SIMPLICIAL OBJECTS
IN ALGEBRAIC TOPOLOGY
J. PETER MAY
The University of Chicago Press
Chicago and London

The University of Chicago Press, Chicago 60637
The University of Chicago Press, Ltd., London
© 1967 by J. Peter May
All rights reserved. Published 1967
Midway reprint 1982
Printed in the United States of America
LCN: 82-51078
ISBN: 0-226-51180-4
INTRODUCTION
Algebraic topology could perhaps be characterized as the
study of those functors from the category of topological spaces to
that of groups which are invariants of homotopy type. From this
point of view, the category of topological spaces which are of the
homotopy type of a CW-complex is equivalent to the category of
simplicial sets (in the literature: complete semi-simplicial com-
complexes) which satisfy the extension condition.
The object of these notes is an investigation of the cate-
category of simplicial sets. There are various technical advantages to
this category. These mainly arise from the fact that, for every
n > 1 and Abelian group n, there exists an explicit canonical sim-
simplicial Abelian group K(n,n) which satisfies n (K(n,n)) = n
and n.(K(n,n)) = 0, i ^ n. These objects are fundamental to
the construction of Postnikov systems and to the study of cohomo-
logy operations.
We will first develop the definitions and elementary proper-
properties of simplicial objects and then begin the study of simplicial
fiber spaces and Postnikov systems. After demonstrating the
"equivalence of categories" described above, we will study fiber
spaces and fiber bundles in some detail, making use of the concept
of "twisted Cartesian product."
We will then study K{n,n)'s and introduce the ^-invariants
of Postnikov systems. This will be done by studying simplicial
fibre bundles with fibre a K {u, n) rather than by use of obstruction
theory. We will conclude by developing the Serre spectral sequence

VI
SMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
by means of Brown's theorem comparing twisted Cartesian products
and twisted tensor products.
Most of the material here is scattered through the literature.
A bibliographical note at the end of each chapter will give refer-
references.
CONTENTS
Page
I. SIMPLICIAL OBJECTS AND HOMOTOPY 1
§1. Definitions and examples 1
§2. Simplicial objects in categories; homology 3
§3. Homotopy of Kan complexes 5
§4. The group structures 9
§5. Homotopy of simplicial maps 12
§6. Function complexes 16
Bibliographical notes on chapter I 24
II. FIBRATIONS, POSTNIKOV SYSTEMS, AND MINIMAL
COMPLEXES 25
§7. Kan fibrations 25
§8. Postnikov systems 31
§9. Minimal complexes 35
§10. Minimal fibrations 40
§11. Fibre products and fibre bundles 43
§12. Weak homotopy type 46
§13. The Hurewicz theorems 50
Bibliographical notes on chapter II 54
III. GEOMETRIC REALIZATION 55
§14. The realization 55
§15. Adjoint functors 59
§16. Comparison of simplicial sets and topological
spaces 61
Bibliographical notes on chapter III 66
IV. TWISTED CARTESIAN PRODUCTS AND FIBRE
BUNDLES 67
§17. Simplicial groups 67
Vll

Vlll
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
Page
§18. Principal fibrations and twisted Cartesian
products 70
§19. The group of a fibre bundle 74
§20. Fibre bundles and twisted Cartesian products ... 80
§21. Universal bundles and classifying complexes ... 83
Bibliographical notes on chapter IV 92
V. EILENBERG-MACLANE COMPLEXES AND
POSTNIKOV SYSTEMS 93
§22. Simplicial Abelian groups 93
§23. Eilenberg-MacLane complexes 98
§24. K(n, n)'s and cohomology operations 103
§25. The i-invariants of Postnikov systems 107
Bibliographical notes on chapter V 116
VI. LOOP GROUPS, ACYCLIC MODELS, AND TWISTED
TENSOR PRODUCTS 117
§26. Loop groups 118
§27. The functors G, W, and E 122
§28. Acyclic models 126
§29. The Eilenberg-Zilber theorem 129
§30. Cup, Pontryagin, and cap products; twisting
cochains 135
§31. Brown's theorem 141
§32. The Serre spectral sequence 149
Bibliographical notes on chapter VI 154
BIBLIOGRAPHY 157
CHAPTER 1
SIMPLICIAL OBJECTS AND HOMOTOPY
§1. Definitions and examples
We introduce the concept of simplicial set and give several
examples here. A categorical definition will be given in the next
section.
DEFINITIONS 1.1: A simplicial set K is a graded set indexed on
the non-negative integers together with maps d.: K -> K and
q, which satisfy the following identi-
V
K_ -* K
ties:
q + 1,
0 < / <
(i)
(ii)
(iii)
a.
s.
*'i
d!
di
a.
s.
J
Sj
Sj
s;
= s. lSi
= sj_1di
= identity
= s di_l
if
if
if
=
if
The elements of Kq are called g-simplices. The d. and s. are
called face and degeneracy operators. A simplex x is degenerate
if x = s.y for some simplex y and degeneracy operator s.;
otherwise x is non-degenerate.
Definition 1.2: A simplicial map /: K -> L is a map of degree
zero of graded sets which commutes with the face and degeneracy
operators; that is, / consists of / : K
difq+l
L and

vm
SIMPL1C1AL OBJECTS IN ALGEBRAIC TOPOLOGY
Page
§18. Principal fibrations and twisted Cartesian
products 70
§19. The group of a fibre bundle 74
§20. Fibre bundles and twisted Cartesian products ... 80
§21. Universal bundles and classifying complexes ... 83
Bibliographical notes on chapter IV 92
V. EILENBERG-MACLANE COMPLEXES AND
POSTNIKOV SYSTEMS 93
§22. Simplicial Abelian groups 93
§23. Eilenberg-MacLane complexes 98
§24. K{n, n)'s and cohomology operations 103
§25. The i-invariants of Postnikov systems 107
Bibliographical notes on chapter V 116
VI. LOOP GROUPS, ACYCLIC MODELS, AND TWISTED
TENSOR PRODUCTS 117
§26. Loop groups 118
§27. The functors G, W, and E 122
§28. Acyclic models 126
§29. The Eilenberg-Zilber theorem 129
§30. Cup, Pontryagin, and cap products; twisting
cochains 135
§31. Brown's theorem 141
§32. The Serre spectral sequence 149
Bibliographical notes on chapter VI 154
BIBLIOGRAPHY 157
CHAPTER 1
SIMPLICIAL OBJECTS AND HOMOTOPY
§1. Definitions and examples
We introduce the concept of simplicial set and give several
examples here. A categorical definition will be given in the next
section.
DEFINITIONS 1.1: A simplicial set K is a graded set indexed on
the non-negative integers together with maps d.: K -> K and
/ q q— 1
s.: Kq -* Kq + 1, 0 < i < q, which satisfy the following identi-
ties:
(i) 6i6i = (9._
(ii) s^. = s.+
(iii) d.s. = s. , d.
d.s. = identity
if ' < 1,
if i < j,
if i > i+
The elements of Kq are called (j-simplices. The d{ and s. are
called face and degeneracy operators. A simplex x is degenerate
if x = sty for some simplex y and degeneracy operator s.;
otherwise x is non-degenerate.
DEFINITION 1.2: A simplicial map /: K -> L is a map of degree
zero of graded sets which commutes with the face and degeneracy
operators; that is, / consists of / : K -> L and
9 9 q

2 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
DEFINITION 1.3: A simplicial set K is said to satisfy the exten-
extension condition if for every collection of n + 1 n-simplices xQ, Xj,
.... x, ,, x. , .,..., x , , which satisfy the compatibility condi-
tion (9.x. = (9._1x., i < j, i ^ k, j 4 k, there exists an (n+1)-
simplex x such that d.x = x. for i 4 k.
EXAMPLE 1.4: We recall that a simplicial complex K is a set of
finite subsets, called simplices, of a given set K subject to the
condition that every non-empty subset of an element of K is itself
an element of K. A simplicial set K arises from K in the follow-
following manner. An n-simplex of K is a sequence (aQ,...an) of ele-
elements of K such that the set {aQ,..., an\ is an m-simplex of K for
some m < n. The face and degeneracy operators of K are defined
<9.(ao,...,an) =
and
si(a0 V = lao'-'ai'ai'ai+l'-'an) *
If the elements of K are ordered and we require K to consist of
those sequences (aQ,..., an) such that aQ < a1 < ••• < an and
{a.,..., a } is an m-simplex of K for some m < n, then there will
be exactly one non-degenerate n-simplex of K for every n-simplex
of K.
Example 1.5: Let An = [(tQ,..., tn)\ 0 < tj < 1, 2tj = 1} c
K"+l jf ^ js a topological space, a singular n-simplex of X is
a continuous function /: An -* X. The graded set S(X), where
S (X) is the set of singular n-simplices of X, is called the total
singular complex of X. S(X) becomes a simplicial set if we de-
define face and degeneracy operators by:
(dif*t0,...,tn_1) = {(to,...,ti_1,Q,t.,...,tn_1)
and
n+l
is a
The following elementary fact will later be used to show
that S(X) determines the homotopy groups of X.
SIMPLICIAL OBJECTS AND HOMOTOPY
LEMMA 1.5: S(X) satisfies the extension condition.
Proof: Since the union of any n+l faces of A
retract of A j, any continuous function defined on such a union
can be extended to A , ,.
n+l
CONVENTIONS 1.6: The word "complex" (unmodified) will always
mean simplicial set. A complex which satisfies the extension con-
condition will be called a Kan complex.
§ 2. Simplicial objects in categories; homology
Recall that a category ? is a class of objects together
with a family of disjoint sets Hom(A, B), one for each pair of ob-
objects, a function Hom(B, C) x Hom(A, B) -» Hom(A, C), ax/8 -» a/8,
and an element 1^ f Hom(A, A), all subject to the conditions
a(/8y) - (a/8)y whenever either is defined and al^ = a = lBa,
a f Hom(A, B). The elements of Hom(A, B) are morphisms with
domain A and range B . The opposite category ? of a category
? has an object 4* for each object A of C and a morphism
a e Hom(B , A ) for each morphism a e Hom(A, B); a /8 is
defined and equal to (/8a) whenever /3a is defined.
A covariant (resp., contravariant) functor F: ? -> 3) is a
correspondence which assigns to each object A e ? an object
F (A ) f 3) and to each morphism a ( Hom(A, B ) a morphism
F(a) f Hom(F(X), F(B)) (resp., F(a) f Hom(F(B ), F(X)) subject
to the conditions FAA) = 1F(-A), 4 f C, and F(o/8) =
F(a)F(/8) (resp., F(aj8) = F(/8)F(a)) whenever a/8 is defined in
C If T: C - C* is defined by T(A) = X* and T(a) = a*,
then T is a contravariant functor; any contravariant functor
F: C -» 3) may be considered as the covariant functor TF : C ->
3) or F71: C -¦ 3). If F and G are covariant (resp., contra-
contravariant) functors C -* 3), a natural transformation A: F •» G is
a function which assigns to each object A of ? a morphism
Hom(F(X), G(A)) subject to the condition that if

5:
5.
1
4 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
aeHom(A,B), then G(a)AU) = A(B)F(a) (resp., G(a)A(/8) =
AU)F(a)).
Now we define a category A as follows. The objects
A of A are sequences of integers, An = @, l,...,n), n > 0.
The morphisms of A are the monotonic maps /x: A -> A , that
is, the maps /x such that fiU) < /x(;) if i < j. Define morphisms
n-1
V An+1 - An< °< ' <"' ^
A) 5/;) = ; if ; < i; 8t(j) = ;+l if / > i,
B) at(j) = ; if ; < i; *,(/) = j-1 a j > i .
Let /i ( Hom(An,Aj, /x not an identity. Suppose ij,..., zg, in
reverse order, are the elements of A not in /x(A ) and /,,...,;,,
in order, are the elements of A such that/x(;) = /^(; + l). Then:
C) ii = 5. ...5; a. ...a. , where 0 < i <••• < i, < m,
1 a 71 't si
0 < /j <••• < jt < n, and n —(+ s = m.
Further, the factorization of /x in the form C) is unique. Having
defined A , we can formulate
DEFINITIONS 2.1: A simplicial object in a category ? is a con-
travariant functor F: A -> C Such functors form the objects of
a category Cs, the morphisms of which are all natural transforma-
transformations of such functors. If F e Cs, the elements of F(An) are
called n-simplices, and the maps d. = FE.) and s. = F(ff.)
satisfy (i)-(iii) of Definitions 1.1. Any simplicial set K deter-
determines a contravariant functor F : A -¦ C, where ? is the cat-
category of sets, by the rules F(An) = K and
f(m) = 8jt...aiidi/..d§i.
where ti is a morphism of A expressed in the form C). Thus a
simplicial set may be uniquely identified with a simplicial object
in the category of sets; Analogously, we will speak of simplicial
groups, simplicial modules, and so forth, depending on the choice
of the category ?.
SIMPLICIAL OBJECTS AND HOMOTOPY 5
Remarks 2.2: Let A = A denote the opposite category of A ,
T ¦ A -¦ A the contravariant functor defined above. The category
Cs could equally well be defined as that of covariant functors from
a to e.
Now suppose that F: C_ -¦ 3) is a covariant functor. By
composition, F induces a covariant functor Fs: Cs -> 3)s. In
particular, suppose that C is the category of sets, D that of Abel-
ian groups, and for A e ?, F(A) is the free Abelian group gener-
generated by A. Then if Kf Cs, FS(K) may be given a structure of
chain complex with differential d defined on Fs(K)n = FS{K ) by
d =
(-D' a,
We denote this chain complex by C(K). If G is an Abelian group,
we define the homology and cohomology of K with coefficients in
G by
ff*(K;G) = H{C(K) ® G) and H*(K;G) = H(Hom(C(K), G)) .
In case K = S{X), these are, of course, the singular homology and
cohomology groups of the space X.
§3. Homotopy of Kan complexes.
DEFINITION 3.1: Let K be a complex. Two n-simplices x and
x' of K are homotopic, written x^x', if d.x = d.x', 0 < i < n,
1 1 — —
and there exists a simplex y t K ,. such that d y = x,
dn + \y = X' and dty = Sn-ldiX = 8n-ldlx'' 0<i <n.
The simplex y is called a homotopy from x to x'.
PROPOSITION 3.2: If K is a Kan complex, then
lence relation on the n-simplices of K, n > 0.
Proof: The relation r^ is reflexive since
dnsnx = x = dn + lsnx
and djSnx = sn_1<?fx, 0 < i < n. Suppose x r^x' and Xr^x''
is an equiva-

6 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
We must prove x' ~ x ". Let y' satisfy
dny'.= x> dn+\y'= x'» and diy'= sn-\dix'- *K n-
Let y" satisfy
dny" - x, <^n+1y"= x',' and diy"- sn_idjx', i < n .
Then the n + 2 In +1)-simplices
are seen to satisfy the compatibility condition. Therefore there
exists an (n + 2)-simplex z such that d.z = dysnsnx', 0 < i < n,
d z = y', and (9 .z = y". It follows that
d-d ,oz= s ,d.x', 0 < i < n,
DEFINITION 3.3: Let L be a subcomplex of K. Two n-simplices
x and x', n > 0, are homotopic relative to L, written
x ~ x'rel L, if d.x = dyx', 1 < i < n, if dQx ~ dQ x' in L,
and if there exists a homotopy y from dQ x to dQx' in L and a
simplex
such tnat
~ x'
^n + l
w
x '
and d.w = s _, dx = s _~d.x', 1 < i < n. The simplex w
is called a relative homotopy from x to x'.
Proposition 3.4: If L is a sub Kan complex of the Kan com-
complex K, then ^ rel L is an equivalence relation on the n-simplices
of K, n > 1.
Proof: The relation ~ rel L is reflexive since if dQ x e L,
then s _ j (9Q x is a homotopy in L from dQx to dox» and if
w = s x, then d.w = sn_151.x, 0 < i < n, and
dw=x = d,,w.
n n + 1
Suppose x /^ x' rel L and x ~ x " rel L. We must prove that
x' ~ x"rel L. Let y' and y" be homotopies in L from dQ x to
dQx' and from dQx to (90x"> and suppose w' and w" are rela-
relative homotopies from x to x' and from x to x" which satisfy
SIMPLICIAL OBJECTS AND HOMOTOPY
dQw
y and dQw" = y". As in the proof of Proposition 3.2,
we may choose z e L , , such that
n+1
diSn-lSn-ld0X'
0 < i < n-1,
5 _ j z = y' and (9 z = y".
Then y = dn , ^ z is a homotopy in L from dQ x' to d0 x".
Now it is easy to see that the n + 2 (n + l)-simplices
SnSn
- lSn Sn X
satisfy the compatibility condition so that there exists v e K
such that d. v
diSnSnX<
1 < i < n, dQv
z, d.
and dn ^ v = w". Let w = d
1 < i < a, dQw = y, dnw = x
v. Then
n + 2
v = w',
and dn+lw =
n-ldix>
Notations 3.5: Let K be a complex, <f> t KQ. <f> generates a
subcomplex of K which has exactly one simplex s _ .... s0 cf> in
each dimension n. We will abuse notation by letting <f> denote am-
ambiguously either this subcomplex or any of its simplices. We call
(K,<f>) a Kan pair if K is a Kan complex. We call {K, L, <f)) a
Kan triple if <f> c LQ and L is a sub Kan complex of the Kan com-
complex K. Simplicial maps of pairs and triples are defined in the ob-
obvious manner.
Definitions 3.6: Let (K, d>) be a Kan pair. Let K, n > 0,
denote the set of all x e K which satisfy d.x = 0, 0 < i < n.
Then we define njji, <f>) = /Cn/(~). nQ(K, <f>) is called the set of
path components of K. K is connected if nQ(K, (f>) = <f> (where
we are letting <f> denote its equivalence class). K is n-connected
if n^K, <f>) = 0, 0 < i < n. Let (K, L, <f>) be a Kan triple. Let
K(L)n, n > 1, denote the set of all x e K which satisfy
dQ x e Ln-1 and d. x = <f>, 1 < i < n. Then we define
nn{K,L,<f>) = K(L)n/(~relL).
Note that nn (K, <f>, <f>) = nn (K, <f>), n > 1. Finally, we define
d: nn(K,L,<f>) - irn_1(L,^), n > 1, by d[x] = [dQx], where
[x] denotes the homotopy class of x.

