of the theorem with /; |- 1
in place of n, and we can conclude that/, induces a C2-isomorphism on
π„+1. But the homotopy groups of Xx are easily obtained from the
homotopy exact sequence of the fibre space: πη{Χχ) = Ζ, π„+,(ΑΊ) = Ζ2, and all
other π„(ΑΊ) are zero. In particular, π„ + ,(5") must be Z2.
The next step is to kill ff" + 3(Xl), obtaining a fibre space F2—> X2^>X{
where the total space X2 will have zero cohomology in dimension // -1 3 and
therefore will have the same homotopy group as S" in dimension η + 2.

114
COHOMOLOGY OPERATIONS
CALCULATION OF n„+k(Sn), к <1
We will compute nn+k(S") for к < 7, assuming η large. Precisely, since
we will obtain Hn+k(X\) up to k= II, we will want to assume η
comfortably larger than 11, in order that the cohomology of the Κ(π,η) spaces
which occur will not contain any cup products in the range under
examination.
We have already begun. The first step is the construction of the induced
fibre space Xx, where in the diagram, L denotes the contractible space of
F, = K(Z2, η + 1) -L, X,
i
В = K(Z,n) -5ii* K(Z2, η+ 2)
paths in K(Z2, η -\ 2). We can compute the cohomology of Xx using the
long exact sequence of Serre, as follows.
tf*(Z2, л+ 1)^ #*(*,)
Υ
^H*(Z,n)
We have already obtained Hn + 2(Xx) = 0 in this manner, but we will need
Hn+\XX) with к < 11. Observe that elements of H*{XX) may be regarded as
being of two kinds: if /*(x) = 0, then xs im (p*), or in fact χ e/>*(coker τ);
otherwise, i*(x)^/0 so that xc(/*)"'(ker τ). Thus in order to compute
H*(Xt), it is sufficient to know H*(Fi), H*{B), and the transgression.
This presents no difficulties, since H*{FX) and H*(B) are known
(Chapter 9), and the transgression is given by t(z„ + ,) = Sq2(in) and the fact that τ
commutes with the squaring operations. Thus we have
T(Sq4e+l) = SqlriiK+l) = SqlSq2(i„)=Sq\iK)
T(Sq2,n+i) = Sq2T(,„+,) = Sq2Sq2(i„) = Sq3Sq\,n) = Sq\0) = 0
and so forth.
The calculation of τ: HJ{F\)^>HJ + \B) is presented, in a condensed
notation, in the first two columns of the table on page 116. This notation is
to be read as shown in the small table here. The cokernel of r contains
Condensed statement Full statement
2<-!„+1 τ(/„+1) = 5<72(ζ„)
3<-l r(Sq\i„+l)) = Sq\i„)
У «-4,2,1 T{Sq*Sq2Sql{ie+l)) = V(0 + W(/„)
elements such as Sq4(i„), represented by a "4" in the first column of the

APPLICATIONS: SOME HOMOTOPY GROUPS OF SPHERES 115
table on page 116, not at the end of any arrow; the kernel of τ contains
elements such as Sq2(in+i), represented by the "2" in the second column
and also by such elements as Sqs(in + i) J- Sq*Sql(in+l), where each sum-
mand has the same image under τ, shown in the table by the common
destination of the arrows originating at the "5" and at the "4,1" in the
second column.
The reader should be able to reproduce this calculation of τ without
peeking. To write down H*{Z,tr, Z2) and H*{Z2, η -J- 1; Z2) requires only a
knowledge of the basis for the Steenrod algebra given in Chapter 3, together
with the results of Chapter 9. The calculation of the transgression will give
useful practice in the Adem relations.
Note that this part of the calculation could be continued indefinitely,
if we assume n sufficiently large. The description of Я*(/г1) and H*(B), and
the exact sequence itself, are valid up to the neighborhood of dimension In.
From this calculation off, we can write down a basis for the cohomology
of Xx. II"{X\) is generated by a fundamental class /„ which is the image of
the fundamental class of В under p*; we use the same notation for both
these classes. Н"+3(Х{) is Z2 generated by a class α such that i*{x) =
Sq2(in+,). The general situation is represented by Hn+*{XX). Here we have a
class p*(Sq\i„)) (we will drop the p* from our notation), and we also must
have a class β such that /'*(/?) = Sq3{i„ +,). Notice that there is some
indeterminacy in the choice of β, since i*(p*(Sq\i„))) = 0. One might try to
identify a canonical choice of/?, for instance by relating it to SV(a). We will
not do this; we will simply assume that β is some class, arbitrarily chosen
subject to the condition /*(/?) — Sq3(i„); whatever statements we make about
β are independent of the choice.
In the third column of the table on page 116, a basis for H" + k(Xi) has
been written for к < 11. J η dimension // I- 10, for example, there is a class
p*(SqX0(t„)), which we write simply as Sqw(i„) (consistent with our writing
/„ for />*(/„)). Then there are three other generators, each of which has a
prescribed image under /'*; each of these elements (denoted λ, μ, ν in the
table) could be chosen in two different ways, since /*(5^'°(/„)) = 0.
As we continue, we will have need of certain Bockstein relations in
H*(XX), which we set forth as a lemma.
Lemma 1
In H*{XX) (coefficients Z2 as always),
1. i/,(o() = /5 — ( )Sq4i„ where ( ) denotes an undetermined coefficient
2. d2(Sq\) = y
3. </,(?) = 0
4. d2(Sq»i„) = κ

116
COHOMOLOGY OPERATIONS
k_ H"+k(Z,n;Z2) H"+k(Z2, η + 1; Z2) Hn+k(Xl;Z2) Hn+k{Z2, η +2; Z2)
0 /„ /„
1
2
3
4
6
7
10
11
12
6,2
p(6,3,l)
<r(9,1 -1-7,2,1)

APPLICATIONS: SOME HOMOTOPY GROUPS OF SPHERES 117
к
~ъ
1
2
3
4
5
6
H"+k(X2;Z2) H"+k(Za,n^ 3;Z2) H"+k(X3;Z2) #"+*(Z2, л + 6)
10
к
"θ"
7
8
9
Д5+4,1>
V'.
C(5,l)
Ζ>(5,2)
H"+k(X*;Z2)
'η
0
tf"+4Z16,« + 7)
H"+k(Xs;Z2)
0
0
0
9
Л+6
The notation is that of the table.
proof:
1. We have i*(a) — Sq2in+i by definition of χ Therefore /*(//,a) =
ί/,ί'*(ο()= diSq2i„+l = S<73!„+). We have also ί*(β) = Sq3i„+l. Since 5i?4/n
is in the image of /?*, i*Sq*in = 0. This proves item (1).
2. We have i/, V. = Sql&fi„ = S^5/„ = r{Sq2Sqii„+,). Thus dlP*Sq4i„ = 0
since />*τ = 0, and, by the "Bockstein lemma" (Theorem 1 of Chapter
11), i*d2p*Sq*in--=dl{Sq2Sqi,n+i)= Sq3Sqli„+l = i*{y\ Since у is the
only generator in W+5(£), the result follows.
3. This follows from (2) since dx — 0 on anything in the image of any dr.
4. Again, as in (2), we use the Bockstein lemma. Since diSqai„ = Sq9i„ =
T{Sqn + Sq*Sq2Sql){i„+l), we have
i*d2p*Sq\ = dx{Sqn + VSi^'X'...) = SqsSq2Sq\in+i) = ί*{κ)

118
COHOMOLOGY OPERATIONS
Since i*(0) = Sq6Sq2in+i and there are no other generators in this
dimension, the result follows.
THE CALCULATION CONTINUES
We have constructed the space Xx so that H*(XX) and H*(S") (with Z2
coefficients) agree through dimension η + 2, and we have a map/,: З1"—> X{
inducing the isomorphisms. The next step is to kill the class α e H"+i(Xl)
in such a way as to produce a space X2 with no cohomology in dimension
η + 3. Recall that this requires killing α in such a way that the transgression
is non-zero on the first class in the fibre above the fundamental class. This
can be accomplished by taking X2 as the induced fibre space over Xx by a
map representing α where the new fibre is K{Z2, η -J- 2), so that the funda-
F2 = K(Z2,n + 2)^X2
I
Xt^> K(Z2,n + 3)
mental class transgresses to α and Sq* of the fundamental class transgresses
to Sq\ct), which is non-zero by part (1) of Lemma 1. If indeed Я"+3(А'2)=0,
then X2 is an improvement over Xx as an approximation to the sphere; we
have πη+2(Χ2) = Z2 from the homotopy exact sequence of the fibre space
/"2—>*2—>*i, and therefore n„+2(S") = Z2.
In order to obtain the cohomology of X2, we must compute the
transgression in the cohomology of the fibre space F2—>X2—>X{. This will be
less routine than the analogous step in the previous stage, because now we
do not know precisely the squaring operations in H*(X{). However, we
know enough to obtain a basis for H*(X2) for a considerable range.
The calculation of this transgression is indicated roughly in the third and
fourth columns of the table on page 116. By construction, the fundamental
class /n+2 transgresses to a. Then, as remarked already, Sq\in+2)
transgresses to Sq\oc) = ί/ι(α), which by Lemma 1 is β plus possibly Sq\in). It is not
essential to know whether or not this second term is present in Sq\<x) (this
depends on the choice of β) because either way we obtain a single generator
for H"+4{X2) from the cokernel of τ, and this class is p^iSq^i^ where
Pi'- X2—>X\- Therefore the first class in H*(X2), after the fundamental
class, will be this one, which we denote by Sq*tm dropping the/^. In fact
this class may equally well be regarded as Sq* on the fundamental class of
X2, using the naturality of Sq4 with respect to p2.

APPLICATIONS: SOME HOMOTOPY GROUPS OF SPHERES 119
Now Sq2(in+2) transgresses to Sq2(<x). We must determine whether Sq2(x)
is γ or zero. But /*(a) = Sq\i„+,), by definition of a; then i*Sq2ct = Sq2i*oc =
Sq2Sq2in+, = Sq3Sqii„+, = i*y. This proves that Sq2a = γ or, what is
essential for us, that r(Sq2i„+,) = γ.
Similarly, r(Sq2Sqli„+2) = δ plus possibly Sqei„, since
/•(We) = Sq2Sq\i*a) = Sq2SqlSq2in+,
= (Sqs+Sq*Sql)in+l
= /*(<5)
so that Sq2Sqla = δ 4- ( )Sqein where ( ) is an undetermined coefficient. On
the other hand, the same argument gives τ(5^3/„+2) = 0+ ( )Sqei„. In
order to know whether or not τ: H" + i(F2) —>#" + 4(ΛΊ) is epimorphic,
that is, in order to obtain its cokcrnel, we must get some information about
these coefficients. In the present case we can argue as follows. We have
Sq3i„+2 = SqlSq2i„+2, and therefore
r(Sq\+2) = Sql(r{Sq\ + 2)) = Sq'y
But Sql(y) = 0 by part (3) of Lemma 1. Thus τ(5^3/„+2)==0. This is enough,
since it now follows, regardless of the coefficient in T(Sq2Sqlin+2), that the
cokernel of τ in this dimension is generated by Sqein (dropping the p% as
usual).
In the next dimension, similar arguments give
T(V.+ 2) = f+( )Sq\
T(Sq3Sq4n+2) = e+( )Sq1in
so that the cokernel is generated by Sqnin (regardless of the unknown
coefficients).
In the next dimension we have
r(Sq5in+2) = n^( )Sq8i„
t(W/„+2) = ^( )Sq\
and we must determine whether the coefficients are the same. But
Sqs + Sq4Sql = Sq2Sq3, by the Adem relation; and we have seen that
т(5<73/„+2) = 0, from which it follows that
T(Sg\+1)+T(St/xSqit„+1) = TiSq2Sq\ + 2)
= V(t(V/„+2))
= 0
which is all we need to know. It shows that the kernel of τ contains
(Sqs +Sq*Sqi)(in+2) and that the cokernel contains Sqsi„.

120
COHOMOLOGY OPERATIONS
By similar reasoning, we can complete a basis for H*(X2) at least
through dimension n + 10; this is given in the first column on the top of
page 117. Perhaps the only other subtlety is showing that r(SqsSq2t„+ 2) — 0.
This follows from SqsSq2 ■— Sql(Sq*Sq2) and from part (4) of Lemma 1,
since dxd2 = 0.
This calculation has verified that H" + i(X2) = 0 as desired, and thus
πη + 2(5") = Ζ2. The next step will be to kill the class Sq4i„ e Hn + \X2). We
will need to know the following Bockstein relations.
Lemma 2
In H*{X2), (1) d3Sq\=--A, and (2) d3Sq8in = D. Here A and D are
defined by i*A = Sqiin+2 and i*D= SqsSq2i„+2 (as indicated in the table).
proof: In both cases the proof is based on the Bockstein lemma of
Chapter 11. In (1) we have, by part (2) of Lemma 1,
d2Sq*i„ = у = r(Sq2i„ + 2) s H" + 5(Л\)
and therefore
i*(d3Sq\) = dl(Sq\ + 2)= Sq\ + 2 = i*A
which implies the result. In the same way, part (2) follows from part (4) of
Lemma 1.
Now our objective is to kill Hn + *(X2). This means killing the class
Sq*in. This class can be realized by a map of X2 into K(Z2, η -'-- 4), but using
this map to induce a fibre space over X2 will not do the trick, because
the fibre will have a class Sqli„+3 which will transgress to zero since
dxSqA\n = 0. The fact that i/, and d2 are zero on Sq*i„ implies that this class
is the reduction (mod 2) of a class with Z8 coefficients. The proper
procedure is to map X2 into Κ(ΖΆ, η -\ 4) by a map corresponding to a class
which reduces to Sq4i„ (mod 2). Then Я" + 4 of the new fibre is generated by
d3 of the fundamental class, which will transgress to d3Sq*i„ — A.
Thus we construct a fibre space induced from the standard contractible
F3=-. K(Za, η-r3)->X3
i
X2~>K(Z8,n^4)
fibre space over Κ(Ζβ, η r 4) by a map having the properties described,
which we might, by abuse of language, refer to as " Sq*i„" for heuristic
reasons.
To obtain H*(X3), we must calculate the transgression in this fibre
space. This calculation is indicated in the first two columns, top of page
117. Most of the details are based on types of arguments which have

APPLICATIONS: SOME HOMOTOPY GROUPS OF SPHERES 121
been used before. We mention that τ(5^4/„+3) = 0 because Sq*Sq*in =
Sq1SqUn -J- SqbSq2in, and Sqx and Sq2 are obviously zero on the
fundamental class /„ of X2 since H*(X2) = 0 in the relevant dimensions. As a
corollary, T(Sqsin+i) is also zero. In a similar fashion τ(5</6/„ + 3) = 0 by the Adem
relation SqbSq4 = Sq'Sq*. The fact that T(Sq5d3in+3) = 0 follows from the
fact that Sq1 D — dxd3Sq*i„ = 0. The other details are left to the reader.
Now we can write a basis for H"+k(X3) for к < 9, as done in column 3,
top of page 117. Notice that we have killed not only H" + \X2) but also Η"+ 5
and Hn + e. This reflects the fact that π„+4(5") and π„ + 5(5") are zero.
We must kill the class Ρ с H" + 1(X3), where i*P= Sq4i„+3. Note that
SqxP is non-zero, since SqxSq*i„ + 3 is also in the kernel of the transgression;
in the notation of the table, SqxP = Q -+- ( )Sq*in. Therefore we construct a
fibre space
F4 = K(Z2, я + 6) -► Xa
I
X3-^K(Z2,n+7)
and this will kill orT#" + 7. It is easy to obtain Hn + k(X4) for к < 8; in fact
the only non-zero classes are /„ and Sqai„ in this range.
Lemma 3
In H" + 9(X4), d4(Sq8i„) is non-zero.
proof: We have shown (Lemma 2) that
d,Sq\ = Z? - r(Sq*d3ie+3) <= H" + 9(X2)
and therefore the Bockstein lemma implies that
i*(d4Sq\) = dt(Sq4d3in + 3)
=-- Sqsd3in+3
where 5 is an appropriate class in H"*9(X3). Since 5 is clearly in the
cokernel of τ: Ηη + 6(Ρ4)->Η" + 9(Χ3), it gives rise to a class S' e H" + 9(X4)
and the result follows from the naturality of i/4.
It follows that we can kill Я" + 8(А,4) by using a map
Xt^£i£.K(Zlb,n-, 8)
corresponding to a cohomology class in H" + (,(X^;Zib) which reduces to
Sq8in (mod 2). This gives us a space Хъ which has the same cohomology as
S" through dimension η -1 8. Moreover it is easy to lift our map /",: S"- > X{
up through the X-, to a map fs: S"1—> A'5 which induces isomorphisms on
cohomology through dimension я Ч 8 and therefore induces isomorphisms

122
COHOMOLOGY OPERATIONS
in homotopy groups through π„+7. The homotopy groups of Xs are easily
obtained using the homotopy exact sequences of the successive fibre spaces
in the construction. These are the same as the homotopy groups of S"
through π„+7. We can therefore state the following results:
Theorem 2
к
nn + k(Sn)
0
ζ
1
z2
2
z2
3
z8
4
0
5
0
6
z2
7
z16
Recall that what is tabulated here is the 2-component of the homotopy
group. Also note that these results are valid for all η such that η > к + 2;
we have assumed η large in the calculation, but nn+k(S") is stable as soon as
n>k-\-2, by the suspension theorem.
The cohomological techniques which we have used to compute
homotopy groups cannot be extended indefinitely. Eventually one reaches an
ambiguity which cannot be resolved by these methods. In particular,
nn+i4(S") must be settled in another way. In Chapter 18 we will develop
more sophisticated algebraic techniques; these will help to explain the
difficulties and the need for other methods.
Figure 1 displays the tower of fibre spaces which we have used in our
calculations.
Fs = K(Zlb,n+7) ^Xs
I
1
F3 = K(Z„ η -+ 3) -^ Хъ —L·^ K(Z2, η + 7)
1
F2 = K(Z2,n \-2) ^X2^±K(Za,n^4)
i
Fx = K(Z2, π^\) ^ Xx —^-+ K(Z2, n+3)
1
В - K(Z,n) —^ K(Z2, η+2)
Figure 1
DISCUSSION
Although the suspension theorem (Theorem 1) leads naturally to the
question of nn+k(S") for /; large (the kth stable homotopy group of spheres),
progress at first was slow. Of course, the result for к = 0 is immediate by

APPLICATIONS: SOME HOMOTOPY GROUPS OF SPHERES 123
the Hurewicz theorem; on the other hand, the value for к = 2 was in
dispute for some time until settled by G. W. Whitehead (using geometric
techniques).
Serre's results drastically changed the situation and facilitated many
cohomological computations, including the ones given here. In Chapter 18
we shall refine the cohomological method and indicate how the results of
Theorem 2 can easily be greatly extended.
The cohomological method is by no means the only possible approach to
the problem. The so-called "composition method" in particular has been
extensively used by Toda and Barratt. In Chapter 16 we shall interpret the
results of Theorem 2 in terms of this method. However, we do not touch on
two of their basic tools, the "EHP-sequence" and J-homomorphism of
G. W. Whitehead.
We have sketched a homological proof of Theorem I. For X= S", the
theorem, due to Freudenthal, greatly antedates the proof we have given. A
modern geometric proof in this case has been given by Bott.
APPENDIX: SOME HOMOTOPY GROUPS OF S3
The computations just carried out for nn + k(S") depend on the assumption
that η is large. We here present a sketch of the computation of n3 + k(S3)
(k < 3), in order to indicate what modifications and complications occur in
the technique. As before, we deal with the 2-components and all cohomol-
ogy is understood to be with Z2 coefficients.
We take В — K(Z,3) as our first approximation to S3. We can write down
H*(K(Z,3)), using the results of Chapter 9; it is the polynomial algebra in
the fundamental class /3 and in sequences Sq\i3) where / is admissible, has
excess less than 3, and does not terminate in 1. Thus we do not find
Sq*i3 or Sqsi3, etc., because the excess rs too large; we find cup products
such as (ii)(Sq2i3); and Sq3i3 appears as (z3)2.
Comparing H*(K(Z,3)) with H*(S3), we see that the first class to be
killed is 5<72/3 (just as the first class to be killed in the stable case is Sc/2i„).
We therefore use a map
B = K(Z,3) -^^ K(Z2,5)
to construct a fibre space over В with total space Xx and fibre У7, = K{Z2A)·
Notice that dxSq2i3 — Sq3i3 =- (z3)2, which is non-zero, so that Z2 is the
correct choice for the first homotopy group.

124
COHOMOLOGY OPERATIONS
There is no difficulty in writing H*(F{); it is a polynomial algebra in /4
and in Sq'u where / is admissible and has excess less than 4.
We next must calculate H*(X{). This is done by means of Serre's
cohomology spectral sequence. As stated in Chapter 8, the E2 term of this
spectral sequence is the tensor product H*(B)® //*(F,). When η is large
and к small, Hn+k(K(Z,n)) and H"+k(K(Z2, η + 1)) do not contain any cup-
product terms and the spectral sequence reduces to Serre's exact sequence;
every element in this range of #*(F,) is transgressive. But in the present
case we must handle the spectral sequence in its general form. Figure 2 is
M(Sqhi)
Sq^Sql и
Sq^SqWi
Sq2n
Stf'u
ч
9
8
7
6
5
4
3
2
1
5
Sq2L3
(i3)(Sq-n)
„3-
10
(S<f-i3)*
Sq*Sq\-i
Figure 2
a diagram of the spectral sequence for F, —> Xx —> 5. The cohomology of
the base and fibre are written out along the horizontal and vertical axes,
and product terms are written at the appropriate locations in the first
quadrant. In Figure 2 the elements of E2 are indicated only by small
circles, since in the scheme of the diagram the name of each term may be
inferred from its location. Only terms with total degree of at most 10 need
be considered for our purposes.
The calculation of the spectral sequence in this range is indicated in
Figure 2. We have i/5(/4) = t(/4) = 5^2/3 by construction. As a corollary,

APPLICATIONS: SOME HOMOTOPY GROUPS OF SPHERES 125
d5(/3 ® /4) = (h)(Sg2i3), since each dr is a derivation (and the products are
consistent in this sense). We have d5(Scj44) r= 0, db{Sqxu) = т(5^'г4) =
■SVCKu)) = Sq3i3 = Оз)2· Several elements transgress to zero; for example,
T(Sq2Sq44)=Sq2Sq\T(u))
= Sq2SqiSq2i3
= Sqsi3+Sq*Sq43
using the Adem relations, and thus r(Sq2Sql /4) = 0. Also, Sq*Sq{u
transgresses to zero, since
z{Sq*Sq'u) = Sq*SqxSq2i3
= V('3)2
=~-(Sq2h)2
which is zero in £10 because (5^2/3)2 = ds(Sq2i3® /4). In this calculation
we have obtained 5^4(/3)2 by the Cartan formula. Another proof that
Sq*SqlSq2i3 = (Sq2i3)2 is obtained by using the Adem relations:
Sq*SqlSq2 = Sq*Sq3 = SqsSq2,
from which the result follows.
We find that the following elements of total degree < 9 survive to Ex of
this spectral sequence:
i3 Sq\ Sq3u Sq2SqUt Sq3Sq44
Sq4Sqli4 t3®Sq2i4.
Recall that the groups Еъ·' for which s + t = η form a composition series
for Hn{Xx). We have no problems with group extensions when we are
working over Z2. Therefore we can easily write a basis for Hn{Xx), η < 9.
We denote the elements of this basis by /3, A, B, C, D, E, and i3A,
respectively. For example, В e H\XX) is defined by i*B= Sq3u e tf 7(F,).
Since i*(Sq'A) = Sq\i*A) = Sq'Sq2i4 = Sq3i4 = i*B, and since
/*:tf7(*i)->#V.)
is monomorphic, we have SqxA = B. Therefore we should kill A by
forming the induced fibre space over Xx by a map of Xx into K(Z2,6)
corresponding to A. We thus have the following diagram:
F2 = K(Z2,5) -> X2
I
F, = K(Z2,4) — *, ^U K{Z2,6)
I
B=K(Z,3)-^K(Z2,5)

126
COHOMOLOGY OPERATIONS
Using the Serre spectral sequence of the new fibre space F2 —> А'г —> A',,
we can calculate H*{X2) at least through dimension 8. This is indicated in
Figure 3. Just as we have seen that SqxA = B, we can show that Sq2A = D.
Note also that i*(Sq2Sql(A))~ Sq2SqlSq2i4- Sq*Sqlu = i*E, while
i*(Sq\A)) = Sq3Sq2i4 = 0. Since i3A = db(i3® i5). this implies that
r(Sq2Sqlis) = Ε while r(Sq3i5) = 0.
Sq^tb
StfSqh
Sq^ib
5?ii
6
A
7
В С
D цА Е
Figure 3
Thus the only survivors to Ey in total degree <8 are /3, C, and Sqiu.
It follows that Hk{X2) is generated by /3 in dimension 3, a class С = р*С in
dimension 7, and a class 5 in dimension 8 such that i*S = Si/3z5. From our
calculations, it follows that π4(53) =- Z2 and π5(53) = Ζ2. To determine
яй(53), we need only determine which Bockstein operator is non-zero on
С Now С was defined by i*C = Sq2Sq1i4e H\F,), and Sq'C=--D--^
T(Sq2i4) ε //8(ΛΊ). Thus the Bockstein lemma implies that
i*d2C
dxSq2ib
Sq3h
and so the second Bockstein d2 is non-zero. It is easy to verify that wc can
kill //7(A'2) by means of a map of X2 into A"(Z4,7) corresponding to a
cohomology class with Z4 coefficients which reduces to С (mod 2). Thus
we have π6(53) = Z4.

APPLICATIONS: SOME HOMOTOPY GROUPS OF SPHERES 127
We will not try to push these calculations any further.
Recall that, by the suspension theorem, the suspension homomorphism
E: ^(S"1)—>πί + 1(5"'4"1) is isomorphic for !<2n— 1 and epimorphic for
/=-- 2n -- ]. Thus π4(53) is isomorphic to π„+1(5") for any η > 3, that is,
π4(53) is stable. Although π5(53) —>7r6(S4) is not an isomorphism a priori,
we can deduce that it is isomorphic from the fact that it is epimorphic and
the fact that π5(53) = Z2 = π6(54). Thus π5(53) is also stable, not by
definition but, as it were, by accident. On the other hand, π6(53) = Ζ4 is
not stable, since we have seen that nn+3(S") = Z8 for η > 5.
EXERCISES
1. Let M„ = S"u2l (?"""', where 2z:S"->S" has degree 2. (Thus Mn is the
(n - l)-fold suspension of the projective plane.) Calculate nn+k(Mn) for large
n, small k.
2. Calculate 7rt(S4) for к < 7 by the methods of the Appendix.
REFERENCES
The suspension theorem
1. R. Bott [1].
2. H. Freudenthal [1].
Generalizations and the EHP-sequence
1. G. W Whitehead [2.4].
Composition-theoretic methods of
calculation
1. M. G. Barratt [2,3].
2. H. Toda [1].
The second stable homotopy group
of spheres
1. G. W. Whitehead [3].

CHAPTER
л-TYPE AND POSTNIKOV SYSTEMS
13
To work inductively in homotopy theory, we would like to have a
sequence of invariants of homotopy type such that the «th invariant
represents the given complex "through dimension w." Unfortunately the
obvious candidate, namely, the «-skeleton of a complex, is not a homotopy
invariant. In fact it is easy to find two complexes which have the same
homotopy type but in which the «-skeletons do not have the same
homotopy type.
A simple example is obtained by giving two different cell structures for
the sphere. Let A" denote S" considered as a complex having one 0-cell and
one «-cell where the /?-cell is attached by the constant map on its boundary.
Let L denote S" obtained from S""1 by attaching two «-cells (as the
"hemispheres" of S" attached to the "equator"). Then К and L are
even homeomorphic as topological spaces, but the (/? — l)-skeletons do not
have the same homotopy type, since K" ' is a point, whereas L"~ ' — S"~'.
We can circumvent this difficulty to some extent by using J. H. C.
Whitehead's theory of «-types. First wc recall (without proof) a useful
technical result, the cellular-approximation theorem, also due to
Whitehead.
Wc will use the letters K,L to denote С И'complexes and А', У to denote
arbitrary topological spaces.
128

n-TYPE AND POSTNIKOV SYSTEMS
129
CELLULAR APPROXIMATION
Definition
A map/: K—>L is cellular \ff(K") <= L" for every «, where K" denotes
the «-skeleton of K. A homotopy F: Κ χ / —>Z. between cellular maps is
cellular if F(K" χ /) <= L" +' for every «.
Theorem 1 (cellular-approximation theorem)
Let K0 be a subcomplex of K. Let/: Λ^—> L be a map such that/ | K0 is
cellular. Then there exists a cellular map g: K^>L such that g ~/rel AT0.
Corollary 1
Let /: K^L. Then there exists a cellular map g: K^L such that
Corollary 2
Let /о,/,: К^L be cellular maps. Suppose/, ~/. Then there exists a
cellular homotopy F: Κ χ Ι —>Z. between/, and/.
«-TYPE
We will define two different notions, «-type and «-homotopy type. The
reader is warned not to confuse them.
Definition
Two maps f,g of X into Υ are n-homotopic if, for every complex К of
dimension at most « and for every map φ of К into X, the compositions
f(p,gq>\ K—> У are homotopic.
It is clear that this defines an equivalence relation among maps of X
into Υ (for each «).
Proposition 1
Let f,g be maps of a complex К into a space X. Then /and # are
«-homotopic if and only if the restrictions of/and g to A^" are homotopic.
proof: Iff,g are «-homotopic, then by definition their compositions with
the inclusion map of K" into К are homotopic, but these arc precisely the
restrictions as stated. To prove the converse, it is enough to know that any
map of an «-dimensional complex into К is homotopic to a map of that
complex into K". This follows immediately from the
cellular-approximation theorem.

130
COHOMOLOGY OPERATIONS
Definition
Two spaces Χ, Υ have the same n-homotopy type if there exists a map
/: X—> Υ and a map g: Y^>X such that the compositions^ and#/are
«-homotopic to the respective identity maps. Then we say that (f,g) is an
«-homotopy equivalence and that g is an «-homotopy inverse of/.
Having the same «-homotopy type is clearly an equivalence relation.
We now come to the definition which will be essential in the sequel.
Definition
Two complexes K,L have the same η-type if their «-skeletons K", L" have
the same (« — l)-homotopy type.
That is, К and L have the same «-type if there are maps between K" and
L" such that the compositions are (« — l)-homotopic to the identity maps
of K" and L".
If we allow « to take the value oo, it is with the understanding that
" oo-homotopic " means homotopic, " K* " means K, and so forth. This is
a convention, not a theorem; it is not true, for example, that two maps
f,g: X—> Fmust be homotopic if/<p and gq> are homotopic for every map<p
of a complex into ^(see Exercise 1). However, the propositions that follow
will hold true when « is allowed to take the value oo.
We use cellular approximation again in the next result.
Proposition 2
If К and L have the same «-type, then they have the same m-type for
every m < «.
proof: We are given that there exist maps/: K" —>Z." and g:L" —> K"
such that the compositions are (« — l)-homotopic to the respective identity
maps. We may assume that/, g, and the homotopies are cellular. Then the
restrictions of/and g to the m-skeletons provide an (m — l)-homotopy
equivalence of the m-skeletons, which proves the proposition.
This proposition is valid with « = oo; that is, if К and L have the same
homotopy type, then they have the same m-type for every finite in. Thus
the «-type is a homotopy invariant of the complex.
Recall the two cell structures for S" given at the beginning of this
chapter. It is easy to verify in this example that the w-type is invariant.
For instance, if« = 3, then K2 = e° andL2 = S2 have the same 1-homotopy
type (since any map of a 1-dimensional complex into S2 is null-homo-
topic); so К and L have the same 2-type.
It is not hard to see that the 1-type of a complex is essentially a measure
of the number of connected components. Whitehead showed that two
complexes have the same 2-type if and only if they have the same funda-

n-TYPE AND POSTNIKOV SYSTEMS
131
mental group. The main subject of this chapter will be Postnikov systems,
which provide a characterization of /?-typc for larger values of n.
Recall that [X,Y] denotes the set of homotopy classes of maps of X
into Y.
Lemma 1
Let (f,g) be an /?-homotopy equivalence of X and Υ and let К be a
complex of dimension /?. Then the induced transformation f#: [K,X]^>
[Κ, Υ] is a set isomorphism.
This is immediate; for the compositions f#y# and g#f# are the identity
transformations.
Theorem 2
Suppose Z., and L2 have the same «-type. Let d\mK<n— 1. Then
the sets [K,LX] and [K,L2] are in one-to-one correspondence.
PROor: By Theorem 1, the inclusions Ц—> L, for /'= 1,2 induce one-to-
one correspondences [К,Ц] —> [K,Lt]. By hypothesis, L" and L\ have the
same (/?— l)-homotopy type. The result follows from Lemma 1.
Corollary 3
If К and L have the same «-type, then the homotopy groups nt(K) and
7Γ;(Ζ.) are isomorphic for all i <n.
The converse does not hold, but we have the following theorem of
Whitehead, which we state without proof.
Theorem 3
Suppose there is a map /: K" —> L" which induces isomorphisms on
the homotopy groups π,(Α"") s& nt(L") for all / < /?. Then К and L have the
same /?-type.
POSTNIKOV SYSTEMS
Now suppose that L is an (/? — l)-connected complex, and let π — n„(L).
Then L and K{n,ii) have the same homotopy groups through dimension /?.
Proposition 3
L and Κ(π,η) have the same (/? г l)-type.
proof: We need a map between these two spaces which induces the
isomorphism of π„. Such a map/: L > Α(π,/?) is given, using the results of
Chapter 1, by the fundamental class /„ t H"(L;n). Wc leave the details to
the reader (see Exercise 3).

132
COHOMOLOGY OPERATIONS
A natural line of inquiry would be to attempt to modify Κ(π,η) in such
a way as to obtain a space having the same (/? + 2)-type as the original
complex L. This leads to the development of Postnikov systems, the main
idea of this chapter.
Definition
Let X be an (n— l)-connected complex, with η >2 (thus X is simply
connected). A diagram of the form of Figure 1 is called a Postnikov system
for X if it satisfies the following conditions:
i
I
<Xm ,±nn+K(nm + i,m+2)
I
1
= Κ(π„η)
1
(*)
Figure 1
1. Each Xm (m > n) has the same (m + l)-type as X, and there is a map
pm: X—>A"m inducing the isomorphisms π,.(Λ') ^π^Χ„) for all i<m
2. All higher homotopy groups of Xm are trivial, that is, π£Χ„) is trivial
for / > m
3. Xm + i is the induced fibre space over Xm induced by km{X) from the
standard contractible fibre space
4. The diagram is homotopy-commutative
In Figure 1, π; denotes π,(Α'); km{X) denotes an appropriate cohomology
class in Hm+2(Xm;nm + l) or the corresponding homotopy class of maps;
and (*) denotes a space consisting of only one point.
We have already remarked that X and Κ(πη,η) have the same (n + 1)-
type. The construction of a Postnikov system is an extension of this result.
EXISTENCE OF POSTNIKOV SYSTEMS
We will now give an inductive construction to prove that for any simply
connected space X there is a Postnikov system of arbitrarily high order.
X<^
ΡΪΤ
xn

n-TYPE AND POSTNIKOV SYSTEMS
133
The induction begins with the above observation that A'and X„ = Κ(π„,η)
have the same (n + l)-type.
Suppose that the following have been obtained through a certain
dimension TV: the finite Postnikov system consisting of the Xm, p,„, etc., for all
m <N; and spaces Yn,.. .,YN with Yn <= Уп + 1 с · · · с γΝ_χ с γΝ such
that the maps pm (n < m < N) are factored as pm = rm о im:
Pm'- X-^+ Ym-^-> Xm
where /'„,: X —> Ym is an inclusion map and rm: Ym —> Xm is a fibre map such
that rm induces isomorphisms л,(Ут) да n;(XJ for all / < m and that /m is a
homotopy equivalence. We thus assume the existence of a diagram,
Figure 2, where each rm is a fibre map, all the inclusions are homotopy
equivalences, and n{{X„) = 0 for / > m.
Yn —*· Xn
I
yN_, ^U XN.X ^-u>> Κ(πΝ, 7V+ 1)
i
Х=УЯ—^ХЯ k"iX) , Д7г,+„я+2)
Figure 2
We will now show how to obtain the spaces A^N + 1 and 7N+, and the
maps /v + 1 and rN+) which satisfy the same requirements.
Let Fm denote the fibre of the fibre map rm: Ym^> Xm. It is clear from
the homotopy exact sequence of this fibre map that
^i(Fm) = 0 i<m
with the isomorphism induced by the inclusion /m: Fm—> Ym. Thus Fm is
w-connected and nm+i(Fm) ^ π„+1(УЯ1) «s π„+,(Α'), which we write simply
asirm+1.
Recall that the characteristic class of this fibre space is an element of
Ят+2(А'т;яш+1), defined as the transgression of the fundamental class of
Fm. We denote this characteristic class for m = N by kN(X), and we take a
map XN— > Κ(πΝ+χ, Ν + 2) from the corresponding homotopy class and
construct a fibre space over Xs as the induced fibre space from the standard
contractible fibre space. The total space of this fibre space will be XN + l. It

134
COHOMOLOGY OPERATIONS
is clear from the homotopy exact sequence that XN + l has the required
homotopy groups. Consider the diagram of Figure 3, where the maps g
and s are yet to be constructed. Now ks{X) is in the image of the
k=k(un+un + \)^x,+1
У 9 Ψ
Figure 3
transgression, and so /·*(£Ν(.Υ)) = 0, and therefore the composition
kNrN: yN —> ΛΓ(π„ +,, N + 2) is null-homotopic. Therefore the map /\ has a
lifting #: Уд, —> Л'д, +,. We will prove in a moment that g may be chosen so
that it induces the isomorphism πν+1(ΓΝ) ъ πΝ+ ,(Α'Ν + 1)(=πΝ + 1). Granting
this, we convert g to a fibre map /·,ν +,: Ул +, —>· Α'Λί+, by the procedure of
Proposition 1 of Chapter 9; then 7V is a subspace of 7N+1 in a natural
way and such that the inclusion is a homotopy equivalence; and iN + l:
.Y—> YN + l will be the composite of this inclusion with /v. Thus the
inductive step of the construction of the Postnikov system is completed,
after we have proved that g can be chosen as asserted.
From the null-homotopy of fcv/\v: 7V —>Κ(πΝ + ι, /V — 2) to the constant
map, we may construct a lifting .9': KN —> XN + i sucn that да' = rs
(Proposition 1 of Chapter 11). Moreover, this у maps fibres into fibres, since
g'(r„ '(*)) ^ p~\x)(xe X); we thus obtain a map
,v':Fv—>ΛΓ=Λ:(πν+1, /V+l)
by restriction. Now s' defines a cohomology class in //v + 1(Fv;7rv + 1)
(which we denote also by s'), and the fundamental class of К transgresses
to kN(X) by construction; so s' transgresses to к%(Х) by naturality. But the
fundamental class / of FN also transgresses to k^(X); in fact this was the
definition of kv(X). Since t(j') = t(/), we have s' = ι +j*(oi) for some
α ε ΗΝ + \ΥΝ;πΝ + ι), by the Serre cohomology exact sequence.
Now the standard contractible fibre space of paths over Κ(πΝ + ι, Ν + 2)
may be taken to be a principal fibre space (see Chapter 11), and then the
induced fibre space A"—> Xi%■ +, —> Xs is also a principal fibre space, with an
action ψ: Λ\+1 χ K—>XN+l. We define the map g: YS—>XS + X by the
composition
g: Y,-lisSlUX^ + l xK^+X„ + i
where —a is a well-defined element of the group Η"^+\ΥΝ;π^ + ]) and

n-TYPE AND POSTNIKOV SYSTEMS
135
hence represents a homotopy class of maps YN —> K. Then pg = pg' = rN
(by condition (1) for a principal fibre space), and we can define a map
s: FN^>K by restriction of y. It follows from the definitions of s and
s' and that of g that s — φ с (У, —У*(я)). But the restriction of ψ to Κ χ Κ
is loop multiplication in Κ— ΩΚ(πΝ+ι, Ν Η 2), which corresponds to
addition in Ял'"1(Гл;лу+1); and so we have, in cohomology, the relation
s = s' —j*(oi). Thus s = ι.
Since s — ι, the map s induces an isomorphism on the (/V-f- l)th
homotopy group. But it is clear that the inclusion maps of the fibres also induce
isomorphisms on this group. From the commutativity of the " square " in
Figure 3, it follows that g also induces an isomorphism on πν+,, which was
to be proved.
This completes the construction of the Postnikov system for X. By
Milnor's result (see Chapter 9) we may assume the spaces Xm are nice.
NATURALITY AND UNIQUENESS
It will be convenient to have a uniqueness theorem for Postnikov systems,
asserting that any two such systems for a given space X are in some sense
equivalent. It will also be useful to have the "functorial" property that a
map from a space A' to a space Υ gives rise to a " map " from the Postnikov
system of X to that of Y. These properties are consequences of the
following theorem.
Theorem 4
Let А', У be simply connected complexes, and let/be a map from Xto Y-.
Let Postnikov systems be given for X and for Y. Then there exists a family
of maps {fm: Xm-^> Ym} with the following properties:
1. f:(km(Y))=f»km(X)£H^2(Xm,nm^(Y))
2. The diagram
л т + 1 * 1 m + 1
Ύ Ύ
У fm * Υ
л т * 1 m
is commutative, and the diagram
X —L+ Υ
4 μ,
Лт * ' т
is homotopy-commutative

136
COHOMOLOGY OPERATIONS
3. If/and g are nomotopic maps from X to Y, then/m and gm are homo-
topic (for every m)
In (1) we mean by f# the coefficient homomorphism induced by the
homomorphism/* : nm+ X(X) —> nm+ ,( Y).
As a corollary, X„, is determined by X, up to homotopy type; for we may
take Υ = X and apply (3) to obtain a homotopy equivalence between the
two versions of Xm.
We omit to prove this theorem; the reader now has all the necessary
tools available.
DISCUSSION
If X is a complex, the «-skeleton X" has the same homology (or
cohomology) as X through dimension η — 1 and has no homology above
dimension n. Evidently the term X„ in a Postnikov system for X plays an
analogous role for homotopy, but of course the situation is much more
complicated and geometrically much less natural.
If X has only finitely many non-trivial homotopy groups, sothatTr^-Y) — 0
for all / > N (for some N), then Xs has the same homotopy type as X and
we can say that X has a finite Postnikov system.
If К is a complex of dimension η (or less), then we can identify the sets of
homotopy classes [K,X] = [K,X„], since X and Xn have the same in -\- I)-
type. This simplification of the task of computing [K,X] will be pursued
further in Chapter 14.
In a certain sense, the Postnikov system provides a complete system of
invariants for «-type for every η (I <«<oo). For example, let X be
(n — l)-connected. Then the (n + I)-type of X is that of Κ(π„(Χ),η), so that
π„ is a complete set of invariants for (n ~- I)-type. (We have proved this
for η > 1; it holds for η — 1, but the details are more complicated.) The
(n + 2)-type of X is that of X„+l, and so it is completely determined by
π„, π„+1, and kn. Here kn ε Я"+2(я„,«; π„+ ,)■ In general, the (n f^ + 2)-
type of X is that of Xn+q+l and thus is determined by the groups
π„, - - .,π„+ί+, and the cohomology classes k„,. . .,£„+,. In our formulation,
it is not quite precise to call the £„'s invariants. Indeed, one may have fibre
spaces of the same homotopy type induced from the same contractible
fibre space by nonhomotopic maps. Nevertheless the £„'s are called the
k-irwariants of the space X.

n-TYPE AND POSTNIKOV SYSTEMS
137
Postnikov systems were indeed discovered (and called " natural systems ")
by Postnikov. Theorem 4 precisely expresses the manner in which the
Postnikov system is a natural system,· this theorem was proved by D. W.
Kahn.
EXERCISES
1. Give an example of spaces Χ, Υ and maps f,g: λ"-> Υ such that/<p ~ gep for
any map φ of any complex К into X but with/,# not homotopic.
2. Give a counterexample to the converse of Corollary 3.
3. Let L be (n - l)-connected and let π = n„(L). Verify from the definitions that
the fundamental class / of L gives a homotopy class of maps L -> Κ(π,η) which
induce the isomorphism of π„.
4. Show that the homotopy classes [K,X„] are in one-to-one correspondence
with the set i*([Kn+\X]) с [K\X]. Here К is a complex, i.K"c A:n+1isthe
inclusion of the skeletons. Л' is a simply connected space, and X„ has the
(n + l)-type of X but with nt(Xn) trivial for ;' > n.
Show that the above correspondence is natural, in the obvious sense, if
Xn is taken to be the «th term in a Postnikov system for X.
REFERENCES
Cellular approximation and η-type Postnikov systems
1. P. J. Hilton [1]. 1. D. W. Kahn [1].
2. J. H. С Whitehead [1]. 2. Μ. Μ. Postnikov [1,2,3,4,].
3. E. H. Spanier [1].

CHAPTER
MAPPING SEQUENCES AND
HOMOTOPY CLASSIFICATION
We will present two constructions of a spectral sequence for [K,X]. One
is based on the Postnikov system for X; the other, on the cell structure
of AT.
THE FIBRE MAPPING SEQUENCE
Let (Ε,ρ,Β; F) be a fibre space in the sense of Hurewicz. If we take the
inclusion^': F—>£of the fibre and convert this map to a fibre map
(Proposition 1 of Chapter 9) and then repeat the process on the inclusion of the
new fibre and continue ad infinitum, we obtain what we will call the fibre
mapping sequence.
The interest of the mapping sequence lies in the fact that it has a kind
of periodicity. Precisely, when we convert / to a fibre map /': F'—>F, the
fibre С is a space of the same homotopy type as ΩΒ. Granting this, then
similarly the inclusion k: G—>F' gives a fibre map k': G' —>F' with a fibre
Η of the same homotopy type as Ω£. Moreover, it turns out that the
inclusion of Η in G' is, except for sign, homotopically equivalent to the
map Ωρ: Ω£—>Ωβ. (For the proofs of these facts, see the Appendix to this
chapter.)
We thus obtain a sequence which may be written
>Ω2β—>Ω£->Ω£->Ωβ^ F^E~> В
138

MAPPING SEQUENCES AND HOMOTOPY CLASSIFICATION 139
where we have put Ffor F', Ωβ for G', etc. Each map in the sequence is a
Hurewicz fibre map. Therefore, if К is any complex, the homotopy classes
of maps of К into the terms of this sequence give an exact sequence
>[K,Cl2B]^[K,ClF]->[K,ClE]^[K,ClB]->[K,F]^[K,E]^[K,B]
Here the last three terms are not necessarily groups, but they are at least
sets with distinguished element, so that exactness makes sense.
We remark that if the original fibre space F—>£—>β is induced by a
map β—> К from the standard contractible fibre space over Y, then we can
add a term at the end of the above exact sequence, so that it terminates
>[K,F]^[K,E]^[K,B]^[K,Y]
In general the set of homotopy classes [K,X] has a natural group
structure if .Y is a space of loops. Thus we can make the following remark.
Proposition 1
[K,X] has a natural group structure if the space X is (n — l)-connected
and the dimension of the complex К is at most 2n — 2.
This follows from the fact that A'and ΩΞΧ have the same (2n — l)-type,
by the suspension theorem.
In the sequel we always assume [K,X] has a natural group structure.
Other sufficient conditions include К being a suspension or A' a loop space
(see Chapter 15).
We will make implicit use of some facts about the behavior of cohomol-
ogy classes and operations under the action of the looping functor Ω.
Consider a cohomology class θ e H"+k(n,n; G) and let 0 also denote a
corresponding map of Κ(π,η) into K(G, η — к). Then CIO is a map of
Κ(π, η — 1) into K(G, η + к — 1), which corresponds to a cohomology class
in H"+k'l(n, и — 1; (7). In this way we obtain a transformation
Hn+k(n,n;G)^Hn+k-\n,n- \;G)
which is called the cohomology suspension and written '( ), so that the
class which we denoted by Ω0 above is written ιθ. It is clear that the above
transformation is an isomorphism in the range of Serre's exact sequence.
We have commutative diagrams
K(Z, η - 1 )-*.'-► K(Z2, η Λ ι - 1)
ΩΚ(Ζ,η) -15sU ΩΚ(Ζ2, η + /)
(and similarly with Z2 replacing Ζ), and in this sense we have \Sq!) = Sq!.
One expresses this fact by saying that the Sq! are stable operations.