Si
Si
SI
ti
Si
!
1
¦
:
] I ij
8 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
THEOREM 3.7: Let (K, L,cf>) be a Kan triple. Then the sequence
of sets with distinguished element <f> is exact, where the maps i and j
are induced by inclusion.
Proof: (i) id = <f>: Consider i[dQx] = id[x], x e K(L)n + r
The n+ 2 (n+l)-simplices -, 4>, • • ¦, 4> >X. are compatible,
hence we may choose z f #n + 2 sucn tnat ^iz = 0» * <'S" + 1,
5n+2z = X. Then 3?902 *3o^tlz = 0> 0 < i <: n+i, a.ni
(9. ^O2 = <^ox' so that dQx <-*> <f> in K.
(ii) Image d = Kernel f: Let i[y] = <f>, y e Ln. Then
in Kn, say
, 0 < i < n, d
n+1
The
n+2 (n+l)-simplices z, <f>,...,<f> are compatible, hence there
exists w e K +2 with dQw = z, d.w = (j>, 1 < i < n+ 1.
' = cf>, 1 < i < n + 1, and do(9 2 w = y, so that
since dQx = <f>, x € Kn
(iii) dj = (f>: dj[x]
(iv) Image ; ~ Kernel d: Let d[x]
x f K(L)
o < i 41
(where - means it = n in the compatibility condition) are com-
com[dQx] ^
There exists z e Ln such that t^z = <f>,
d z = (9nx. The n+1 n-simplices z, $,..., (f>, -, x
1 < i < n-1, <?
patible, say dQy = z, d-t y =
Thus x ~ dn y rel L. Since d.dny = <f>, 0 < i < n,
n+1
y = x-
(v) jj
Consider ji[y], y t Ln. Then + 1 n-sim-
n-simplices —, <f>,...,<f>, y are compatible in L, say z e ^n+1 satis-
satisfies d. z = (f>, 1 < i < n, dn j z = y. d;. c
hence dnz is a homotopy between 0 and d>
relative homotopy between <f> and y.
(vi) Image i = Kernel;: Let j[x] ¦¦
n+ 1
= <f>,0< i < n,
cL y, so z is a
X €
K
Choose w e Kn + 1 such that df w - cj>, 1 < i < n, dn + l w =
= x,
and (9
Q
eL . The n + 1 n -simplices z
SIMPLICIAL OBJECTS AND HOMOTOPY 9
are compatible in L, say do» = z and dji> = <?, 1 < i < n. The w + 2
(w + l)-simplices sn-iz, <t>, . . . , $, v, w, - are compatible in K, say
dot = s«_iz, dit= <f>, 1 < * < » — 1, 3J = », an+1< = w. Then
is a homotopy
hence [x] = ii
§ 4. The group structures.
DEFINITIONS 4.1: Let (K, cf>) be a Kan pair. Write x e a if x
represents a. Suppose a, /S ( nn(K, 0), n > 1, and let x t a,
y e /8. The n+1 n-simplices <?,..., cf>, x, -, y are compatible,
say djZ = <f>, 0 < i < n — 1, (9 _ , z = x, 5 , 1 z = y. We
define a/8 = [dn z].
LEMMA 4.2: a/S is well defined.
Proof: Suppose z' also satisfies d.z' = 0, 0 < i < n-1,
<?n_jz'= x, and<9n+1z'= y. By the extension condition, there
exists w € Kn + 2 such that d.w = 0, 0 < i < n-2,
^w is then a homotopy from d z to d z'. Suppose y r^> y't
say d{w = <f>, Q < i < n, dnw = y", <?n+1w' = y. Choose z'
such that (9n_1 z' = x, ^n + ^ z' = y', and (9fz' = <f>, 0 < i < n-1.
By the extension condition, there exists u e K „ such that
(9. u = 0, 0 < f < n - 1,
(9n_j u = sn_j x, <9n u = z', and d ~u — w •
Then (9n^n + j u = (9nz' (9n + l(9n+ j u = y, and dn_-idn + 1u = x,
so the same choice of z may be taken for the two choices of repre-
representatives for /S. Similarly a/3 is independent of the choice of the
representative for a-
PROPOSITION 4.3: With the above multiplication, nn(K,(fi) is a
group, n > 1.
Proof: (i) Divisibility: Let x t a, y e /S, a and /8 in
nn (K, <f>). By the extension condition, there exists z e K . such

10
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
SIMPLICIAL OBJECTS AND HOMOTOPY
11
= x.
that d{ z = <f>, 0 < i < n -1, dn z = y, and dn +
Then fd _ i z]a = /S. Right divisibility is proven similarly.
(ii) Associativity: Let x e a, y t /8, z € y, where a,
/S, and y are in n (K,<f>). Using the extension condition, choose
'n-VWn+l'Wn + 2 sucn that &i wj = 0' 0 < I < n - 1,
w
dn - 1 Wn -
!- 1 "n+1
d w ,,
n n — 1'
Applying the extension condition again, choose v e Kn + 2 such
that dj.u = cf>, 0 < f < n-1, and djti = w., i = n-1, n + 1,
n+2. Then 5JI_15flu = x, an + 15nu = ^wn + 2- and
(9;u = ^>, 0 < f < n-1. Therefore:
Proposition 4.4: nn(K,(f>) is Abelian if n > 2.
Proof: Let w, x, y, z e Kn-
(i) Suppose vn+1 f Kn+1 satisfies C,vn+1 = ^,
0 < i < n-2, <?n+1vn+1 = 0, 5n_2vn + 1 = w,
Then [y][w] = [x]: Choose vn _ j e Kn +1 with faces
djvn_1 = <?, 0 < i < n-2, <?nvn _ x = x, and
5n+lvn-l = w>
and let t = dn __ x vn _ x . Let v. = <f>, 0 < f < n - 2, and let
v r. = s w, v,r, = s t^. The v. satisfy the extension
condition, say d.r = v., i ^ n. Let vn = (9nr.
d.vn - 4>, 0 < i < n-2, <9n_1vn = t, (9nvn = y,
and c?n + 1 vn = ^>. Therefore [*][</>] = [y]; but by the choice of
vn ,, [t][w] = Lx], hence [y][w] = [x].
(ii) Suppose vn f Kn+i satisfies d.vn = <f>, 0 <i < n~2,
Then [^[y] = [z]: Choose vn_1 t Kn+1 with faces d.v _. = <p,
0 < i < n-2, ^v^j = w, and <9n_1v_1 = 0 = <9
and let t = (?nvn_1. Let vf = 0, 0 < i < n-2, and let
The v. satisfy the extension condition, say d.r = v., i ^ n+1.
Let Vl = ^n+l1-- 5iVl = *' ° * f * "~2, <3n_lVn+l = ^'
dnvn+1 = y, and (9n+1vn+1 = z. Therefore [f][z] = [y]; but by
the choice of vn_1 and(i), [t][w] = [0], hence [w][y] = [z].
(iii) Suppose vn+2 f Kn+1 satisfies d.vn+2 = «^>,
0 < i < n-2, 5n_2vn+2 = w, <3n_1v+2 = x, dnvn+2 = y, and
(9n+1vn+2 = z. Then M^UM = [y]: Choose vn_2 f Kn+1
with faces divn_2 = $, i + n-2,n + 1, <5n+1vn_2 = w; let
0 un-
f = 5n-2vn-2- Choose vn_i e
with faces djvn_1
less i = n-2, n, or n+1, d ov , = t, d ,.v - = x; let
v = dnvn_y Let v. = 0, 0 < i < n-2, vn = sny. The v. sat-
satisfy the extension condition, say dtr = vjt i ^ n+1. Let
n-f-1 n+1 / n+1 *
0 < i < n-2, dn_lVn+1 = u, dnvn+1 = y, and (9n+1vn+1 = z.
Therefore [u][z] = [y]. Combining, we find [h']~1[x][z] = [y].
(iv) Set z = <? in (iii). Then [h'1~1[x] = [y]. But applying
(i)to vn+2 of (iii) in this case, we find [y] = [xltw]. Therefore
for any [x] and [w], [w]~ [x] = [x][w]~ , and this implies the re-
result.
Definition 4.5: Let (K, L, <p) be a Kan triple. Suppose
a, 0 e nJ^K,L,<t>), n > 2, and let x e a, ye /S. We have
ldox][dQy] = [^..jz], where d. z = 0, 0 < i < n-3,
<9n_2z = <9qx, and (9nz — d^y. The n +1 n-simplices
^» 4*,••••$, x,—f y are compatible, say 5,w= f 1< i<n-2,
dQw = z, d ,w = x, (9 jiy = y. We define a/S = [d w\.

12
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
SIMPLICIAL OBJECTS AND HOMOTOPY
13
SI
5:
LEMMA 4.6: a/8 is well defined.
PROPOSITION 4.7: With the above multiplication, rrJ,K, L, 0) is
a group, n > 2.
PROPOSITION 4.8: nJ,K, L, <f>) is Abelian if n > 3.
The proofs of the above statements are analogous to those
for the absolute groups and will be omitted.
Now maps of Kan pairs or triples induce homomorphisms of
homotopy groups in the obvious manner. Further, d is a homomor-
phism of groups by construction. Thus we find:
PROPOSITION 4.9: n is a functor from the category of Kan pairs
(resp., triples) to that of groups, n > 1 (resp., n > 2), and to the
category of sets, n = 0 (resp., n = l). d is a natural transforma-
transformation of functors, n > 1. In particular, the exact sequence of Theo-
Theorem 3.7 is functorial and is an exact sequence of groups up to
n^K, 0).
§5. Homotopy of simplicial maps.
DEFINITIONS 5.1: Let / and g be simplicial maps from a complex
K to a complex L. Then / is homotopic to g, written / ~ g, if
there exist functions h{: Kq -> L9?l, 0 < i < q, which satisfy:
a) <y>0 = /, <Viftq = ^
(ii) d.h. = h^d. if i < j
(iii)
d.h. =
sihj =
if
if
if
h is called a homotopy from / to g, ft: ( c* g. UK' and L' are
subcomplexes of K and L and / and g take K' into L', then
h is a relative homotopy from / to g if h is a homotopy from / to
g, if h(K') CL', and if /i|K' is a homotopy from /|K' to g\K\ K'
is said to be a deformation retract of K if the identity map of K is
homotopic relative to K' to a map from K onto X', which extends
the inclusion map K' -> K. Two complexes K and L are said to be
of the same homotopy type if there exist maps /: K -> L and
g: L -» K which satisfy /g ~ 1 and gf ~ 1.
LEMMA 5.2: Let /,/': K -> L and ^, g': L ^ M. Then:
(i) / c* /
(ii) If A: / ~ / ', then g ° h: g°i~gof
(iii) If /i: g ~ g", then ftof: g°/~g'°/.
Proo/: If A/x) = jr^U), then A: / a* /. (ii) and (iii) are
clear.
We will prove that homotopy is an equivalence relation on
maps from a complex K to a Kan complex L in the next section.
The following proposition implies that homology is an in-
invariant of homotopy type on the category of simplicial sets.
PROPOSITION 5.3: Suppose / and g are maps from K into L.
Then if / c~ g, /* = g*: H*(K) -> ff*(L).
Proof: Let ft: / ~ g. Define s: C(K) -> C(L) by
s(x) = 2 (-1)'/>/*), x e K
i= 0
Then it is easily verified that ds + sd = C(t) - C(g), which im-
implies the result.
We now wish to express our definition of homotopy groups in
terms of homotopies of maps. We first define the standard simpli-
simplicial n-simplex A[n].

14
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
SIMPLICIAL OBJECTS AND HOMOTOPY
15
§!!
tl
Definitions and notations 5.4: For each n > 0, a simpli-
cial set A[n] is given by the contravariant functor Tn defined by
T"(A ) = Horn (A , A ) and Tn(u)(\) = \u whenever A^i is de-
fined in the category A . The face and degeneracy operators are
of course Tn(8t) and Tn(at). Note that S. and ajt 0 < i < n,
induce simplicial maps
8.: A[n-1] -» A[n] and of A[n + l] -> A[nl.
Identifying A t Horn (Am, AR) with its image MAm), an m-simplex
of A[n] is a sequence (aQ,..., am) of integers a.,
0 < aQ < ¦¦• < an < n, and
dSaO'-'an) = (aO'-"'ai-l'ai+l'-'aJ'
si(a0'-'aJ = (ao,...,a.,a.,...,aj .
We let An denote @,l,...,n) e A[n]n. We denote by A[n], A°[n],
and A2[n] the subcomplexes of A[n] generated by {chAjO < i\,
cLA , and {<9.A II < i\ respectively.
U n l n ' ~~
Now suppose K is a complex, x e Kn. Clearly there is a
unique simplicial map x: A[n] -» K such that x (Ar) = x. If
x e Kn t then x: (A[n], A[n]) -> (K, </>) and if L is a subcomplex
of K and x e K(On» then
x:
iK,L,<f>) .
Lemma 5.5: Let (K, ^) be a Kan pair, and suppose x, y e Kn .
Then x ~ y if and only if F a y rel A[n].
Proof: Suppose x ~ y, say z e ^n+1 satisfies 5^ = 0,
0 < i < n, 5nz = x, and 5n+1 z = y. We must define h: x ^ y
rel A[n], and it clearly suffices to define h{ on An . We do this by
A.(A ) = s.x, 0 < i < n, and h (A ) = z. Conversely, suppose
ft: x ^ y and let z{ = A/(An), 0 < / < n. Then (Jj.z^. = 0 un-
unless i = ; or i = j+ 1, and dQzQ = x, dtz. = d.z._1 for
1 < r < n, dn+1 zn = y. We will prove that if z e Kn+V d{z
for i ? r, r + 1, and dfz e Kn, df+1 zt^, then dfz ~ <9
This will prove the result, since it will imply
z.
Thus suppose t < n, and let a = dfz, b = <?r+1z, where d.z =
if i ^ r, r + 1. By the extension condition, there exists w t K
such that d.w = <f> if 0 < i < r and if r + 3 < i < n + 2,
<?r+ ! w = ®r+ i b, df+2w - z> an^ ^T+Zw ~ sr^' Then
d{drw = 0, i fi r+l,r+2, and dr+1dfw = a, <?r+2<?rw = 6.
Iterating the process, we find a ~ 6.
LEMMA 5.6: Let (K, L, <p) be a Kan triple, and suppose
x, y f K(L)n .
Then x ~ y rel L if and only if x ^ y rel (A°[n], A1 [n]).
The proof is similar to that of Lemma 5.5 and will be omit-
omitted. Lemmas 5.5 and 5.6 imply
PROPOSITION 5.7: Let (K, L, <?) be a Kan triple. Then:
(i) Homotopy is an equivalence relation on maps
(A[n],A[n]) - (K,<t>),
and Trn(K, <P) may be identified with the set of equiva-
equivalence classes of such maps,
(ii) Homotopy is an equivalence relation on maps
and nn (K, L, <p) may be identified with the set of equiva-
equivalence classes of such maps.
Together with (iii) of Lemma 5.2, Proposition 5.7 implies:
Proposition 5.8: If f,g: (K,L,<f>) -> (K',L',<f>') and / =* g

$1
$"<
Si
5!
y
si
16
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
SIMPLICIAL OBJECTS AND HOMOTOPY
17
ill
rel(L', <?'), then /* = g*: nn(K,L,<t>) -» nn(K',L', <f>'), and simi-
similarly for the absolute groups. In particular, the homotopy groups are
invariants of homotopy type on the categories of Kan pairs and of
Kan triples.
§6. Function complexes
In this section, we reformulate the definition of homotopy of
maps and show that maps K -> L are vertices of a complex LK, the
paths of which are homotopies between maps.
DEFINITION 6.1: The Cartesian product K x L of two complexes
K and L is defined as {KxL) = K xL with face and degener-
degeneracy operators given by
d. (x, y) = (d. x, d. y) and si (x, y) = (s. x, s. y) .
NOTATIONS 6.2: Let / = A[l] and / = A [1] • @) will denote
any simplex of A1 [1], A) any simplex of A°[l].
PROPOSITION 6.2: Let / and g be simplicial maps K -» L. Then
t an g it and only if there exists a simplicial map F: Kxl -» L
such that F(x, @)) = g{x) and F (x, A)) = /(x), where x is any
simplex of K.
Proof: Suppose h: f an g. For x t K define F (x, @)) =
g(x), FU,(D) = /(x), and
F{x,Sq_1...si+lS._l...SoA1) = dj+1h}{x),
0 < i < q-1. Then it is easily verified that F is a simplicial map.
Conversely, given F define
h.U) = F(s.x,sq...Sf+1s._1...s0A1).
x t Kq, 0 < i < q. Then A: / ~g.
We will call F a homotopy from g to /, F: g ~ /. Similar-
Similarly we may prove
- PROPOSITION 6.3: Let f,g: (K, K') -> (L,L'). Then / ~ 6
¦ rel L if and only if there exists F : Kxl -» L such that F: g ~ {,
\ F(K'xl)cL', and F\K'xI: g\K* c* f\K\
I DEFINITION 6.4: The function complex LK of maps from a complex
K to a complex L is defined as follows:
(i) (LK) = Hom(Kx A[g], L), the set of simplicial maps
' F: KxA[g] -» L
(ii) d.f = /(lxSp and s^ = /(Ixct^, where 1 denotes
the identity map of K.
(L, Lr ' is similarly defined, with g-simplices the elements
of Horn ((Kx A[q], K'xA[q]), (L, L')).
Now a homotopy F: f ~ g between maps K -» L is just a
path A-simplex) of L which starts at 5j F = / and ends at
dnF = g. It follows that homotopy is an equivalence relation on
maps K -» L whenever L is a Kan complex; and similarly for
maps of pairs, etc. We will prove that LK is a Kan complex if L
is. To do this, we need a combinatorial description of q-simplices
of L similar to that of 1-simplices given by the original definition
of homotopy of maps.
DEFINITION 6.5: A (p, g)-shuffle is a permutation n of
iO,..., p+ q—1] which satisfies n(i) < v{j) if 0 < i < j < p-1 or
if p < i < j < p+q-1. Let /z^ = nU-1), 1 < i' < p, and let
k = w(/ + p-l), 1 < j; < q. n is determined by /z or p-, and we
write n = (/z, v).
U y t K is non-degenerate and y denotes the subcomplex
of K generated by y, then the non-degenerate (p+ q) —simplices of
yx A[q] are {(s ... s^ y, s ... s A )|(/z, i/) is a (p, q)-shuffle}.
q 1 "p ' 1
This motivates the introduction of shuffles.

18
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
Let i t {0,..., p+g}. We classify (p, g)-shuffles relative to
i as follows, (jjl, v) is of type I if
(i) i < nv or
(ii) i, i-1 f \vv...,vq\, or
(iii) i = p+g, i-1 = vq.
(/*, v) if of type II if
(i) i < vv or
(ii) i, i-1 t [fjLv...,iip\, or
(iii) i = p+g, i—l-u.
(/*, pO is of type III otherwise; that is (/*, v) is of type III
if
(i)
< i < p+q, and either
(ii) i t \y.v..., np\ and i-1 t \vx vq\,
or
(iii) i t \vv...,vq\ and i-1 e \nv..., fip\.
To each shuffle (ft, u) and each i we associate a new
shuffle (JT,"?) and an index r as follows: If i.y.,v) is of type I,
(ji.u) is a (p, q — l) -shuffle. Let & be that integer such that
i =
in cases (i) and (ii) and let k = q in case (iii). Let
SIMPLICIAL OBJECTS AND HOMOTOPY
19
Next, to each shuffle (fj., v) and each i we associate a
(p+1> q) -shuffle (/i, 7/) and a second index f as follows. If s is
the largest integer such that u. < i for ; < s, then p.. = ft. for
j < s, }is = i, and /I = /*•_!+ 1 for ; > s. r is defined as
the number of v. such that i > i/..
P|;FINITION 6.6: Let F: KxA[q] - L be a simplicial map. If
(jii, jy) is a (p, g) -shuffle, define
KP
t- by
If (/i, j^) is a (p, g-1)-shuffle and i f \0,...,q\, define
by
If q
1 and we let h. =
v. = k- for j < k and let ia = i>. , - 1 for Ar < ; < g—1. This
defines (JT,"P), and r is that integer such that jl. = u. for ; < r
and /T. - fi. — l for r < / < p.
If (/z, i/) is of type II, (jZ.'v) is a (p— 1, q) -shuffle. Let
k be that integer such that i = y.k in cases (i) and (ii) and let
k = p in case (iii). Let Ji. = u. for ; < k and let JZ. = /i.+1~l
for k < j < p—1. This defines (/I, P), and r is that integer such
that v. = ja for ; < r and v. = i/. -1 for r < ; < q.
If (/z, v) is of type III, ^\L,"v) is a (p, g)-shuffle. Incase
(ii), let i = \i. i—l = v& and define JI: = y.. for ; ^ r, JIr = i—1.
In case (iii), let i = v , i — 1 = u, and define JI. = u. for ;' ^ r,
s r J J
( . where vx = i, then the
proof of Proposition 6.2 states that A is a homotopy and that given
a homotopy h we can construct a map F. A tedious verification
proves the following generalization.
PROPOSITION 6.7: Let F: Kx&[q] -.L bea simplicial map.
Then
(i) di\fi.v) =" A(ju» if ^'^ is (Reshuffle
of type I and index r with respect to i;
(ii) di*(/*.»/) = h(fI.F)di-T if 0«,v) is a (p, g)-shuffle
of type II and index r with respect to i ;
(iii) djh^^ = dih(jr,?) if ^'^ is a (p, g)-shuffle of
type III with respect to i;
Ihll

Si
?
I ' 1 ! :
: :l
20 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
(iv) s./j(/xv) = h^j>)s._t if (//,v) is a (p, q)-shuffle
and f is the second index of (u, p) with respect to i.
SIMPLICIAL OBJECTS AND HOMOTOPY
21
of functions K
L q indexed onthe
Conversely, a set \h, VA of functions K
(p, g)-shuffles and satisfying (ii)-(iv) determines a unique simpli-
cial map F: KxA[q] -> L by the first formula of Definition 6.6.
We will need one simple lemma to prove that LK is a Kan
complex if L is.
LEMMA 6.8: Let L be a Kan complex. Suppose given r + 1
q-simplices x. ,...,x. of L, where t < q and
0 < in < ¦•¦ < i < q+1 and suppose d. x. = d- , x. , s < t.
— U t — ' a 't ' t—1 * S
Then there exists x e L
such that d. x = x.
0 < s < r.
J _i_ 1 h>UVU LUUl. 1/ • A — A •
Proof: li r = q, the statement is true by the extension con-
condition, and therefore the statement is true for q = 0. Assume q > 0
and the result holds for q' < q and assume r < q and the result
holds for r' > r. Let v e {0,..., q+1}, v { \iQ,..., ij and let u
be maximal such that i < v. Define / = i if i < v,
•7u+ 1
x = x
= V,
and i =
's-1
ju+ i
such that x.
if u+l < s < r+1. We wish to find
., x. satisfy the hypothesis, for
it+ l
then an application of the induction hypothesis on r gives the de-
desired x. But the (g-l)-simplices d .x. , s < u, and d x. ,
v -1 'a s
u + 1 < s < r, satisfy the hypothesis, hence induction on q gives
the desired xy.
THEOREM 6.9: If K is a complex and L is a Kan complex, then
LK is a Kan complex.
Proof: Let FQ,..., Fk_v Fk+l,—, Fq be compatible
(g-l)-simplices of LK, and let \h,' A be the functions indexed
on the (p, q-1) -shuffles determined by the F. , i ? k. We will de-
define functions fi( u) indexed on the (p, q) -shuffles and satisfying
(i)-(iv) of Proposition 6.7. It will follow that the corresponding
map F has d.F = F., i ? k. We order all (p, q)-shuffles (q fixed)
by letting an (r, q)-shuffle precede a (p, q)-shuffle if r < p and by
letting a (p, q)-shuffle (/z, v) precede a (p, q)-shuffle (^', v') if
p. = (ij' for i < ;' and fj. < [i '. We proceed by induction on the
(p, q)-shuffles. The first such shuffle is the unique (o, q)-shuffle.
If y e KQ, choose z e L such that d.z = h,' _ny, / 4 k.
This is possible by the extension condition. Define h, y = z.
Now suppose that h, ' * is defined for {fj.',u') < {[i.v).
Case 1: (/z, v) is the first (p, q)-shuffle, so that /z. = i-1,
1 < i < p, and v{ = i+p-1, 1 < i < q. (/z, ^) is of type III with
respect to p, and (/z, i/) < (/T.^), since JL. = i-1 for i < p and
/Ip = p. Thus if y t Kp, dph,^y has not yet been defined, but
all other faces have been except d .h, if k > 1. If y is non-
degenerate, we may apply the lemma to define h. (y). If y is
degenerate, we may use (iv) of Proposition 6.7 to define h. . (y).
Case 2: fif = i —1 and v = i, re |l,...,p|, s e |1,..., q).
(li, v) < (]i,v), where {~fj.,i>) is the associated (p, q)-shuffle rela-
relative to i, hence d. h. . has not been defined and we may proceed
as in case 1.
Case 3: (^, v) is the last (p, q)-shuffle, so that fj.. = i + q — 1,
1 < i' < P> afld H = i-l» 1 < *' < 9- Suppose also that Jt < q.
Now d.h. A = /iA -\, where (u, j7) is associated with (u, v)
relative to k, and h? ^. has not been defined. Thus we may pro-
proceed as in the previous cases.
Case 4: Cases 1, 2, 3 complete the proof except when k = q. In
this case, we simply reverse the ordering of the (p, q)-shuffles, p
fixed, and proceed exactly as above starting with case 3, continuing