140
COHOMOLOGY OPERATIONS
MAPPINGS OF LOW-DIMENSIONAL COMPLEXES INTO A SPHERE
In Chapter 12 we computed some homotopy groups of spheres by a
construction which may be called the "mod 2 Postnikov system" for
S". In fact, up to dimension и + 2, this is a Postnikov system, with no
apologies about mod 2, according to the following results.
Proposition 2
Let ρ be an odd prime. Then the least integer η such that π„(53) has a
non-zero element of order ρ is η = 2p.
For an indication of the proof, see Exercise 1.
Corollary 1
nn+k(S") contains no elements of odd order for к < 2.
proof: From the proposition, π„(53) has no elements of odd order for
η < 6. By the suspension theorem, π„ + 1(5") ^ π4(53) and this group must
be Z2, since the 2-component is Z2 and there is no odd torsion. Also,
nn + 2(S") во π6(54), and this is the homomorphic image of π5(53) = Ζ2; so
π„ + 2(5") contains no elements of odd order.
Thus, through dimension η -\ 2, the Postnikov system of S" (for η large)
is exactly the mod 2 system previously constructed.
We will use this Postnikov system and the fibre mapping sequence of
certain maps to study the classification of maps of a complex К into S".
Suppose first that К has dimension n. Then [A^S"] = [Κ,Κ(Ζ,η)] since
K(Z,n) and S" have the same (n + l)-type. Since [K,K(Z,n)] is in 1-1
correspondence with H"(K;Z), we recover the classical Hopf classification
theorem for [K,S"].
Suppose next that К has dimension η ~\. We have [A",5"]= [A',A'„ + 1]
where Xn + X is the appropriate stage in the Postnikov system for S".
Consider the diagram below where the map Ωβ—> F, is essentially Ω(5^2) = Sq2.
Ωβ= K(Z, и - 1)—► F, = K(Z2, n + \)-U.X„ + l
i
В = Κ(Ζ,η)-^. Κ(Ζ2, η 4- 2)
Mapping Κ into the successive spaces in the diagram, we obtain an exact
sequence
[K,CiB]-U [K,Ft]-L> [Κ,Χη + ι]-ϊ+ [K,B]-±> [K,K(Z2,n + 2)]
in which each term is a group. (We recall that Κ(π,η) can be thought of as
ΩΚ(π, n + l) and [Κ,Κ(π,η)] = ΗΠ(Κ;π).) Thus we know [K,Xn + l] up to a

MAPPING SEQUENCES AND HOMOTOPY CLASSIFICATION 141
group extension, since we have a short exact sequence
0 ->coker/-> [К, *„ + ,]—> ker &—>0
where coker/= H" + l(K;Z2)/Sq2H"-\K;Z) and where ker к is the kernel
of Sq2: Hn(K;Z)^H"+2(K;Z2).
We will push this method one step further and consider [A",5n] where К
has dimension η + 2. Here [K,S"] = [K,X„ + 2] where X„ + 2 is the next stage
in the Postnikov system for S". By converting certain maps to fibre maps
according to the method described at the beginning of the chapter, we can
expand the Postnikov system (up to X„ + 2) into the diagram of Figure 1.
a2X„^^*QF2-K(Z2,n+ I) >ilX„,2
K(Z, η-2) -^» 0.F, - K(Zi, n) —£i—> L1X„ +, ^—* F2
i Ь
(*) *K(Z,n-\) = >K(Z,n- l)-^/7,
i
(*)
Figure 1 Diagram for analysis of [K, 5"] — [A\A"„ + 2] (dim К = я + 2)
Here each symbol (*) denotes a one-point space, and we have used the
remark that \Sc/2) — Sq2. The map 1 was defined in Chapter 12 and
satisfies the relation /*a - S?2(».. + i) с W" + J(F, ;Z2).
To study [Κ",λ^ 2], we map К into the following portion of Figure 1 :
J
*„ + 1-^*(Ζ2,Λ + 3)
and obtain a short exact sequence
0 —>coker (Ωα)* —> [*,*„+ 2] -> ker a* —>0
Now я# : [A-,.^,] — >[K, K(Z2, η -\ 3)] is the zero homomorphism, since
[K,K(Z2.n I 3)] is classified by Hn^3(K;Z2), which vanishes by the
assumption on the dimension of K. Thus kera, is the entire group
[КъХпц\ which we have described previously.
It is much more complicated to describe coker (Ωα)#. Let A denote the
kernel of Sq2: H"'\K;Z) + Hn+\K;Z2) and let С denote the cokernel of
■K(Z2,n-\ 2) >X„<.2
K(Z2,n-\-\)-^X„ + 1—ϊ-»· K(Z2, η + 3)
—> K(Z,n)-^+B= K(Z,n)-^*K(Z2, η+2)
I
(*)

142
COHOMOLOGY OPERATIONS
Sq2: H"(K;Z2)^H" + 2(K;Z2). (The reader should identify these cohomol-
ogy groups with the appropriate locations on Figure 1.) We can define a
transformation
ί/3 = (Ωα)(Ωρ)-1:/1 >C
In terms of the figure, we use the portion
QFt -= K(Z2,n)^ ΩΧΠ+, — F2 = K(Z2, η+ 2)
K(Z, η - 1)^- K(Z, η - 1)—► F,
An element of A corresponds to a map of К into K{Z, η — 1) which can be
lifted to a map into ΩΧη + ι and thus continued to F2, resulting in an
element of [K,F2] = Hn + 2{K\Z2). This element is not well-defined there,
since the lifting may vary, but is well-defined as an element of the quotient
group C, since the lifting may only vary by the effect of a map of К into
Ω77, and the composite HF, —>ΩΧΠ+, —> F2 is the Sq2 used to define C.
We claim that coker (Ωχ)# is isomorphic to coker аъ; we omit the proof,
which is a tedious verification. Granting this, we obtain a composition
series for [K,X„ + 2] with the following terms: (1) ker Sq2 <= H\K;Z);
(2) coker Sq2 = ΗΠ + \Κ;Ζ2)/Sq2Hn' \K;Z); (3) coker d3 as above. Of
course (1) and (2) represent [K,Xn + i].
THE SPECTRAL SEQUENCE FOR [K,X]
We hope that by now the reader can see what is coming. Abandoning any
dimensional restriction on К and no longer assuming X a sphere, we can
subsume the above cases into a general framework to obtain a spectral
sequence for [/C-Y].
We obtain the exact couple simply by mapping the complex К (of any
dimension, now) into the diagram of Figure 1 (extended upwards and to the
left according to a Postnikov system for A'). We will denote the initial stage
with the index 2. The D2 terms are homotopy classes of maps of К into
the terms of the Postnikov system for the space X and into the derived
towers, i.e., into the first, third, and fifth columns of Figure 1. The E2
terms come from maps of AT into the fibres. The bigradation will come from
the plane coordinates implicit in the diagram.
Explicitly, we have an exact couple
E2

MAPPING SEQUENCES AND HOMOTOPY CLASSIFICATION 143
where D-™ = [Κ,ΩρΧη_ρ] and E2P·" = [K,a'Fq-„-„]. All the D's and £"s
are groups. It may be more convenient to use the natural equivalence
[Κ,ΩΥ] = [ΞΚ,Υ] to obtain alternate expressions for these groups. For
example,
£2-'·« = [K,a>F^p_„] = [K,W(K(nm(X),m))] m = g-p
= [SpKK(nm(X),m)]
= H'-'(S'K;nt_p{X))
Thus £2-'·" + 2 = [Κ,ΩΓ,]= [Κ,Κ(Ζ2,η)]= H" + i(SK;nn+l(X)).
The map ?2 is induced by the projections in the towers and has bidegree
(0,-1); thus, typically,
/2: D2-"" + 1^D2^: [K,WXQ_p+l)^[K,Cl'XQ_p]
The mapy2 is induced by the ^-invariants and has bidegree (1,0):
j2: /?,-'·«->£2-'+'-«. [K,n'Xq_p]^[K,a>-lF4-p+l-K]
The map k2, of bidegree (0,0), comes from the fibre inclusions:
k2: E;'*^D;'*: [K,il'FQ-p_„]^[K,WXQ_p]
Then of course d2 =j2k2: E2P'4^>E2':' + Uq, This comes from "two steps
sideways," for example, from [ΛΓ,Ω/7,] to [K,F2]. Observe that the d2 is
induced by a map of Κ(π,η) spaces, so that it corresponds to a primary
cohomology operation.
We leave it to the reader to trace the " path " of a higher differential on
the diagram of Figure 1; dr has bidegree (1, r — 2) for r > 2.
If К is finite-dimensional, each column of the diagram is finite and we can
replace X by X4 for q = dim K. We obtain a composition series for
[Κ,ΩΓΧ] = [SrK,X] where the successive quotients in the series are the
terms E'"·" with —p = r. The filtration is defined by taking F} to be the
kernel of the map [K,X]^>[K,Xj] induced by p};X^X}\ we have
Fj-JFj -^ £°J and 0cf.cf..|C ... cfu= [κ,Χ] (assuming X
connected). The filtration is thus obtained by means of the Postnikov system
for X.
The reader should reexamine the example [K,Sn] where К has dimension
/? -l 2 to see how it fits into the general scheme.
THE SKELETON FILTRATION
AND THE INCLUSION MAPPING SEQUENCE
The above theory can be dualized. Let К be a complex and L a sub-
complex, with /: Z,—> К the inclusion. Letybe the inclusion of К in К u CL.

144
COHOMOLOGY OPERATIONS
Then, if A' is a simply connected space,
[L,X]+£-[K,X]*£-[Kv CL, X]
is an exact sequence (the verification is easy); in fact Z,—> A—>A u CL is a
"co-fibration." We have assumed that (K,L) is a pair of complexes so
that А и CL will have the same homotopy type as KJL.
We now consider^': A—> А и CZ. and form (A u CL) и САГ, which is the
" co-fibre " of the inclusion map/ It is not hard to see that (A u CL) и СК
has the homotopy type of the suspension SL. The next step produces' a
complex of the same homotopy type as SK, and the map is homotopically
equivalent to Si: SL—>SK. In this way we obtain the inclusion mapping
sequence which may be written, up to homotopy equivalence, as
L-UK-U K/L-> SL^U. SK-ϊύ S(K/L)->S2L-> ■ ■ ■
Mapping the terms of the above sequence into a simply connected space
X, we have the exact sequence
[L,X] +- [K,X] <- [K/L,X]«- [SL,X] <- [SK,X]«- [SK/SL,X] <
where the first three terms are sets with distinguished point, the next three
are groups, and all the rest are abelian groups.
We remark that it is possible to give a suitable definition of the cell
structure of the complex SK so that S(K/L) = SKJSL and, less obviously, so
that Sr(K") = (5rA)p+r (where K" is the p-skeleton of A).
Fix ρ for the moment and note that Kp/Kp~l is a wedge of p-spheres.
According to the preceding remarks about SrKp, we can identify
S'iK'/K"-1) = ΓΚ'/ΓΚ'-1 = (SrKY+rKSrK)p+'-'i
which is a wedge of (p + /-)-spheres, and we can then identify the maps of
this wedge into X with a cochain group,
[(SrK)p+'/(SrK)p+r-\X] ъ Cp+r(SrK;np+r(X)) « Cp(K;np+r(X))
Thus if we take K= Kp, L = Kp~\ and let ρ vary, the resulting sequences
fit together to form an exact couple, with a typical Dx term given by
[SrK",X] and a typical Ei term by [S\KPIKp-x),X] *, C"(K;np+r(X)).
The reader should draw an appropriate diagram. The differential
dx: C(K;np+XX))^ Cp+l(K;np+r(X))
turns out to be the usual cohomology coboundary operator, up to sign. A
typical £2 term has form Hp{K;np+r{X)).

MAPPING SEQUENCES AND HOMOTOPY CLASSIFICATION 145
With a suitable bigrading chosen on £, and /?,, one finds that the E2
term of the resulting spectral sequence is isomorphic to the E2 term of the
spectral sequence obtained from the exact couple of the fibre mapping
sequence. By Exercise 2 of Chapter 13, the same holds for the D2 terms. The
filtration, instead of being based on the kernels of the maps [λ',Α']—>
[K,Xj], is based on the kernels of the maps [K,X]^[KJ\X]. These filtra-
tions coincide and we obtain the same spectral sequence for [K,X].
DISCUSSION
We have given two constructions of the spectral sequence for [K,X].
Without precisely defining duality, we view these constructions as dual.
We think of fibration and co-fibration as dual notions. The " building
block " using fibrations is the Eilenberg-MacLane space; for co-fibrations,
it is a wedge of spheres (having only one non-vanishing cohomology
group).
The inclusion mapping sequence is often referred to as the Puppe
sequence; it was first studied extensively by Barratt. The existence of the
spectral sequence for [K,X] is perhaps more obvious from this point of
view. On the other hand, the Postnikov approach shows how the ^-invariants
of X and cohomology operations in К affect the result; namely, they
determine d2. Subsequently we will introduce secondary cohomology
operations; the reader will then immediately be able to interpret the next
differential.
APPENDIX:
PROPERTIES OF THE FIBRE MAPPING SEQUENCE
Lemma IA
Let (Ε,ρ,Β; F) be a Hurewicz fibre space. Let Ζ denote the subspace
{(еф): p(e) = fi(0)} of £ χ В'. Then there is a map A:Z—>£' such that
λ(β,β)(0) = e and ρλ(β,β) = β.
Here ρ denotes the map £—>5 and also the induced map E'^B1. It
will sometimes be convenient to use the notation Ε * В' for Ζ. Mapping
spaces like E' are always understood to have the compact-open topology.
proof: We have a map Z—>£ taking (β,β) into e; this projects to a map
Ζ—> Β taking (β,β) into p(e) = β(0). We have a homotopy of this latter map,

146
COHOMOLOGY OPERATIONS
taking Ζ x / into В by taking (β,β; ή into /?(/). Applying the covering
homotopy theorem, we obtain the map λ of the lemma.
Lemma 2 A
The map E'^>E' given by ω—>;.(ω(0),ρω) is homotopic to the identity
map, and the homotopy is " vertical," that is, the image of any point remains
in its fibre throughout the homotopy.
The proof may be left as an exercise.
Lemma ЗА
Let (Ε,ρ,Β; F) be a Hurewicz fibre space. Convert the inclusion/: F—>£
into a fibre map as in Proposition 1 of Chapter 9. Let G be the fibre of
/: f—>£. Then G has the homotopy type of ΩΒ.
proof: Recall that F' = {(/,ω): j{f)| = ω(0)} с f χ £' and that/ is
defined by /(/,ω) = ω( 1). Let e0 denote the base point of the pair (E,F).
Then (7 = (/Wo), so that
G =-- {(/,ω): ω(0) =j(f), 41) = fo}cf
We have an obvious map φ: (7—> ΩΒ: (/,ω)—>ρω. We obtain a map
ι/': Ω2?—>(7 as follows. Given a path /? e B1, let ε denote the inverse of the
path λ(β0,β~'). Then put φ(β) = (ε(Ο),ε). This defines a map of ΩΒ into G,
since ε(Ι) = /(έ>0,/Γ')(()) = έ>0.
We must prove that the compositions φψ and φφ are homotopic to the
identity maps. Now ψφ(β) = <ρ(ε(0),ε) = ρε = β, so that φφ is actually the
identity itself. In the other direction, φψ(/,ώ) = φ(ρω) = (y(0),y) where
y_1 = λ(ε0,(ρω)~ι). Now (/,ω)ε(7; so ω(Ι) = έν Thus л{е0,{рт)~1) =
?.(а>~1(0),р(а>~*)); but the mapco—>y_1 = ;.(ω"'(0),/)(ω"1)) is homotopic to
the map ω—>ω-1, by the previous lemma. Thus the map ω—>y is homo-
topic to the identity map ω—>ω. It follows easily that the map φφ: (7—>
G: (/, ω)—>(y(0),y)is homotopic to the map(/,co)—>(ω(0),ω) = (/ω), i.e.. to
the identity. This proves the lemma.
Since the composition φφ:ΩΒ- >ΰ—>ΩΒ is the identity map, we have
actually proved that ΩΒ may be identified with a deformation retract of G.
THE MAP FROM Ω£ TO ΩΒ
Now consider the diagram below, where (Ε,ρ,Β; F) is the original
Hurewicz fibre space. The inclusion j: F^E is converted to a fibre map/
with fibre G, and the inclusion k:G^>F' is converted to a fibre map

MAPPING SEQUENCES AND HOMOTOPY CLASSIFICATION 147
k' with fibre H. The four inclusion maps indicated by vertical arrows are
homotopy equivalences.
Ω£-2£-> ΩΒ F-ύ £-£* В
-I \/
ΩΕ G—U F'
Я—i-^(7'
Lemma 4A
The maps / and Ωρ are homotopically equivalent except for sign.
The proof requires no deep thinking; it consists principally of tracing
the definitions through the diagram. We present it here for the curiosity
value of the result that the homotopy equivalence is not the natural one
but must be obtained by reversing all the paths in ΩΕ. Precisely, we will
prove that the following diagram is homotopy-commutative where Г is the
homotopy equivalence which reverses the parametrization of every path
and a, /?, and у are the retractions given by Proposition I of Chapter 9
(for β) and Lemma ЗА (for a and y). The need to use Τ in this lemma may
be viewed as the source of an exasperating (— 1)" which appears frequently
in algebraic topology. In this book we usually work mod 2.
ΩΕ-^ΩΒ
1 ί-
ΩΕ G
•1 ί'
Let a typical " point" of Η be {g,q>') where g = (/,ω) e G and ψ' is a
path in F'. Since (#,<?') e H, we have (p'(0) = g, φ'(\) = (e0,e*) where e0 is
the base point in F<= Ε and e* is the constant path there. Since g e G, we
have ω(0) =/and ω(Ι) = e0.
The two routes around the diagram give the following calculations for
the image of (#,<?') in ΩΒ:
yW(g,<p'))) = уШ<р')) = v(g) = K/>) = ΩΜω)
QpWsrf))) = QpOWW))) = τ(ΩΡ(Ω/(φ')))
where the last Γ denotes an operator on B. Now it is readily seen from the
definitions that pf takes a point (e,e) e F' = F * E' to the point ρ(ε(1)) e B.
The path ψ' e (/")' is a continuous deformation of (/,ω) to the base point

148
COHOMOLOGY OPERATIONS
(e0,e*) of F'. Geometrically speaking, we take (/,ω) and move the point/
along the fibre F to e0, while the path ω is free (except that its initial point
moves with/) until at the end of the deformation ω is collapsed to e*. Then
Ωρ(Ω)'(φ')) is the track in В of the endpoint of this path. If we write
ψ'{ί) = (/„ω,) e F', then we must compare the path /—>ρ(ω(?)) with the
path r—>/>(ω,(Ι)). Consider the square I1 mapped into £as indicated in the
diagram. The horizontal segments are mapped by ω, and the whole is a
/-
ω
ft ! ω,
(1)
continuous map /2—>£ which essentially represents ψ'. When we project
this square into В by means of p, the left edge and bottom edge and the
upper-right corner are mapped to the base point b0 = p(e0). This square
then yields a homotopy between /—>ρ(ω(Ι — /)) and /—>ω,(Ι), represented
in the diagram by imagining the square being swept out by a radial line
rotating about the upper-right corner. This homotopy completes the proof
of the lemma.
EXERCISE
Let/: S3 -> K(Z,3) generate #3(S3). By the construction of the fibre mapping
sequence, we may assume a diagram X —2-> S3 —£* K(Z,3), where X is the fibre
of/. Converting ρ to a fibration, we have a fibre space (X,p,S3) where the
total space A4s 3-connected and ρ induces isomorphisms π;(Υ) ν π,(53) for
all (' > 3. Use this to show that the smallest integer η such that π„(53) contains
an element of order p, where ρ is an odd prime, is я = 2p. (Find the coho-
mology of the fibre, write the cohomology spectral sequence, and apply the
universal coefficient theorems and the Hurewicz theorem mod С .)
REFERENCES
The fibre mapping sequence The inclusion mapping sequence
I. F. P. Peterson [I]. I. M. G. Barratt [I].
2. D. Puppe [I].

CHAPTER
PROPERTIES OF THE
STABLE RANGE
From time to time we have considered various phenomena which exhibit
particularly regular behavior in a certain range of dimensions depending
on the connectivity of the spaces in question. Perhaps the most
fundamental example is the suspension theorem (Theorem 1 of Chapter 12),
which asserts that the suspension homomorphism Ε: π,(ΛΓ)^π;+1(5ΑΓ) is
an isomorphism, provided i <2n — 2, where X is (n — I)-connected.
Stated briefly, the theorem asserts that the suspension homomorphism Ε is
an isomorphism " in the stable range."
In this chapter we examine various constructions which have a stable
range and we point out exactly what nice property holds and in what range.
In future chapters when we wish to restrict attention to the stable range,
we merely utter the magic phrase and usually leave to the reader the task of
working out exactly what dimensional hypotheses are necessary.
NATURAL GROUP STRUCTURES
We first recall the ways in which a set of homotopy classes [K,X] may
have a natural group structure; namely, К a suspension or A' a space of
loops. Speaking precisely, the pinching map SK ^SKv SK induces a
group operation on [SK,X], natural with respect to maps of X, while the
multiplication ΩΧ χ ΩΧ^ΩΧ induces a group structure on [Κ,ΩΧ],
149

150
COHOMOLOGY OPERATIONS
natural with respect to maps of K. On the other hand, we recall also that the
sets [SK,X] and [Λ^,ΩΑ'] are naturally equivalent; this equivalence is a
group isomorphism as well. This observation makes trivial the proof of
the following result.
Proposition 1
Let/: L->К and g: JT-> У. Then the induced maps (Sf)* : [SK,X]^
[SL,X] and {Clg)#[K,ClX] —> [Κ,ΩΥ] are group homomorphisms.
For our mapping sets to be abelian groups, we consider mapping sets of
the form [S2K,X], [SK,aX], and [Κ,Ω2Χ]. We ask the reader to verify that
each has an abelian group structure, that the three are naturally isomorphic
as groups, and that maps SL^>SK or ΩΧ^> Ω У induce group
homomorphisms (see Exercise I).
Now suppose A'is (n — I)-connected. The identity map of S2 X induces a
map h: Χ—> Ω252Χ; by the suspension theorem, h induces an isomorphism
in homotopy through dimension 2n — 2. Thus by Theorems 2 and 3 of
Chapter 13, dim К < In - 2 implies that h# : [K,X]-^> [Κ,Ω2Ξ2Χ] is a one-
to-one correspondence. Hence by transport of structure we have the
following.
Proposition 2
Suppose Xis, (n — l)-connected and dim К < 2n — 2. Then [K,X] has an
abelian group structure, natural with respect to maps f:L—>К and
g: .Y—> У, where У is (n — l)-connected and dim L < In — 2.
Briefly stated, in the stable range [K,X] has a natural abelian group
structure.
EXAMPLES: COHOMOLOGY AND HOMOTOPY GROUPS
Recall by Theorem 1 of Chapter 1 that we have a one-to-one
correspondence between [Χ,Κ(π,η)] and Η"{Χ\π). Viewing Κ(π,η) as ΩΚ(π,η + 1),
we obtain a group structure on [Χ,Κ(π,η)], natural with respect to maps of
X. The correspondence is indeed an isomorphism of groups—in fact,
[Χ,Κ(π,η)] is probably the most suitable definition of Ηη(Χ;π).
Now let 0 e Η"(π,η; G) be a cohomology operation of type (π,η; G,q). Is
0: Η"(Χ;π)^H"(X;G) necessarily a group homomorphism? In general,
the answer is no. (Consider, for example, the operation taking и to u2 on
two-dimensional classes in the integral cohomology ring of S2 χ S2.)
However, Proposition 1 tells us that the answer is yes, provided 0 can be written
in the form Ωψ for some φ: Κ(π, η + \)^>K(G, q + 1), i.e., provided θ is
in the image of the cohomology suspension.

PROPERTIES OF THE STABLE RANGE
151
We say that 0 is additive if 0: Η"(Χ;π) —> Hq{X;G) is a group homomor-
phism for every X. Summarizing the above remarks, we state the following.
Proposition 3
Suppose 0 is in the image of the cohomology suspension. Then 0 is
additive.
We next consider the import of Proposition 1 for homotopy. Since n:(X)
is just [5',^], the group structure in π; arises from viewing S" as a
suspension for i > 1; π, is abelian for i > 2 since S' is then a double suspension.
The naturality with respect to maps g: X^> Υ is just the statement that
g# : π^-Υ)—>π,·(7) is a homomorphism.
Now let /: S^—>S". Then composition with/induces a natural map,
also denoted /, from π{Χ) to πρ(Χ). By analogy this transformation is
called a primary homotopy operation. Need such a primary homotopy
operation be additive? In general, the answer is no, but Proposition 1 tells
us that the answer is yes if/ is a suspension element.
As a very important special case, take Xalso to be a sphere, say, X= S".
Suppose/) < 2/ — 1, so that/is perforce a suspension. Then the composite
g o/ (where g: S"'—>S") is a bilinear function of the variables and thus
defines a pairing n^S") ® πρ(5"')—> np(S"). This pairing is clearly compatible
with suspension, that is, E(g of) = E(g) о E(f). Therefore the pairing is
well-defined on the stable stems.
We denote the stable &-stem by Gk. Thus Gk = nn+k(S") for η sufficiently
large. Let G be the graded group with Gk as its &th term. We now may state
the following theorem.
Theorem 1
The composition operation endows G with the structure of
commutative graded ring with unit.
By the above remarks, and by the observation that the identity map of the
«-sphere serves as unit in G0, only the commutativity need be proved. We
omit the proof of this result of Barratt and Hilton but remind the reader
that commutativity for a graded ring involves a sign.
In Chapter 17 we will compute the product structure in the 2-component
of С in the range of the calculations of Chapter 12, namely, through degree 7.
CONSEQUENCES OF SERRE'S EXACT SEQUENCE
We now recall Theorem 2 of Chapter 8 and its analogue for cohomology.
Suppose p: E^>B is a fibration with fibre F. If В is (p— l)-connected

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COHOMOLOGY OPERATIONS
and Fis (q — I)-connected, then we have the exact sequence
Hp+Q.t(F)^ Hp+q.t(E)^^ HpU.t(B)^ HP+Q.2(F)^ ■ ■■
and similarly in cohomology.
Speaking roughly, wc would say that Serre's exact sequence is valid in
the stable range.
We now apply Serre's exact sequence to the cohomology suspension,
which we define for an arbitrary space Xas follows. Let и e Η" + )(Χ;π) be
represented by the map u: Χ^Κ(π, η + 1). Then the cohomology
suspension of u, 'we Η"(ΩΧ;π), is the class represented by the map Пи.
To study the cohomology suspension, consider the contractible fibration
over Xwith fibre ΩΧ. In the range of Serre's exact sequence, the suspension
and the transgression are inverse homomorphisms (see Exercise 2). We
will obtain the following.
Proposition 4
Suppose X is m-connected. Then the cohomology suspension
'( ): Нк*\Х)-+Н\С1Х)
is an isomorphism if к < 2m — I.
proof: Wc apply Serre's exact sequence with В — X, F — ΩΧ, and Ε
contractible. Then ρ = m — 1 and q = m. The isomorphism holds for
к <p — q — 2 = 2m — I.
Corollary 1
Let θ e Hk(n,m; G). Suppose к < 2m — 1. Then we may write 0 = ιψ in a
unique way. Hence 0 is additive.
Briefly put, in the stable range a cohomology operation is additive and
may be uniquely desuspended.
For we Hk(Y;n), we denote by E(u) the total space of the fibration
induced over У by и from the contractible fibration over K(n,k).
Proposition 5
Suppose X is m-connected and η <2m— 1. Let us Ηη(ΩΧ;π). Then
E(u) has the homotopy type of a loop space.
proof: By Proposition 4, we may write и = xv for some υ с Η"+ι(Χ;π).
Then E(u) and ΩΕ(ν) have the same homotopy type.
Moreover, by the results of the Appendix to Chapter 14 together with
Proposition I. we note that for each K, the map [Κ,Ε(ιή]^>[Κ,ΩΧ] is a
homomorphism.

PROPERTIES OF THE STABLE RANGE
153
Corollary 2
Suppose X is (n — l)-connected and π{Χ) = 0 for /"> 2/7 — 1. Then X
has the homotopy type of a space of loops.
proof: Consider a Postnikov system for X with typical term Xj. We
work by induction on/ Now Xj+1 is of the form E(u) for a certain class
и e Ηρ(Χ};π) for some π, where ρ < 2n— 1. By the induction hypothesis,
Xj is a space of loops (on an «-connected space, since X is (n — 1
^connected). Therefore Proposition 5 applies and E(u) = XJ +, is a loop space. But
X= X2n, and thus the corollary is proved.
We next consider the relation between the fibre and cofibre of a map.
Suppose p: £—>β is a mapping. By the standard construction, we may
assume ρ is a fibre map with fibre F; by another standard construction, we
may assume ρ is an inclusion with "cofibre" BJE.
There is a natural map /: SF—> BJE, induced from the following
diagram, since SF is the cofibre of F—> *.
F^E
ϊ Ϊ
*—>β
Theorem 2
Suppose В is (p— l)-connccted and F is (q — l)-connccted. Then the
natural map / induces an isomorphism from n,(SF) to π^Β/Ε) for
i <p+q-2.
proof: We implicitly assume that ρ and q are sufficiently large to assure
that SF and B/E are simply connected. It is thus enough to prove that /
induces an isomorphism in homology for i<p+q— 1. To do so, we
merely compare the Serre exact homology sequence for the fibration,
replacing F by SF with a dimension shift, with the exact homology
sequence of the pair (B,E). With the help of/, we then have a homomorphism
of long exact sequences, with the terms in question sandwiched between the
common terms for £ and B. The result follows from the five-lemma, noting
that the last term to be sandwiched is Hp+q-2(F) = Яр+,_1(5/г).
Briefly put, in the stable range the cofibre and fibre agree, except for a
dimension shift.
We are already familiar with the following basic facts (see Chapter 14):
cofibre commutes with suspension and fibre commutes with loops. Indeed,
these are the properties which give rise to the cofibre (inclusion) and fibre
mapping sequences, respectively. But Theorem 2 tells us that fibre and
cofibre agree (except for the dimension shift) in the stable range. Thus in

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COHOMOLOGY OPERATIONS
the stable range cofibre commutes with loops and fibre commutes with
suspension.
We leave to the reader the task of precise formulation of the numerology
for these statements. Nonetheless we will use them, especially in Chapter
18. For example, we feel free to use the fibre mapping sequence of the pair
(B,E) of form
• · · -> [ΑΓ,Ω'£]-> [Κ,Ω'Β]^ [K,Clr(B/E)]^> ■ ■ ■
which is valid when r and dim К are sufficiently small.
DISCUSSION
What we have done in this chapter is to establish a stable range in which
many things are well-behaved. In Chapter 16 we will study a new sort of
cohomology operation in the stable range; in Chapter 17 we will apply
these operations to gain new information about the composition ring G of
stable homotopy groups of spheres. Stability becomes even more crucial
in Chapter 18, where we introduce a totally different tool for the study of
stable mapping groups.
We based our comparison of fibre and cofibre on the natural map
/: SF—>5/£. The adjoint of/is another natural mapg: F—>Ω(β/£).The
study of g would be a dual method of approach to the subject. Our proof of
Theorem 2 uses in a key step the exact homology sequence of the fibration
in the stable range. A completely dual proof of the corresponding results
would require an exact homotopy sequence of the cofibration £—>5—>
B/E. Thus we would need, in the stable range, an excision property for
homotopy in order to identify π[Β,Ε) with π^Β/Ε). That such an excision
property holds in the stable range is a consequence of the triad theorem
of Blakers and Massey.
It is possible to formulate precisely a " stable category" in which stability
conditions are built in. The objects in this category are known as spectra;
they were first introduced by Lima. Subsequently Spanier, G. W.
Whitehead, and Kan have put stable homotopy theory, via spectra, on a firm
footing.
The use of spectra leads, as one might well imagine, to enormous
simplifications in the formalism; no longer need every statement be qualified
by an awkward range of dimensions which have to be kept track of. On the
other hand, there is a great deal of technique needed to establish the theory.

PROPERTIES OF THE STABLE RANGE
155
For the applications we give, the mass of technique outweighs the gain;
thus wc do not introduce spectra here. However, for more sophisticated
applications they seem to be indispensable.
EXERCISES
1. The mapping sets [S2K,X], [SK,&X], and [Κ,Ω2Χ] are in natural one-to-one
correspondence. The first and third have group structures; the second has
two. Prove that these group structures all coincide under the natural set
isomorphisms and are abelian. Prove also that maps SL -> SK and ΩΛΓ-> Ω Υ
induce group homomorphisms.
2. In the range of Serre's exact sequence, the groups Hk+ l(X) and Hk(£lX) are
isomorphic. Prove that the cohomology Suspension and cohomology
transgression are indeed inverse isomorphisms, hint: Recall that the coboundary
operator for a pair may be defined by mapping the inclusion mapping
sequence (Chapter 14) of the pair into an appropriate Eilenberg-MacLane
space and use the second definition of transgression (Chapter 8).
REFERENCES
Spectra
1. J. F. Adams [4].
2. D. M. Kan [1,2].
3. — and G. W. Whitehead [I].
4. G. W. Whitehead [5].
Commutativity of the composition
ring G
I. M. G. Barratt and P. J. Hilton [I].
The triad theorem
I. A. L. Blakers and W. S. Massey [I ].

CHAPTER
HIGHER COHOMOLOGY
OPERATIONS
We are moving toward a more intimate knowledge of the homotopy
groups of spheres. Before launching into the homotopy theory, we need to
develop some more sophisticated tools. In this chapter we will describe two
kinds of higher cohomology operations which will be useful in the
homotopy calculations of the following chapter.
To simplify matters we will restrict our attention to the "stable range."
In each specific context the precise limits of the " stable range " will be more
or less evident from the context; sometimes we will make them explicit.
FUNCTIONAL COHOMOLOGY OPERATIONS
The first new kind of cohomology operation is the functional operation.
We suppose that we are given a map/: У—> X (this is the " function ") and
a primary operation Oof type (G,n; n,q). For stability, we assume^ <2n — 2,
so that 0 is in the image of the cohomology suspension and thus 0 is a
homomorphism when considered as a transformation of additive groups
tf"(,·(?)-> H"( ;π).
Since every map is homotopically equivalent to an inclusion map, we
may as well assume that /is an inclusion map and У <= X.
Let и be a cohomology class in H"(X;G) satisfying the two conditions
f*u = 0eHn(Y;G) and 0(h) = 0 e tf"(A». Consider the following
diagram, in which each row is exact.
156

HIGHER COHOMOLOGY OPERATIONS 157
Diagram 1
tf"-' (Y;G) -i* Hn(X, Y; G) -Л* Hn(X;G) -£+ H"{ Y;G)
he β β
Я""'(A» -JU //"-'(Υ;π)-U H"(X, Υ; π)-Л+ Η"(Χ;π)
Since f*u = 0, we can find и e H"(X, Y; G) such that j*(u') = u. Since
0(u) = 0, j*0(u') = 0j*(u') = 0(h) = 0, and so we can find и" е Я""'(У;тг)
such that S(u") = O(u'). Of course u" is not uniquely determined by u; what
is uniquely determined by и is the projection of u" in B"~l(Y;n)/Q, where
the indeterminacy Q is the sum ιθ(Ηη~ι( Y;G)) +/*(#""'С-*») of
subgroups of Η"~ι(Υ;π). To see this, suppose that υ and v" were chosen in
place of и and «". Then У*(и' — υ) = 0, and so u' — υ = Ob for some
b e Я""'(Y;G). Therefore d(u" - v") = О(и') - 0(«') = 0(db) = δ^θψ)), so
that (и" - υ" - l0(b)) is in the kernel of δ and thus и" - и" =/*(а) + 'б(б)
for some a f Я"- \Χ\π) and some йеЯ"" '(Y;G).
We define the functional operation 0f by putting УДи) to be the coset
и + Q. Thus θχ has as its domain a subgroup of H"(X;G), namely, the
intersection of ker/* and ker Θ; and it takes values in the quotient group
H"~'(Y;n)/Q, where (? = im Ό + im/*. Like the operation Θ, it goes from
cohomology with coefficients in G to cohomology with coefficients in π;
note, however, that it raises dimension by only q — η — I and not, like Θ,
by q — n. Like the homomorphism/*, 0f goes from the cohomology of X
to that of Y.
These functional operations are natural in the following sense. Suppose
we have the following commutative diagram and suppose that и е H"(X,G)
Y-UX
\ Iе
Y'-L+X'
satisfies f*u = 0, 0u = 0, so that 0f(u) is defined in H4~l(Y;n)/Q. Then
obviously (/')*(£*(«)) = 0 and 0(£*(н)) = О, so that ΘΓ is defined on
£*0), taking its value in Hq~l(Y';n)JQ'. Then we claim that f?*(0/(M)) =
0г(^*(и)) or, more precisely, that the left-hand member of this "equation "
is contained in the right-hand member when both are considered as subsets
(cosets) of Η"'ι(Υ';π). in fact, the indeterminacy of η*(0Ι(ιι)) is clearly
f*(G) = n*f*(H«- ι(Χ\π)) + n\l0{H'-\Y;G)))
while that of θΓ{ξ*{ύ)) is
Q' = (fT(H"-l(X'-n)) + lO(Hn-l(Y';G))

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COHOMOLOGY OPERATIONS
so that clearly r\*{Q) is contained in Q''. To verify that η* maps any
representative of Ojiu) to a representative of ηΓ{ξ*{ύ)) is no more than an exercise
in diagram-chasing, which we leave to the reader.
One of the important applications of functional operations is in
establishing that a given map is essential. It is a classical pattern of proof in
algebraic topology to show that a map is homotopically non-trivial, i.e.,
essential, by showing that it is algebraically non-trivial. Thus if a map
/:У—>X induces a non-zero homomorphism /*: H*(X)^> H*(Y), then
clearly/must be essential. The following proposition represents a
considerable sharpening of this method.
Proposition 1
Let/be a map У—> X. Suppose there exists a primary operation 0 and a
cohomology class и such that θ/и) is defined and non-zero. Then / is
essential.
Of course to say that 0f{u) is non-zero means to say that it is not the zero
coset.
Under the conditions of the proposition, we say that the operation 0
detects the map/.
To prove the proposition, suppose that/is null-homotopic. We may as
well assume that/is an inclusion map. Then the fact that/~0 means that
X yjfCY has the homotopy type of the wedge Χ ν SY, and the
cohomology sequence of the pair (A', Y) splits: H"(X, Υ; π)^Η\Χ\π) φ Η4~\Υ;π).
The splitting map r* gives a commutative diagram
Hn(X,Y;G)+±-Hn(X;G)
i Iе
Η<(Χ,Υ;τ:)^Η'>(Χ;π)
where θ is an arbitrary primary operation of type (G,n;n,q). Then, if
ueH"(X\G) satisfies f*{u) = 0 and 0(u) = 0, we can represent 0f(u) by
zero, since we can take и = r*(u) and then O(n') = 0(r*(u)) = r*(0(n)) = 0.
This proves the proposition.
As an example, let / be the map S"^1—>S" which is the (n — 2)-foId
suspension of the Hopf map η: S3—+S2, and let 0 be the operation Sq2.
Recall (from Chapter 4) that Sq2 is non-zero in the complex K-^ S" <ufen+2.
IMow К is precisely MJS"+\ where Μ is the mapping cylinder of f. Then Μ
has the homotopy type of Sn. We take Υ =-- 5"+1, X= M~S", and of
course G=n — Z2. Let и be the generator of H\X;Z2). Obviously
Sq2u = 0 and/*» = 0. Therefore Sqj-{n) is defined. The diagram defining it
is as shown below. (All coefficients are understood to be in Z2). We have

HIGHER COHOMOLOGY OPERATIONS
159
H"-'( Y) —> H"(K) -J-+ #"(*)-> H"( Y)
i -I I
Hn+l(X)^Hn+l{Y)^UHn+2(K)-+Hn+2(X)
Н"(К) ъ H"(X) = Z2, since H\ Y) = 0 for к Φ η + 1; thus we must take
и to be the generator of this group. Then Sq2u is non-zero. Finally, since
Hk(X) = 0 for к -/- η, the δ shown is an isomorphism and we must take u"
to be the generator of Hn^\Y). The indeterminacy is zero, and we have
proved that Sq2f{u) is non-zero. Thus, in the language introduced above, Sq2
detects/(as an essential map). Note that/* is zero in all dimensions.
In the same way, the other Hopf maps v: S" + 3—>S" and σ:5" + 7^5"
give non-trivial functional operations Sqt and Sql, respectively.
ANOTHER FORMULATION OF 9f
We now introduce another (equivalent) definition of the functional
operation 0f. Let 0, / and и be as before. We can use θ to define a fibre
space £over K(G,n) induced from the standard contractible fibre space over
K(n,q), as indicated in the following diagram.
Diagram 2
F=K(n,q-\)^E
Y^U X^ В = K(G,n) -?- K(n,q)
To say that 0(u) = 0 is to say that the map и can be lifted to give a map
й: Х^Е such that рй = и. То say that/*(w) = 0 is to say that the
composite uf is null-homotopic. This means that uf: Κ—> Ε is in the kernel of
p« : [ Y,E] —> [ Y,B], which is the same as the image of /„: [ K,F]—> [ Y,E],
so that there exists a map //": K->Fsuch that iu" is homotopic to uf. The
corresponding cohomology class и" е H"~l( Υ;π) will represent the image
of и under the functional operation 0f, which we shall denote 0f for the time
being to distinguish it from the 0f previously defined, it is easiest to
evaluate the indeterminancy in the definition of 0f by considering the following
diagram.
Diagram 3
H" ~ '(-*»—> [A\£]—> Hn(X;G) -^ Η"{Χ;π)
#"-'( Y;G)-^+ Н"-\ Υ;π)^ [ Υ,Ε]^ Hn(Y;G)

160
COHOMOLOGY OPERATIONS
Here the vertical maps are/* and each row is exact (cf. Chapter 14). Since
we are in the stable range, Diagram 3 is a diagram of groups and homo-
morphisms. Now the definition of Oj{u) which was given above is easily
traced on this diagram; formally, it bears the closest possible resemblance
to the definition of 0f{u). In particular, it is evident, chasing the diagram,
that the indeterminacy is f*(H"-l(X;n)) + 19(Hn-l(Y;G)), which is
exactly the same as the indeterminacy Q in the definition of θχ. Thus
Oj(u) may be considered as a well-defined element of the quotient group
H"-l{Y;n)IQ.
We leave it to the reader to verify that such operations 0f are natural
in the same sense as we proved before, using the first definition of
functional operations. This fact would be immediate from the next proposition,
but we want to use it in the proof, so it must be established directly.
Proposition 2
The functional operations 9f and 0f are identical.
We will prove this by the method of universal example. For the universal
example, take X = Ε (where Ε is the total space introduced above),
Y=F= K(n,q- 1), /= /·. F—>£, and u = p e H"(E;G). Then in Diagram
2, we may take и and u" to be the appropriate identity maps, and thus it is
obvious that the identity map (or fundamental class) /„_, e Η4~\Υ;π)
represents Bf(u). To see that;,_, also represents 0/и), consider the
following portion of Diagram 1:
Hn(E,F;G)^+H"(E;G)
H"-l(F;n)-UH"(E,F;n)
Now in the stable range the cohomology exact sequence of Serre for this
fibre space over В = K(G,n) yields isomorphisms
H\E,F; )ъНк(В;)
for whatever coefficients. Under this isomorphism, the class ρ f H"(E,F; G)
corresponds to the fundamental class ine H"(B;G) and the coboundary
δ: Hq~\F\n) —>H"(E,F; π) corresponds to the transgression t: H"~\F;n) —>
Η"(Β;π). [f we choose p' e Hn(E,F; G) such that7 V) = p, then O(p')
corresponds to 0((„). Thus, showing that ;,_, represents 0f(u) comes down to
showing that τ(ί^_,) = 0(ι„). But by Formula 1 of Chapter 11 this is
immediate from the construction of E. Thus the proposition is established for the
universal example.

HIGHER COHOMOLOGY OPERATIONS
161
The general case follows easily by naturality, as in the following
calculation:
ef{u) = 9f(u*(p)) = (Охи)*(в{р)) by naturality of 6f
= (Bfu)*(di(p)) by the universal example
= Oj(ii*(p)) by naturality of 0f
= Bj{u)
This completes the proof of Proposition 2.
SECONDARY COHOMOLOGY OPERATIONS
The other new kind of cohomology operation that we will be using is
the secondary operation. We suppose that we are given a two-stage Post-
nikov system as in the following diagram.
Diagram 4
F=K(n,q- \)-L>E^K=K(H,m)
X^UB = K(G,n) -Л K{n,q)
Here θ is a given primary operation of type (G,n; π,α). Suppose also we are
given a cohomology class ψ in Hm(E;H). The secondary operation Φ
will be defined on a class и е H"(X;G) provided that the composition
0u is null-homotopic, i.e., that 0(u) = 0. In this event we can find a
lifting #: X—>£ such that рй=и. We can therefore consider the class
ΰ*(ψ) e Hm(X\H). The indeterminacy is due to the choice of й; the kernel
ofp#: [X,E]^>[X,B] is the image of i# [X,F]^*[X,E], and so it is easy
to see that ΰ*(ψ) may vary by the addition of an arbitrary element of the
image of />: Hq~\X\n)^Hm(X\H). Thus the coset Ω*(φ) + im (/"» is
well-defined. We thereby obtain a secondary operation Φ which has for its
domain ker 0 с H"(X;G) and which takes values in coker (Ί*ψ), a quotient
group of Hm(X;H).
We have made use of the stability assumption in the assertion that the
indeterminacy is the subgroup im (/*</>). What is needed is that q and m are
not too large with respect to /; or, precisely, that q and m are each less than
2/7 — 1. Without such an assumption, the evaluation of the indeterminacy
becomes a thorny problem.