1
I I
22
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
SIMPLICIAL OBJECTS AND HOMOTOPY
23
with case 2 where now fiT = i and vs = i-1 for some i, r, and s,
u v)
and finishing with case 1 where now d n^(u v) w*^ not
been defined. Similarly we can prove
THEOREM 6.10: If L' is a sub Kan complex of a Kan complex L,
then (L,L')(KfK/) is a Kan complex.
As we noticed before, these results imply
COROLLARY 6.11: Homotopy is an equivalence relation on maps
into Kan complexes or Kan pairs.
We remark that L is a functor, covariant in L and contra-
variant in K. The following two properties of this functor will be
needed later.
LEMMA 6.12: Let K, L, and M be complexes. Then we have a
natural transformation of functors yc. L x M -» M defined by
/*(/,*)(*, u) = g(f(x, u), a), x e K, u t A[n], / e LK, g e ML.
In particular, K is a simplicial monoid which operates on L
from the left and on KL from the right.
LEMMA 6.13: Let K, L, and M be complexes. Then there exists
a natural equivalence of functors i//: M
KxL
(MK)L.
Proof: Given /: KxLx A[n] -¦» M, we define
y, v){x, u) = f{x,T{u)(y,v)),
where
x e K , y ( L j u e A[g] , v e A[n]4
and where
r is the contravariant functor on A* which defines Lx A[n].
Thus we are regarding u as an element of Hom(A--, A ) in the cat-
category A* so that T(u): L xA[n] -» L xA[n] . Next, given
p LxA[n] -> MK, we define
<?(gX*,y,v) = i(y,v)(x,t±.J, x e K^ y i Lj v i A[n]
It is easy to see that ift and <f> are simplicial maps and are inverse
isomorphisms.
In particular, i//\,mkxl>. gives an isomorphism of sets
^Q: Uom(KxL,M) -» Horn(L, WK). Taking L = AT, we find that
^fir (/), where /: KxL -» L is the projection, is an injection L -> LK
Explicitly,
to(fXy)(x,u) = y(u), y(Ln, z e K, a t A[n] .
M!

!! h*
ill
24 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
Bibliographical Notes on Chapter I
Simplicial sets were first defined by Eilenberg and Zilber
[15], who called them complete semi-simplicial complexes. The
categorical definition of simplicial objects is due to Kan [33], and
is also used by Cartan [6]. Kan [25] first recognized the possibil-
possibility of defining the absolute homotopy groups of complexes satisfy-
satisfying the extension condition using only the simplicial structure. Our
definition follows Kan [30]. A slightly different formulation is given
in Moore [52]. The proof that njji, <fi) is Abelian, n > 2, is due to
Moore [52]. The treatment of homotopy between maps and the proof
that LK is a Kan complex if L is, are both essentially as given
in Moore [52]. Alternative proofs of the latter fact may be found in ¦
Cartan [6], Gugenheim [18], and MacLane [41]. Note that this theo-
theorem is the simplicial analog of Milnor's result [50] that if X has the
homotopy type of a CW-complex and Y is compact, then X has
the homotopy type of a CW-complex. In the literature, L is often
denoted by Hom(K, L), emphasizing the analogy with category theo-
theory rather than that with topology.
CHAPTER II
FIBRATIONS, POSTNIKOV SYSTEMS,
AND MINIMAL COMPLEXES
§7. Kan fibrations
DEFINITION 7.1: Let p: E -> B be a simplicial map. p is said to
be a Kan fibration if for every collection oi n+l n-simplices
x0'""' Xk-V Xk+V''xn+1 °* ^ wnich satisfy the compatibility
condition d.x. - d._lxj, i < j, i ? k, j ? k, and for every
(n+l)-simplex y of B such that d.y - pKx^, i 4 k, there exists
an (n+l)-simplex x of E such that d.x = x., i ? k, and
p(x) = y. E is called the total complex, B the base complex, and
(E, p, B) is called a fibre space. If 0 denotes the complex gener-
generated by a vertex of B, F = p~l((f>) is called the fibre over ^.
If if/ is the complex generated by a vertex of F, then the sequence
(F,^)-U (E,if,)-^ (B»
is called a fibre sequence.
REMARK 7.2: Let E be a complex, 0 the complex generated by
a point. Then the unique simplicial map E -+ <f> is a Kan fibration
if and only if E is a Kan complex.
PROPOSITION 7.3: Let p: E -> B be a Kan fibration. Then F is
a Kan complex.
Proof: Suppose xQ,..., xki_v xfc+ v..., xn+ 1 ate compatible
25

24
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
¦Hi:
Bibliographical Notes on Chapter I
Simplicial sets were first defined by Eilenberg and Zilber
[15], who called them complete semi-simplicial complexes. The
categorical definition of simplicial objects is due to Kan [33], and
is also used by Cartan [6]. Kan [25] first recognized the possibil-
possibility of defining the absolute homotopy groups of complexes satisfy-
satisfying the extension condition using only the simplicial structure. Our 1
definition follows Kan [30]. A slightly different formulation is given
in Moore [52]. The proof that nn(K, cf>) is Abelian, n > 2, is due to
Moore [52]. The treatment of homotopy between maps and the proof
that LK is a Kan complex if L is, are both essentially as given
in Moore [52]. Alternative proofs of the latter fact may be found in
Cartan [6], Gugenheim [18], and MacLane [41]. Note that this theo-
theorem is the simplicial analog of Milnor's result [50] that if X has the
homotopy type of a CW-complex and Y is compact, then X has
the homotopy type of a CW-complex. In the literature, L is often
denoted by Horn (JK, L), emphasizing the analogy with category theo-
theory rather than that with topology.
CHAPTER II
FIBRATIONS, POSTNIKOV SYSTEMS,
AND MINIMAL COMPLEXES
§7. Kan fibrations
DEFINITION 7.1: Let p: E -> B be a simplicial map. p is said to
be a Kan fibration if for every collection of n+ 1 n-simplices
xO>---'xk-Vxk+V-'xn+l oi E which satisfy the compatibility
condition d.x. = d._1x., i < j, i ^ k, j ^ k, and for every
(n +1)-simplex y of B such that d.y = pCx^.), i ^ k, there exists
an (n +1) -simplex x of E such that d. x = x., i ^ k, and
p(x) = y. E is called the total complex, B the base complex, and
(E, p, B) is called a fibre space. If 0 denotes the complex gener-
generated by a vertex of B, F = p~1(<j>) is called the fibre over <f>.
If if/ is the complex generated by a vertex of F, then the sequence
is called a fibre sequence.
REMARK 7.2: Let ? be a complex, <t> the complex generated by
a point. Then the unique simplicial map E -» <t> is a Kan fibration
if and only if E is a Kan complex.
PROPOSITION 7.3: Let p: E -» B be a Kan fibration. Then F is
a Kan complex.
Proof: Suppose x-,..., x...,x. .,..., x . are compatible
25

26
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
il'i
n-simplices of F. Using our convention that <f> denotes ambiguous-
ambiguously any simplex of (f> C B, p{x.) = <j>, and, since p is a Kan fibra-
fibration, there exists x e #n +1 suc^ ^at d{x = x{, i ^ k, and
p(x) = <?. But then x e F ,, since by definition F = p~ @).
LEMMA 7 A: Let p: E -» B be a Kan fibration and assume p is
onto. Suppose given r+ 1 q-simplices x{ ,...,x. of E, where
r < q and 0 < iQ < ••• < if < q + 1 and suppose
d. x = (9 x. , s < t .
's '< 't-1 's
Assume further that y e ^q+1 satisfies d. y = p(xf ), 0 < s < r.
Then there exists x t E j such that (9^. x = x; , 0 < s < r,
and p(y) = x.
The proof of the lemma is identical with that of Lemma 6.8,
of which it is a generalization.
PROPOSITION 7.5: Let p: E . B be a Kan fibration.
(i) If E is a Kan complex and p is onto, then B is a
Kan complex,
(ii) If B is a Kan complex, then E is a Kan complex.
Proof: (i) Suppose E is a Kan complex, and let
be q-simplices of B such that
k> J ^ k •
Using the lemma, choose xQ e E such that p(xQ) = yQ, choose
x t ? such that p(xx) = y1 and (JqXj = dQxQ, and so forth,
until xa..., xk_v xk+ 1,..., xq+ x have been chosen satisfying
p(x.) = y., i ? k, and d.x. = d._*x., i < j, i p k, j ^ k.
Then we choose x t E 1 such that ^.x = x^ i / k, and we
let y = p(x). Obviously d.y = y., i ^ k.
FIBRATIONS, POSTNIKOV SYSTEMS, ETC.
27
(ii) Suppose B is a Kan complex, and let
X0'*"' Xk-V Xk+ I'""' xq+ 1
be g-simplices of E such that cKx. = d._lx., i < j, i ? k,
j 4 k. Choose y t Bq+y such that d.y = pU^, i ? k. Then
we may choose x e E x such that p(x) = y and d.x = x.,
itk.
Now let (F, iff) '-» (E, iff) -? (B, 0) be a fibre sequence
of Kan pairs. Regarding p as a map (E, F, ^f) ^ (B, j>, ?), p in-
induces p#: rrn(E, F, iff) - 7rn(B, </.), Let y f B , n > 1. Since p
is a Kan fibration, there exists x e E such that d.x = i/r.
1 < i < n, and p(x) = y. (?ox t Fn_1, since pEQx) = dQy = 0.
Define q: nn(B,j>) -> nn{E,F,iff) and d#: nJiB.cf,) -> n^^F.iff)
by q[y] = [x] and <9#[y] = [dQx\. It is easily verified that q
and dft are well defined and are homomorphisms if n > 2. Further
p^q = 1, gp^ = 1, and the following diagram is commutative:
irn(F, ^) -^ rrn(E, iff) 1* rr^F.
> ffn(F, ^) -^ rrn(E, ^) -^ rrn(B, 0) -»...
Thus we have proven
THEOREM 7.6: Let (F, iff) - (E, ^) -> (B, ^>) be a fibre sequence
of Kan pairs. Then p#: nn(E, F, iff) ¦+ nn(B, ^>) is an isomorphism
and the following is an exact sequence:
•••-. na+1{B,<f,)-^ nn{F,if,)-U rr^E, iff) ^ n^B, +) -,-. .
Next we define maps and homotopies of maps of fibre spaces.
DEFINITION 7.7: Let (?,p, B) and (E', p', B"') be fibre spaces.
A map (/, /): (E, p, B) -* (E'p', B') is a commutative diagram

Si
Si
SI
m
28
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
E'
B
f
B'
of simplicial complexes. A homotopy (F, F): (/, /) ^ (g, g) is a
commutative diagram
F
ExI
E'
px 1
Bx/-
B'
of simplicial complexes, where F: f s* g and F: f c* g. (E, p, B)
is said to be the same homotopy type as (?', p', B') if there exist
(/, /): (E, p, B) ^{E',p',B') and {g, g): (E', p', B') - (E, p, B)
such that (/, f)(g,g) =1 and (g,g)(f,f) = 1. A sub-fibre space
(E, p, B) of (E", p',B') is said to be a deformation retract of
(E', p' B') if the identity of (E', p', B') is homotopic relative to
(E, p, B) to a map of (?', p', B') onto (E, p, 2?) which extends the
inclusion map (E,p,B) -» (E", p', B').
Using Lemma 7.4 and the machinery developed to prove
Theorem 6.9, we can prove the following generalization of that
theorem.
THEOREM 7.8: If p: E -> B is a Kan fibration, then so is
pic. ?*c ^ jjK^ wjjere # is any complex. If F is the fibre over
<f> C B, then FK is the fibre over cf>K S 0.
COROLLARY 7.9: Let {E,p,B) and (Ej, pv BJ be fibre spaces
and suppose that B. is a Kan complex. Then homotopy is an
equivalence relation on maps (E, p, B) -> (E-, pi; Bj).
Proo/: Given (f, /), define F(x,z) = /(x) and
F (y, z) = /(y),
e En, y e
FIBRATIONS, POSTNIKOV SYSTEMS, ETC.
29
Then (F,F): (/, /) ~ (/",/). Now suppose iF,F): (/,/) at(g,g)
and (G, G): G, /) ~ (h, h). We must prove (g, g) c* (h, h). BjBis
a Kan complex, hence there exists U e i^1BJ such that d. V = G
and d2V = F. Let ff = dQ*7. Then ff: ^~i. Since
is a Kan fibration, there exists V e (Ej^ such that d. V = G,
d2V = F, and PlB(P) = C/°(pxl). Let ff = d0 7. Then
ff: g ~ A and px o ff = PlE(ff) = H ° {px 1) as desired.
To prove the next corollary, we must first rectify a small
omission.
DEFINITION 7.10: Two vertices x and y of a complex B are in
the same path component of B, written x ~ y, if there exist
1-simplices Zy..., zn such that x is a face of Zj, y is a face of
z , and z. has a face in common with z., ,, 1 < i < n. ~ is
an equivalence relation on vertices of B, and we define
nQ(B) = BQ/(~).
If B is a Kan complex, then obviously nAB) = nJ,B, 0), where
cj> is any vertex of B.
COROLLARY 7.11: Let p: E -> B be a Kan fibration and let F,
and F/ be the fibres over two vertices cf> and if/ of B. Then if
0 and i// are in the same path component of B, F • and F,^ are of
the same homotopy type.
Proof: p <t>: E <t> ^> B <f> is a Kan fibration. As shown
in Remarks 6.10, there is an injection B -» B 4>, and if we let
K = (p^)^) and q = pF01K, then q. K -> B is a Kan fi-
fibration. An element x e Kn is a map Fj x A[n] -> E for which
there exists a commutative diagram F, x AM -^-» E.
A[n]
-+ B

30
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
FIBRATIONS, POSTNIKOV SYSTEMS, ETC.
31
m
"'V
I! ,
I I.
m
il
In particular, an element of KQ is a map from F > into another fi-
fibre. Now we may assume that there exists z e B. such that
d. z = <f> and dQz = iJj. If x f KQ is the inclusion F, -» E, then
there exists y t K. such that g(y) = z and d. y = x. It follows
that doy is a map F, -* F. . Similarly, let K' = (pFt//)~1(B),
q' = pF"A | K': K' -+ B. If x': F^ -» E is the inclusion there ex-
exists y' e Kj' such that g(y') = z, d^y' - x', and cLy' is a
-> Ffk- Since q : K ^ B is a Kan fibration, there exists
L such that dQw = y, d1 w = y'°Woyx 1), and qr(w) =
It follows that gCc^w) = 0, ^^2^ = x>
map
d2 w
Since
/) CF
,,
~ ^ in
2
we may prove that dQy ° d^y' c^ 1 in F/, and therefore F# and
Fi are of the same homotopy type.
COROLLARY 7.12: (Covering Homotopy Property). Let p: E -> B
be a Kan fibration and let K be any complex. Let t: K -» E and
f = p° f: K -> B. Suppose F: K x / ^ B satisfies d^F = i.
Then there exists F: Kxl -» E such that p o F = F and dj F = /.
A proof similar to that of Theorem 6.9 also gives
THEOREM 7.13: Let f: K -> L be an inclusion of complexes and
let E be a Kan complex. Then E': EL -> EK is a Kan fibration.
COROLLARY 7.14: (Homotopy Extension Theorem). Let K be a
subcomplex of the complex L and let E be a Kan complex. Let
/: L -» E and let / = 1\R: K -> E. Suppose F: Kxl * E sat-
satisfies d- F = /. Then there exists F: Lx/-» E such that
F|Kx/ = F and ^F = f .
Theorems 7.8 and 7.13 are both special cases of a more gen-
general result, which we now state.
DEFINITION 7.15: A commutative diagram of complexes and maps
A S >B
D
is said to be a fibre square if for any q+1 compatible q-simplices
xO'-'Xk-Vxk+V-'xq+l of A andforany y ? Bq+1 and
z t C +1 such that c^y = eCx^ and ^.z = /(x^, i ^ A, and
= h{z), there exists x e ^q+\ suc^ ^at e^ = y> ^x^ = z'
d. x = x2., i ^ L
THEOREM 7.16: Let p: E ^ B be a Kan fibration and let i: K ~> L
be an inclusion of complexes. Then
is a fibre square.
COROLLARY 7.17: (Covering homotopy extension theorem). Sup-
Suppose given (/, /) : (L, &,L') -+ (E, p, B), where |j is any map of
complexes and p is a Kan fibration. Let K' C L' and
K = g-^K') C L
(here K' may be empty). Further, let G: Kxl -^ E and
F: L'xl -* B be given satisfying ,dj F = /, dx G = /|K, and
p o G = F ^glKF.^Then there exists F: Lx I ^> E such that
F|K x / = G, p o F = F o (g x 1), and dx F = 1.
§8. Postnikov systems
In this section we define the natural Postnikov systems

32
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
FIBRAT1ONS, POSTNIKOV SYSTEMS, ETC.
33
Si
Si
if
llHil
1 ! !
associated with Kan complexes and fibre spaces. The definition of
the k- invariants will be given later, after the concept of twisted
Cartesian product has been developed.
DEFINITIONS 8.1: Let A[q]n denote the n-skeleton of A[q], that
is, the subcomplex generated by all simplices of dimension < n.
Define an equivalence relation 5 on q- simplices of a complex K
by xT-y if x|A[q]n = y|A[q]n (where xCAq) = x, jKA^) = y).
Noting that x-y implies d^-d^ and s.x^s.y, we can define a
complex K(n) by K(?> = K /B), where the face and degeneracy
operators are induced from those of K. Note that K^ = K if
n >^ q, and define KS °°^ = K. Let p" denote the natural map of
K(n) onto K(m), m < n < oo. We will usually write p for p^.
PROPOSITION 8.2: Let K be a Kan complex. Then
(i) K(n) is a Kan complex;
(ii) pnm: K(n> -. K(m) is a Kan fibration.
Proof: (i) follows from Proposition 7.5 and the case n = oo
of (ii). Let x0,..., xfc_1,x|c+1,..., xq+1 be compatible q-simplices
of K(n) and suppose y e K^ satisfies d.y = p(x.), i ^ Jt. If
9 < m> K(q") = Kq so that if z f K(q| j represents y, dyz = x^
i ^ A. Then if we let x = z, d{x = xjt i ^ k, and p(x) = y.
Thus assume q > m. Suppose first that n = °c Then there exists
x e K j such that 5. x = x^, i ^ k. Since any face of x of di-
dimension < m is also a face of some x., x^z if z represents y
and therefore p(x) = y. Now if n < oo, we know that KSn' is a
Kan complex, and the same argument applies.
DEFINITION 8.3: Let (K, <f>) be a Kan pair. Define En(K,<f>) to
be a fibre over 0 of p: K -» K^). Thus En(K,(f>) is the sub-
subcomplex of K consisting of those simplices of K all of whose faces
of dimension less than n are at 0 . E (K, 0) is called the nth
Eilenberg subcomplex of K based at (f>.
THEOREM 8.4: Let (K, <f>) be a Kan pair. Then
(i) p*: nq(K^n\cf>)^ nq(KSm\<l>) for q < m;
(ii) nq(K(m\<f>) = 0 for q > m;
(iii) i: nq{Em+1(K<-n\<f>), 0)-^» itq(K<-n\ <f>) ior q > m;
(iv) nq(Em+1(K(n\<f>),<f>) = 0 for q < m and for q > n.
Proo/: Em+1^(nW^q = ^ if q < m. This proves part
' df (iv) and the homotopy exact sequence of the fibre space
(K<n), p, K(m)) then gives (i). Suppose x ( a e ^q(K(m), 0), q > m.
By definition, d.x = cj> for all i and, again by definition, this im-
implies that 0 is a representative for x. Therefore a = 0, and (ii) is
proven. Using the homotopy exact sequence, (ii) implies (iii). The
last part of (iv) now follows.
The special cases n = oo and n = m+1 ate of particular
importance.
DEFINITION 8.5: If K is a Kan complex, let X" denote
(K(n+ 1}, p, K(n)). The sequence of fibre spaces X = U0,..., X",...)
is called the natural Postnikov system of K. We let F(n+ ^(X) de-
denote the fibre En + x (K(n + 1), 0) of the fibre space Xn .
DEFINITION 8.6: If (K, <?) is a connected Kan pair such that for
some n > 0, nn(K> <f>) = it and n(K, cj>) = 0 if q ^ n, then K is
said to be an Eilenberg-MacLane complex of type {n, n).
The fundamental property of the natural Postnikov system
is given by the following corollary of Theorem 8.4.
COROLLARY 8.7: Let (K, <f>) be a Kan pair, X its natural Postni-
Postnikov system. Then F^n'(X) is an Eilenberg-MacLane complex of
type (nn(K, 0), n), n > 1.