162
COHOMOLOGY OPERATIONS
We remark that if φ is such that i*q> = 0, or equivalently if φ is in the
image of p*\ Hm(B\H) —>Hm(E;H), then Φ is essentially a primary
operation, in the following sense. Let φ = p*0' where Θ' s Hm(B;H); then we
have a commutative diagram of maps X—> К — K(H,m) with φΰ — θ'и, and
consequently Ф(и) = ϋ*(ψ) = u*(0') = в'(и), with zero indeterminacy, and
so Φ is nothing but the restriction of 0' to ker Θ.
We will therefore assume from now on that ϊ*φ is non-zero.
There is an obvious naturality formula for secondary operations: if/is a
map У—> X, then /*(Ф(и)) = Ф(/*(г/)), modulo the indeterminacy of the
right-hand side. To see this, consider Diagram 4 augmented by the map
/: Y—>X. The class of the map φϋ represents Ф(и) е Hm(X;H)/Q, where
Q = ((pi)[X,F]=\m (ί*φ). Then /*(Ф(и)) is represented by <pw/and has
indeterminacy Q' =f*Q =/* im (i*<p). We could equally well represent
Q' as (<p/)(im/*), where/*: Hm(X;H)^Hm(Y;H) or equivalently/* =
f* : [X,F] —* [ Y,F]. Now uf is a lifting of uf, and thus obviously the class
(^w/'also represents Ф(/*(г/)). The indeterminacy of this term, however, is
Q" = (φ/)[7,.Γ] = im(/V)»and clearly Q' с ρ". Thus the naturality formula
must be interpreted as an equality in. Hm(X;H)IQ".
SECONDARY OPERATIONS AND RELATIONS
In a very precise sense, a secondary operation corresponds to a relation
between primary operations. Suppose first that a secondary operation Φ is
given, as above. Under suitable stability hypotheses, the composition
(φι): K(n,q— 1)—>K(H,m) has a unique representation as V where
ф е Hm+1(n,q;H). We therefore have the following diagram.
Diagram 5
F= Κ(π, q-l)-U £^> K(H,m)
i
В = K(G,n) -Ϊ+Κ(π,ς) -±> K(H, m + l)
where (φί)= V- The diagram
tf^-'CF;*)—i-+Hq(B;n)
'4 1*
Hm(F;H)-^Hm+1(B;H)
is commutative (under suitable sign conventions); we omit the proof of this

HIGHER COHOMOLOGY OPERATIONS
163
general lemma. Granting this, we have
0 = τ(/*(φ)) = τ(φ/0,_,)) = τ(>(ί,_,)) = Φ<ι,- ,) = φΟ
and we thus obtain the relation φΟ = 0 between the primary operations θ
and ι/'.
Conversely, suppose we begin with a relation φΟ = 0. Using Θ, we
construct the fibre space i7—>£—>/? as before. Since τ(ιφ(ιη_ι)) = φθ = 0,
V must be in the image of /*, so that we have ιφ = ΐ*(φ) for some
ψ e Hm(E;H). This completes the reconstruction of the original diagram,
and we can define a secondary operation Φ. Note, however, that φ is not
unique; we may vary φ by adding an arbitrary element in the image of
p*. Thus the relation ф0 = 0 does not determine Φ uniquely; but any two
O's thus obtained will differ only by a primary operation, according to the
previous remark.
Observe the relationship of degrees:
φΟ: H\X; G) ->Hm + \X\ H)
whereas Φ goes from a subgroup of H"(X;G) to a quotient group of
H"\X\H).
Before giving an example of a secondary operation, we remark that the
theory can readily be generalized by replacing the space K(n,q) by a cartesian
product of such spaces, Y\;K(nhq^. Then the loop space, F, is similarly a
product. The operation 0 becomes, in the general formulation, an ordered
«-tuple (0,,. . .,0„), since a map of В into Π.Λ^ί (where Kt — K(nhq$) is
just an «-tuple of maps 0t: B^>Ki.
Our example will be of this kind. Consider the Adem relation
R: Sg3^1 4- Scj2Sq2 = 0. From this relation we can obtain a secondary
operation, as follows. We take the diagram
F= K(Z2,n) χ Κ(Ζ2, η + \)^E
1
K(Z2,n) <5«'·5«2>>Κ(Ζ2, η + 1) Χ Κ(Ζ2, η+2)
The role of χφ is played by the pair of operations (Sq3,Sq2). The co-
homology class (Sq3in)® 1 + \®(Sq2in+{) ε Hn+\F) (coefficients in Z2)
transgresses to zero, where the symbol 1 denotes the unit in the appropriate
cohomology ring. Therefore this class is i*q> for some ψ e W" + 3(£). In this
example, ψ is-uniquely determined, for it is a routine matter to verify that
p* is zero on W" + 3(5); in fact H"+3(B) is generated by Sq2Sq4D =
r(Sq2i„) and by Sq3iB = r(Sqiin+i). Then if we H\X;Z2) is a class such that
Sqxu and Sq2u are both zero, Ф(и) ib a well-defined element of the quotient

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COHOMOLOGY OPERATIONS
group H"+ 3(X;Z2)/Q where Q is the image of the map [X,F] ->tf"+ \X\Z2)
induced by Sqz + Sq1: F—>Λ^(Ζ2, η -f 3). This operation Φ, discovered by
Adem using entirely different methods, was one of the first secondary
cohomology operations to be studied; it can be used to show that the
composition S" + 2—>S"+1—>S" is essential, where both maps are the
suspensions of the Hopf map η: S3—>S2.
THE PETERSON-STEIN FORMULAS
There reader has no doubt observed some similarity between the defining
diagrams for 9r and Φ. These two sorts of operations are indeed closely
related; the following formulas tell how.
Throughout this section we retain the notations and assumptions of
Diagrams 4 and 5; that is, we have our cohomology classes θ, φ, and φ
satisfying V — 'V and F, E, B, i, and ρ are as in the diagrams. Φ is the
secondary operation determined by ψ. We consider a map/: У—> X and a
cohomology class и е H"(X;G).
Now suppose that/*w = 0, which is to say that the composition uf is
null-homotopic. Then the naturality formula shows that /*(Ф(и)) = 0 in
Hm(X;H)/Q", where Q" = im ιψ. We can sharpen this result by reducing
the indeterminacy from Q" to Q' =f* im 1ф by the following formula.
Theorem 1 (first Peterson-Stein formula)
/*(*(«))= VO/и) in Hm(,X;H)lQ'.
proof: 0f{u) is represented by a map u": Y—>F such that iu" ~ uf. The
indeterminacy of 0f[u) is Q* = im Ό + im/* с //"(XjW). Then (<pi)(9fu)
is represented by the composite (φ/и"), and its indeterminacy is (</>/)(£*) =
im Ό + 'm '·/'/*=/* Im V = 6'> since ι/Ό = 0. This shows that (<pi)(9fu)
is equal to the class (<p/V) (modulo Q'). But in the proof of the naturality
formula we saw Ша1/*(Ф(и)) is equal to the class (<рй/) (modulo Q'). Since
iu" ~ uf this proves the theorem.
The following observation will be used in the last two results of this
chapter.
Lemma I
In Diagram 5, with (φι) = хф and ι/Ό = 0 as usual, we have ϊ*φ = ϊ*(φρ0)
modulo /*(im (ιφ)), that is, modulo the image of the composite
Η" -\Ε;π) '* , Hm(E;H) -il» Hm(F;H)
[E,F]^l^[E,K(H,m)]-^[F,K(H,m)]

HIGHER COHOMOLOGY OPERATIONS
165
proof: To define φρ0, we replace ρ by an inclusion map and use the
following diagram, which, in the stable range, is isomorphic in a natural
A
Hm(E;H)-^Hm(B,E;H)
way to the next diagram so that φρθ is represented by any class χ e Hm(E;H)
H"-l{F\n)-L+№{B;n)
Ή
Hm{E\H)-±>Hm(F;H)
such that i*x = 1ф(у) with τ(γ) = 0. Now Ε is the fibre space induced by 0,
so that τ(ί,_,) = θ, and we can conclude that ϊ*{φρ0) = '(/*(/„_,) = ϊ*φ
modulo the indeterminacy. The indeterminacy is
i*(p*Hm(B;H) +ιφΗ"-\Ε;π))
Since i*p* = 0, this reduces to the indeterminacy stated in the lemma, and
the result is proved.
Theorem 2 (second Peterson-Stein formula)
Suppose that in the diagram below the composite 9vf is null-homotopic.
Then
Φ(/*Ό = Φχ(0ν) s Hm( Y;H)IQ, where Q = (ψΟΑΥ,Π -Ι /*[ВДЯ,т)].
F=K(n,q-\) -\E^+ K(H,m)
- - *" " " 1"
Y^CxJi+B = K(G,n) -£> Κ(π, q) -*» K(H, m + 1)
We have used the letter ν instead of и to emphasize that we do not assume
here that either Ov orf*v is zero. It is still understood that (φί)= 1ф.
proof: Since 0i/~O, 4>(f*v) is defined and is represented by cpw, where
w. Y—>E is a lifting of (vf). The indeterminacy is (</>/')#[Y,F]. Now
consider φχ(0ν) = φχ(ν*0). By the naturality formula for functional
operations, фх(и*0) =- \ν*(ψρθ), modulo the indeterminacy of ij/f(v*0\ which
is precisely the Q of the theorem. Thus we must compare cpw, that is,
\ν*φ, with \ν*{φρ0). By the preceding lemma, ϊ*ψ = ϊ*{φρ0) modulo
/*(im СФ)), which implies that q> = φ β modulo (im (1ф) — im (p*)) where
p# : [B,K(H,m)]^> [E,K(H,m)], We shall see that when we apply w*, this is

166
COHOMOLOGY OPERATIONS
absorbed into Q, which will complete the proof. In fact, w*<p and \ν*(ψβ)
can only differ by an element of the sum
w*(\m(^)) + w*(\m(p*))
The first term is the image of the composite
[£,F]-!±* [E,K(H,m)]^U[YMH,m)]
which is the same as the composite
[E,F] -Ju* [Y,F]-!±+[ Y,K(H,m)]
and thus the first term is contained in (φί)#[ Y,F], the first term of Q. The
second term is the image of w*p*, which is the same as /*!v*', so that this
term is contained in/# [X,K(H,m)], the second term of Q. This shows that
\ν*ψ and \ν*(φρ0), which represent Φ(/*ν) and ψχ(0υ\ respectively, are
equal modulo Q, which proves the theorem.
We could summarize the above proof by the calculation
Φ(/*ι>) = <b(w*p) = \ν*(Φ(ρ)) = w*tp = \ν*(φρ0) = ψχ(υ*0) = ψχ(0ν)
by simply ignoring any problem of indeterminacy. However, the
evaluation of the indeterminacy is not only the hard part of the proof but also
the main content of the theorem itself.
We now prove an important result which overlaps with the " Bockstein
lemma" of Chapter 11.
Theorem 3
Let (E',r,B'; F') be a fibre space, let 0 be a primary operation of type
(G,n; n,q), and let we H"(B';G) and vs Hq~\F';n) be classes such that
φ) = 0(u) e Η\Β'\π). Suppose that Our ~ 0 and that ψθ = 0, Then
j*m0u))=l*Ki:)=J*(4r*u))
in Hm(F';H)/Q, where Q =7*(im Сф)), in other words Q is the image of
the composite
[£\Fl-!M£',*(//,/w)l-iMF,*(//,#»;)l
(here F, Ε, Β, φ, i, and 0 are as in Diagram 5),
proof: For the universal example, take (E',r,B'; F') to be identical with
(Ε,ρ,Β; F) and let и and ν be the identity maps of B= K{G,n) and F =
K(n,q— 1). Then we are required to show that ί*(φρ0) = i*<p — /*(Φ(/?))
in Hm{F\H)jQ0 where (?0 = '*(im (Ί/0). This is precisely the assertion of
Lemma 1, together with the remark that ψ = Φ(ρ) modulo im (Ί//).
To prove the general case, we map it into the universal example, as in
Diagram 6.

HIGHER COHOMOLOGY OPERATIONS
167
Diagram 6
F= K(k, q-\) '—pE^-> K{H,m)
F 1 ► E'' B = K(G,n) -£» K(n,q)-A» K(H,m + 1)
5'
The existence of the map/: E'—> £ such that #/"= w is implied by the fact
that 0(ur) ~0. Then К/У) — "('У') 'S trivial, so that/gives a map g: F'^F
with ig=jj- We now have the following calculation:
]*(фг(0и)) = }*(фХи*9))=]*/*(фр0) by the naturality
formula for functional
operations
= д*1*(ф„0) = д*1*(р^д*1*(Ф(р)) by the result for the
universal example
=7*/*(φ(ί0) =7*φ(/ν) =7*Ф(г*м) using the naturality
formula for secondary
operations
Here every equality is to be interpreted as an equality in the quotient group
Hm(F';H)IQ. We should check the indeterminacy throughout the above
calculation. To begin with, фг(0и) has indeterminacy
Q = 'фН"-\Е';п)+г*Нт(В';Н)
and j*Q' = Q. When we pass \ο/*{ψρ0), we can only decrease the
indeterminacy. When we apply the formula from the universal example, the
indeterminacy \s g*Q0; we leave it as an exercise for the reader to verify
thatg*Q0^Q.
We have shown that j*{\jjr(0u))=--j*{<${r*n)) modulo Q; it remains to
compare ιψ(ν), which is the same as <piv, whereas the middle term in the
above calculation is g*i*(p, that is, q>ig. Now we do not claim that g = v, but
we see that x{g) — t(;); for, by naturality, z{g) = и*(т(/ч_,)), and of course
t(i4_ ,) = 0(in) by definition of E, so that u*(t(i4_ ,)) = u*0{in) = 0u, which by
hypothesis equals t(r). Thus g = ν modulo the image of j*, and thus
<pig= φϊν modulo Q. This completes the proof.
If, in the present theorem, we take 0 = Sql —ψ and φ = the second
Bockstein operator d2, we recover the special case r ■= 1 of the Bockstein
lemma of Chapter 11.

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COHOMOLOGY OPERATIONS
DISCUSSION
Of our two definitions of functional operations, the first, 0f, due to
Steenrod, is older and conceptually simpler. However, the formulation 0f,
due to Peterson, is much more convenient for the study of secondary
operations.
The main results of this chapter, Theorems 1, 2, and 3, were proved by
Peterson and Stein. The usefulness of these theorems, as we shall see in
the sequel, lies in the reduction of the calculation of secondary operations
to that of primary and functional primary operations.
We have given a constructive definition of secondary operations. Adams
has formulated an axiomatic approach; Theorem 1 becomes his key axiom.
As the reader undoubtedly suspects, there exists a full-blown theory of
higher-order operations, based at least intuitively on cohomology classes
of higher-stage Postnikov systems. From the constructive viewpoint, such
operations were formalized by Peterson; Maunder has successfully
generalized the Adams axiomatic scheme. Such higher-order operations
have been crucial in many recent calculations, as the reader who persists to
Chapter 18 will believe.
EXERCISES
1. Verify directly the naturality formula for 0f.
2. Interpret the Hopf invariant Η: π2„_,(5")->·Ζ as a (non-stable) functional
cohomology operation.
3. Let K= S" <jfen + k for some/: S"*k_1 -» S". Suppose there exists a
secondary cohomology operation Φ defined and non-zero on the generator of
H"(K). Then prove/is essential. (We say Φ detects/.)
4. Let ueHp(X), ve H"(X), w e Hr(X) (all with integer coefficients) have
the properties uv = 0 and vw = 0. Let u, υ and \v be cochain representatives,
and let a, b be such that uvj ν — δα and ν υ w = 6b. Prove:
i. ии£ + (— 1)р+1(яи и) is a cocycle
ii. The cocycle of (i) projects to an element of the quotient group
HP+" + r-1(X)/(Hp+,l-i(X)w + uH'l + r-1(X))
which depends only on Й, ν and w
This clement is denoted </7, v, w) and is called the Massey triple product
of the three cohomology classes.

.HIGHER COHOMOLOGY OPERATIONS
169
REFERENCES
Functional cohomology operations
1. F. P. Peterson and N. Stein [1]
2. N. E. Steenrod [4].
Secondary operations
1. J. F. Adams [2].
2. H. Cartan [6].
Higher-order operations
1. С R. F. Maunder [1].
Functional higher-order operations
1. F. P. Peterson [1 ].

CHAPTER
COMPOSITIONS IN THE STABLE
HOMOTOPY OF SPHERES
In this chapter we will take a closer look at the stable homotopy groups
of spheres in the range computed in Chapter 12. As an example of the sort
of question we ask, consider the Hopf map η: 53 — > S2. Under suspension,
η gives rise to a map Ε"η: S"+3—>,S"+2 for each positive integer n. This
element, also denoted by η, is the generator of the stable 1-stem. We have
seen that the stable 2-stem is a cyclic group of order 2. Is it generated by the
composition Ε"~2η c £"~ V S"+2 — >S", denoted η2, or is this composition
null-homotopic?
We will study such compositions as η2 as well as compositions of a
somewhat different nature, the so-called secondary compositions, which we
now introduce.
SECONDARY COMPOSITIONS
Suppose we are given homotopy classes/?e [X, yjandae [Y,Z], as shown,
X-L> Y^L>Z
1 y "
K= YVpCX
and we wish to extend α (or rather a representative of a) over K= Υ \JP CX.
It is an easy exercise to verify directly that such an extension can be found
170
17

COMPOSITIONS IN THE STABLE HOMOTOPY OF SPHERES 171
if and only if the composite α 0 β is inessential, that is, if α 0 β = 0 e [X,Z ].
Jn fact, we have an exact sequence
[X,Z ] «-£1- [ Y,Z ] *-<!- [ Υ и, CX, Ζ ] «-ΙΪ- [SX, Z)
as in Chapter 14. This shows that α can be extended over all of
K= Υ kjp CX, that is, α is in im (('*), if and only if α is in ker (β*), that is,
α ο β = 0.
If α is a homotopy class such that i*(S) = a, we call α an extension of a. Of
course, this is consistent with the usual sense of the word "extension."
Working in the stable range, we may assume that all the sets of homotopy
classes under discussion have natural abelian group structure. Then it is
clear from the above exact sequence that any two extensions (in the present
sense) of α differ by the image under/'* of a map of SX into Z. We should
make explicit that the cone CX с к is the " upper" cone, i.e., is a quotient
space of lx [0,1], while SX is represented by C7u K, where CY is the
"lower" cone, i.e., a quotient space of У χ [—1,0], with Ух Ос с У
identified in the natural way with Υ <= К. (We have remarked in Chapter 14
that SX and С Υ u К have the same homotopy type.)
We will now introduce a kind of companion or dual to the notion of
extension as defined above. Let у be a homotopy class of maps W'—> X.
Then a representative of у induces a map CW—> Υ νρ CX — К (where C.W
is the "upper" cone) in an obvious way. We ask when a map Cy: CW—> К
admits an extension to a map
y:{SW,C_W)^{K,Y)
that is, to an extension of Cy over all of SW with the property that the
extension у maps all of the " lower" cone of SW into Υ с к. It is easy to
see directly that this is possible if and only if the composite β 0 у: W—> Υ is
null-homotopic. When this occurs, we say that у is a co-extension of y.
Note that if we take any co-extension γ and follow it by the
identification map A^—> SX obtained by collapsing Ус a; to a point, we obtain
Sy: SW^> SX. (We could just as easily call this map Ey instead of Sy. Both
notations are in widespread use.)
The choice of co-extension is seen to be equivalent to the choice of the
null-homotopy of /?0y: W'—> Y. Thus in the stable range any two co-
extensions of у differ by a map of SIV^> Y, since such a map represents in
a natural way the difference between two null-homotopies of β 0 y.
Now suppose we are given homotopy classes
W-^X-L, Y^L,Z

172
COHOMOLOGY OPERATIONS
such that both β ο γ and α ο β are zero. Then we can form an extension α of
α and a co-extension у of у. The composite defines a map
SW^K= Y^ipCX^Z
and it remains to see what the indeterminacy is; that is, the above
composite varies with the choice of α and y. In the diagram below, the unmarked
SW^K= Уи. CXJ+Z
arrow SX—> Ζ represents the indeterminacy in the choice of a, while the
unmarked arrow SW—> У represents the indeterminacy in the choice of y.
Recalling the assumption of stability, the diagram makes obvious that the
indeterminacy of α о у is ol#[SW, Υ] + (Sy)*[SX,Z]. Thus the above
composition jof: SW^> Ζ defines the secondary composition or Toda bracket
<a,/?,y> as a uniquely determined clement of the quotient group [SW,Z]jQ
where Q = a# [SW, Υ] -^ (Sy)# [SX,Z ].
For heuristic purposes we can appeal to Figure 1. The co-extension γ
maps the upper cone of SW by Sy into CX c K, maps the " equatorial" W
by β о у, and maps the lower cone of SW by the null-homotopy of β о у. The
extension α maps Υ into Ζ by α and maps CX с к into Ζ by the null-
homotopy of α ο β.
SW
Figure 1
Now suppose W, X, Y, and Ζ are spheres of dimensions sufficiently large
so that α, β, and γ are in the stable range. We would then expect the stable
class of <a,j8,y> to depend only on α, β, and γ viewed as elements of G
and not on the dimensions of the chosen representatives.

COMPOSITIONS IN THE STABLE HOMOTOPY OF SPHERES 173
While this is not quite true, it is true up to sign, and we may define a
stable secondary composition as follows. Let α e Gh, β e Gk, and у е Gf.
Suppose α ο β and β ο γ are zero. Let a, b, and с represent α, β, and у as
follows:
O" + h + к + / £\ c« + h + к b ^ c« + h a ^ en
Then the class of (-1)"~'<йДс> in Gh + k+J+l/(oc „ (7k + j + 1 +Gh + k + l ο β) is
independent of η and defines the stable secondary composition <a,/?,y>.
THE 0-, 1-, AND 2-STEMS
We now begin our detailed analysis of the stable Ar-stem, Gk, that is, the
group n„ + k(S") for к small and η large (and of course working in the 2-
component only).
We have in the 0-stem an infinite cyclic group generated by the identity
map /: S" —>5". We may say / is detected by its induced cohomology
homomorphism /*, since /*: Hn(S") —> Hn(S") is non-zero. We also observe
that 2/ is detected by Sql; that is, Sq^a") =-- e" +' in the complex S" u2, e" +'
and the functional operation SgJ, is non-trivial. This result may be
generalized as follows.
Proposition 1
2Ί is detected by the Bockstein operator dr, which is non-zero in
5"u2r,e" + 1.
The proof is left as an exercise.
The stable 1-stem (7, is the cyclic group Z2 generated by the Hopf map η,
detected by Sq2.
We recall that the stable 2-stem G2 is also cyclic of order 2; the next
result identifies a generator and answers the question posed at the
beginning of this chapter.
Proposition 2
The composition η2 is non-zero and hence generates G2.
We shall give two proofs of this.
first proof: We will derive a contradiction from the assumption that η2
is null-homotopic.
Suppose then that η2 — 0. For
W^+X-U. У
we take
С"· + 2 η ^ en + 1 η с"

174
COHOMOLOGY OPERATIONS
so that K= Уи„ СХ becomes K= S" u„ f" + 2. We form a co-extension
and consider the complex L= Κνή c" + 4.
For simplicity we use the symbol ek to denote not only the cell in
question, as a cell of A^or L, but also to denote the corresponding generator of
H\K) or H\L). In general cohomology will be understood to have
coefficients in Z2.
Now the Hopf map»; has the property that Sq2(a") = e" + 2 in the complex
K. The same relation must hold in L, by naturality of Sq2 with respect to
the inclusion /: К c L, since /'* is isomorphic in dimensions η and и +2.
We have observed that pr\ = Sr\ where p: /l—>S"' + 2 is the map which
pinches S" to a point. Thus the diagram below is commutative, where we
■I J-
on + 3 >. О" + 2
have written >; for Sty as usual. Now we verified in the last chapter that
Sq2: Я" + 2(5" + 2)^Я" + 3(5" + 3) is non-zero, with zero indeterminacy.
Applying the naturality formula for functional operations to the above
commutative diagram, we deduce that 5^?(e"+2) = e" + 3, again with zero
indeterminacy. (Here we might be precise and identify e" + 2 and e"+ 3 as the
generators of #"+2(/Q and Я"+3(5П + 3), respectively; but this is clear from
the context, and we will generally leave these matters implicit.)
Now if Sq?(e"+ 2) = enX \ then Sq2(e" + 2) must be e"+4 in the complex L.
To see this, recall that Sq\ is defined by the following diagram, since L is
Hn+2(L)^Hn + 2(K)
Sq2\
Я" + 3(5" + 3)^/У" + 4(/-)
precisely the space obtained by converting ή to an inclusion ή: S"+i—>
A^u4(5" + 3 χ /) and then pinching the subspace identified with 5"+3 to a
point, and thus L plays the role of K/S" + 3 (or of the pair (K,S" + 3)).
We have thus shown that, in the complex L,
Sq2Sq2(an) = Sq2(en+ 2) = en + *
so that the composition Sq2Sq2 is non-zero. But this is impossible, in the
light of the Adem relation Sq2Sq2 -- Sq3Sql. Obviously Sq\a") = 0; in the
first place, σ" is the reduction (mod 2) of a class with integer coefficients and

COMPOSITIONS IN THE STABLE HOMOTOPY OF SPHERES 175
Sql is the Bockstein; in the second place, there is nothing that Sq\a") could
possibly be, since H"+l(L) = 0.
Thus the assumption that η2 = 0 leads to a contradiction based on the
hypothetical co-extension ή. This proves the proposition.
It is fruitful to think of the contradiction as arising from the relation
Sq2Sq2 = 0, which holds on any class which is in the image of the reduction
map from integral coefficients to Z2 coefficients.
The above proof could be summarized as follows: Suppose η2 = 0; then
we can form a co-extension fj: S"+3 —>K= S" u, en+1 and form the
complex L = A" u^ <?"+4. By the naturality formula for functional operations,
Sqjj(en+ 2) = en + 3 (with zero indeterminacy,) and therefore Sq2(en+2) = en + *
in L. Thus Sq2Sq2 is non-zero in L, which is impossible. Thus»;2 is
nonzero.
We have given the above proof in complete detail; future proofs will look
more like the above summary than like the full-length proof which preceded
it.
second proof: To illustrate another technique, we give a second proof,
using the first Peterson-Stein formula. We will use the complexes K =
5"+1 и, еп + 3 and L= S" u„2 e" + 3 and the map /: #—>L which is given
by η: 5"+1 —>S" and by the identity on the interior of en + \
Since such maps are used frequently, it might be well to be precise on this
occasion and explain exactly how/is defined; then, in the sequel, the brief
description given above will be considered sufficient. Recall that К is
obtained from the disjoint union K0 = 5"+1 и е" + 3 by identifying certain
points; under the identification mapp:K0^K, p(x) = p{y) \ϊη{γ) = χ, where
у s S" + 2 = bdy(e" + 3). Similarly L is the identification space obtained from
the disjoint union i0 = 5"ue"+! and the map q: L0^L such that
#(.y) = q(y) if η2(,ν) = x, where ye S"+2 = bdy(en + 3). We have an obvious
map/0: K0^L0 given by η and the identity map of e" + 3. We consider
the composite qf0: K0^>L. This map is constant on each ^-equivalence
class of points of K0, i.e., the diagram below, considered as a diagram of
\ Ϊ-
K--UL
sets and transformations of sets, may be completed (in a unique way) by a
transformation /, to yield a commutative diagram. It follows that / is a
continuous map (this is the fundamental property of the identification
topology of K), and this is the map/which we will use.