•¦rli;>
II
|{1
34
SIMPUCIAL OBJECTS IN ALGEBRAIC TOPOLOGY
FIBRATIONS, POSTNIKOV SYSTEMS, ETC.
35
Now we generalize the constructions above to fibre sequen-
sequences. For the remainder of this section, (F, i//) ' -» (E, i/r) p-» (B, (f>)
will be a fibre sequence of connected Kan pairs. We will define the
natural Postnikov system of the fibre space (?, p, B).
DEFINITIONS 8.8: Define an equivalence relation " on q- simplices
of E by x 2 y if x|A[qln = y |A[q]n and p(x) = p(y). Define a >
complex E(n) by E^ = #/(-), where the face and degeneracy
operators are induced from those of E, and let E*-00' = E. Let p"
denote the natural map of E(n) onto ?(m)» m < n < «>, and write p i
for pj\ Note that if EQ = i/r, then ?@) S S, since x 2 y if I,
p(x) = p(y).
COROLLARY 8.11: F(n>(?) = F<n>U), where X is the natural
Postnikov system of the fibre F, and therefore F(n)(?) is an
Eilenberg-MacLane complex of type (nn(F, ip), n), n > lv
REMARK 8.12: The important part of the homotopy exact sequence
of ?" is the following:
0.
This is essentially just the exact sequence
0 - P*(nn + 2(E,if,)) + 7Tn + 2(B,<f>) + nn+
PROPOSITION 8.8: Under the above hypotheses
(i) E^ is a Kan complex;
(ii) p: E^ -» B is a Kan fibration;
(iii) p": E(n) -» E(m) is a Kan fibration.
THEOREM 8.9: Under the above hypotheses
(i) the fibre over cf> of p: E<n>- B is F(n), the nth
total space in the natural Postnikov system for the
fibre F;
(ii) the fibre over </r of p^: E(n> -. ?(m> is
o.
§9. Minimal complexes
DEFINITION 9.1: A Kan complex K is said to be minimal if
dtx = d{y, i ? k, implies dkx = dky.
(iii) p*: n
(iv) p*: ^
DEFINITION 8.10: Let en denote (E(n+1), p, ?(n)). The sequence
of fibre spaces ? = (?0,..., ?",...) is called the natural Postnikov
system of (E,p,B). Let F(n+1)(?) denote the fibre of ?n.
LEMMA 9.2: A Kan complex K is minimal if and only if x ~ y
implies x = y.
Proof: (i) Suppose K is minimal and x ~ y, x, y e K
Then there exists z e K , such that d.z = s .d.x, i < a,
5qz = x and dq+1z = y. Now ^sqx = s^d.x, i < q, and
5qsqx = x. By assumption, this implies dq+1z = dq?ls x,
,(E(m), </r) for q < m and for S that is> 7 = x-
(ii) Suppose x ~ y implies x = y. Let d.x = d.y,
i ? k, x, y ( Kq?l. We must prove d^x = ^y, and it suffices
to prove dkx ~ dky, By the extension condition, if k < q there
exists z e K^ such that d.z = s d.x, i < q, i ^ k,
dq±lz = x' and 5cn-2z = y- Then 5icz: dkx ~ dky- If*=9+-1»
nq(B,<f>) for q
"

I:
5:
nil
« -
'lii
m\:;.
36
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
there exists z ( Kq+2 such that d{z = s ^ x,
x, and dq+1z =
Then
dq+Jx
LEMMA 9.3: Let K be a complex. Suppose x, y t K satisfy
d.x = d¦ y, all i, and x = sm z, y = snw. Then x = y.
Proof: If m = n, z = d x = dm y = w, and x = y. As-
Assume m < n. z = d x = dmsnw = sn_ 1 dmw, hence
x = sasn _ j dm w = sn sm dm w . «
Thus sm dm w = dn x = dn y = w, and, finally, x = snw = y.
Now if K is a Kan complex, we define a minimal subcomplex
of K as follows. Choose a vertex in each component of K. These
vertices are the elements of MQ. Suppose M * has been defined,
0 < <?' < 9- Consider the homotopy classes of q- simplices all of
whose faces are in M'_y and choose one q- simplex in each such
class, choosing a degenerate representative whenever possible.
These q- simplices are the elements of M . By Lemma 9.3, M is a ,
subcomplex of K. We will prove that M is a retract of K, and since
any retract of a Kan complex is clearly a Kan complex, it will follow
from Lemma 9.2 that M is a minimal complex.
DEFINITION 9.4: A deformation retract K of a complex L is said
to be a strong deformation retract of L if some homotopy F: L x / -> L
defining K as a deformation retract satisfies F(x, y) = x, x e K^.
all 9.
THEOREM 9.5: M is a strong deformation retract of K.
Proof: We will define h. satisfying the conditions of Defini-
Definition 5.1, with dQhQ(x) = x, and dq+1hq(x) e Mq for x c Kq and
with fyU) = SjX for x e Mq.
(i) If x e KQ, let hQ(x) be a 1-simplex satisfying
FIBRATIONS, POSTNIKOV SYSTEMS, ETC.
37
dohQ(x) = x and ^ AQ(x) e A/o, and let AQ(x) = sQx if x f MQ.
(ii) Assume h. has been defined on K ', 0 < q' < q, and
let x e K If x is degenerate, say x = s.y, define
A/x) = s.h.^iy) if j < i- 1 and A.(x) = s.+ 1 A.(y) if ; > ;_ 1,
checking that if x = sk z is also true, the two resulting definitions
agree. If x e Mq, let /k(x) = s^. Otherwise, let hQ(x) be a
(q+ l)-simplex satisfying dQhQ(x) = x and d.hQ(x) = hQ(d._ j x) if
/ > 1. There exists such a simplex by the extension condition.
(iii) Assume h. has been defined on K . 0 < i < ; < q,
and let x f Kq be non-degenerate, x / Mq. By the extension con-
condition, we can choose h.(x) satisfying d^ix) = h. Ad.x) for
/ < j, d.h.U) = djh^U), and d.hjU) = A^W^jx) for i > / + 1.
(iv) It remains to define h (x), where x ( K is non-
degenerate and x / M . Choose y t K such that
diy =
9+1
<
and
Now if y < q,
and
Therefore there exists a unique z t M such that z ~
the extension condition, there exists w e K
_
y. By
such that
— _ H f
and
Define hq(x) = 5q+2 w. It follows that d.hq(x) = d.y, 0 < f < q,
an(* dq+1hq(x) = z e M as desired.

38
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
F1BRAT10NS, POSTNIKOV SYSTEMS, ETC
39
I
i
1
II
¦ II
LEMMA 9.6: Let f a± g: L -> M, where M is a minimal complex.
If / is an isomorphism, then so is g.
Proof: Let h: L -> M be the given homotopy.
(i) Suppose g{x) = g(y), x, y e L . We must prove that
x = y. Now dq+1hq(x) = g(x) = g(y) = dq+1hq(y). If q = 0,
then by the minimality of A/ /(x) - dQhQ(x) = dQh0(y) = /(y),
hence x = y. Assume the result for q' < q, q > 0. Then
di x = d.y. If i < q,
By the minimality of M, d h (x) = d h (y). Now
h (
hence
Iterating, we find fix) = dQhQ(x) = dQhQ(y) = /(y), and therefore
x = y.
(ii) Let y e M . We must find x e L such that g(x) = y.
If q = 0, there exists n J/, such that cL z = y. Let x satisfy
/(x) = dQz. dQh0(x) = fix) = dQz, hence
g(x) = d1hQ(x) = ^z = y.
Assume the result for q' < q, q > 0. Let x. ( L _j be the unique
element satisfying ^(x^ = 5^. Choose z e ^q+\ such that
dq+lzq= y and dizq = hq-l^x?' ' < <*• By induction on q-;,
we may choose z e A/ j such that d.z. = h^^x.) if i < j,
dj+1zj = dj+1zj+l, and 5i z^. = h.U^J if i > j+1. Now let
x ( L satisfy f(x) = dQzQ. If i > 0,
/(,9.x) = ^^ozo = dodi+izo = 5dAo(xi) = /(xP '
and fE0x) = dodxzQ = ^^^j = aoaoZl = dohQ(xQ) = /(xQ).
Therefore d.x = xy for all f. Now d{hQ(x) = c^-Zq, i ^ 1, hence
cL/io(x) = ^1zQ. Similarly, we find d.hix) = d; z. for all / and
j and, in particular, g(x) = d i^Jx) = ^q+ i zq = ^> as desired.
PROPOSITION 9.7: Let /: M -> W be a homotopy equivalence,
where M and M' are minimal complexes. Then / is an isomor-
isomorphism.
Proof: Let g: W ¦* M satisfy fg se 1 and g/ ~ 1. By
the lemma, fg and g/ are isomorphisms, hence so are g and / (al-
(although g need not be f~1).
THEOREM 9.8: Any two minimal complexes which are deformation
retracts of a Kan complex K are isomorphic.
Proof: Let M and M' be two such complexes with retrac-
retractions r: K -» A/ and r': K -. A/' and let i: M C K and i': ATC K.
Since it ca 1 and i 'r' ~ 1 on K, (f 'i)(«) ~ r 'i' = 1 on A/' and
(ri ')(r'/) ~ ri = 1 on A/, hence the result follows from the proposi-
proposition.
CONVENTION 9.9: From now on, when we speak of a minimal sub-
complex of a Kan complex K, we mean one which is a strong defor-
deformation retract of K.
We now discuss briefly a more general concept than that of
minimal complex which applies to the relative case.
DEFINITIONS 9.10: A sub Kan complex M of a Kan complex K is
said to be admissible if for every x e K such that d.x e M ,
for all i, there exists y e M such that x ~ y. If (K, L, <f>) is a
Kan triple, an admissible subcomplex M of K is said to be rela-
relatively admissible if cf> C M and M f\ L is an admissible subcom-
subcomplex of L.

h
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THEOREM 9.11: Let (K, L, <f>) be a Kan triple and let M be a rela-
relatively admissible subcomplex of K. Then (A/, A/ D L, 0) is a strong
deformation retract of (K, L, <?).
The proof is essentially the same as that of Theorem 9.5.
REMARKS 9.12: A Kan triple (K, L, <f>) is said to be n-connected if
n^K, L, 0) = 0, 1 < i < n. The n-th relative Eilenberg subcomplex
En(K, L, (f>) of K is defined by x t En(K, L, <f>) q if and only if
x|A[ql° = 0|A[g]° and Image(xlAtq]") C L. If (K, L, 0) is
n-connected, then E ^(Kt L, <f>) is relatively admissible.
§10. Minimal fibrations
In this section, we outline the construction of minimal sub
fibrations of Kan fibrations.
DEFINITIONS 10.1: A Kan fibration p: E -> B is said to be mini-
minimal if p(x) = p(y) and c^.x = di y, i ^ k, implies d.x = d.y. If
p is minimal and B is a minimal complex, then (E, p, B) is said to
be a minimal fibre space.
LEMMA 10.2: Let p: E -» B be a Kan fibration. Then if p is min-
minimal, each fibre of p is a minimal complex.
LEMMA 10.3: Let (E, p, B) be a fibre space. Then if E is a mini-
minimal complex and p is onto, (E, p, B) is a minimal fibre space.
Proof: Clearly p is a minimal fibration. We must prove that
B is a minimal complex. Suppose x, y e B and z: x ~ y. Let
x = p(w). Since p is a Kan fibration, there exists u e E 1 such
that d.u = s _. d.w, i < q, d u = w, and p(u) = z. Now
u: d u <^> d . u, hence since E is minimal, d u = d .^u.
p{d ju) = d 1 z = y and p(d u) = p(w) = x, and therefore
x = y.
DEFINITION 10.4: Let p: E -. B be a map. If x, y e E , we say
p(z) = s p(x).
that x is p-homotopic to y, written x &) y, if x ~ y and there
exists z e Eq+i suc^ ^at z: x
LEMMA 10.5: If p: E -> B is a Kan fibration, then
equivalence relation on the g-simplices of E.
i LEMMA 10.6: A Kan fibration p: E -» B is minimal if and only if
is an
implies x = y.
Now if p: E -» B is a Kan fibration, we can define a sub-
subcomplex E' of E by a procedure precisely parallel to the construc-
construction of a minimal subcomplex of a Kan complex, the only difference
being that the relation of p-homotopy rather than that of homotopy
is used.
LEMMA 10.7: p: E'-> B is a Kan fibration.
It follows immediately from Lemma 10.6 that p: E' -» B is
a minimal fibration.
DEFINITION 10.8: A deformation retract (E', p, B') of a fibre space
(E, p, B) is said to be a strong deformation retract if some homotopy
(F F) defining (E', p, B ') as a deformation retract satisfies
and
F(x,y) = x, x ( Eq',
lq) all q,
Iq, all q.
F(x,y) = x, x ( Bq, y
THEOREM 10.9: (E' p, B) is a strong deformation retract of
(E,p,B).
More generally, we can prove
THEOREM 10.10: Let p: E -> B be a Kan fibration and let B' be
a strong deformation retract of B. Then there exists E' C p~ (#')
Such that p': is * -» fl' is a minimal fibration, and (E", p',B') is a

42
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43
strong deformation retract of (E, p, B). H
COROLLARY 10.11: Let (E, p, B) be a fibre space of Kan com- ft
plexes. Then there exists a minimal fibre space {E',p',B') which %
is a strong deformation retract of {E, p, B). fi
--V
LEMMA 10.12: Suppose given a strong homotopy '%
(f, l) ^ (/, 1): (E'tP',B) -*(E,p,B), I
where p is a minimal fibration. Then if f is an isomorphism, so f§
is 6- f§
PROPOSITION 10.13: Let (f, 1): {E',p', B) -» (E, p, B) be a if
strong homotopy equivalence, where p and p' are minimal fibra- j|
tions. Then / is an isomorphism. |f
THEOREM 10.14: Any two minimal fibrations with base complex B |g
which are strong deformation retracts of a given fibre space (E, p, B)p
are strongly isomorphic (in the sense that the isomorphism on B is |
the identity). |
i
LEMMA 10.15: Let (/,/) ^(g,g): (E',p',B') -. (E,p,B), where f
(E, p, B) is a minimal fibre space. Then if (f, /) is an isomorphism,^
so is (g,g).
Proposition 10.16: Let (f,f): {E',p',B') -> (E,p,B) be a ho- j
motopy equivalence, where {E',p',B') and (E, p, B) are minimal j
fibre spaces. Then (/, f) is an isomorphism. \
THEOREM 10.17: Any two minimal fibre spaces which are deforma-j
tion retracts of a given fibre space of Kan complexes are isomorphic'
CONVENTIONS 10.12: By a minimal sub fibration p: E' ~> B of a \
Kan fibration p: E ^ B we will understand one such that (E' p, B) ]
is a strong deformation retract of (E, p, B). Similarly, a minimal sub
fibre space (E', p', B') of a fibre space (E, p, B) of Kan complexes
will mean one which is a strong deformation retract.
§11. Fibre products and fibre bundles
In this section we define fibre products and begin the study
of fibre bundles. In particular, we will prove that every minimal fi-
fibration is a fibre bundle.
DEFINITION 11.1: Let p: E -> B and /: A -> B be maps. De-
Define El = |(e, a)|p(e) = /(a)} C Ex A and let pl: El -> A and
/: El -» E be given by pf(e, a) = a and 7(e, a) = e. The commu-
commutative diagram
- B
is called the fibre product of p and /. p is called the map in-
induced from p by /.
The following lemma results immediately from the definition.
LjEMMA 11.2: If p is a Kan fibration, so is pf. If p is a minimal
% fixation, so is p .
If p is a Kan fibration, then {El, p{, A) is called the fibre
space induced from (E, p, B) by /. Fibre products satisfy the fol-
following universal property.
Proposition 11.3: Let
E' ? .E
! B' 1 >B
i be a commutative diagram. Then there exists a unique map

i! ;;t
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Si
45
iiifl
h: E' -> El such that g = f° h and p' = pf ° h.
Proof: The result is obvious: we must define .
Me') = (K(e'),p'(e')).
PROPOSITION 11.4: Let {g, f): (?', p', B') -> (E, p, B) be a map
of fibre spaces. Suppose that p' is onto and that for b' e B'
^|p'-1F') is a one-to-one correspondence onto p!/^')). Then ¦"
the map h: E' -> E( such that g = f ° h and p' = pf ° A is an
isomorphism.
Proof: Define ;: E -> E' as follows. Suppose (e, b') e E',
Choose any x e E ' such that p'(x) = 6'. Then
pg(x) = /(?>') = p(e).
By assumption, there exists a unique e' e E ' such that |f(e') = e
and p'(e') = p'(x) = ?'. Define ;(e,4') = e'. Then /»/i=l
and A ° ; = 1.
Next we show that homotopic maps induce fibre spaces of
the same homotopy type.
PROPOSITION 11.5: Let p: E -» B be a Kan fibration and let
/ ~ |f: /I -» B. Then there exists a strong homotopy equivalence
between (E{, pf, A) and (Eg, pg, A).
Proof: Let F: Ax I -» B be the given homotopy and let
(E*, p*, Ax/) denote the fibre space induced by F. Identify El
with p*~1(/lx@)) C E* and let x: Ef C E*. Identify E«
with p*~1(A x A)) C E* and let x': E^ CE*. For notational sim-
simplicity, let q: K^L denote the Kan fibration (p*)E': (E*)E'-» Ux/)*
and form q'\ K' -* L' using E^ instead of E . Now p xl e L.
and ^jlp^xl) = q(x), hence there exists y ( K, such that-
q(y) = pfxl and 51y = x. Then dQy is a map E( -> E^. Simi-
Similarly, there exists y' f K! such that ^'(y') = p^x 1 and
ft y' = x'. dxy' is a map Eg -» Ef. Again, since g is a Kan
fibration, there exists z ( K2 such that q(z) = sQ(p/xl), cLz = y
and dlz = y'oEQyx l). Let w = d2z. Then
dow = x and d1w = dxy'° dQy .
q(w) = so^1(pf x 1) = soq(x) and therefore the image of w is
contained in Ef and pf ° wKe, ^) = pf(e), e ( E{q, k ( I . Thus
w gives a strong homotopy d.y' ° cLy ~ 1 in Ef. Similarly, we
may obtain a strong homotopy ^Q y ° dj y' o=; 1 in Eg, and this
implies the result.
COROLLARY 11.6: Let p: E -> B be a minimal fibration and let
f ~ g: 4 -> B. Then there exists a strong isomorphism between
(fV.iO and (E* p*,i4).
Proof: This follows from Lemma 11.2, Proposition 10.13,
and the result above.
COROLLARY 11.7: Let (F,i//) -> (E, i/r) -> (B, <f>) be a fibre se-
sequence and suppose that B is contractible. Then there is a strong
homotopy equivalence (g, l): (E, p, B) -> (FxB, p*, B) where
p*(f, 6) = b. If p is minimal, (g, 1) is a strong isomorphism.
Proof: If >: B -» cj>, then by assumption 1 ~ /: B -* B.
Since FxB = E; and E1 = E, the result follows.
At this point, we are ready to define fibre bundles.
DEFINITION 11.8: A map p: E -> B is called a fibre bundle if
p is onto and if for every map F: A[n] -+ B, the induced map
p : E -> A[n] is strongly isomorphic to p*: F x A[n] -> A[n],
where p*(/, k) - k and where F is a given complex called the
fibre of the bundle. If F is a Kan complex, then p: E ^ B is
called a Kan fibre bundle.

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;¦ 1
I rk
iirl'V
iiiilll
Lemma 11.9: A Kan fibre bundle is a Kan fibration.
Proof: Let xQ,..., x|c_1, xfc + v..., xq+ 1 be g+1 compatible
g-simplices of E and suppose d.b = x., i ^ k, b ( B .. Let
a(b): F x A [q+l] -> E be a strong isomorphism, say
)
There exists y e Fg+\ such that c^y = yt, i ^ k, and if
q
then p (x) = b and d. x = x., i 41 k.
Lemma ll.lO: A[nl is contractible for all n.
Proof: Define A~(A ) = snA , /i,(A ) = s~A , and <¦
h.(kn) = si-x ••• so5x ••• d._x An, 1 < / < n. Then if
;: A[n] -+ @) C A[n], it is easily verified that h: 1 ^ j.
THEOREM 11.11: Every minimal fibration with a connected base
complex is a Kan fibre bundle. <
Proof: Let p: ?-» B be the given minimal fibration and let
F: A[n] -> B. By Lemmas 11.2 and 11.10, p^.is minimal and A[n]
b — *v
is contractible, hence by Corollary 11.7 (Ep , pb, A[nl) is strongly
isomorphic to a product (F, x A[n], p*, A[nl). 6 is homotopic to a
map 0: A[n] -» S,where ^ is a vertex of B, and thus by Corollary
11.6 (FfaxA[n], p*, A[n]) is strongly isomorphic to (Fjx A[n], p*, A[nlL
Now F i is the fibre over <f> and since, by Corollary 7.11, any two fi- "f
bres are isomorphic, the result follows.
COROLLARY 11.12: Every Kan fibration with a connected base
complex contains a Kan fibre bundle as a strong deformation retract.
§12. Weak homotopy type
In this section we will use the earlier material of this chap-
chapter to find sufficient conditions for Kan complexes and Kan fibra-
tions to be of the same homotopy type.
We note first the following corollary of Lemma 10.3.
LEMMA 12.1: If K is a minimal complex, then each (K(n), p, K(m))
is a minimal fibre space.
Since a connected minimal complex K has only one vertex
(j,, we will denote nq{K, (f>) by nq(.K).
LEMMA 12.2: Let K be a connected minimal complex. Consider
the fibre space (K, p, K{q~l)), q > 1. If b e K^~l), let
Hb = |x|p(x) = b\,
that is, if x e Hb, then y e Hb if and only if d.y = <9. x for all f.
Then the homotopy classes of elements of Hb are in one to one
correspondence with the elements of n (K).
Proof: By Lemma 12.1 and Theorem 11.1, p: K -> K(-q~1)
is a Kan fibre bundle. Let ^ denote the vertex of K(q-1). De-
Define b and ^: A[g] -» JO9" ' in the obvious way. Then
(E-A, p\ A[qr]) is strongly isomorphic to (Eq(K)x A[ql, p*, A [n]).
where E (K) is the qr-th Eilenberg subcomplex of K. Therefore
the sets of homotopy classes of elements of
|(x, A^lx i Hb\ C Eb
and of elements of {(y, A )| y e Hi } C ? (K)xA[g] are in one-to-
one correspondence. Noting that H j = E (K) , the latter set is
nlElK)) — n (K). Since the former set is that of homotopy classes
of elements of ff. , the result is proven.
DEFINITION 12.3: A map f: K -> L is called a weak (or singu-
singular) homotopy equivalence if f*: n (K, (f>) -> K (L, f(j>)) is an iso-
isomorphism for all q.
THEOREM 12.4: Let K and L be connected minimal complexes.
Thena weak homotopy equivalence /: K -* L is an isomorphism.

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ffeiiifh
Proof: Let (K(n+ x>, p, K(n)) and (L<n+*>, p', L<">) be
the n-th terms in the natural Postnikov systems for K and L. By
Lemma 12.1, both of these are minimal fibre spaces. Since
Kq = K<"> if q<n,
it suffices to prove that /^: J^n) -+ Z/n) is an isomorphism for
all n. /@): K@)-^—* L@) is clear, so assume that
f(n-l). K(n-1)^ L(n-i)f fl > 0
Consider the commutative diagram
l« . r(»)
(i) Suppose f(n)(x) = f(n)(y), x, y f K(qn). We must
prove that x = y. /^"^pU) = /^"^pCy), hence p(x) = p(y).
This implies x = y if q < n—1. Suppose q = n. Then by Lem-
Lemma 12.2, the homotopy classes [x] and [y] can be taken to be ele-
elements of ;7n(K) = 7rn(K(n)), hence by naturality /*[x] = f*[y\.
Since /* is an isomorphism, this implies [x] = [y], that is, x ~ y.
Then by minimality x = y. Assume the result for q' < q, q > n.
Then 6Lx = dty for all i and since v (K(n)) = 0, x = y follows
from Lemma 12.2.
(ii) Let y t L(?>. We must find x e K(qn) such that
f("Xx) = y. If q < n — 1, the result is obvious. Suppose q = n.
[y] can be taken to be an element of irn(L) = nn(LSn>). Since /*
is an isomorphism, there exists x e Kn such that /#[x] = [y].
Then f^Kx) *j y, hence l^Kx) = y. Now assume the result for
q' < q, q > n. Let f^n\x.) = d.y. By the extension condition,
there exists x e K^ such that dx = xj, i < q. Then
5./(")(x) = d.y, i< q,
49
= 0, it
|| hence by minimality d i^n\x) = d y. Now since 77
$ follows from Lemma 12.2 that /(n)(x) = y.
;Jj , The following theorem is an immediate corollary.
§ ffjEOREM 12.5: Let K and L be connected Kan complexes. Then
|§ {he following conditions are equivalent.
8 i: (i) K and L have the same homotopy type.
% (ii) There exists a weak homotopy equivalence /: K -» L.
if ¦";;'.'• (iii) K and L have isomorphic minimal subcomplexes.
it?
~s? i Now we outline the parallel development for Kan fibrations.
^ I^^MMA 12.6: Let p: E -» B be a minimal fibration, where E, B,
I': laid the fibre F are connected Kan complexes. Then each of the
I laii fibrations p': E(n) -» B and p'\ E(n) ^ E(m> is minimal.
'^ Proof: Let u, v e E represent x, y e E(nh Suppose
1 # dtx = dty, i + k, and p'(x) = p'(y).
% i|e latter implies p(u) = p(v). If g < n+l, 5.x = 5.y implies
J Iff = d/v an<l, by the minimality of p, dku = dkv. If q > n+l,
^ ^:^x = d.dky implies dku = dkv on A[q-l]n, hence
1 DEFINITION 12.7: A map (/,/): (E',p',B')-> (E,p,B) of fibre
ft' Spaces of Kan complexes is called a weak homotopy equivalence if
it induces an isomorphism of the homotopy exact sequence of
¦ (E',p',B') onto that of (E, p, B).
THEOREM 12.8: Let (E', p'B') and (E, p, B) be minimal fibre
spaces of connected Kan complexes with connected fibres. Then a
: weak homotopy equivalence (f, /): (E', p',B') -> (E, p, B) is an
isomorphism.

! i
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In
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¦ I
1
1
{¦
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THEOREM 12.9: Let {E',p',B') and {E,p,B) be fibre spaces of ^ aRL) = C{K)/C{L), so that HjiC(K,L)) = HJiK.L).
connected Kan complexes with connected fibres. Then the follow- ^ = Q> where y denotes the image of y in Cn(K, L), and we
ing are equivalent: define h, ^ L> +) _> H^K> L) by ft(/3) = jyj The mapg ft ar
(i) (?', p', B') and (E, p, B) have the same homotopytype. cglled the Hurewicz homomorphisms.
(ii) There exists a weak homotopy equivalence
(/,/): (E',p',flO - (E,p,B)
(iii) (E',p',B') and (E, p, Z?) have isomorphic minimal
sub-fibre spaces.
§13. The Hurewicz theorems
> LEMMA 13.3. The maps h are well-defined and are homomorphisms
?i ofgroups.
S Proof: We give the proof for the absolute case, that for the
^relative case being similar. Suppose x, y e Kn and z: x ~ y.
%,Then 5(z) = (_l)n(X-y), hence h is well-defined. Suppose next
fthat x, y represent a, /8 e 7^(/f, 0) and w e Kn+1 satisfies
In this section, we obtain some of the classical results com-vd.H' = 4>> 0 < i < n —1, dn _ 1 w = x, and dn + 1w = y, so that
paring homotopy groups with homology groups. Throughout this secfajS ,= [dn w\. d(w) = (-l)n + (x~dnw + y) and therefore
tion, homology means homology with integer coefficients. |' h(aB) = ft(a) + h(B)
PROPOSITION 13.1: Let K be a complex. Then HQ{K) = FinQ(K)X-
,r. ?,PROPOSITION 13.4: The Hurewicz homomorphisms define natural
the free Abelian group generated by rrAK). t*
., \ , . , . ^"transformations of functors. If (K, L, 0) is a Kan triple, then the
Proof: The map KQ -» KQ/\~) of sets induces an epimor- ^
.,,. . . „/„ -> -/ /«w ^.1 i r v '- following is a commutative diagram of exact sequences:
phism CoU0 =Z0(ff) = F(K0)-» F(wo(J0). Clearly if x, y e KQ, ;,.
then x ^< y if and only if there exists a 1-chain w such that ,-...-» n- , .(K, L, 0) -> 77 (L, 0) -> rr (K, 0) -> n (K, L, J>) -> • • •
bXw) = x-y. This implies the result. v, L L , L
From now on, we consider the reduced homology groups '
Hn(K) = Hn(C(K)) of a pair (K, 0), Here C(K) = C(K)/C@). ^
Clearly ffn (K) = #nU0, n > 0, and HQ(K) has one less free gen-.
., „ / r,\ - Proof: The result is clear from the definitions, recalling that
erator than HAK). ' '" . &
zd:.Hn+ t(K, L) - tfn(L) is defined by ^iy I = {5(y)}, y e Z (K, Q.
DEFINITIONS 13.2: Let (K, L, 0) be a Kan triple. Suppose x re- "
presents a e nJ,K,4>), and consider x as a cycle of C^K). De- n'ThE0REM I3.5 (Poincare): Let (K, 0) be a connected Kan pair,
fine h: rrJiK, 0) -» ffn(/f) by ft (a) = ix}, where ixl denotes the homol(/Then ^e maP h '• ^K> ff>)/\.n^K, 0), tt^K, 0)] -* H^K) induced by
gy class of x. Similarly, suppose y represents j8 e nn (K, L,j>). .h is an isomorphism.
H
1CK, L)
fl_
f, L)

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53
ill
Proof: We may assume K = E^K.fi). Then |§
CAK) = ZAK) = FUC.-<f>). §
Let j: ZA^K) -> nA.K)/[nA.K), nA.K)] be the natural epimorphism ff
induced from K, -> K, /(~). If x e CO(K), then §
11 2 |
<Xx) = dQx —djX+d^x ||
and by definition of the group structure in nAK), jd(x) = 0. ThusjS
j induces ;': HAH) -> nAlO/WAK), nAK)]. Clearly ;'oft'= 1 1
fi
and ft 'o; ' = 1, and this completes the proof. H
THEOREM 13.6(Hurewicz): Let (K, <f>) be an (n - l>connected Kai(|
pair, n > 2. Then fff(K) = 0, 0 < i < n-1, and h: ttJJO=EJ^
Proof: We may assume K = E (K, <?). Then K has only one sim- §
plex in each dimension < n, hence H .(K) = 0, 0 < i < n —1, andll
C (K) = Z (K) = F(K -d>). K -> K /(^) induces a natural ep§f
morphism ;: Z (/f) -> nn(K). As in the proof above, it suffices toys
prove j d (x) = 0, x e K +1. If n = 2, this follows immediately |K
from step (iii) of the proof of Proposition 4.4 on page 11; if n > 2,/ff
a tedious but essentially similar computation gives the result. ll
COROLLARY 13.7: If K is a 1-connected Kan complex and |j
H.(K) = 0 for all i > 0, I
then K is contractible. ¦-§.
Similarly, we can prove
THEOREM 13.8: Let {K, L,4>) be a Kan triple, where K and L are
1-connected Kan complexes. Then {K,L,^>) is (n — l)-connected if
and only if H .{K, L) = 0, 0 < i < n— 1, and in that case
h: nn(K,L,<f>) - Hn{K, L)
is an isomorphism.
; The hypothesis that/is an inclusion can be eliminated in the
following result, but some technical work, or use of geometric realiza-
realization, is required since the natural simplicial mapping cylinder of a
map of Kan complexes need not itself be a Kan complex.
Theorem 13.9 (Whitehead): Let/: K —>L be an inclusion of 1-con-
1-connected Kan complexes and let n > 2 be an integer. Then the fol-
following are equivalent:
(i) /*: n^K, (f>) -» TTjiL, /@)) is an isomorphism for
i < n and an epimorphism for i = n.
(ii) /*: H .00 -» H^L) is an isomorphism for i < n and
an epimorphism for i - n.
Proof: Since we have assumed that / is an inclusion, this
follows immediately from the theorem above.

54
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
i !i;i i:
Bibliographical Notes on Chapter II
The concept of simplicial fibre space is due to Kan [25],
[30]. Complete proofs of Theorems 7.8, 7.13 and 7.16 may be found
in Cartan [6], in Gugenheim [18], in Kan [33], and in MacLane [41].
The original definition of Postnikov systems is of course
that of Postnikov [54]. An alternative semi-simplicial treatment is
that of Heller [21]. We have followed Moore [52],[53]. The fibre
spaces {K, p, K^n') are the simplicial analogs of "construction II"
of Cartan and Serre [4]. The Eilenberg subcomplexes of the total
singular complex of a space were defined by Eilenberg in [10].
Eilenberg-MacLane complexes were introduced and studied in the
series of papers [13].
The construction of the minimal subcomplex of the total sin-
singular complex of a space is due to Eilenberg and Zilber [15]. Our
treatment is essentially that of Moore [52], [53]. The theorem that
every minimal fibration is a Kan fibre bundle is due to Barratt,
Gugenheim, and Moore [l]. The equivalence of weak homotopy type
and homotopy type for Kan complexes is of course the analog of a
theorem due to Whitehead [62] on CW-complexes.
CHAPTER III
GEOMETRIC REALIZATION
§14, The realization
Throughout this chapter, A will denote the topological
n-simPlex, An = KtQ,..., tn)\ 0 < t. < 1, X t{ = 1} C R"+1. We de-
deA A
fine maps 8.: An_j -> An and a.: . Afl+1
An by:
<Ti(t0'-'tn+l) = (t0>->ti + ti+l>-
A point of An is called interior if n = 0 or if 0 < t. < 1 for all i.
Note that every un e An can be uniquely expressed in the form
u = 8. ...8j un_q> where un_ e ^n is an interior point and
0 < it < •¦• < iq < n.
Now let K be a complex. Give K the discrete topology
and form the disjoint union K = Un>Q(KnxA ). Define an equiv-
equivl lt in K by:
p alferice relation
and
(si*n'Bn+l)
kn (Kn> Un-1
kn < Kn>
n-1>
is called the geometric
The identification space T(K) = g
realization of K. The class of (kn, ufl) in T(K) will be denoted by
\kn, u \. If (: K -* L is a simplicial map, then / induces the con-
55

5"
5.
S
S,
ilt IIJ i
I
t
f
M*|M'i;;iJ:ii
54
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
Bibliographical Notes on Chapter ii
The concept of simplicial fibre space is due to Kan [25],
[30]. Complete proofs of Theorems 7.8, 7.13 and 7.16 may be found
in Cartan [6], in Gugenheim [18], in Kan [33], and in MacLane [41].
The original definition of Postnikov systems is of course
that of Postnikov [54]. An alternative semi-simplicial treatment is
that of Heller [21]. We have followed Moore [52],[53]. The fibre
spaces {K, p, K^n') are the simplicial analogs of "construction II"
of Cartan and Serre [4]. The Eilenberg subcomplexes of the total
singular complex of a space were defined by Eilenberg in [10].
Eilenberg-MacLane complexes were introduced and studied in the
series of papers [13].
The construction of the minimal subcomplex of the total sin-
singular complex of a space is due to Eilenberg and Zilber [15]. Our
treatment is essentially that of Moore [52], [53]. The theorem that
every minimal fibration is a Kan fibre bundle is due to Barratt,
Gugenheim, and Moore [l]. The equivalence of weak homotopy type
and homotopy type for Kan complexes is of course the analog of a
theorem due to Whitehead [62] on CW-complexes.
f CHAPTER III
| GEOMETRIC REALIZATION
ft §14, The realization
| s| Throughout this chapter, An will denote the topological
I |pmplex, An = \(tQ,..., g| 0<f.<l,S<. = l)c Rn+1. We de-
I f|f maps S.: An _ % -> An and a:. An+1 -> An by:
A point of An is called interior if n = 0 or if 0 < t. < 1 for all /.
Note that every un e An can be uniquely expressed in the form
0" = 8j ... 8. un „, where un__ ( ^n_9 *s an interior point and
0 < i1 <¦¦¦ < i < n.
Now let K be a complex. Give K the discrete topology
and form the disjoint union K
alence relation ~ in K by:
and
U ^n(K xA). Define an equiv-
Kn> Un-1
n-1'
kneKn> Un+1 f
The identification space T(K) = /(/(») is called the geometric
realization of K. The class of (k , u ) in T(K) will be denoted by
\k \ If /: K -> L is a simplicial map, then / induces the con-
55
\kn, u

56
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
GEOMETRIC REALIZATION
57
tinuous map T(/): T(K) -» T(L) defined by *
r(/)|*n,un| - |/(*n),uJ. ']
It is clear that T thus defined is a covariant functor from the cat- fi
*l
egory of simplicial sets to that of topological spaces. 4*
We will prove that T{K) is actually a CW-complex. We re- J)
call the definition. A cell complex X is a Hausdorff space which §
is the disjoint union of open cells en subject to the requirement J
that for each cell en, if en denotes the closure of en and A V
the boundary of A , then there exists a map /: A -» en such *
that /I (A - A ) is a homeomorphism onto e" and /(A ) is con- "
tained in the union of the cells of dimension less than n. A sub- v
complex Y of X is a union of cells of X such that e" C Y im- \
plies en C Y. A cell complex is closure finite if each e"" is con-,,
tained in a finite subcomplex and has the weak topology if a subset^
is closed provided its intersection with each e~n is closed. A clo-^
¦4
sure finite cell complex with the weak topology is called a CW- V(»
complex.
THEOREM 14.1: T{K) is a CW-complex having one n-cell for each '
non-degenerate n-simplex of K. ]
A point (A: , u ) of K is said to be non-degenerate if
k 1 K is non-degenerate and u e A is interior. The theorem
n n n n
will follow immediately from the following lemma.
LEMMA 14.2: Every point {k , u ) e K is equivalent to a unique
non-degenerate point.
Proof: By formula C) on page 4, every k e K can be
uniquely expressed in the form k - s. ...s. k , where
H y r " Jp ; 1 n-p'
k e K is non-deeenerate and 0 < j', <¦••<»' < n. The in-
n—p n—p ° — ' 1 Jp
dices j which occur are precisely those such that k e s.K , .
r J n ] n—1
Define A: K -¦ K by
kn-p non-degenerate, 0 < j1
Similarly, define p: K -» K by
un-q interior,
0
The composition A°p carries each point into an equivalent non-
degenerate point. Uniqueness follows since x » x' implies
Next we prove that, under mild hypotheses,
T(KxD = TUQx T(L) .
Let tt: KxL -» K and 77': KxL ^ L be the projections and de-
define r?: ZtKxL) -* T(K)x HL) by v = Tin) x Ktt').
THEOREM 14.3: 77: T(KxL) -» T(K)xT(L) is one-to-one and on-
onto. If T(K)xT(L) is a CW-complex, then 77 is a homeomorphism.
Proof: If z e T(KxL) has non-degenerate representative
(Jt x^n>"'n)> then T(n){z) - \k , wn\ has non-degenerate represen-
representative \{kn,wn) and T(n')(z) = |?n, wj hfes non-degenerate repre-
representative A(? , w ), where A is as defined in the proof of Lemma
14.2. We define an inverse function ^ T(K)xT(L) -> T(KxL) as
follows. Let (x, y) e T{K) x T(L), where x and y have non-degen-
non-degenerate representatives (kg,ug) and (?fc, vfc). If ufl = UQ,...,tg) and
= Un,...,^), define
m
= 2
i=0
t.
and
= 2 t:
Let r <•••< rc = 1 be the sequence obtained by arranging the dis-
distinct elements of (uml U {vn\ in order of magnitude, and define

#¦
||
lll
58
t"
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
- r. ,, 0 < i < c, r = 0. Clearly 0 < f/' < 1 and
GEOMETRIC REALIZATION
59
f/' = rc =
hence wc = {t'Q',...,<c") is an interior point of Ac. Let '1<--<Jc_a
be those integers i such that r, / |o»l and let /><-</,_, be
those integers j such that r. / {v"|. Then {ij and |ypj are dis- ,
joint, ua = af ...oi wj and *b = "j^-Oj _b"c- Define
Clearly
1"c
and
_
= y, so that 77°^ =1. Taking 2 as above, |
§15 Adjoint functors
In this section» we define adjoint functors and develop a
few of their properties.
DEFINITION 15.1: Let T: fi . S and S: S - fl be covariant
functors. We say that S and T are adjoint, or that T is an ad-
fint of S and S a coadJ°int of T, if there exists a natural equiva-
jehce of functors 0: HomgD, S(B)) - Homcg (T(A),B) (where
lach side is considered as a functor Q* x S -> C, C the category
of sets).
We notice first the following
: S "* ^ and T: a "* S be covariant functors.
_ |(t x
- |U,x
= 2
Finally, we observe that 77 is continuous on each product cell of |
T(K) x T(L) and, if T(K) x T(L) is a CW-complex, this implies j
that 77 is continuous, hence that 77 is a homeomorphism. I
REMARK 14.4: The proof above parallels that for simplicial com- I
plexes given in [14, p. §8]. The hypothesis that T(K) x T{L) is a I
CW-complex holds if K and L, are both countable [47, p. 2721 or if f
either T(K) or T(L) is locally finite (i.e., every point is an inner f
point of a finite subcomplex) [62, p. 227]. I
COROLLARY 14.5: A simplicial homotopy F: Kxl -» L induces i
a topological homotopy TtF) ° jj: T(K)xT(l) -> T{L). I
COROLLARY 14.6: If K is a countable simplicial monoid, group, 3
or Abelian group, then T(K) is a topological monoid, group, or Abet;
ian group. \
0) The correspondence between natural transformations
Jetting cf> correspond to $ if $(fi) = ,?A )
for / e HomQ (A, S(fi)) is one-to-one.
(ii) The correspondence between natural transformations
xf/i. Honug (T (A), B) ¦* HomQ (A, S(B)) and ?: 1q -> ST obtained
by letting if/ correspond to V! if ?D) = 1//^t(A') and
^r(^) = S(g) o WA)
for g e Homtg (T(A), B) is one-to-one.
PROPOSITION 15.3: Let 0: Homg {A, S(B)) -¦ Homcg (T(A), B)
correspond to $: TS ¦* l<g. Let
iff. Hom<ft(T(A), B) - HomQ(A,S(B))
correspond to .$: Iff -* ST. Then
(i) ifr o 0 = 1 if and only if S > STS —
identity natural transformation of S into S.
S is the