176
COHOMOLOGY OPERATIONS
We now give a second proof that η2 is essential.
Let A^, L, and f: K^L be as just above. In the commutative diagram
below, where the vertical maps are the inclusions, we have Sq2(a") = σ"+1
K-UL
Τ Τ
Sn+1-i+S"
(with zero indeterminacy) and therefore, by naturality, Sqj(a") = a"+1
(with zero indeterminacy), where in this last formula σ" and σ" + 1 denote
cohomology classes of L and K, respectively. Now we can define a
secondary operation Φ based on the following diagram, which is part of the
F= K(Z2, n + l)-U Xn,, -i* K(Z2, n+3)
1'
В = K(Zjt)-U K(Z2, n + 2)^L, K(Z2, n+4)
Postnikov system for S", with θ = Sq2, φ = Sq2, and φ equal to the class α
of Chapter 12. Then, by the first Peterson-Stein formula (Theorem 1 of
Chapter 16),
/*(φ(σ")) = Sq2{Sq}{an)) = Sq2(an+l) = en+3
with indeterminacy Sq2(f*(Hn+l(L))) = 0. But if/*(Φ(σ")) = е" + 3, then
Ф(а") = е" + 3 e Hn + 3{L), also with zero indeterminacy. However, if
L = S" νη2 е" + 3 has this non-zero secondary operation Φ(σ") = е" + 3, then
the attaching map η2 cannot be null-homotopic (see Exercise 3 of Chapter
16).
Notice again that it is the relation Sq2Sq2 = 0 (on classes with integer
coefficients) that provides the key step, and in fact we have proved the
following corollary.
Corollary 1
Let Φ be the secondary operation associated with the relation Sq2Sq2 = 0
(valid on cohomology with integer coefficients). Then Φ detects η2.
There is another question which we may ask about the 2-stem. Since the
1-stem is of order 2, the composition of >7 and 2/ is null-homotopic which-
every way we form it; thus the diagram
gives rise to a secondary composition (2ι,η,2ι}. This bracket takes its
value in a quotient group of [S(S"+ l),Sn] = π„+2(5"), and in fact the
indeterminacy is readily seen to be
(2/)#Κ+2(5")) + (2/)#(π„+2(5"))

COMPOSITIONS IN THE STABLE HOMOTOPY OF SPHERES 177
which is zero, since the 2-stem is a group of order 2. Thus the bracket
<2z,f/,2i> must be either η2 or zero. We can settle this question by a direct
argument.
Proposition 3
<2.^2.> = ^2бя„ + 2(5").
proof: Let/represent the bracket, and let K= S" u7 en + 3. Let Ψ denote
the secondary operation associated with the relation Sq2Sq2 -l· Sq3Sql = 0;
this operation was described in some detail in the last chapter. We will show
that Ψ is non-zero in K, which will imply that /Ms essential, from which the
result follows.
Let L = S" u„ e"+2, and let 2/ be a co-extension of the map 2/: S"+1—>
S"~l. Then 2i: S"+2—> L and we can form the complex M=Lu^e" + 3.
Now Sq3 = SqlSq2 is non-zero in M\ in fact this is the essential property of
Μ for our purposes. It is clear that Sq2(a"M) = e" + 2, and the fact that
Sql(e"+2) = e"+3 follows by naturality from the obvious fact that Sq1 is
non-zero in M/S" = S"+ 2 u2, e"+ 3.
We now map Μ into К by a map # which takes L ^ Μ into 5" by the
extension 2i, maps the boundary of e" + 3 by/, and maps the rest of e"+i =
C5" + 2 by C/.
Since g maps 5" с Μ to 5" с ί by 2/, we have Sq\{<fK) = σ"Μ. On the
other hand, Sq) is clearly zero.
By the first Peterson-Stein formula,
д*(ПО) = V(S#M)) + Sq>(Sql(a"K))
= 0-lS?3OM
and the indeterminacy is zero; thus Ψ(σ£) = ί'κ"1"3 with zero
indeterminacy, and this completes the proof.
Corollary 2
Let Τ be the secondary operation associated with the relation Sq2Sq2 +
Sq3Sql = 0. Then Ψ detects >/2.
It is instructive to compare the above operation Τ with the operation Φ
used previously. We may view Φ as defined on classes in #"( ;Z2) which
are in the image of reduction mod 2 from integral cohomology and which
are in the kernel of Sq2. The operation Τ is defined on a larger subset but
with larger indeterminacy. Whenever both operations are defined, they
coincide, modulo the larger indeterminacy. In the particular case of

178
COHOMOLOGY OPERATIONS
K=S" u^2 e"+3, we see that both Φ and Τ are defined and non-zero
modulo zero.
We sketch an alternate proof that <2i,j/,2i> = η2. Suppose that 2/ о 2/ is
null-homotopic; then we can form a co-extension 2i of 2/. This will bea map
of 5"+3 into S" u3, C(S" u,e" + 2). In this latter complex, Sq2Sql is
nonzero. If we now attach e" + 4 by 2/, Sql(Sq2Sqx) is non-zero, but Sq2Sq2 is
necessarily zero, which contradicts the Adem relations.
THE 3-STEM
We now turn our attention to (73, the stable 3-stem. We have seen that
the 2-primary part of this group is Z8, and it is natural to conjecture that
the Hopf map v: 5"+3 —>5" is a generator.
Proposition 4
The Hopf map ν generates the 2-primary component of G3.
proof: By the suspension theorem (Theorem 1 of Chapter 12), the
group na(Ss) is the stable group Z8 and the suspension homomorphism
Ε: π7(54) —> π8(55) is an epimorphism. It thus suffices to prove that Ev is
not divisible by 2 in Z8.
Consider the Hopf invariant homomorphism H: ^(S4)—>Z. If
/: S1 —>S4 then #(/), reduced mod 2, is nothing other than Sqj(\), but
Sq* commutes with suspension. Thus every element in the kernel of the
suspension Ε has even Hopf invariant.
Suppose Ev is divisible by 2; say, Ev — 2x Since Ε is an epimorphism, we
may write a = Εβ. Thus E(v — 2β) = 0. Thus ν — 2β has an even Hopf
invariant, which is impossible, since Я is a homomorphism and ν has an
odd Hopf invariant.
We next consider the composition η3 in G3. To study η3, we will use the
following relation between secondary operations. Let К be a complex and
let и be an integral class, и <= H\K;Z), such that Sq2u ■■ 0. Let Φ be the
secondary operation based on the relation Sq2Sq2 — 0 for integral classes
(Φ detects η2).
Lemma 1
Sq2{<X>{u)) = d1{Sq\u)) ε H"+5(K;Z2)/SqlH" + 4(K;Z2).
proof: It is quite natural to look for this relation, since, in the
calculations (Chapter 12) of the Postnikov system of 5", we found that d2Sq*(i„)
vanished in H*(X2) because it was equal to т(5^2(г„ + 2)) — Sq2(x) where a is
the class which gives rise to Ф.

COMPOSITIONS IN THE STABLE HOMOTOPYOF SPHERES 179
This will serve as the universal example for the proof; taking K= Xv
and м = />=/„ e #"(A\;Z), we have Sq2u = 0, and by our calculations
(Chapter 12) we have
Sq2(<t>(i„)) = d2Sq\ln)
with zero indeterminacy.
Now in the general case, Sq2u = 0; so we can find a lifting v. A"—> XL such
that pv = u, which is to say that и = v*{p). The lemma follows by natu-
rality. The total indeterminacy would be
V(S92""+,(K;Z2))^s<7'(//"+4(K;Z2))
but the first term is contained in the second, since Sq2 =Sq2 Sqi(Sq2Sqi).
(When K= Xu the fact that SqxHn + * = 0 follows from the calculations of
Chapter 12.)
We can now evaluate the composition η3.
Proposition 5
In the stable 3-stem (73, η3 = 4v.
proof: If η2 ο η were null-homotopic, then we could form a co-extension
ή: 5" + 4->А:=5"и,ге" + 3 and form the complex L^K^e" + s. We
have been before that the operation Φ of the lemma detects η2 and Φ(σϊ.) =
έ>ΐ+1 with zero indeterminacy. On the other hand, Sq2(e"L + *) = e" + s, since
L/S" is just S"+3 u, e"+s and here Sq2 is certainly non-zero. Thus we have
shown that in L, Sq2(<b^")) = en+S. But Sq\an) is obviously zero, and
d2Sq*(a") = 0 with zero indeterminacy. This contradicts Lemma 1.
Therefore η3 is essential.
But η is of order 2, since the 1 -stem is Z2, and therefore »73 is also of order
2. The 2-component of the 3-stem is Zg, and the only non-zero element of
order 2 in this group is 4v. Therefore η3 must be 4v. This completes the proof.
The reader may well suspect that the relation Sq2(i> = d1Sq* between
secondary operations gives rise to a tertiary operation and that this
tertiary operation detects 4v. This suspicion is indeed correct, but we do not
pursue it.
We can form the bracket (η,ΐι,η}, since 2η = 0. We conclude our
discussion of the 3-stem by evaluating this bracket.
Proposition 6
<»7,2i,»7> = 2v modulo 4v.
The indeterminacy of this bracket is the subgroup of Zs generated by
4v = η3. Thus the proposition asserts that the bracket contains precisely the
elements 2v and 6v.

180
COHOMOLOGY OPERATIONS
To prove the proposition, suppose for some value of r, 0 < r < 3, we
have 2rv e (η,2ι,η}. Recall that the bracket is the composition
Sn+3-i+K=Sn+l \J2len+2-i+S"
and consider the composition
We are assuming this composition null-homotopic. Using an extension of
ν ν ή as attaching map, we may construct a complex
L = S" Uf't! Uf't3 Uf" + 4 uf'ts
The following relations hold in the cohomology of L: Sq2a" ~e" + 2;
Sqle" + 2 = e" + 3; Sq2e"+3 = e"+5; Sq*an = en + *; and dren + * = en+s. From
the Adem relation Sq2SqiSq2 = Sq^Sq* + Sq4Sq' = 0, it follows that r=\.
This completes the proof.
We remark that we may use the above relation, written for example as
SqlSq4 +Sq*Sql + (Sq2Sq')Sq2 = 0, to define a secondary operation Θ.
The reader may easily verify that Θ detects 2v.
Note also the hierarchy which has appeared in the 3-stem: ν is detected
by the primary operation Sq*; 2v, by the secondary operation Θ; and 4v,
by a tertiary operation. A similar phenomenon appears in the 0-stem—we
may view dr as an r-ary operation and even view /* as a 0-ary operation (the
induced homomorphism).
THE 6- AND 7-STEMS
We next consider Gb, where the 2-component is Z2. It is natural to ask
whether the generator is v2.
Proposition 7
v2 is essential and hence generates the 2-component of G6.
proof: In H*(K(Z,n)), Sq*Sq*in = SqbSq2i„. Let Ξ denote the associated
secondary operation. We define a map / from S"+ 3 uv e"+ 7 to S" uv2 e"+ 7
using v. S"^3—>S". Then, by the first Peterson-Stein formula,
/ *(Ξ(σ")) = Sq\Sq*{a»)) -\ Sqb{Sq}{a"))
= en + 1 +0
(with zero indeterminacy), and thus Ξ is non-zero in S" u,;e"t7, which
proves that v2 is essential.
For another proof, see Exercise 2.
We now state the value of a secondary composition in the 6-stem.

COMPOSITIONS IN THE STABLE HOMOTOPYOF SPHERES 181
Proposition 8
<Л,М> = v2 mod zero.
The proof is left to the reader as Exercise 3; it is of course sufficient to
show that the bracket is nonzero.
We turn next to the 7-stem. According to the results of Chapter 12, G-,
is cyclic of order 16 (in the 2-component). In a manner analogous to
Proposition 4 and by a similar proof which we omit, we have the following.
Proposition 9
The Hopf map σ generates the 2-component of C7.
Thus the generator of the 7-stem is detected by the primary operation
Sqa. Further, a hierarchy similar to that in the 3-stem occurs; namely, 2σ
can be detected by a secondary operation; 4σ, by a tertiary; and 8σ, by a
quaternary operation. The only secondary composition we will consider in
the 7-stem is the following.
Proposition 10
<v,8/,v> = 8σ modulo zero.
That the indeterminacy of the bracket vanishes is obvious for
dimensional reasons. The evaluation of the bracket is similar to the proof of
Proposition 6; however, we first need a lemma, valid for η sufficiently large.
Lemma 2
Let и be an «-dimensional cohomology class—that is, let и е H"(X;Z)
for some space X. Suppose Sq2u and Ф(и) are zero. Then Sq^diSq^u =
d3Squu modulo the total indeterminacy.
proof: In the calculations of Chapter 12, we have proved this lemma in
the special case и — ι„ in H"(X2;Z) (see the table on page 117 and Lemma 2
of Chapter 12). But this is the universal example; the general case follows
by naturality.
proof of proposition 10: Proceeding as in the proof of Proposition 6,
suppose we have Τσ in <v,8/,v> for some r, 0 <r <4. We then may
construct a complex
L = S" Uf'Huf"ts Uf" + 8 Uf't9
with the following properties: 5/σ" = e" + 4;i/3e" + 4-=e" + 5;5^V+5=e" + 9;
StfV^ e" + 8; and dre" + * -- e"+9. By Lemma 2 it follows that r--=3. This
completes the proof.
Of course, a quaternary operation which detects 8σ may be based on the
relation of Lemma 2.
We summarize the results of our computations in an omnibus theorem.

182
COHOMOLOGY OPERATIONS
Theorem 1
The 2-components of the groups Gk, к < 7, are given by the table below.
STEM
0
1
2
3
4
5
6
7
GROUP
Ζ
z2
z2
z8
0
0
z2
zl6
GENERATOR AND RELATIONS
г
>?
>72 = <2г,»7,2г>
ν; 2ν = <»7,2i,»7> mod η3 = 4v
v2 = <4,v,4>
σ; <ν,8/,ν> = 8σ
DISCUSSION
We have considered only the simplest sort of Toda bracket. It is not
possible to express 2σ as a bracket without considerably complicating the
allowable construction.
However, we have evaluated many important compositions and
secondary compositions; in particular, we have obtained all the significant data
through the stable 6-stem and much in the 7-stem. But of course we have
only scratched the surface. Toda especially has taken the point of view
of calculation by means of composition. In his book will be found many
additional results obtained oy composition methods, as wejl as a detailed
exposition of the properties of the secondary composition.
Nor does the theory stop with secondary composition. Oguchi has
thoroughly analyzed tertiary composition; Spanier has indicated the general
framework of higher compositions.
Spanier has also shown that both the functional and secondary cohomol-
ogy operations of Chapter 16 fit into the framework of secondary
composition. The Peterson-Stein formulas become special cases of some general
properties of the secondary composition.
Some care need be exercised with the indeterminacy, though, since the
usual definition of a secondary operation has only " half" its natural
indeterminacy as a secondary composition. Thus, if we have the relations
ψθ = 0 and 0u = 0, we might define Φ(;/) = (υ,Ο,φ}, but this secondary
composition has larger indeterminacy than does our definition of Ф(и). The
reason for this is that our definition includes the choice of the specific
cohomology class φ.

COMPOSITIONS IN THE STABLE HOMOTOPY OF SPHERES 183
EXERCISES
1. Prove Proposition 1.
2. Give an alternative proof of Proposition 7 by imitating the first proof of
Proposition 2, that is, assume v2 null-homotopic and form a coextension
v2: S"+7 ->S" uve" + 4. Then obtain a contradiction to the Adem relation
Sq*Sq'i = Sq6Sq2 + Sq1Sql.
3. Prove Proposition 8. hint: Imitate the alternative proof of Proposition 3.
4. Use Adem relations in similar fashion to find some non-zero stable elements
in stems higher than 7.
REFERENCES
Compositions Higher compositions
1. F. P. Peterson [1]. 1. K. Oguchi [1].
2. H. Toda [1]. 2. Ε. Η. Spanier [3].
Secondary compositions
1. M. G. Barratt [2,3].
2. Ε. Η. Spanier [2].
3. H. Toda [1,2].

THE ADAMS SPECTRAL
SEQUENCE
CHA PTER
18
In Chapter 14 we constructed a spectra! sequence for [ Y,X] based on a
Postnikov decomposition of X (or a cell decomposition of Y). The spectral
sequence starts at H*( Υ;π^ Χ) and thus requires knowledge of the homotopy
groups of X; il is not a useful tool with which to compute those homotopy
groups.
In this chapter another spectral sequence, constructed by Adams, is
described. This spectral sequence differs from that of Chapter 14 in the
following important respects. First of all, it yields information only about
the 2-component—i.e., it yields a composition series for the 2-component
of [ Y,X]. Secondly, this spectral sequence is valid only in the stable range—
for example, if X is (/? — l)-connected and dim Υ <2n — 2. Thirdly, the
starting point, the £2-term, depends only on the cohomology H*{X\Z2)
and H*{ Y;Z2), viewed as graded modules over the Steenrod algebra Λ.
Taking X and Υ to be spheres of suitable dimensions, we would thus
expect the Adams spectral sequence to give information about the
2-component of stable homotopy groups of spheres. This is in fact the case, and
in recent years the Adams spectral sequence has been the most important
tool for computing these groups.
184

THE ADAMS SPECTRAL SEQUENCE
185
RESOLUTIONS
In Chapter 12, we calculated some homotopy groups of spheres by
constructing a Postnikov system. In this method of computation, the
cohomology of K{Z,ri) is killed, one step at a time, by passing to successive
fibre spaces induced by maps into Κ(π,η) spaces. Among the
cohomology classes of K(Z,n) which had to be killed by separate maps were the
classes given by Sq2, Sq*, and Sq* on the fundamental class. Since Sq* is
indecomposable whenever / is a power of 2 (Theorem 1 of Chapter 4), it
is not surprising that these classes must be killed and it is natural to wonder
whether it would not make sense to kill all three of these classes at the
first stage of the construction. We can achieve this by a map of K(Z,n) into
a product space K2 X K4 χ Ks where Kt = K{nh η + /'), with an appropriate
choice of the groups π(.
As usual we will concern ourselves only with 2-primary cohomology
and homotopy, and by H*{X) we mean the reduced cohomology of X
with Z2 coefficients, etc. For technical reasons, it turns out to be convenient
to take all the groups π, equal to Z2 and moreover to start with K(Z2,n) in
place of K{Z,n) at the base of the tower. We therefore formulate our new
kind of tower, called a complex, in the following way.
Definition
A complex X over X, where A' is a (nice) simply connected space having
finitely generated homotopy groups, is a diagram (finite or infinite) of the
form
ψ
ςΐΚ3_, -±-i* Χ, 9- > К,
1
ωα:0 —!ϋ-+ i, —^-> к,
X0=X-«L>K0
where each A", is a generalized Eilenberg-MacLane space
and each Xs+, is the (nice) space induced over Xs by the map gs from the
standard contractible fibring over A^s.

186
COHOMOLOGY OPERATIONS
A complex X will be called (n — \)-connected if every Xs is (n~ 1)-
connected (s > 0) and nsJ > η for all (sj).
The case X = S", in which one might expect to collect into the map g0
all the indecomposable squares, suggests that the rank rs should be allowed
to be infinite. On the other hand, we will be studying [Y,X] for dim У
finite and bounded by a bound depending on the connectivity of X. Thus
we can assume that the rank rs is finite for every s, without much loss of
generality, and at considerable gain in technical simplicity.
Recall that in the Postnikov system for the sphere we endeavored to
kill all the cohomology of X by the successive choices of the Ar-invariants.
This idea underlies the following notion.
Definition
A complex X over X is acyclic if the induced map g* maps H*(KS)
onto H*(XS) for every s >0.
An acyclic complex over X is called a resolution of X.
However, it will be sufficient to study complexes which are acyclic " up
to a certain dimension." It is therefore convenient to have the notion of
an N-acyclic complex, by which we mean a complex for which cfc is onto
H\XS) for every s and for all к in the range 0 < A: < N. Such a complex
may be called an N-resolution.
Before giving a characterization of acyclic complexes in terms of the
induced maps p*, we make some observations on the cohomology of Ks.
We know by Theorem 2 of Chapter 9 that H*{KSj) looks exactly like the
Steenrod algebra Л (applied to the fundamental class) in the range from
dimension nSi to dimension 2(«sJ). We may therefore assert that, in an
(n— l)-connected complex, through dimensions < 2n — 1 H*(KS) is
isomorphic to the free .Α-module with the fundamental classes {isj} as
basis.
Proposition 1
Let N <2n— 1. Then an (n — l)-connected complex is TV-acyclic if and
only if pi is the zero homomorphism for every s > 1 and к < N.
We will usually formulate such propositions as the above without giving
details about the range of dimensions for which the assertions are valid;
instead, we will say "in the stable range" and leave the precise
interpretation to the reader. For example, Proposition 1 in this form reads: In the
stable range a complex is acyclic if and only if pks is the zero homomorphism
for every s > 1.
The proof of Proposition 1 is easy. If the complex is acyclic and we have
a class ueHk(Xs), then u = gks{v) for some ν ε H\KS). But in the stable

THE ADAMS SPECTRAL SEQUENCE
187
range, H*(KS) is a free ^-module, and since the squares commute with the
transgression, » = τ('ι;), χν e //'"'(Ω/ς). It follows that /?sk+1(u)=0, by
Serre's exact sequence. Thus p* is zero. The argument can be reversed to
obtain the converse.
We will often apply the functor [ Y, ] (where У is a nice space) to the
objects and maps of a complex. Recall that [Y,K(Z2,n)] = H"( У), and
observe that we obtain a similar classification theorem for generalized Eilen-
berg-MacLane spaces: [Υ,Κ*] = Σ] H"',J(Y). In particular, if a map Y^>KS
pulls all the fundamental classes back to the zero of H*{ Y), then it is null-
homotopic. Note also that 2[ Y,KS] — 0 for any Y.
By a " map of complexes " we mean the obvious thing: a family of maps
fs: Xs —> Ys commuting with the projections.
Proposition 2
Let У be a resolution of У and let X be a complex over X. Then any map
Υ — > X can be covered by a map of complexes.
proof: In the diagram below we have q* = 0, since У is a resolution.
Ρ,+ 1
,s+il
fs
X<^+K<
Thus q*f*gs = 0, which implies that the map gsfsqs+l: Ys+l^>Ks is null-
homotopic, and this gives us the existence of a map/s+l completing the
diagram. Starting with/0=/, we construct the desired map of complexes
by induction.
Suppose that Ή is only an TV-resolution; then if some nsj is greater than
TV, the above argument does not go through. However, in such a context, we
are only interested in the behavior of the various spaces up to
dimension /V, and so we lose nothing by replacing ЭС by a complex in which every
nsJ is less than or equal to jV (we simply omit KStJ if n,yJ is too large).
THE ADAMS FILTRATION
We now would like to use a resolution of X to study [Κ,Α']. To do so,
we will need the mapping sets [У,Л\] to have natural abelian group
structures. Thus to study [Κ,λ'], where A' is (n— I)-connected, we assume
dim Υ < 2n — 2 and take an (n — l)-connected /V-resolution of X, with N
large.
These hypotheses of stability will not be stated at every juncture. They
are there, nonetheless.