60
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
GEOMETRIC REALIZATION
61
II
(ii) 0 o xfj = 1 if and only if T J-t—» TSr ^— 2* is
the identity natural transformation of T into T.
Proof: We prove (i), the proof of (ii) being similar. Let
/ ( HomQ (A, S(B)). The following diagram is commutative:
iff
It follows immediately that S$(B) ° <PS(B) = ls(fl) implies
if/ ° <?(/) = / and, taking / = 1S(B\, that if/ ° <?(/) = / implies
THEOREM 15.4: Let S, S': % - Q and T, T': Q -* S be co-
variant functors. Let 0: Hom(j(i4, S(B)) ~» Hom<g(T(i4), B) be a
natural equivalence with inverse i// and let
0': KomQ(A,S'(B)) -> Honvg (T'U), B)
be a natural equivalence with inverse i//'. Suppose that r. T -* T
(resp. a: S -* S') is a natural transformation. Then there exists a
unique natural transformation a'. S -» S'(resp. r: T'•* T) such
that the following is a commutative diagram:
Hom<g(r04), B)?
Homtg(rU),B)
Homg (A, a (B))
Homtg (T \A), B)
Homg (A, S '(B))
If t (resp. a) is a natural equivalence, then so is a (resp. r).
Proof: Given / e Homg (-4, S(B)), the commutativity of the
diagram (*) is seen to imply a(B) ° f = t//'(cf>tf) o t(A)). In particu-
particular, taking A = S(B) and / = lS/fi->, we must have
(i) a(B) = ifr'(<f>(lSiB)) ° tS(B)) = ^'(«(B) o TS(B)).
Thus given t, a must be defined by (i). It is easily verified that a
so defined is a natural transformation. Similarly, for
g e Hom<g(rU),fl),
the commutativity of the diagram (*) implies
gor(A) = tf>'(a(B)o^(g)).
Taking B = T(A) and g = 1 _,. >, we must have
(ii) rU) = 0'(
Given a, r must be defined by (ii) and naturality is easily verified.
The last statement follows from the uniqueness: if r has inverse
r~ , then if a' is induced from r~ , aa'and a'a are induced from
zr~ and t t, hence aa' and a'a are both indentities.
Note that the previous theorem implies that any two adjoints
of a functor S are naturally equivalent and any two coadjoints of a
functor T are naturally equivalent.
§16. Comparison of simplicial sets and topological spaces
Let § denote the category of simplicial sets and 3* that of
topological spaces. We claim that the realization functor T: § -> J
is adjoint to the total singular complex functor S: J -» S. Thus de-
define <f>: Hom^(K,S(X)) -» Homy(T(K),X) and
if,: Homy (.T(K),X) -» Hom§(K,S(X))
by

62
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
GEOMETRIC REALIZATION
63
ill1'
A) 0(/)|*n,un| = f(kn)(an);
B) *<4)(kn)(un) = *|*n,un|.
It is easily verified that <? ° t// = 1 and ^r ° 0 = 1. The corre-
corresponding natural transformations <E>: TS -¦ 1<t and 'P: lc -» ST
are given by
C) •(*)!*„, un| = *„(«,„);
D) ^(K)Un)(un) = |*n,uJ.
Note that *PUO: K ¦* STiK) is an inclusion of complexes and
$(X): TS(X) -» X is a surjection of spaces. By Proposition 15.3,
we find
E) (S$ o VS)U): S(X) -> S(X) is the identity map;
F) (*roTV)(/f): T(K)^T{K) is the identity map.
We claim further that 0. and iff preserve homotopies. If
fc^g: K ^ S(X) and if F: Kxl -* S(X) defines the homotopy, then
by Corollary 14.5, T(F): T(f) <^T(g). Since 0(F) = $SU) o TiF),
we find 0(F): 0(f) ca <f,{g). Similarly, if / c* g: T(K) - X and
F: T(K)x Til) -> X defines the homotopy, then S(F): S(/) ^ S(g).
Since ^(F) = S(F) o ViK), we find ^(F): ifiil) ^ifiig). Summa-
Summarizing, we have the
THEOREM 16.1: The natural equivalences <f> and i/r define S ¦
and T as adjoint functors. 0 and i/r induce one-to-one corre-
correspondences between the homotopy classes of simplicial maps
K -> SiX) and of continuous maps T(K) -» X. In particular,
PROPOSITION 16.2: Let K be a complex, X a space. Then
(i) ?(K)*: ff*(K) - H*(ST(K)) = ff*(J(K)) is an iso-
isomorphism;
(ii) $00*: H*(TS(X)) -» ff*U) is an isomorphism.
Proo/: (i) Let D(K) C C(/f) denote the chain subcomplex
generated by all the degenerate simplices of K. We will prove
later that CN(K) = C{K)/D{K) is chain homotopic to C(K). If K(n) de-
denotes the n-skeleton of K, then the inclusions kS0^ C /f^1) C---
induce a filtration of CN (K). The resulting spectral sequence
\ET\ converges to H*(K) and satisfies E^ q= Hp+^J^,K^X>).
Clearly E1 = 0, q 4 0, and E* o is the free Abelian group gen-
generated by the non-degenerate p-simplices of K. Therefore
Ez - E°°
P.o ~ cp,o = >m
Again, renew) . root-), ^ the inclusions
sr(jc(°>) c src^1)) c -
give rise to a spectral sequence |F} which converges to
H*(T(K)) and satisfies E^. ^rO^)), rQtCP-l))) Now
the homology of a CW-complex can be computed from the cellular
structure, that is, from the subcomplex of the total singular com-
complex generated by the attaching maps, and it follows that the map
E, -+ E induced by *<*) is an isomorphism. Therefore the
spectral sequences are mapped isomorphically, and
H(K) = E2 -
P p, o
Ep,o
Now we consider the maps of the homology and homotopy
groups induced by the natural transformations <E> and 1P.
as was to be proven.
(") !UTSW) - H*(STS(X)) and HM = H,{SX). By
OX VS(X)*: H.tSX) .- H4STSW) is an isomorphism. Since
(SO oos): S(X) , S(X) is the identity, the result follows

s
s.
5
S
5
5
5
64
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
GEOMETRIC REALIZATION
;,-.¦¦
-i.t'
ill
LEMMA 16.3: Let (K, 0) be a Kan pair. Then
?(K)*: ^(#,0) -» n^STUCSUcf,)) - i
is an isomorphism. :Kg
Proof: We may assume that K is connected and minimal. §j|
Then n^K, 0) is a group with one generator for each non-degener- |§
ate 1-simplex and one relation for each non-degenerate 2-simplex. |1
T(K) is a CW-complex with one 0-cell and therefore tt^TUQ, r@)) f|
is a group with one generator for each 1-cell and one relation for :|
each 2-cell. The result follows. i
T:t
DEFINITIONS 16.4: Let K be a Kan complex having just one ver- ;|
tex 0, and let n = n^{K, 0). Define a complex K by Kn = K x n if
with face and degeneracy operators defined on K by s|
(i) d.(x, a) = (d.x,a), i < n, j|
dn{x,a) = Wnx, t^-1 x]-1^, where 6^~x denotes if
dn iterated n —1 times, and §
(ii) s.U,a) = (s.x,a). /|
It is easily verified that K is a Kan complex and that the projec- Sf
tion p: K -> K is a Kan fibration. The fibre F over 0 satisfies f
nn(F, 0) = 0, n > 0, and jtq(F, <f>) = n, where 0 = @, e), e the f
identity of n. If x represents a e n-^K, 0), then choosing (x, /3) |
such that djCx, /3) = 0, that is, such that [x]^ = e, we find that %
d(a) = [dQ(x, 0I = [@, a)] f rrQ(F, 0). Therefore
d: TTjCK, 0) - no{F, 0)
is an isomorphism. By the homotopy exact sequence, K is
1-connected and satisfies rrn{K, 0) = nn(K> <fi> n > I. K is
called the universal covering complex of K.
65
REMARKS 16.5: Recall that a map p: X -» Y of topological
spaces is called a Serre fibration if p has the covering homotopy
property for polyhedra and that this is true if and only if p has
this property for each An- It is easily seen that p is a Serre fibra-
fibration if and only if S (p) is a Kan fibration. Further, if p: K -> L
is a Kan fibration, then it follows from Corollary 7.12 that T(p) is
a quasi fibration; J. C. Moore (unpublished) has proven that if p is
minimal, then T(p) is a Serre fibration.
THEOREM 16.6: Let (K, 0) be a Kan pair and let X be a topolog-
topological space with base point xQ. Then
(i) TO)*: nn(K,<f>) -» nJLSTilO, STty)) is an isomorphism
for all n, hence ^(K) is a homotopy equivalence,
(ii) $U)*: Trn(TS(X), TS(xQ)) -* nn(X, xQ) is an isomor-
isomorphism for all n, hence $(X) is a homotopy equiva-
equivalence if X is a CW-complex.
Proof: (i) We may assume that K is connected and minimal.
Let K be the universal covering complex of K, p: K -> K the pro-
projection, and F the fibre over cf>. Observe that p is minimal. By
Lemma 16.3, Proposition 16.2, and Theorem 13.9
=
and trivially *P(F)*: nn(F,i) S rrn(ST(F), ST@)). By naturality,
? induces a map of the homotopy exact sequence of the Kan fibra-
fibration p to that of the Kan fibration ST(p). and the result follows,
(ii) This follows from (i) and the fact that
(S$°?S)U): S(X) ^ S(X)
is the identity.
Finally, we extend the definition of homotopy groups to the
category of simplicial sets.

66
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
DEFINITION 16.7: Let (K, 0) be a pair. Then we define
rrn(K,4>) = na(ST(K),ST(<f>)),
and similarly for the relative groups.
mm-
lii
J;#.;-:---v;':i
, %
'\
yiih
It
Bibliographical Notes on Chapter III
The construction of the geometrical realization and the deri-
derivation of its properties presented here is due to Milnor [48].
The concept of adjoint functor and the results of section 15
are due to Kan [26]. In [28], Kan has developed a procedure for as-
associating to a complex K a Kan complex E°^K in such a manner
that T(E°°K) has the same homotopy and homology groups as T(K).
This gives an alternative, and self-contained, procedure for extend-
extending the definition of homotopy groups to the category of simplicial
sets. Another possible definition, also due to Kan, will be present-
presented in Chapter VI.
The CW-hypothesis in Theorem 14.3 and the countability
hypothesis in Corollary 14.6 can be eliminated by working in the
category of compactly generated spaces.
CHAPTER IV
^y TWISTED CARTESIAN PRODUCTS
,'n AND FIBRE BUNDLES
H( The concept of twisted Cartesian product will dominate the
Immaterial to be presented in the next three chapters. We will intro-
SHuce this notion after obtaining certain special properties of sim-
fplicial groups. Then we shall use the notion to study the classifi-
¦ Ration of fibre bundles.
f §17. Simplicial groups.
Wf Recall that a simplicial group, or group complex, G is a
Icontravariant functor from A to the category of groups. Thus
leach Gn is a group and the face and degeneracy operators are
Shomomorphisms. We let en denote the identity of Gn and e the
^complex consisting of all of the en.
THEOREM 17.1: Every group complex G is a Kan complex.
Proof: Suppose x0, ..., xk_x, xk+1,..., x<r+1 t Gq satisfy
d/Xj = c^_! x{, i < j, i, j A k. We must find xeGq+i such that
<9;X = Xj, i / k. We first find ueGq+1 such that dia=xi if
j < k. If k = 0, this condition is vacuous. Assume k > 0. We will
find ur such that (9<ur = x/, i < r. Let u° = sox0; proceeding
inductively, suppose, ur~1 has been defined, 0 < r < k—1. Let
yr-1 = s/C^u11)*,.) and ur = ur~V~1- A simple calculation
67

66
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
II
I
in
m
««¦H>.v •
mim
fi
iSiKs::
DEFINITION 16.7: Let (K, <j>) be a pair. Then we define
nn(K,<j>) = j,a{ST{K),ST{<f>)),
and similarly for the relative groups.
Bibliographical Notes on Chapter ni
The construction of the geometrical realization and the deri-
derivation of its properties presented here is due to Milnor [48].
The concept of adjoint functor and the results of section 15
are due to Kan [26]. In [28], Kan has developed a procedure for as-
associating to a complex K a Kan complex E°°K in such a manner
that T(E°°K) has the same homotopy and homology groups as 7\K).
This gives an alternative, and self-contained, procedure for extend-
extending the definition of homotopy groups to the category of simplicial
sets. Another possible definition, also due to Kan, will be present-
presented in Chapter VI.
The CW-hypothesis in Theorem 14.3 and the countability
hypothesis in Corollary 14.6 can be eliminated by working in the
category of compactly generated spaces.
CHAPTER IV
TWISTED CARTESIAN PRODUCTS
AND FIBRE BUNDLES
The concept of twisted Cartesian product will dominate the
material to be presented in the next three chapters. We will intro-
introduce this notion after obtaining certain special properties of sim-
simplicial groups. Then we shall use the notion to study the classifi-
classification of fibre bundles.
§17. Simplicial groups.
Recall that a simplicial group, or group complex, G is a
contravariant functor from A* to the category of groups. Thus
each Gn is a group and the face and degeneracy operators are
homomorphisms. We let en denote the identity of Gn and e the
complex consisting of all of the en.
THEOREM 17.1: Every group complex G is a Kan complex.
Proof: Suppose x0, ..., xk_lt xk+1,..., xq+1 e Gq satisfy
djXj = dj_1 xf, i < j, i, j A k. We must find x e Gq+1 such that
djX-Xf, i / k. We first find utGq+1 such that <3,-u = x1- if
i <k. If k = 0, this condition is vacuous. Assume k > 0. We will
find ur such that djur = xi, i < r. Let u° - soxo; proceeding
inductively, suppose ur~l has been defined, 0 < r < k-1. Let
yr-1 = s/GJX)^) and ur = ur~Vr~1- A simple calculation
67

ill
68 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
'§&
proves that dtyT : = e if i < r. It follows that d. uT = x. if i < r|
as desired, and we let u = u*"~ . Next we find vr e G j such f|
that d{vT = x. if i < k or if i > q— r+1. Let v = u or v = e
if it = 0. Proceeding inductively, suppose v
0 < r < ,_*+!. Let z^
r-l
9+1
sq_r + 1(«qL_r+2
has been defined,
r~1)-1x )
' Xq-t+2'
and vr = vr-1 z1". Then ^z*" = eq if i < k or if i > q-r+2.."
It follows that d^ = x. if i < k or if i > q—r+\, as desired.
Finally, letting x = vq~k+x, we find d.x = xy, / ^ A. X
If G is a group complex or a Kan monoid complex, we denote jj
77n(G,e) by nn(G). |
PROPOSITION 17.2: Let G be a Kan monoid complex and suppose f
x,ye Gq, that is, c^.x = eq-1 = d(y for all i. Then [x][y] = [xy]. ^
Proof: Let z = s^jX • sqy. Then 61. z = e^j, i < q-1, *|
5_jZ=x, <3z= xy, and d .z = y. The result follows from t
the definition of the group structure. r^
PROPOSITION 17.3: Let G be a Kan monoid complex. Then n (G) f
is Abelian, q > 1. I
Proof: Let w = sqx-sq_iy> x,y e Gq. Then d. w = eq_lr|
i < q-l, dq_lw = y, dqw = xy, and dq+1 w = x. Therefore J.
[y][x] = [xy]. Since [xy] = [x][y] by the previous proposition, the
result follows.
There is a useful alternative definition of the homotopy
groups of a group complex, which we now develop.
PROPOSITION 17.3: Let G be a group complex and define
G = G fiKero1,, n-HKerd , .
q q 0 q-l
Then
(i)
Vl'^,. 9>0.
TWISTED CARTESIAN PRODUCTS AND FIBRE 'BUNDLES
69
(ii) Image
Gq C Ker dq. G
q+1 . Gq C Ker dq. Gq-*Gq_vq> 0.
dq+i(Gq+1) is a normal subgroup of Gq and of G
Proof: Let x e Gq+y Then
= *Jr *q>
0 < i < q, and this proves (i) and (ii). Let z e Gq and define
dw = eq for i < g,
i,i"'= z- ^1 x • z. This proves (iii).
w = sqz-x-sqz
hence w e Gq+1, and
d iw - z'^a+\
Now G is a (not necessarily Abelian) chain complex with
respect to the last face operator. Define n' (G) = H (G). Then
we find
Proposition 17.4: nq{G) = 7r'q(G) for all q.
Proof: Gq = Z G , hence if x e G , x represents an ele-
element of 77'(G). If x, y e G and z: x ~ y, then s x-1-z t G j
and d As x • z) = x~ y, so that x and y represent the same
element of n '(G). Using Proposition 17.2, it follows that there is
a natural epimlrphism of groups i: v (G) -> n'(G). Suppose
i[x] = 0. Then there exists z e G jSuch that d j
x, hence
z: e ~ x and [x] = 0 in ^q(G). This completes the proof.
PROPOSITION 17.5: A group complex G is minimal if and
only if d \ : G j —• G is zero for all q.
Proof: Suppose G is minimal and x t G *. Then
d.x
and therefore d
dq+V Gq+1 -* Gq is zero for a11
eq. Conversely, suppose
let x, y f Gg+1 satisfy
4X = °iy» i ^ *• Oberve that oLlxy) = e , i ? k. If
& = g+ 1, let z = xy. If A < g and g - it is even, let
z =

70
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
[ !
i I'
7 I
r\
ill
mas.
fa-:.
If it < q and q — k is odd, let
In all cases, d. z = e , i < q+ I, and <3q+1z = ^^y" )¦
By assumption, d +1 z = e , so that c^x = dky, as was to be
proven.
The result above states that a group complex G is minimal
if and only if the corresponding chain complex G is minimal.
§18. Principal fibrations and twisted Cartesian products.
We will first define principal fibrations and then twisted
Cartesian products. It will turn out that the former concept is a
special case of the latter.
DEFINITIONS 18.1: A group complex G is said to operate from
the left on a complex E if there exists a simplicial map
(f>: G x E -> E such that i^(e . x) = x and
<j>(g1,<j>(g2,x)) = <f>(g1g2,x) .
We will denote <j>{g, x) by gx. G is said to operate effectively if
gx = x for all x e E implies g = e . G is said to operate
principally if gx = x for any one x e E implies g = eq. Note
that G operates principally on itself from both the left and the
right. If G operates principally from the right on E, then we de-
define a quotient complex B of E by identifying x and xg for all
x e E and g c G . The map p: E -» B is called a principal
fibration with base B and structural group G.
LEMMA 18.2: Any principal fibration is a Kan fibration.
Proof: Let p: E -> B be a principal fibration with group
G. Suppose xo,---'x*-l'x*+l'---'xq+l f Eq satisfy
I
I
*
TWISTED CARTESIAN PRODUCTS AND FIBER BUNDLES 71
diXj = dj-\xi> » <h hi t k,
and b e Bq+1 satisfies d.6 = p(x;), i 4 k. Then if
p(y) = b, y e Eq+1, d.y = x.g. for some g. e Gq, i 4 k.
If i < ']> h i ? k, we have
Since d(x. = d._1x. and G acts principally, d.g. = a_xgj
and therefore there exists g e G +1 such that d.g = g., i ^ it.
Let x = yg~x. Then d.x = x., i 4 k, and p(x) = 6, as de-
desired.
DEFINITIONS 18.3: Let F and B be complexes and G a group
complex which operates on F from the left. A twisted Cartesian
product, or TCP, with fibre F, base B, and group G is a com-
complex, denoted by either F xf B or ?(r), which satisfies
E*/ \ J? \s H
n n n
and has face and degeneracy operators
(i) A (/,?>) = FU, d. b), i > 0;
(ii) <90 (/, 6) = (rU) • <90 /, <90 b), T{b) e G;
(iii) s. (f,b) = (s^/, S/6), i > 0;
r: 5^ -* Gq_1 is called the twisting function. The requirement
that ?(r) be a complex is equivalent to the following identities on
r:
(T)
0
0
r(sQb) = 6q, b e Bq.
If F = G, then E(t) is called a principal T C P, or PJ C P.
The term TCP will also stand for the projection p: ?(r) -» B.