188
COHOMOLOGY OPERATIONS
Any complex over X defines a filtration of [ Υ,Χ] in a natural way. We
put FS[Y,X] equal to the image of [Y,XJ]—>[Y,X] under the composite
projection X,— >X. We clearly obtain a decreasing filtration, PTl <= Fs,
with F° = [ Υ,Χ]. We can define Г* [ Υ,Χ] as the intersection f)/=0 Γ[ Υ,Χ].
Proposition 3
If F* is defined in terms of a resolution and F* is defined in terms of
any complex whatsoever, then Fq <= Fs for every s.
This is immediate by Proposition 2: we have a map of the resolution into
the other complex, and the result follows from the definition.
As an obvious corollary, all resolutions of X define the same filtration
on [У,^]. Assuming that factually has a resolution, we thus obtain a
canonical filtration on [У,^].
Proposition 4
Every X admits an jV-resolution for each N.
Here we assume that X is (/; — l)-connected; the /V-resolution we obtain
will likewise be (n — l)-connectcd. To assure that rs < oc for every s, we
have assumed that nt{X) is a finitely generated group for every /. We will
show how to construct the first stage of the resolution in such a way that
Xx has all the properties required of X; this will indicate how to build the
desired resolution inductively.
Let {.Xjj} be a set of generators for H'(X), i < N, as an Λ-module. Since
all r4(X) are finitely generated, so are the H'{X), by the Hurewicz theorem
mod (%c. We form the complex \\imj K(Z2,n,j), where n;j is the dimension
of Xjj, and map X into this complex by a map g0 such that </*(',.>) — v;,y.
Let ΑΊ be the induced fibre space over X() = X by #„. From the homotopy
exact sequence, it is clear that Xx is {n — l)-conncctcd and that its
homotopy groups are finitely generated. The proposition now follows by induction.
Henceforth, the symbol F"[Y,X] will always be understood to refer to
this canonical minimal filtration of [Κ,-Υ].
We remark that the abo\e construction is not designed to guarantee that
the higher stages in the resolution will have higher connectivity than X. In
fact, our resolution of K(Z,n) will simply have A', = K(Z,n) for every s.
It gives the filtration of [S",K(Z,n)] = Ζ by powers of 2, that is Fr[S",K(Z,n)]
consists of the classes divisible by 2r. But note that F* = 0 in this case.
EVALUATION OF F*
We now turn to the general problem of the evaluation of Fy[Y,X].
Under the usual assumptions on У and Χ, [Υ,Χ] is an abelian group.

THE ADAMS SPECTRAL SEQUENCE
189
Therefore we can speak of a map У —> X as being " divisible by η " where η
is any non-zero integer; this means that there exists a e [Y,X] such that
я a is the homotopy class of the given map.
Theorem 1
A class a e [ Y,X] lies in Fy[ Y,X] if and only if a is divisible by 2r for
every positive integer r.
We will prove this theorem in two parts: first, if a is divisible by 2s, then
xc FS[Y,X]; second, if α is not divisible by 2r, then there is some t for
which a docs not lie in F'[Y,X].
The first part is easy; the essential point is that 2[Y,K,] = 0. We argue
by induction on j, assuming the statement valid for F"~' in all complexes
over any nice space. (The assertion is trivial when s = 0.) If a = 2s/?,
consider 2β e [ Y,X]; this lies in the kernel of {gQ)# : [ Y,X] —> [ Y,K0], and
therefore there exists a lifting w: Κ—>ΛΊ with p{u= 2β. (As usual we use
the same letter for a map and for its homotopy class.) Now
ζ = 2*β = 2>-\2β)=Τ-\Ριιή = ρχ{2^ιή
By the induction hypothesis, 2s"hi lies in Fs~l[Y,Xt], so that this map of
Υ into X{ factors through Xs, which is to say that a e F'[ Y,X], as was to
be proved.
The second part of the proof requires the construction of a certain
complex. Let ι denote the class of the identity map X—> X. Whether or not
[Α',Α'] is a group depends on the space X, and wc wish to consider the class
2r(i). Since we work in the stable range, we can validate this without much
trouble, say by assuming X to be (/? — l)-connectcd and then considering
[Α' ,Α'] instead of [X,X], where X is an appropriate skeleton of X, for
example, the (2« — 3)-skeleton; ι then denotes the class of the inclusion.
The reader should be able to handle the details of such matters.
Let U be the fibre space induced over SX—or S(X')—by the mapping
S(2ri). The fibre is ΩΞΧ, which agrees with X in the stable range, and we
consider the diagram
X^UX- ΩΞΧ-ί->υ
i
SX St2'">SX
which gives rise to an exact sequence
[ γχ] _Л> [ γ,Χ] _Ll» [ y,i/] -£i+ [ Y,SX] ^U [ Y,SX]

190
COHOMOLOGY OPERATIONS
under a suitable restriction on the dimension of Y. Thus [ Y,U] is a group
extension of ker (20 by coker (2r), that is, the sequence
0 -> coker (20 -> [ Y, U ] -> ker (20 -> 0
is exact. Since 2'(coker (20) = 0 and 2'(ker (20) = 0, it follows that
22r[Y,U] = 0. We use this fact to construct a complex over U, killing its
homotopy groups. If X is (n — l)-connected, then so is U. Let π0 be the
first nontrivial homotopy group of U, occurring in dimension ηυ>η.
Then π0 is an abelian group and there is a canonical epimorphism π0—>
π0®Ζ2. Let h0: U —> K0 be the corresponding map under the equivalence
Horn (π0, π0®Ζ2)^Η"°(ϋ; π0®Ζ2)^>[1/,Κ0]
where K0 = Κ(π0 ®Ζ2, η0). Let £/, be the induced fibre space over U
defined by Λ0. From the homotopy exact sequence of this fibre space, it
is clear that π„0(£/,) is isomorphic to 2(π„0(ί/)). But 22r(nm(U)) is zero for
all m in the stable range. Thus we may proceed with the above construction,
killing n„0(U) in not more than 2r steps. We can then continue to build the
complex over U, killing the next non-trivial homotopy group, and we can
annihilate n*(U) up through dimension m in a finite number of steps. Thus
if the space Υ has finite dimension m, we reach a t such that [ Y,U,] = 0.
With this machinery, it is easy to complete the second part of the proof
of the theorem. If α e [K,X] is not divisible by 2r, we cover the map /:
X—> U by a map of a resolution of X into the above complex over U.
Then α is certainly not in F'[Y,X] for any t such that [У,£/,] = 0, since
from the diagram below, it is clear that a e F'[ Y,X] implies α e ker (/#) =
2r[Y,X]. This completes the proof of Theorem 1.
/
/
Ys+X_uu
i
Suppose that [K,X] is a finitely generated abelian group; then it is
expressible as a direct sum of a free abelian group and of finite cyclic
groups of prime-power order. Clearly the elements infinitely divisible by
2 are just the elements of the odd-primary part of the group.
Our hypotheses assure that [Κ,Χ,] will be a finitely generated abelian
group for every s > 0. Indeed, У is a finite complex with dim Υ < 2n — 2
and each X, is (n — l)-connectcd with finitely generated homotopy groups.
Then the Postnikov spectral sequence of Chapter 14 converges to [Κ,Α^]
and has only finitely generated groups in its initial term.
Under these hypotheses, then, Fr[Y,X] is the direct summand of [Y,X]

THE ADAMS SPECTRAL SEQUENCE
191
consisting of torsion elements of odd order; in particular, it has no element
of order 2. Further, [Y,X]/F~[Y,X] is the 2-component of[Y,X].
THE STABLE GROUPS {Y,X}
For Υ in the stable range with respect to X, we now have defined a
filtration on [У,Х] by means of a resolution of X. It would be tempting
now to consider, in the manner of Chapter 14, an exact couple with typical
D-term of form [S"Y,Xq] and typical £-term [S"Y,Kq], with suitable
bigradings. Indeed, basically this is what we are going to do in order to
construct the Adams spectral sequence.
However, some more preliminaries are still necessary. First of all, we
need an algebraic detour so that we will recognize E2 for what it is. We
take this detour in the next section.
More importantly, the above candidates for D- and £-terms of our
proposed spectral sequence take us back out of the stable range. Dim Υ <
2n — 2 does not imply dim Sp Υ < In — 2.
However, if we replace η by η + m, for m sufficiently large with respect
top, dim Υ <2n — 2 does imply dim Sp+mY<2(n +m) — 2. This
observation suggests replacing the (n — l)-connected X with the (n +m— 1)-
connected SmX and Υ with Sm Y. We then would be studying a term in
a spectral sequence for [Sm Y,SmX], but since dim Υ < 2n — 2, this group
is independent of m anyway.
We thus are led to consider a spectral sequence with typical Z)-term of
form [Sp+mY,mXq] and typical £-term [Sp+mY,mKQ], where mXQ and mKQ
are the appropriate terms of a resolution of SmX and where m is sufficiently
large, depending on ρ and q. Of course, to justify this procedure we must
show that the stable portions of the different exact couples obtained for
different m agree under suspension.
Granting this (see Proposition 5), we now notice that the assumption
dim Y<2n — 2 is no longer essential. Assuming only that У is a finite
complex, the groups [SmY,SmX] are independent of m for m sufficiently
large. The common value is denoted {Y,X} and is called the (abelian)
group of 5-maps of У into X.
We now state the result which shows independence of the choice of
large m.
Proposition 5
Let X be an (n — l)-connected resolution of X. Then there exists an
«-connected resolution '.T. of SX such that SXS and lXs have the same
(2n — j)-type for each s.

192
COHOMOLOGY OPERATIONS
Thus for Υ such that dim Υ < In - 2 - s, we have [ Y,XS] --- [S Y,SX,] --=
[SY,lXs]. Hence form sufficiently large, depending on ρ and q, [Sp+m Y,mXq]
is independent of in.
The proof of Proposition 5 is by induction on .9. The assertion is trivial
for s = 0. We sketch the induction step, supposing ' X, has been constructed
of the same {In — лО-typc as SXt. Thus we may choose' Ks so that Ω(' Ks) — Ks
through dimension 2n — s — 2; that is, for / < 2n — s — 2, π,/C, = π,·+,'K„
with the image of '#s+1 corresponding under suspension to the image of
g's. For such a choice of *gs and 'A"5, calculation verifies that ' Xs+l satisfies
the induction hypothesis.
Of course the notations 'A', '#, etc., have nothing to do with the co-
homology suspension of Chapter 14.
THE DEFINITION OF EXT t {M.N)
In this section we take the necessary algebraic detour so that we will
recognize E2 of our spectral sequence. Our goal is to define the symbol
ExLt (M,/V), where Μ and N are graded modules over the Steenrod algebra
A.
This symbol is defined by a standard construction of homological
algebra. We will now sketch the pertinent algebraic theory. The main theorem
of this chapter, on the existence and fundamental properties of the Adams
spectral sequence, will tie this algebraic material to the above material on
complexes, resolutions, and nitrations.
Let R be a graded ring with unit. Recall from Chapter 6 that a graded left
/?-module Μ is a sequence of unitary left /?0-modules Λ/, together with an
associative action /г./? ® Л/—> M. We will suppose Mt — 0 for /<0. A
homomorphism of degree t between two graded left /?-modulcs is a sequence
of left /iu-homomorphisms/,: Mq ~>Nq + , respecting the actions μ%ι and/\.
Such a homomorphism may be abbreviated/: M-+N. Wc denote by
Hom'R {M,N) the group of homomorphisms Μ > N of degree —t. Λ
sequence
0 «_ Μ ^~ C„ *^- C, ^— C, * +i- C4«-. · ·
of graded left /?-modules and homomorphisms is called a projective
resolution of Μ if the following conditions are satisfied:
1. Each Cs— \C\ ,} is a projective graded left /?-modu!e
2. All the homomorphisms {ds\, г are homomorphisms of degree zero
3. The sequence is exact

THE ADAMS SPECTRAL SEQUENCE
193
It is a standard theorem of homological algebra that such projective
resolutions exist and are unique up to chain homotopy.
If {Cs} is a projective resolution of M, and N is a graded left /?-module,
then in the sequence
0 -+ Hom'R (C0,/V)^* Horn;, (Ct,A0-> -'-^ Horn; (C„N) · · ·
we have d*+ld* = 0. The homology of this sequence at Horn!, (CS,N) is
denoted Ext^' (Μ,Ν); the groups Ext«f (Λί,/V) form a bigraded group.
Later we shall make some observations about how to calculate such things.
For the moment we content ourselves with the remark that Ext£,r (Μ,Ν) ъ
Hom'R (Μ,Ν), since the Horn functor applied to the exact sequence
gives an exact sequence
0-> Horn'» (Μ,Ν)-£+ Horn1, (C0,N)-^-> Hom'R (C,,/V)
for each t.
Under certain conditions, Ext as defined above can be given a
multiplicative structure so that Ext'·' becomes a bigraded algebra. In particular,
if R is an augmented algebra over a field K, then Exts'(K,K) becomes a
bigraded algebra over K. (Here К is made a graded /?-module, concentrated
in gradation 0, by means of the augmentation.) We will be interested in
the case K = Z2, Л = -4 = the mod 2 Stccnrod algebra. In this case, R
is a commutative Hopf algebra and the algebra Ext_.t (Z2,Z2) turns out to
be commutative. We will not define the multiplication in Ext.
PROPERTIES OF THE SPECTRAL SEQUENCE
We are now in a position to formulate the main theorem on the Adams
spectral sequence.
Theorem 2
Let A' be a space such that n-[X) is finitely generated for every />0,
and let У be a finite complex. Then there exists a decreasing filtration F*
of { Y,X\ and a spectral sequence {Er,dr\ with the following properties:
1. Each £, is a bigraded group, and dr is a homomorphism of bigraded
groups, </;■': £,*·'--»£,; + ,' + '-1 such that drdr - 0
2. E2 is naturally isomorphic to Ext.t (H*(X),H*( Y)) (cohomology with
coefficients in Z2 as usual)

194
COHOMOLOGY OPERATIONS
3. There is a natural monomorphism £ΛΊ —>£"/■' whenever r >s, and
£»' = f)r>s E?'' (where Ex will be defined in the proof of this assertion
below)
4. E^ is naturally isomorphic to FS{S"SY,X}IFS+S {S"SY,X}
5. F^ {Y,X} consists of the subgroup of elements of finite odd order
Some parts of this theorem have essentially been proved already. Before
completing the proof of properties (1) to (5), we state here for completeness
some further results for the case X—S", Y= S°, which pertains, by (4)
above, to the stable homotopy of spheres. In this case, by (2) above, E2 =
ExUZ2,Z2).
6. Every £, is endowed with a multiplicative structure compatible with
the bigrading, and dr is a derivation
7. The product in £2 is the usual product in Ext (which is associative and
commutative)
8. The product in £,+ , is induced from that in Er by passing to quotients
(and hence is also associative and commutative)
9. The product in £x may equally well be regarded as induced from that
in £,, using (3) and (8), or as obtained by passing to quotients in the
composition ring of stable homotopy of spheres, using (4)
This theorem has become one of the cornerstones of stable homotopy
theory. It separates the problem of calculating the stable homotopy groups
of spheres, for example, into three distinct steps: the purely algebraic
problem of calculating Ext^ (Z2,Z2); the determination of the differentials
in the spectral sequence; and the reconstruction of the group extensions
and products which are lost in passing to quotients—as in (4).
We now proceed with the construction of the spectral sequence and the
proof of properties (1) to (5).
The spaces X and Υ are not assumed to have any connectedness
properties, since we are interested only in stable calculations. For the most part,
we will be working in the proof with S" Υ and S"X, instead of Υ and X.
Then of course we are dealing with spaces which are (n— l)-connected.
We begin by taking a resolution of S"X, in which we denote the stages
of the resolution by "Xs (s > 0) and the corresponding generalized Eilen-
berg-MacLane spaces by "Ks (s > 0). Recall that by Proposition 5 a
resolution of S"+mX for the stable range can be assumed to look like the mth
suspension of our resolution of S"X.
For any non-negative m we obtain a filtration on the homotopy classes
[Sn+mY,S"X] in the manner which has been discussed earlier. Together

THE ADAMS SPECTRAL SEQUENCE
195
with our previous remarks about the stability of the resolution, this defines
the stable filtration of the theorem and proves (5).
Let Y' denote an appropriate suspension of Y. Then the homotopy
classes of maps of Υ into the various spaces in the resolution of S"X (and
their loop spaces) form a commutative diagram
i i
> [S2 y':k0] -> [S υ',-xj -> [S γ',-κ,] -> [ Y':x2] -> [ y';k2]
i i
[SY',snx] --»[s y':k0] -> [ y'sxj -> [ υ',-kj
i
[Y',SnX]^[Y',nK0]
as in Chapter 14. From this diagram we can form a bigraded exact couple
with
Ζ?ϊ'=[5" + '-,Κ,"ΑΓΙ]
(with the convention that "X0 = S"X). The maps i,j,k of the exact couple,
in the notation of Chapter 7, are induced by the projections, the " k-
invariants," and the inclusions of the fibres. The map / has bidegree
(—1,-1); the тару has bidegree (0,0); the map k, and hence also the first
differential d=jk, has bidegree (1,0). (We remark that the bigrading used
here is not the same as that used in Chapter 14.)
We have defined E\·' only for non-negative s, but we can put El' = 0 for
s <0. This is tantamount to continuing the resolution below S"X by a
string of one-point spaces.
The rth differential of the exact couple may be roughly described as
joi~r+l ok, and it obviously has bigrading (r, r—l). We have thus
verified (1).
It remains to prove (2), (3), and (4). We will now undertake the proof of
(2).
Lemma 1
[Sn + '~sY,nK,] is naturally isomorphic to Нопгд1 (H*(nKs),H*(S"Y)).
Every map between two spaces induces a map on the cohomology level
which is an Λ-module homomorphism of degree zero. In this way we obtain
a transformation [Sm Y,nKs) -> Ηοιτβ (H*(nKJ,H*(SmY)). We have proved
in Chapter 1 that this is an isomorphism in the case when "Ks is a Κ(π,η)
space. It is not hard to establish this isomorphism in the present case.

196
COHOMOLOGY OPERATIONS
We omit the details. The result then follows from the observation that
Hom°4 (H*("Ks)M*(SmY)) ** Нотк4 (H*("Ks),H*(Sm~k Y))
(using the suspension isomorphism) by putting m = η + t — s and к = ι — s.
Thus i/,: E\-'->E\+ut may be considered as a map
Hom.V (H*("K,)M*(S" Г))->Нот:.Г,_| (//*("/:,+,),Я*(5" Υ))
and it is clear from the definition of i/, and the proof of Lemma 1 that in
this interpretation dx is simply the map induced by going from ΩΑ", to
Ks+l in the (expanded) resolution.
Now Ext54' (H*X,H* Y) = Exts4' (H*(SnX),H*(Sn Y)) by the suspension
isomorphism, and by definition this latter group is the homology of the
sequence {Hom'.t (CS,H*(S"Y))} where the {C5} form a projective
resolution of H*(S"X) in the algebraic sense. We claim that such a resolution
is obtained from {H*("KS)} simply by modifying the gradation. In fact,
the expanded resolution over S"Xgives, on the cohomology level, a diagram
(below) with each p* trivial, each g* epimorphic, and hence each /* mono-
"t 1
//*(Ω2("*,))«^-( )«-^//*(Ω("Χ,Μ))«^( )«_il//*(»*j+2)
+ A
ρ* Ι ρ*
morphic. The i-g-p sequences are exact, coming from the mapping
sequences. It is easy to verify from this that the sequence
H*(n2("Ks))<-H*(n("Ks+l))^HXKs+2)
is exact. But in the stable range this is exactly
H*-\"K,) *ii-Η*' '("*<+,) *^- H*("K5+ 2)
with the reinterpretation of dx given above In this way wc find that
0 «_ H*(S"X) 4- H*("K0) *±- Н*{ЯКХ)<£- H*("K2) *
is a projective resolution (in the stable range) of H*(S"X), except that the
homomorphisms all have degree I instead of degree zero as required for a
resolution. Wc thus obtain a resolution if we replace.each H*("KS) by a
module Cs isomorphic to it but graded so that each element of C, has
gradation less by s than the corresponding element of H*("K<), Wc then have
that ExtY (H*X,H* Y) is the homology of the sequence
{Hom:4(C„//*(S"n)},
but this is the same as the sequence (HomV (//*("Л:ч),//*(5" Г))}, and
we have seen that the homology of this sequence is just E,.

THE ADAMS SPECTRAL SEQUENCE
197
This proves (2). Observe, as a corollary, that E2 depends on X and Υ
only through H*X and Η* Υ as ^-modules.
In order to prove (3), we will follow much the same route as was taken
in Chapter 7 in the discussion of the homology exact couple of a filtered
complex.
Lemma 2
£/■' is isomorphic to the quotient
a im: [Y':x,rx,+r]^[Y':xsrx,+i]
В im:[SY':Xs-r+irXs]^[Y':XJ"Xs+l]
where the map in the numerator A is induced by the natural map "XJ"Xs+r
—> "XJ"X,+, (each projection A',.., —> Xs being interpreted as an inclusion),
and the map in the denominator В is from the exact fibre mapping
sequence obtained by mapping Y' into successive pairs from the triple
("A'J_r+1,"A's,"A'5+1); here У denotes S"+'"Y.
We will not give the details of the proof of this lemma because it is
formally the same as the proof of the corresponding result in Chapter 7
(Proposition 3), with the appropriate modifications. The essential
observations are the exactness of the sequence arising from the triple, since we
are in the stable range, and the identification of "XJ"X,+ l with "K, in the
stable range.
We now define £x as follows.
Definition
F,, __ \m:[Y',"Xs]^[Y',"Ks]
x im: [SY'^X/'X,.] -+[Y',"KS]
with the notations and conventions described above. The map in the
denominator comes from the triple (S"X,"X„"X,+ l).
This definition is motivated by the formal relationship with Er as
characterized in Lemma 2 and by part (3) of the theorem, the proof of which is
now well under way.
Although an η occurs in this definition of Ε, (as indeed in all the
definitions which we have given), it does not play an essential role, since we
confine our interest to the stable range, in which the results are
independent of «.
We can now complete the proof of (3). Recall that E\·' = 0 if s <0,
and thus the same is true for all Er. Since dr has bidegree (/·, r — I), no

198
COHOMOLOGY OPERATIONS
differentials map into Es/ if r > s- Therefore there is a canonical mono-
morphism £/+1—>Es;' whenever r > s, and the symbol
f]r>s E?
has a meaning.
We denote by
ai:F*[r,"*,]->Fx[r,\r,_l]
the restriction of (ps)# where p5: "A^ —> "A^-i-
Lemma 3
Every a5 is monomorphic.
This follows from the exact sequence
because, since [Y',"XJ\ is a finitely generated abelian group, nothing in
F^IY' "Xs] is of order 2, whereas everything in im (;'#) = ker ((ps)#) is of
order 2. Thus the intersection of the domain of a5 with the kernel of (ps)#
is zero, which proves the lemma.
Part (3) of the theorem will follow from the next lemma.
Lemma 4
In the notation of the diagram
[Y\"XJ"Xs+r]-+ [Y',S("Xs+r)]^
^^[Y',"XJ-X,+ ll-^[Y',SCXs+l)]^
and with the notations and conventions which have gone before, the
following statements are equivalent:
i. im(/x)= f]im(fr)
ii. im(/x)= c-'iOimig,))
iii. д-Щ=8-1(()\т(дг))
iv. 0=im(^)n(nim(sfr))
v. 0 = teT(p*)nF*>[Y',SCXs+l)]
vi. Each as is monomorphic
proof: We leave to the reader the verification that the diagram is
commutative and that its rows are exact. The equivalence of (ii) with (iii)
and of (iv) with (v) follow from exactness and from the definition of the
filtration. The equivalence of (iii) with (iv) is elementary set theory. The
equivalence of (v) with (vi) follows from the definition of a5. It remains

THE ADAMS SPECTRAL SEQUENCE
199
to prove the equivalence of (i) with (ii). Now im (/r) = д '(im (gr)) by a
simple " diagram-chase." Therefore
Π im(/r) = H З-'От (#,)) = β-'(Π im Ы)
and the equivalence of (i) with (ii) follows.
This proves the lemma.
Now we have already proved (vi), in Lemma 3. Thus (i) has been
established. But (i) implies part (3) of the theorem. For, if we denote by Br
the denominator of the quotient in Lemma 2, we have, by that lemma,
whereas, by definition,
im (/,)
00 Q
Boo
with 50C = limr(fir) under an obvious convention. (Br becomes constant
when r>s.) From (i) of the last lemma, part (3) of the theorem follows.
It remains to prove (4). From the diagram below and a standard argu-
[Y',"Xs+i] [SY',S"XrXs]
[Y\s-xf \y':ks]
ment, already used in Chapter 7 (see Exercise 1 of that chapter), we have
the isomorphism
„, \m:[Y\"Xs]->[Y',S"X]
tx **im:[Y',"Xs+l]^[Y',S"X]
which proves (4).
This completes the proof of (1) to (5) and establishes the fundamental
properties of the Adams spectral sequence.
MINIMAL RESOLUTIONS
When the algebra Л is as complicated as the Stcenrod algebra,
Exl+(H*X,H* Y) may be very complicated and may present a considerable
task in computation. In fact, the computation of Ext t(Z2,Z2) has been
the subject of much recent research.

200
COHOMOLOGY OPERATIONS
We will make some observations on the calculation of ExtA(M,Z2), where
Μ is a left „4-module, before specializing to the case M= Z2. Although we
are using A to denote the mod 2 Steenrod algebra, the following remarks
are valid if A is any connected algebra over Z2 (and in fact can be
generalized much further).
The augmentation of A is a map ε: Λ:—>Ζ2, of which the kernel contains
all of A except the multiplicative unit. This kernel is called the
"augmentation ideal" and is denoted Ι(Λ).
If Μ is a left Λ-module, we write J(M) for f(A)M, that is, for the set of
all finite linear combinations £ а,т; where a, e 1(A) and m, e M. This is
a sub-^t-module of M.
Definition
An .Α-module homomorphism/: M—> TV is minimal if ker (/) is contained
in J(M).
A projective resolution is called a minimal resolution if all the homomor-
phisms in the resolution are minimal.
Proposition 6
Every graded left ^-module Μ admits a minimal resolution.
It is enough to construct one stage, i.e., to find a free graded left
.Λ-module С and a minimal epimorphism d: C—>M. (The proof is then
completed by induction.) Let {g;} be a minimal set of generators for Μ as
an ^-module. Then let С be a free ^-module on the symbols {ht} where
each A, has the same grading as the corresponding #,. There is an ^-module
homomorphism of degree zero, d: C—> M, which takes A; to g, for each /.
Since {gt} is a generating set for Μ as an Λ-module, d is obviously an
epimorphism. To see that d is minimal, suppose that the kernel of d
contains an element b = Y_alhi not in J(M); then we may suppose that a, is
not in Ι(Λ), that is, that a, is the unit of A. Then 0 = d{b) = d(£ яД.) =
^fl;ij, but this implies that gi =£,*, а-,д,, which contradicts the choice
of {gi} as a minimal set of generators.
Proposition 7
Let
0«-Л/*^-Со«-^С,« *^CS<
be a minimal resolution of M. Then the homomorphisms
d*: Ногг& (^„Z^HonV.t (C„Z2)
are all zero homomorphisms (for every s > 1).

THE ADAMS SPECTRAL SEQUENCE
201
proof: First observe that, by the definition of Z2 as a left .,4-module,
Ι(Λ)Ζ2 = 0.
If g e Нотд (Cs_i,Z2), then d*(g) is defined by the equation
d*{g){c)^g{ds{c)) csCs
and we wish to show that this vanishes for all g and с Now ds-tds = 0 for
every s>2 (and ci/,=0), so that d,(c) e ker (i/,_,) (and i/,(c) e ker (ε)).
By the minimality hypothesis, thisimpliesi/s(c) eJ(Cs-t) (and i/,(c) sJ{M)).
But # is an Λ-module homomorphism, and so # must annihilate J(Cs-i)
(and J(M)), by the opening remark of the proof. This shows that indeed
d*(g)(c) = 0 for every ceCs(s> 1), so that d*{g) = 0, but this holds for
all g, so that d* = 0 (for every s > 1). Thus the proposition is proved.
The usefulness of minimal resolutions is now apparent from the
following consequence.
Corollary 1
If the {Cs} form a minimal resolution of M, then Ext^' (M,Z2) =
HomU (CS,Z2).
This is immediate from the definition of Ext and from the last
proposition.
SOME VALUES OF EXTj (Z2,Z2)
In order to see how the machinery works, we look at the Steenrod algebra
Λ and take Μ = Ζ2. This is the E2 term of the Adams spectral sequence for
X= Υ = S° and is therefore related to the stable homotopy of spheres.
To begin the calculation of Ext^ (Z2,Z2), we observe that C0 in a minimal
resolution of Z2 over Λ should be a free Λ-module on one generator, so
that C0 я* Л and c: C0^>Z2 is just the augmentation of Λ. Thus Ext0^'
(Z2,Z2) vanishes for ti~0, while Ext°t0 (Z2,Z2) = Z2, generated by the
unit 1 of Ех1д (Z2,Z2).
The next term, Cb should be a free .4-moduIe with one generator for
each element of a minimal set of generators of the Λ-module ί(Λ). Such a
minimal set is given by {Sq2'} (i >0), since these elements generate Λ as
an algebra and are indecomposable in Λ. Thus C, should be a free Λ-
module with one generator in gradation t = 2' for each / > 0. Each such
generator gives a homomorphism of graded left .-t-modules of degree (—/)
from C, to Z2, and thus Ext^·' (Z2,Z2) = Ногпд {CUZ2) has a generator
for each t which is a power of 2. The usual notation for these elements in
Ext is ht.

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THE ADAMS SPECTRAL SEQUENCE
203
It is considerably more difficult to write down the structure of C2 and
thus of Ext^' (Z2,Z2). We will not go any further with the details but will
be content with a statement of results.
Recall that ЕхГд (Z2,Z2) has a multiplicative structure under which it
is an algebra over Z2. The following results arc proved by purely algebraic
methods. In any finite range of dimensions, they can be verified by
calculating in a minimal resolution, but this method has many limitations and
has been superseded in practice by deeper algebraic methods.
Theorem 3
For s = 1, Ext= Ext^' (Z2,Z2) has as Z2-basis the elements h, e Ext1·'
with t = 2'. For s=2, Ext is generated by the products hthh subject only
to the relations /i;/ii+, = 0 (/ > 0). For s = 3, the products h,hjhk are subject
only to the additional relations И$+ 2 = 0 and h] = h}- ,Ai+, and the
relations implied by A|Ai+i =0.
However, there are other generators for Ext with s = 3, the first of
which is a generator c0 in bigrading (j = 3, /=11). A complete set of
generators of Ext^' (Z2,Z2) for / — s < 17 is presented in Figure 1.
In Figure 1 it is to be understood that there is a non-zero generator
hs0 e Extss for every s > 0, but otherwise all generators in the range t — s <
17 are explicitly shown. (It is not hard to see that Exts,r = 0 if / < s.)
We might remark that to obtain this much of Ext by hand calculation
of a minimal resolution requires several days.
APPLICATIONS TO HOMOTOPY GROUPS
We have arranged Figure 1 so that the bigradings of Ext for fixed / — s
appear in a column. This arrangement is motivated by the assertion of
Theorem 2 that these are the groups which are associated to the stable
(/ — j)-stem of the homotopy of spheres. In fact, parts (4) and (5) of
Theorem 2 assert that the groups ££', with / — s fixed, are the components of
a normal series for the 2-primary component of the stable homotopy group
7I,_S.
By part (1) of Theorem 2, dr raises s by r while decreasing (/ — s) by 1.
We will show now that the calculation of the differentials in the spectral
sequence presents no difficulty whatsoever in the range / — s < 14. We will
thus be able to write down a normal series for the 2-component of the
first 13 stable homotopy groups of spheres, assuming Theorem 2 (parts
(1) to (9)) and the calculation of Ext for the range of Figure 1.

204
COHOMOLOGY OPERATIONS
The differential i/r(Ai), if defined, must be hr0+' or zero, since it goes from
ErU2 to £;+lr + 1. But the relation /!„/?, =0 implies that this differential
is zero; for dr(ho) = 0 for dimensional reasons (since Er — 0 when t<s
and dr is a derivation (part (6) of the theorem), so that
0 = </Γ(0) = </Γ(Μι) = Μ4(Α,))
whereas h\ is non-zero for all s. Thus dr(ht) cannot be hr0+x and must be
zero. Note also that A, cannot be in the image of any differential, since
Er vanishes for s < 0.
This shows that the 2-component of the stable 0-stem is infinite.
Moreover, it is possible to identify the image of h0 in Er with 2v. S"—> S", and
on this basis it can be shown that multiplication by h0 represents a non-
trivial group extension in homotopy. Granting this, we deduce that the
2-component of the 0-stem is an infinite cyclic group Z. The 2-component
of the stable 1-stem is clearly Z2, generated by the image of hx in Er. We
showed earlier that this generator is the (stable) Hopf map η.
Now dimensional considerations are enough to rule out any non-zero
differentials originating in 3 < t — s < 7. Thus the 2-stem is Z2 generated
by A2, that is, by η2 (cf. part (9) of the theorem), the 3-stem is Z8 generated
by the image of h2 (which therefore corresponds to the Hopf map v), and
the 4-stem and 5-stem are empty. The relation η3 = 4r, proved in Chapter
17, appears already in Ext as h\ = h20h2. The 6-stem is Z2 generated by
h22, i.e., by v2.
Since d2(hih3) lands in £2,n, which is a non-trivial bigrading with a
generator hgh3, dimensional considerations do not settle this differential.
However, both dr(ht) and </,(Λ3) are zero for all r>2, and since dr is a
derivation, this differential too is zero. Thus the 7-stem ib Z16 generated
by h3, which corresponds to the Hopf map a.
In the 8-stem there are two survivors to £x, namely, Л,/г3 and an element
which we have denoted c0, not expressible as a product of the h,. It is
perhaps not immediately clear whether the group extension here is trivial,
i.e., whether the 8-stem should be Z4 or Z2 + Z2. However, the fact that c0
and h0{hih3) (=0) have the same filtration degree s =r- 3 implies that the
extension is trivial and the 8-stem is Z2 — Z2. For (9) asserts that the
products in £x are obtained from those in the homotopy composition
ring by passing to quotients according to the degree s, and here, since there
is no change in filtration, there can be no alteration of the product.
We caution the reader, however, not to assume too much about the
relationship between the products in E„ and the homotopy products. For
example, the fact that hth3 survives the spectral sequence does not mean

THE ADAMS SPECTRAL SEQUENCE
205
that the corresponding homotopy element is the composition product of
the Hopf maps η and σ which are the images of A, and /?3. [n fact, "the
corresponding homotopy element" is not well-defined. What we can say
about the 8-stem is this. There are three non-zero elements, each of order 2.
One of these elements projects to zero in ElJ0; thus its projection to
El;n is defined and equal to <■„. The other two non-zero elements project
to /?,/;3 in El;10. One of these elements is ησ; the other is ησ plus the
element which projects to c0.
There obviously cannot be any non-zero differentials originating in the
range 9<t- s < 14. We find that the 9-stem (or its 2-component, as
always) is some group extension of three Z2 groups, and the argument
indicated above for the 8-stem applies here as well to show that the
extensions are trivial. The 10-stem is Z2 and the 11-stem is Z8.
Following Adams, we have used the symbol Ρ in the notation for the
9-stem and those following. There is a complicated and irregular
periodicity in the stable homotopy of spheres, which is in evidence near the
uppermost edge of the table of Ext, of which Figure 1 shows a portion. For our
purposes we may simply regard Ρ as an operator which goes from £"■■' to
E"s+4,t+ 12
The 2-components of the 12- and 13-stems must be zero since there are
no non-7ero elements in E2. However, after this point the work begins.
There arc several possible differentials between the 15-stem and the 14-stem,
and indeed some of them turn out to be non-zero. (It is remarkable that the
topology of spheres has waited so long before intruding into the algebraic
picture.)
In the 14-stem we find that E2 contains h] and also h0h\. If both these
elements survived to Ey, then it would follow that σ2: 5"+14->5" + 7- >S"
was of order 4 (at least). However, the composition ring is in general
anti-eommutativc, and since σ is in an odd stem (the 7-stem), it follows
that σσ — —σσ or 2<r2 — 0. Thus these two elements of E2 cannot both
survive to Ey, and the only possibility is that </2(/?4) =/?0/?3. In this way
Adams obtained a new proof of Toda's theorem that there is no element of
Hopf invariant one in the 15-stem, since elements of Hopf invariant one
appear in filtration degree s — 1 and project to //,-.
There is also a non-zero d3 at this point, namely, r/3(/i0/;4) — h0d0 (and of
course t/3(/iu/;4) - hldn), where dQ is a new kind of generator of Ext
appearing in Ext4,18. It can be shown that the only other non-zero
differential originating in the portion of Ext shown in Figure 1 is c/2(<o) — /?f^o-
Thus the 14-stem is a group extension of two Z2 groups; the 15-stem is
Z32 -1 Z2 and the 16-stem is an extension of two Z2 groups.

206
COHOMOLOGY OPERATIONS
DISCUSSION
The above results give some idea of the workings of the machinery
which we have set up and serve to suggest that the Adams spectral sequence
can be a very useful weapon for the calculation of stable homotopy groups.
As we write this, the complete description of the E2 term in the above
spectral sequence is not known, and no effective procedure is known for
obtaining the differentials or reconstructing the group extensions, so that
this technique is the subject of much current research.
This spectral sequence is of course due to Adams, who proved Theorem
2 on its existence and basic properties. We have not defined the
multiplication in Ext (which is often called the cohomology of the Steenrod algebra
and denoted H*(A)), much less proved statements (6) to (9) of Theorem 2.
The reader will find a proof in Adams' paper [1].
In this fundamental paper Adams also made the first important
calculations of H*(A), namely, those of Theorem 3. They are not extensive, and
further, very little was known at that time about the differentials.
Since then a great deal of progress has been made. Adams has proved
various general structural theorems on Η*(Λ), while May and others have
introduced and refined new computational techniques, making possible
extensive calculation in a relatively short time.
The differentials too have been subjected to attack. In particular, Adams,
Maunder, May, and Mahowald have among them settled many cases,
concerning both specific low-dimensional elements and certain periodic
families of elements recurring in arbitrarily high stems.
REFERENCES
General properties of Ext
1. H. Cartan [6].
2. __and S. Eilenberg [I].
3. R. Godement [I].
4. S. MacLane [I].
Construction of the spectral sequence
1. J. F. Adams [1,4].
2. H. Cartan [6].
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algebra
1. J. F. Adams [1,2,4].
2. A. Bousfield, et a!. [I].
3. J. P. May [I].
4. M. С Tangora [I].
Differentials in the Adams spectral
sequence
1. J. F. Adams [2,4].
2. M. E. Mahowald [I].
3. and M. С Tangora [I],
4. С R. F. Maunder [2].

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1. Gruppentheoretisrher Beweis des Satzes von Hurwitz-Radon, Comm. Math.
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Fadell, E.
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Hopf, H.
1. (jber die Abbildungen der dreidimensionalen Sphare auf die Kugelfldche,
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Ни, 5.-7".
I. Homotopy theory. Pure and Applied Mathematics VIII, Academic Press, New
York, 1959.
Husemoller, D.
I. Fibre bundles, McGraw-Hill, New York, 1966.
Kahn, D. W.
I. Induced maps for Postnikov systems. Trans. Am. Math. Soc. 107 (1963),
432-450.
Kan, D. M.
1. Semisimplicialspectra, Illinois J. Math. 7(1963), 463-478.
2. On the k-cochains of a spectrum, Illinois J. Math. 7 (1963), 479-491.
Kan, D. M., and G. W. Whitehead
I. The reduced join of two spectra. Topology 3 (1965), suppl. 2, 239-261.
MacLane, S.
I. Homology, Springer-Verlag, 1963.
Mahowald, Μ. Ε.
I. On the Adams spectral sequence, Canad J. Math, (in press).
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Massey, W. S.
1. Exact couples in algebraic topologv I, II, Ann. Math. 56 (1952), 363 396; 57
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2. Products in exact couples. Ann. Math. 59 (1954). 558-569,
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I Cohomo/og) operations of the n-th kind, Proc. London Math. Soc /J (1963).
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May, J. P.
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Meyer, J.-P.
I. PrincipalJibrations, Trans Am. Math. Soc. /07(1963), 177-185
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I. The topology of rotation groups. Ann Math (2) 57 (1953), 90 113.

210
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Milnor, J.
1. Lectures on characteristic classes (mimeographed notes), Princeton
University, n.d.
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3. On spaces having the homotopy type of a CW-complex, Trans. Am. Math.
Soc. 90(1959), 272-280.
Milnor, J., and J. C. Moore
I. On the structure of Hopf algebras, Ann. Math. (2) 81 (1965), 211-264.
Oguchi, K.
I. A generalization of secondary composition and its application, J. Fac. Sci.,
Univ. Tokyo, 10 (1963), 29-79.
Peterson, F. P.
I. Functionalcohomology operations, Trans, Am. Math. Soc. 86 (1957), 187-197.
Peterson, F. P., and N. Stein
1. Secondary cohomology operations: two formulas, Am. J. Math. 81 (1959),
281-305.
Postnikov, Μ. Μ.
1. Determination of the homology groups of a space by means of homotopy
invariants (in Russian), Dokl. Akad. Nauk SSSR, 1951, Tom 76, 3, 359-362.
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1951, Tom 76, 6, 789-791.
3. On the classification of continuous mappings (in Russian), Dokl. Akad. Nauk
SSSR, 1951, Tom 79, 4, 573-576.
4. Investigations in homotopy theory of continuous mappings, Am. Math. Soc.
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Puppe, D.
1. Homotopiemenge und ihre induzierten Abbildungen, J, Math. Z. 69 (1958),
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Serre, J.-P.
1. Homologie singuliere des espacesfibres, Ann. Math. (2) 54 (1951), 425-505.
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2. Secondary operations on mappings and cohomology, Ann. Math. (2) 75 (1962),
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Phil. Soc. 53(1957), 57-62.

INDEX
Adams filtration, 187-188
Adams spectral sequence, 193-199, 203-
206
Adcm relations, 22, 29-31
table of, 3 1
Addition formula, 6
Admissible sequence, 27, 51
Algebra, 45-46
Hopf algebra, 49-50
over a Hopf algebra, 52-53
Augmentation ideal, 47, 200
Bockstein exact couple, 60-62
Bockstcin homomorphism, 22-23
Bockstein lemma, 106, 166
Bockstein operator dr, 61, 104-105, 173
Borel's theorem, 88, 91-92
Butterfly lemma, 71-72, 199
C-thcory, 94-95
6„ approximation theorem, 100-101
Carrier, 14
Cartan formula, 22, 24-25
Cell complex, homology of, 67-68
Cellular approximation, 129, 137
Characteristic class. 103. 109, 110
Class of abelian groups, 93-95
Co-algcbra. 49-50
Co-augmentation, 49
Coboundary formula
for cup-/ products, 16
for obstructions, 6
Co-extension, 171
Co-fibration, 144, 145, 153-154
Cohomology of Κ(τ,ι\)
of K(Z,\), 83
of K(Z,2), 84-85, 90
of K(Z,<j), 90
of K(Z=,1), 14, 26-29, 85-86
of K(Z?,2), 86-88
of K(Z~.q), 88
of K(Z,m,q), 90
Cohomology of the Stcenrod algebra, 206
Cohomology operation, 2, 10, 156-169
additive, 151-152
decomposable, 36, 47
functional, 156-161, 168
primary, 2. 10
secondary, 161-169
Cohomology suspension, 139, 150-152,
155
Complex over a space, 185-186
acyclic, 186
/i-connccted, 186
Composition, secondary, 170-172
higher. 182
Composition ring. 151, 155
Cone. 19
Convergence conditions, 66
Covering. 75
spectral sequence of, 97. 101
Cross section, 109-110
Cup ί products. 15-16
212

INDEX
213
CW -complex, 11
Decomposable, 36, 47
Degree, of an operation, 27, 163
of a homomorphism, 192
Derived couple, 59-60
Detect, 158, 168
Diagonal map, 49
of Λ. 47-48
of Λ", 53-56
Difference cochain, 6
Double complex, 68-70
EHP-scquencc, 123, 127
Eilcnberg-MacLane space, 10
See also Κ(τ,η)
Equivariant chain map, 13
Exact couple, 59
Excess, 27
Ext, 192-193. 199-203, 206
Λ table of ExtA(Z-,Z,), 202
Extension. |7l
Fibre. 74
Fibre homotopy equivalence, 102, 1 10
Fibre map, 73
everv map equivalent to, 84
Fibre mapping sequence. 138-139, 145—
148
Fibre space, 73, 84
characteristic class of, 103. |09, 1 10
exact homotopy sequence of, 74
induced, 102-103
principal, 108-110
relative, 98
Scrre's exact homology sequence of,
77-78, 81, 151-152
spectral sequence of, 76. 80-81
Filtered complex, 62-70
Frame, 39
Functional operation, 156-161
Fundamental class, 3, 103-104
Homomorphism (of graded objects). 45
of degree t, 192
Homotopy exact couple, 70-72
Homotopy groups of spheres, 111-122.
182, 203-206
of V, 123-127, 148
Homotopy operation, primary. 1 5 1
Homotopy type, n-, 130
Hopf algebra. 49-50
Hopf construction, 38
Hopf invariant, 33-38, 168
Hopf maps, 34, 74. 158-159, 170, 173.
178. 181
Hopf-Whitncy theorem, 7
Hurewicz homomorphism, 2, 71
Hurcwicz theorem, 2-3, 71, 78
mod C, 95-97
relative, 79
mod C, 95, 98
Hurwitz-Radon theorem, 44
Inclusion map, every map equivalent to,
79
fnelusicn mapping sequence, 143-144,
148
Indeterminacy, 105, 157
Λ'(τγ,η), 2-5, 10
K(Z.l), 83
KUP-). 75
X(Z.,1), 12-14
See also Cohomology of Κ(τ,η)
λ-invariant, 136
Length (of a sequence), 27
Loop space, 75, 152-153
of 5", 76-77, 82
Massey triple product, 168
Milnor basis, 56
Milnor's theorem, 91, 92
Module (graded), 45, 52
Moment (of a sequence), 47
Moore paths, 108-109
ii-connccted space, 2
/i-homotopic maps, 129
м-homotopy type, 130
V-rcsolution, 186
м-typc 130. 136
Natural group structure, 139, 149-150,
155
Nice space, 2
Normal cell. 42
Obstruction Iheory, 6-10
Orthogonal group, 40, 75
/^-component, 93, 190-19)
Path space, 74-75
Periodicity theorem (Bott), 75, 82
Peterson-Stein formulas, 164-166
Pontrjagin square, 20-21
Postnikov system, 13 1-137
for V. 122, 140
Projective space, complex, 75
real. 1, 12. 40
Puppe sequence, see Inclusion mapping
sequence

214
INDEX
Resolution, minimal, 199-201
of a module, 192-193
Resolution of a space, 186
N-resolution, 186
S-maps, 19|
Secondary compositions, 170-172
Secondary operations, 161-169
Serre-Cartan basis, 47
Scrre's exact sequence, 77-78, 81, 151-
152
Simple system of generators, 88
Skeleton filtration, 143-145
Spectra, 154-155
Spectral sequence, 60, 70, 72
comparison theorem, 90-91
Squaring operations, see Steenrod squares
Stable homotopy group, 122
Stable range, |49
Steenrod algebra, 46-58
basis, 47, 56
diagonal map, 47
dual, 50-57
Steenrod squares, 12, 16-20, 22
Stem, 151, 173
Stiefel manifold, 40
Suspension, 19, I 11-112
homomorphism (in homotopy), | |2
isomorphism (in homology), 19
theorem, 1 11-112, |23, 127
See also Cohomology suspension
Tensor algebra, 46
Toda bracket, 172
Tor, 69-70
Transgression, 78, 80
in cohomology, 81, 155
Triad theorem, 154-155
Type (α,/З), map of, 38
Universal coefficient theorem, 3
Universal example, 106
Vector field, 39, 44
Whitehead product, 34, 38
Whitehead's exact sequence, 70-71
Whitehead's theorem, 79
mod C, 95, 98