72
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
ii
l
PROPOSITION 18.4: Let p:?(r)-»B bea TCP with group G
and suppose the fibre F is a Kan complex. Then:
(i) p is a Kan fibration;
(ii) p is minimal if and only if F is a minimal complex;
(iii) if F = G, p is a principal fibration.
Proof: (i) Suppose x0,..., xk_v xk + l,..., xq+1 e E{r)q
satisfy d.x. = d._ix., i < j, i, j 4 k, and b ( B j satisfies
dtb = p(xp, i ? k. If xt = (fifb.), we have d.b = b., i + k.
If it = 0, then the /. are clearly compatible, and we may choose
/ e Fq+l such that d.f = f., i 4 0. If k > 0, then it is easily
verified that [r Cb)~" *] •/0 and the {., i > 0, i ? k, are com-
compatible, and we may choose f e F ^ such that dQf = [r(b)~ ] • fQ
and d. f = /., i > 0, i'4 k. In both cases x = (f, b) satisfies
d.x = Xj, i ^ k, and p(x) = 6, as-desired.
(ii) If cj> is any vertex of B, then / -» (/, <f>) defines an
isomorphism of F with the fibre over <f>. Therefore if p is mini-
minimal, F is a minimal complex by Lemma 10.2. Conversely, sup-
suppose F is minimal and let x = (/, b) and x'= (/',?»') satisfy
d.x = djX', i ? k, and p(x) = p(x'). Then b = b' and
dJ = d.f if i' ? k, hence dkf = b\i\ It follows that
dkx = dkx',
as desired.
(iii) Since F = G, G operates on the right of E{r) via
(?i> b)?2 - (iii?'^' ^ear^y & may be identified with the quo-
quotient complex of E(t) obtained by identifying x and xg, and the
result follows.
Now we wish to obtain a converse to (iii) of the proposition.
We need a preliminary result.
I
TWISTED CARTESIAN PRODUCTS AND FIBER BUNDLES 73
DEFINITION 18.5: Let p: E -> B be a simplicial map. A pseudo-
>i cross section of p is a function a: B -> E satisfying pa = 1B,
Act = ad. if i > 0, and s.a = asf if f > 0. If dQa - adQ,
then a is called a cross section. Observe that any PT C P has
| the canonical pseudo-cross section a defined by o(b) = (e , b),
Wb t B , and that a and the twisting function r are related by
fdQa(b) = a(dQb)-r(b).
I LEMMA 18.6: Let p: E -* B be a Kan fibration, where p is onto.
Let B'C B and suppose a': B' -> p~ (B') is a pseudo-cross
| section (B' may be empty). Then there exists at least one pseudo-
f cross section a: B -* E such that o\B' = a\
| Proof: If b e BQ, let a(b) be any element x t EQ such
? that p(x) = b, choosing o'(b) if b e Bq. Now suppose a has
I been defined on each B . q < n, and n > 0. Let b e 2?n. If
6 ? B^, define ctF) = a'(b). Assume b / B\ If b is degener-
degenerate, say b = sfy, define ct(?») = 5^G). If 6 = s z is also true,
then it is easily verified that s.a(y) = s.a(z). Finally, suppose b
is non-degenerate. There exists x e En such that p(x) = b and
Mx = a(dtb) for i > 0. Let a(b) = x. Clearly pa(b) = x and
ld.a(b) = a(d.b), i > 0, and this procedure does define a pseudo-
cross section.
Together with the preceding lemma and Lemma 18.2, the fol-
following proposition implies that every principal fibration is a PT C P.
PROPOSITION 18.7: A principal fibration p: E -» B with group G
and pseudo-cross section a may be identified with the PT C P
p: E(t) -> B with group G and twisting function determined by the
formula dQa{b) = a(dQb)r{b).
Proof: We note first that pa(dnb) = dnb --

B-P
Ml
i
if A
74 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
implies dQo{b) = cr(do6)rF) for some r(b) e G. Now define a
bijection of graded sets iff. GxB -> ? by ifr(g,b) = a(b)g and
give GxB the structure of complex induced by i/T" . Then it is
easily verified that i//~HE) = G xf B, as desired. Note that
if/a = a, where a on the left is the canonical pseudo-cross section
of ?(r).
REMARKS 18.8: Clearly a principal fibration (or fibre bundle) in
the category of topological spaces passes via the functor S to a
principal fibration in the category of simplicial sets. This gives
one motivation for the study of PT C P's.
§19. The group of a fibre bundle. >
In this section, we show how to associate a group to a :
fibre bundle. The theory here is roughly analogous to that of co-
coordinate transformations in the topological case. As an easy con-
consequence of this theory, we shall find that every fibre bundle may
be regarded as a TCP.
We need some preliminary observations about the operations
of groups on complexes. In Lemma 6.12, we showed that K is a
monoid complex if K is a complex. The product is defined by
Ug)(x,u) =/(g(x, u), u), where i, g e {KK)n, x ( Kq, and u e A[n]q
If ? is another complex and / e EK, then the same formula defines
an operation from the right of KK on EK. If g e (KK)n, then G
determines g: Kx A[n] -» K x A[n], where g{x, u) = (g(x, u), u);
g is an isomorphism if and only if g is invertible. We let
A(K) C Kk denote the set of invertible elements. A(K) is a group
complex. Observe that KK and A(K) operate from the left on K
via f-x = /(x, Afi), x e Kn, i e (KK)n. If a group complex G
operates on K, we define a homomorphism v: G -> A(K) by
TWISTED CARTESIAN PRODUCTS AND FIBER BUNDLES 75
v(|)(x, u) =
, where g f Gfi, x e
is that simplicial operator such that u = ?A . Clearly
and ?
,r-t g . x =
y5 is a monomorphism if and only if G operates effectively. Replac-
Replacing G by G/Ker v, we may assume that operations by groups are
j|§|§§§^ys effective.
^ Now let p: ? -> B be a fibre bundle with fibre F (Defini-
41-8). For each b e Bn, choose a strong isomorphism
:1:
J ||f
|f
a(b): F x A[n]
||||||||$} is called an atlas of p. If for each b e Bn we are given
1 **'* ( ^(F)n, then 5F) = a F) ?F) defines another atlas {5F)}.
l^lijiyersely, if {aF)I and {5F)} are atlases of p, then letting
|N
%
, we have 5F) = a(b)g(b).
Itr Given an atlas {a F)}, let fi(b) = So aF): FxA[n] -» ?,
j||ere S: ?F -* ? covers 6 : A[n] -* B, be Bfi. Since
HI aF)(/,u) = @F)(/, v), v) ,
||J6)} and {/SF)} determine each other. We regard {?F)} to be
^i|o|itained in ? , and then face and degeneracy operators and oper-
from the right by elements of A(F) are defined on elements
I, although {|8F)} need not be closed under these opera-
. If {5F} and {aF)} are atlases related by 5F) = a(b)g(b),
giln jS(fc) = p(b)g(b).
II' The atlas {aF)J is said to be normalized if
f; i8(s.6) - s.pib)
|or all i and 6. If we choose an a(b) for each non-degenerate 6
|i(d define j8(s.6) = s./3(b) for all f and 6, we obtain a normaliz-
||||d?atlas. The definition is consistent since if s.b = s.b', i < j,
*c|hen, using induction on the dimension of 6,

76
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
s.p(b') = s.pis.djb) = s.Sip(d.b) = s.pis^^.b) = s.pib).
From now on, we assume all atlases to be normalized.
The difference between d.p(b) and pid.b) is more signi-
significant, and we investigate this now. Recall the definition (page H]
of St: A[n-1] -* A[n]. If b e Bn, then since
a(A)(lxS.): Fx A[n-1] -* E^ :
is an isomorphism onto Image (lxS^), lxS;: E ' -> E , there
exists a strong isomorphism a. (A): Fx A[n-l] -> E ' such
that the following diagram commutes:
p Ar -, a(A)
F x A[nJ
By definition, Hjb o a.(A) = ^.^(A). Define ry(A) e A(F)n_1 by
O] a/A): F x A[n-l] -» F x A[n-1]. Since
))r7(A), dtp{b) = p{d.b)T.(b). {r.(b)\ is called
the set of transformation elements associated with the atlas [a{b)\,
a.F) =
DEFINITIONS 19.1: Let p: ? -* B be a fibre bundle with fibre F
and let G be a subgroup complex of A(F). An atlas {aF)} of p
is called a G-atlas if all of its transformation elements are in G.
Two G-atlases {a(b)\ and {5F)} are said to be G-equivalent if
for all b ( B, a{b) = a(b)g(b), where g(b) ( G. p: E -* B
together with a G-equivalence class of G-atlases is called a G-
bundle. Observe that p is necessarily an A(F) -bundle. Let
p: E -* B and p': E' -» B' be G-bundles with the same fibre F.
A map (/,/): {E, p, B) -> (JE',p',B') is said to be a map of
TWISTED CARTESIAN PRODUCTS AND FIBER BUNDLES 77
G-bundles if for all b e B and any G-atlases \a{b)\ and {a'(?>')}
in the given G-equivalence classes of atlases,
hp(b) = p'(Hb))j{b),
where if/(b) t G. There results a concept of G-equivalence of
- G-bundles. The concept of A(F)-equivalence is identical with
that of isomorphism of fibre bundles with a given fibre.
; An atlas \a(b)\ of a fibre bundle p: E -> B is said to be
' regular if j\F) = en_1 for all i > 0 and b e Bn, n > 1; this
means that d.fiib) - fiid.b), i > 0.
LEMMA 19.2: Let p: E -> B be a fibre bundle with fibre F and
let G be a subgroup complex of A(F). Then there is a regular
G-atlas in every G-equivalence class of G-atlases of p.
Proof: Let \a(b)\ be a G-atlas of p. We will define a
G-equivalent regular G-atlas {a(b)\. If b e BQ, let a(b) = a(b).
Suppose a (b) has been defined for b e B _., n > 1. Let
b e B . If b is degenerate, a(?>) is already defined and has the
desired property. Suppose b is non-degenerate and
a(d.b) - dd^g., i > 0.
If 0 < i < ;, then an application of the induction hypothesis gives
(i) at ap{b) = a.p(d. b) ¦ d.r.(b) = atpid.b) ¦ o.g.)~1 ¦ d. t.(b)
.)-1 ¦ d.r.(b)
Clearly G operates principally on the G-submodule of E gener-
generated by {p(b)\ and therefore d.{g-x rf(b)) = d^ig^r.ib)).
Choose g e Gn such that d{g = g~1Ti(b) for f > 0 and let
a(b) = aitig-1. Then:

m
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
78
B) ^^
This completes the proof.
If G' C G is a sub-group complex and there exists a
G-atlas in the given G-equivalence class of G-atlasesof a G-bundle,
then we say that the group of the bundle can be reduced to G'.
Note that the resulting G -bundle need not be uniquely determined.
LEMMA 19.3: Let p: E -» B be a G-bundle. Suppose G'CG is
a subgroup complex which is a deformation retract of G. Then the
group of p can be reduced to G'.
Proof: Let {a(b)} be a regular G-atlas in the given G-equi-
G-equivalence class of G-atlases of p. We will define a G-equivalent
regular G'-atlas {5FI. Let aib) = a(b) if b t BQ and suppose
5(fc) has been defined for b e B , , n > 1. Let b t B . If b
is degenerate, a(b) is already defined and has the desired proper-
properties. Suppose b is nondegenerate and 5F16) = aidjb)/*., i > 0.
Arguing as in A) of the previous lemma, we find that if 0 < i < j,
then
A) d.d./3(b) = d._ldip{b)^i_li){d.i^X .
Similarly we find, for 0 < j < n,
B) dQd^{b)
- ^o mvd^ ro{b)Yl d._, gor?;_ x{d0 b)]-1 ro(a b)id0 g.)-1
It follows that d;g. = d._1gt, 0 < i < j, and
C) F._1E06)-1F0E.6) = («9;._1g0)-1<9._1r0F).a0g., 0 < j < n.
Choose g e Gfi such that dtg = g{, i > 0, and let a'(b) = a(b)g .
Then d. P '(b) = J3 id. b), "i > 0, and
j|g|revious lemma, we may replace fi"{b) by fi(b) such that
"fete) = en , for i>0and tD) ( G' ..
TWISTED CARTESIAN PRODUCTS AND FIBER BUNDLES 79
Let u = gQ1 rQ(b) dQ g and observe that by C) and by
dogj = dj_1 dQg, we have <9._1 u e G^, 0 < j <„.
Now let h: G -> G be a homotopy such that A(G') C G' and
dQhQ(x) = x and dq+1hq{x) ( G'q if x ( Gq. If n is odd, let
k = A0(r/)~1A1(u)A2(r/)~1 ••• An_1(u)~1 and if n is even, let
*= h^-H^h^u)-1 ¦•An_1(u). Define a"(b) = a'{b)k.
Then d^"{b) = fUd.tidjk if i > 0 and
- ''- By the identities satisfied by A, it follows that
i > o.
SThus
' anc*' arguing as in the proof of the
ssi
From now on, we assume all atlpses to be regular and we
!p|||et r{b) denote the transformation elements tq{b) of an atlas
lSSlaF)} of a G-bundle p: E -> B. We call r a twisting function of p.
L»' s This notation is justified by the following result.
I:
gTHEOREM 19.4: Let p: ? - B be a G-bundle with fibre F and
^^|et laF)! be a G-atlas of p. Then the transformation elements
pr(A) define a twisting function r and (?, p, B) is isomorphic to
|(F xf B, p, B).
I Proo/: Define f: Ffi x Bn - En by f (/, 6) , #?,)(/, Afl).
;|^ is clearly an isomorphism of sets and:
0,
0 .
s.
||.:;.Thus if g~1 is used to define a structure of complex on the set

80
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
TWISTED CARTESIAN PRODUCTS AND FIBER BUNDLES 81
F x B, we obtain $~ 1(E) = FxTB. This proves the result.
Observe that the G-bundle p: E -* B with atlas {aF)j de-
determines the isomorphism ?.
§20. Fibre bundles and twisted Cartesian products
In this section we first define maps of TCP's with fixed
fibre and group. We then examine in detail the relationship between
T C P's and fibre bundles and prove that every TCP is a fibre
bundle.
DEFINITIONS 20.1: Let p: E(T) -* B and p': E(t') -* B' be
TCP's with group G and fibre F. A simplicial map
6: E{t) - E(r')
is said to be a map of TCP's if 6{i, b) = {if/(b)f, n(b)), where
if/: B -* G is a m a. p of s^ts. Observe that p'8 = 77 p. The re-
requirement that 6 be a simplicial map is equivalent to the following
identities on if/ •
T'77(b)dQiff(b) = f{d Qb) rib)
(U) dt\lA.b) - iffid.b) if i > 0
stip{b) = if/is.b) if f > 0.
6 is said to be a strong map if B = B' and n = 1B, and in this
case we say that r and r' are equivalent twisting functions. Ob-
Observe that every strong map is necessarily a strong isomorphism.
LEMMA 20.2: Let p: E -* B and p': ?'-> B be G-bundles with
fibre F. Then p and p' are strongly G-equivalent if and only if r
and t' are equivalent, where r and r' are any twisting functions of
p and p'.
Proof: Suppose p and p' are strongly G-equivalent. Then
there exists a simplicial isomorphism A: E -> E' such that
'A = p and
p'(b)if,(b), iff(b) f G, where {aF)j and
are any G-atlases of p and p'. If r and t' are the twist-
twisting functions defined by \a{b)\ and \a'ib)\, then it is easily veri-
J^ed that ^r satisfies the identities (U) (with n = 1B). Thus r and
M;.ate equivalent. Conversely, suppose r and r' are equivalent
S§id if/: B -* G satisfies the identities (U). Then if x e E is
Uritten as x =¦ fi(b)U, An), 6 = p (x), and we define
(I A CO = p'{b)if/(b)(f,An) = p'(b)bf,(b)f,An),
%|follows that A is a strong G-equivalence of G-bundles.
p§| Before proving that every TCP is a fibre bundle, we need
s||jie following definitions.
S|pEFINITlONS 20.3: Let G operate on the right of a complex E
rfipd on the left of a complex F. Then F xG E is defined to be the
IfjUotient of F x E obtained by identifying {gi, x) and (/, xg) for all
|Q§ I, and g. If ? = G xf B, then F xQE may be identified with
|| xf B and is called the TCP with fibre F associated to G xf B.
vlf p: E -* B is a principal G-bundle, then for some r E is iso-
inorphic to G xf B, and therefore G operates on the right of E.
;S(The operation is actually independent of the choice of atlas.) In
this case, (F xG E, p*, B), where p*(f, x) = p(x), is said to be the
(J-bundle with fibre F associated to p. If \a{b)\ is an atlas of p,
then {a*(b)} is an atlas of p*, where
P*(bX(,u) = (f,p(b)(eq,u)), u e A[n]q, b c Bn> i t Fq.
Now observe that if F xf B is a T C P and n : A -> B is a
simplicial map, then E(j)n may be identified with F x A, where
7
t ° 77, and 77:
E(r) is a map of TCP's.

82
SIMPUCIAL OBJECTS IN ALGEBRAIC TOPOLOGY
11
THEOREM 20.4: Let FxfB be a TCP and suppose A~ n: A -> B.
Then E(t) is strongly isomorphic to E{t) .
Proof: Suppose first that F = G, that is, ?(r) is a PTCP.
Then by Proposition 11.5, there exists a strong homotopy equiva-
equivalence 6: ?(r)A -* E(r)n. Clearly we may write dig, a) = iip{a)g,a).
Thus 6 is a strong map of TCP's, hence a strong isomorphism, and
the twisting functions r^ and r77 are equivalent. The latter state- -f-
ment implies the result for arbitrary F, and the theorem is proven. ;L
COROLLARY 20.5: Every TCP is a fibre bundle. ",-
Proof: Let F xT B be a TCP. Choose a base point cj> e B''
and let b e B . ~b <^d>: A[n] -> B, hence ?(r) is strongly iso-*
morphic to ?(r)^\ Since />" = t ° <j>, the latter is F x A[n]. This .,M
proves the result.
REMARKS 20.6: An atlas for F xf B may of course be defined by >\
p(b)(f,An) = (f, b). The twisting function of this atlas is r. In
other words, a TCP with group G is essentially just a G-bundle *
with a chosen atlas. ^
If p: E -> B is a G-bundle with given atlas \a(b)\ and
n: A -> B is a simplicial map, then p77: En -> A is a G-bundle with'
atlas {an{a)\, where nfFia) = fiinia)). If p is principal, then so ",
is p77 and 77 is equivariant, that is, n(xg) = nix)g for x f En,
g € G.
COROLLARY 20.6: If p: E -> B is a G-bundle and A~77: A -» B,
then p : ? -* 4 and p77: E77 -* 4 are strongly G-equivalent
G-bundles.
Finally, we note that since every TCP is associated to a
PTCP and every TCP is a fibre bundle, we have
TWISTED CARTESIAN PRODUCTS AND FIBER BUNDLES 83
PROPOSITION 20.7: Every G-bundle is associated to a principal
(j-bundle which is uniquely determined up to strong G-equivalence
pf G-bundles.
The converse is obvious: If p: ? -> B is a principal
G-bundle and G operates on F from the left, then there is a
unique G-bundle with fibre F associated to p.
¦ §21> Universal bundles and classifying complexes
y- We will construct a P T C P, or G-bundle,
1 Wy\j) = Cr Xt(q\ " V.tW
*• ^for each simplicial group G. W(G) will play the role of a classi-
classifying complex for G.
^DEFINITION 21.1: A PTCP G xT B is said to be of type (W) if
J-Bo has one element bQ and if dQ: e x B -> ?(r) . is an iso-
- Vorphism of sets for all q > 1. We let S: E(r)n_^ -> e x B de-
'¦... note the isomorphism of sets inverse to cL , so that dQS is the
9-1 •
',< identity on ?(r)
We will prove that, up to natural isomorphism, there exists
,"bne and only one PTCP of type (W) with group G. Before proving
^"existence, we obtain some of the properties of PTCP's of type
, >)¦
Lemma 21.2: If G xf B is a PTCP of type (W), then:
(ii) a+1S(x) = S(d.x) if x e E(T)q, q > 0;
(iii) S = s_ on e x B ;
(iv) s.+1S{x) = S(s.x) if x e ?(r)q, g>0.
Proo/: If x e E(r)q, q > 0, then x = {T(b),dQb) for some
. If i > 0.

SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
84
S(Ax) =
and
S(dQx) = S{T{dob)doT(b),dodob) - S(rx F), A^ 6) = (eq,d1b). '
On the other hand, AjSU) = di(eq+1,b) = (e^oLi) if i > 0. This
proves (i) and (ii), and the proof of (iv) is similar, (iii) follows im-
immediately from the definition. •
Lemma 21.3: If G xf B is a PTCP of type (W), then B, and there,
fore E(t), is a Kan complex.
Proof: Let bQ,..., bk__v bk+1,..., bq+l e Bq satisfy
d.bj = d._1b., i < j, i,j 4- k.
We must find b such that d.b = bjt i 4 k. We break the proof in-
into two cases.
(i) k > 0: Here we let x. = (r(*i+1), dQb.+1). Then
Ax^. = ^_jX/; i < i, i, / ^ *-l, and p(xp = ^06i+1 = <5fV
Since p is a Kan fibration, there exists x e E(r) such that E.x
= x., i ? k~\, and p(x) = 6Q. Let (en+1,?») = S(x). Then
EO6 = pE0S(x) = p(x) = bQ and, if f > 0, i ^ k, we have
6L6 = pd.Six) =» pSCol^jx) = p(en,bt) = 6/ .
(ii) A: = 0: Here we note that 6LrF.+1) = d._1r(b.+1)
if 0 < i < j, so that there exists g f G such that
Let (eq+vb) = S(& c), where c
(c)
) and
o .
. Further:
p S(r E0 b2), dQ dQb2) = 50 b2 ,
). Now
and
TWISTED CARTESIAN PRODUCTS AND FIBER BUNDLES 85
Ac
= PS(r(dob.+1),dodobj+1) = <90
Therefore dx b
and A b =
c)^og, 5Qc)
._x g, d,_t c)
6f), 5,,^) = 62. if i > 1.
LEMMA 21.4: If G xfB is a PTCP of type (W) and G is a mini-
minimal complex, then (E(r), p, B) is a minimal fibre space.
Proof: By Proposition 18.4, p is a minimal fibration. We
must prove that B is a minimal complex. Thus suppose b,ceB~
and A, b = d{ c, i 4s- k. We must prove dk b = dk c. Let
x = (rF), do?) and y= (r(c), Ape).
if ; + w *,
S(Ax) = (eq_vd.+1b) = (e^^A^c) = S^y),
hence A^x = A,y. Therefore d.r(b) = ^r(c) if 0 < i ^ i-1 and
r«o6)Cor(&) « rEocMor(c) if it ^ 1. Thus if it > 0, then
<9 rF) = Ar(c) for i 4 k~\, and, since G is minimal,
dk.xr{b) . ^.^(c).
It follows that ^6 = pS(r(dkb),dQdkb) = 5fcc If it « 0,
A/•(*>) = A,r(c) for i > 0,
hence dQr(b) = A-,r(c) and therefore r(dQb) = r((90c). Then
dQb = pS(rE06),50506) - 50c.
PROPOSITION 21.5: If G xfB is a PTCP of type (W), then E{t)
is contractible.
Proof: Taking cf) = (eQ, bQ) as base point, (i) and (ii) of
Lemma 21.2 imply that dC{S) + C{S)d is the identity map of

86
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
III
hi
fits
P§|l
1111
im
i
i
i
i
Itltl
llfl
It! ft
Kill
C(?(;•)), and therefore Hn(E(r)) = 0 for all n > 0. Since E(t) is
a Kan complex, Corollary 13.7 implies that E(t) is contractible
provided that ttAE{t),c?>) = 0. Suppose x e E{r)y that is,
5q X = (f> = (9. X.
Then <9n S(x) = x, 5, S(x) = S{dn x) = S(<?), and
aos(x) = sid.x) = su).
Clearly S(<f>) = sQ (f>, and therefore [x][sQ <?] = [sQ <?]. Thus [x]
= 0, as was to be proven.
Before proving the next, and fundamental, property of P T C P's
of type (W), we need a definition.
DEFINITION 21.6: If G and G'are group complexes which oper-
operate on E and on E' from the right and if y: G -» G' is a simpli-
simplicial homomorphism, then a simplicial map 0: E -* E' is said to be
y-equivariant if 6{xg) = 6{x)y{g) for all x t Eq, g e Gq. If
G = G' and y = 1«, 0 is said to be equivariant.
THEOREM 21.7: Let GxrB and G'xr'B' be PTCP's and let
y. G -* G' be a simplicial homomorphism. Then if G'xf'B' is
of type (W), there exists a unique y-equivariant map Q: E(r) -» E{r')
such that d(eq x Bq) C eq x B'q, q> 0.
Proof: 6 is determined, if it exists, by its values on e xB.
q > 0, and since BQ' = b^, 6(eQ, b) = (e^, b?) defines 6 on E(r)Q.
Again, if 6 exists, then for b e b . q > 1, there exists n (b) e B'
such that Oie^b) = (e^nib)) = SdQ{eq>rr(b)).
and, requiring dQ 6 =
the inductive formula
(i) d(e.b)-
we find that ^ is necessarily given by
q>l.
TWISTED CARTESIAN PRODUCTS AND FIBER BUNDLES 87
It remains to prove that 0 so defined is a simplicial map. d $
d is true by construction and if 6 ( B,, then
Now suppose d.d = 6d., 0 < i < j. Then we find
d. + 1
'• on eqx Bq, q > 1. Next, S = sQ on e x B , g > 0, hence
'- sn6 = Sddr.sn - 6sn on e.xBn. Finally, assume s.d = 6s,,
r^ 0 0 0 0 Q Q i v ^'
": 0 < i < j. Then

; j.
i
I on eqx Bq, q > 1.
- COROLLARY 21.8: Any two PTC P's of type (ff)with group G
''* >are naturally isomorphic.
* , Proof: If E(r) and ?(r') are PTCP's with group G, there
' exist natural maps d: E(r) -* E(r') and d': E(t') -> E{r). By
^uniqueness, 66' and 0'0 are the respective identities.
For any simplicial group G, we now construct a PTCP
v~i_ G < ,qi W (G) which is of type (W). Let H'.(G) have one element
I [] and define WJG) = Gn_x x Gn_2x--x GQ, n > 0. Write ele-
L*v ments of Wn(G) in the form [gn_1,...,gQ\, gi e G.. Define face
," and degeneracy operators on W{G) by s0 [ ] = [eJ, by
I d.[gQ] = [ ], i = 0 or 1,
and if n > 1 by
'" (i) do[gnr...,gQ] = [gn_v...,g0];
(ii)

88
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
Define T{G) on Wn (G), n > 0, by
(v) ^G)[4-l'-'^ = 4-l-
We could verify directly that W (G) is a complex and that
r(G) is a twisting function, but the following inductive description J§i
of W (G) and W(G) will make these facts obvious. Define an iso- i|
morphism of sets A.: Wn+liG) -> Wn (G) by |
(vi) \[gn,...,g0] = (gn,[gn_V:.,go\), n>0. |
Let ct(G): Wn(G) -> W (G) be the canonical pseudo-cross section, Jjg
ct (G)(vc) = (e , w). Now knowing the face and degeneracy operators!!
on W (G), those on W (G) are of course determined by the require- J|
ment that W(G) = G XwqxH'(G). Knowing the face and degeneracy ?|
operators on W (G), those on W - (G) are determined by: j|f
(vii) dQ = pA; di + 1 = y1 Bjk] sQ = k^aiG); ;|
si+l = A SiX- I
Finally, define S = ay1: H^(G) -> en+1 x %+1(G), that is |
(viii) S(gn,[gn_v...,g0]) = (en+v[gn g0]). I
Clearly dQS is the identity of H^(G) and SdQ is the identity on *:$
Given a simplicial homomorphism y: G -> G', define
(ix) W(y)[gn,...,g0} = [y(^,..,y^)].
With this definition, W becomes a functor from the category of sim-
simplicial groups to that of simplicial sets. W(G) will be called the
classifying complex of G, W(G) the universal G-bundle. We now
proceed to justify this terminology.
If TWISTED CARTESIAN PRODUCTS AND FIBER BUNDLES 89
f: LEMMA 21.9: Every PTCP G xfB is induced from a simplicial
1 map v: B -> W(G).
( Proof: Let 0: ?(r) -» W(G) be the equivariant map obtain-
1 ¦;ed in Theorem 21.7 and let n: B -> W(G) be defined by
|; ^(e^fi) = (e^, nib)) .
§:Then dQ6 = 6dQ implies t = ?-(G) ° ?r = r(GO7, so that G xr B
i is induced from n and 6 = n . We remark that n is given explicit-
|; ly by the formula:
|A) nib) = [Tib),Tidob),...,Tid?b),...>Tid%-1b))> where
| Bq = 50--- 50, i factors of 50 .
%; The following remarks will lead to a simple proof of the
I: classification theorem.
|REMARKS 21.10: By Propositions 18.4 and 18.7, a PTCP is just
I" a principal fibration with a given pseudo-cross section. From this
? point of view, a map n: B -> WiG) induces a principal fibration
-with given pseudo-cross ov, namely the canonical pseudo-cross
=section of Gxf(G,ff B. on is compatible with ct(G) in the sense
¦that n o71 = ct(G) ¦ n. We define a morphism of principal fibrations
I with group G to be an equivariant simplicial map. Then a strong
isomorphism of PTCP's with group G may be regarded as an auto-
automorphism of a single principal fibration.
LEMMA 21.11: Let p: E -> B be a principal fibration with group
G, let B' C B, and define E' = p (B'). Then any morphism
0': ?' -> H'(G) may be extended to a morphism 6: E -> WiG).
Proof: 6' defines a pseudo-cross section a': B' -> E'.
By Lemma 18.6, o' may be extended to a: B -> E. By Proposition
18.7, a defines a twisting function r: B -> G and the resulting
equivariant map 0: E = ?(?•) -> W(G) extends 0'.

90
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
Iff
-.f
'vi.
THEOREM 21.12: 77 c* A: fi -> W(G) if and only if the PTCP's
W{G)n and W{G)^ are strongly isomorphic.
Proof: If tt c^ A., then W(GO7 is strongly isomorphic to
W(G)^ by Theorem 20.4. Conversely, suppose W{G)n and W(G)X
are strongly isomorphic. Regard W(G)n and W(G) as copies of
the same principal fibration E. Let G operate on E x / by
(x, u)g = (xg, u), so that pxl: E x / -• 6 x / is a principal fibra-
fibration. Identify W(Gf with Ex@) and W(G)X with Ex(l). Then
E' = W(G)n U W{G)^ is a sub principal fibration of E x I, and
77: W(G)n -. W(G) and X: IP(G)A -. W(G) together define a morphism
0': ?' -¦ W(G). By the previous lemma, 6' can be extended to
6: E x 1 -* W(G). If 6 induces ^ on B x I, then i/r is a homotopy
from n to X.
Combining Lemma 20.2, Proposition 20.7, and Lemma 21.9
with the theorem above, we obtain the classification theorem:
THEOREM 21.13: Let F be a complex on which G operates effec-
effectively from the left. Then the assignment to any map
rr: B -* W{G)
of the TCP (or G-bundle)with fibre F associated with the PTCP
(or principal G-bundle) induced from n defines a one-to-one corre-
correspondence between the homotopy classes of maps B -» W(G) and
the strong isomorphism classes of TCP's (or strong G-equivalence
classes of G-bundles) with fibre F and base B.
REMARKS 21.14: Given a G-bundle p: E -> B, the corresponding
homotopy class of maps n- B -> W(G) defines a homomorphism
tt*: H*{W{G), A) -> H*{B, A). If A is a ring, tt* is a morphism of
rings, and the image of tt* is called the characteristic subring of p.
TWISTED CARTESIAN PRODUCTS AND FIBER BUNDLES 91
For example, if G = S@(n)) and A = Z2 we obtain the Stiefel-
Whitney classes, and if G = S(V(n)) and A = Z we obtain the
Chern classes.

92 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
Bibliographical Notes on Chapter IV
The material of section 17 is all due to Moore [52]. The
definition of T C P adopted here is a special case of that of Moore
[52], and gives what is called a regular TCP in [l]. The compari-
comparison of principal fibrations and PTCP's follows Cartan [6]. Of
course, the simplicial concepts of principal fibration and principal
bundle are equivalent, but the connection with topology is more
easily seen using principal fibrations. Nearly all of the material
of sections 19 and 20 is due to Barratt, Gugenheim, and Moore []].
The complexes W(G) were first constructed (without the
introduction of W(G)) by Eilenberg and MacLane [13]. The con-
construction of G xrfG\W@r) is due to MacLane [40]. The axio-
axiomatic definition of PTCP's of type (W) is due to Moore [5, 52].
The classification theorem as stated here is developed in
[1], but our proof follows Cartan [6],
CHAPTER V
EILENBERG-MACLANE COMPLEXES
AND POSTNIKOV SYSTEMS
In this chapter, we first prove that the category of simpli-
simplicial Abelian groups is isomorphic to that of chain complexes. We
then introduce the minimal Abelian group complexes K(ji, n). A
study of their properties leads to a proof that every Abelian group
complex is of the homotopy type of a product of K(jt, n)' s. We also
obtain the well-known characterization of cohomology operations in
terms of the cohomology of Eilenberg-MacLane complexes. Finally,
we will obtain the A-invariants of Postnikov systems by means of a
study of fibre bundles with fibre a K(ji, n).
§22. Simplicial Abelian groups
In this section, we investigate the category of simplicial
Abelian groups. We will prove that this category is equivalent to
that of (Abelian) chain complexes. An incidental result is the nor-
normalization theorem for simplicial Abelian groups. Finally, we will
compare the homotopy and homology groups of a simplicial Abelian
group.
We first develop an alternative definition of the homotopy
groups of an Abelian group complex G. Let A(G) denote G re-
regarded as a chain complex with differential d defined on Gn by
n
d = ? (— l)'dj. Let N(G) denote the chain complex G defined
93

m.
92 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
Bibliographical notes on Chapter IV
The material of section 17 is all due to Moore [52]. The
definition of T C P adopted here is a special case of that of Moore
[52], and gives what is called a regular T C P in [l]. The compari-
comparison of principal fibrations and PTCP's follows Cartan [6]. Of
course, the simplicial concepts of principal fibration and principal
bundle are equivalent, but the connection with topology is more
easily seen using principal fibrations. Nearly all of the material
of sections 19 and 20 is due to Barratt, Gugenheim, and Moore [l].
The complexes W(G) were first constructed (without the
introduction of W(G)) by Eilenberg and MacLane [13]. The con-
construction of G xr(GJ(G) is due to MacLane [40]. The axio-
axiomatic definition of PTCP's of type (W) is due to Moore [5, 52].
The classification theorem as stated here is developed in
[1], but our proof follows Cartan [6].
CHAPTER V
EILENBERG-MACLANE COMPLEXES
AND POSTNIKOV SYSTEMS
In this chapter, we first prove that the category of simpli-
simplicial Abelian groups is isomorphic to that of chain complexes. We
then introduce the minimal Abelian group complexes K(n, n). A
study of their properties leads to a proof that every Abelian group
complex is of the homotopy type of a product of K(n,ri)'s. We also
obtain the well-known characterization of cohomology operations in
terms of the cohomology of Eilenberg-MacLane complexes. Finally,
we will obtain the ^-invariants of Postnikov systems by means of a
study of fibre bundles with fibre a K(jt, n).
§22. Simplicial Abelian groups
In this section, we investigate the category of simplicial
Abelian groups. We will prove that this category is equivalent to
that of (Abelian) chain complexes. An incidental result is the nor-
normalization theorem for simplicial Abelian groups. Finally, we will
compare the homotopy and homology groups of a simplicial Abelian
group.
We first develop an alternative definition of the homotopy
groups of an Abelian group complex G. Let A(G) denote G re-
regarded as a chain complex with differential d defined on Gn by
n
(9=2 (—l)'<9j. Let N(G) denote the chain complex G defined
;=;0
93

94
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
in section 17, so that Wn(G) = Gn D Ker 50 D .. . D Ker (9n_i- For
convenience, we redefine the differential d on N(G) by
d(x) = (-1)" dn, x e Nn(G). Then tf(G) is a chain subcomplex of
i4(G) and there results a natural homomorphism
THEOREM 22.1: Let G be an Abelian group complex. Then
i: rrn(G) -> Hn(/4(G)) is an isomorphism for all n.
Proof: Filter A(G) by xfFMn(G) if xeGn and c?,x = 0
for 0< i < min (n,p). Then FM(G) = A(G) if p < 0 and
Fp4n(G) = Wn(G) if p > n, so that HFPA(G) = N(G). Each
p
FP+M(G) is a chain subcomplex of FPA(G). We will prove that
each of the inclusions ip: FP+1A(G) C FPA(G) induces an iso-
isomorphism on homology. It will then follow that i is an isomor-
isomorphism. We first define an epimorphism of chain complexes
tp: FpA(G) ^ Fp+1A(G) by
x if xeFpAn(G) where n<p + l
-spdpx if xeFpAn(G) where n>p + l.
Clearly /p(x) e FP+1A(G) and /" ° ip is the identity map of
FP+1A(G). A simple calculation proves that c?/p(x) = /p(c?x) for
all x. We will complete the proof by obtaining a chain homotopy
tp between ip °/p and the identity map of FPA(G). Thus define
0 if xtFpAn(G) where n<p
(-l)pspx if xtFpAn(G) where n>_p.
Then rp(x) t FpAn+l(G) and it is easily verified that
dtp(x) + t" d(x) = x - (ip ° /p)(x), as desired.
COROLLARY 22.2: Let G be an Abelian group complex. Then
A(G) = N(G) © D(G), where D(G) is the chain subcomplex of A(G)
generated by the degenerate elements of G.
f x
1 x~s
tp(x) =
K(tt, n)'s AND POSTNIKOV SYSTEMS
95
in~l
Proof: Let i = i° o . .. o in~l : yVn(G) -, An(G) and let
/ = f" o . .. o /° : i4 n(G) ^ A^n(G). ioi is the identity map on
N(G) and therefore i4(G) = N(G) © Ker /. It suffices to prove
Ker / = D(G), and Ker / C Dn(G) is clear from the definition of /.
Let x = 2
.=
f D(G). We must prove /(x) = 0, and we have
=/
"~1
n—1
1 o..
n1
where x1 = ^ s,y/, y/ ^
= -.. =/
" J
o...
where x'=
n-1
.2_
!
/, y/=y/~1-s^d-^y!~x
- =... = /"-'(s^j y^1) = 0, as claimed.
COROLLARY 22.3: (Normalization Theorem). Let G be an Abe-
Abelian group complex and let n: A(G) -> AN(G) = A(G)/D(G) be the
projection. Then 77 is a chain equivalence.
Proof: Identifying AN(G) with N(G), we find rr = /. But
i o /: A(G) -»/4(G) is a chain equivalence by the proof of the theo-
theorem.
N is a functor from the category u of simplicial Abelian
groups to the category C of chain complexes. We now show that
: there exists a functor F: C -> u such that T oN and # o F are
the identity functors of <J and C Thus let X be a chain com-
complex. Define V(X) as follows:
:'¦ n-1
(i) Vn(X) = Xn © 1q ^2^ oj!k... Oj j Jr, where cr/fc... a; 1 Xr is
the Abelian group whose elements are symbols cry . .. aJx x,
x ( Xt, with the addition defined by
ajk... ahx + ajk... ohy = ajk... ah(x+y),
and where the sum is taken over all sequences \j:\ such that
n >>fr > ... >y, > 0.
.. (ii) 5i:Tn(^)-.rn_,W is defined by

96
iii
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
(a) dnx = d(x) and <9,x = 0 if i < n, x e Xn
(b) If k = n — r and x e Xr, then
when
Sj ... Sj 1
is expressed in the canonical form de-
derived from formula C) on page 4.
(iii) s,: rn(X) - Fn+1(X) is defined by
(a) s,x = OjX, x f Xn
(b) If k = n — r and x e Xr, then
s,aiV . . a7lx = ahfc+1... ohlx if s/S^
shv
where stSj ... Sj 1 is expressed in canonical form.
It is easily verified that V(X) is a simplicial Abelian
group. Now we can state
THEOREM 22.4: The functors N:Q^C and r: C -. fl satisfy
T o A^ = l(j and A^ o T = 1@. Further, if A: G -> G' is a simplicial
homotopy, h: [ cz g, then there exists a chain homotopy
s: N(G) - A^(G'), s: #(/) ^ ^).
Conversely, if s: X -> A" is a chain homotopy, s: / ^?,then there
exists a simplicial homotopy h: F(X) -» r(*'), ft: T(/) ^ T(g).
Proof: That F o ^V = l^j and N oV = 1C is an easy con-
consequence of Corollary 22.2 and the construction of P. Given the
homotopy h: G -> G', h: I c^ g, define s = 1 ._0(~l)'h{ on An(G).
Then s: A(f) ^ i4(^). s(D(G)) C D(G') and, identifying N(G)
and /V(G') with /4^(G) and ^^(G'), we obtain a chain homotopy
s: #(/) =^ N(g) by passing to quotients. Conversely, let s: X -*X'
be a chain homotopy, s: i ^- g. Then (9s + s<3 = / — g. Define
KpX n)'s jI/VD POSTNIKOV SYSTEMS
h: TCJO-rCJT) as follows:
(i) If x f Jn, then
ftn(x) = <rn/(x) - ansd(x) - s(x)
ftn_i(x) = o-n_i /(x) - ffn sd(x)
ft,(x) = ct,/(x) if i <n-l.
(ii) ft; is defined inductively on the symbols ctj ... crj.x by
hj(crjk...crhx) = ajkhi_1(crjkl...ajix) if jk<i-l
fi/o-j, ,...ct;,x) if ;t>i—1.
97
A simple calculation then proves that ft is a homotopy from P(/)
to V(g), as desired.
We have now proven that the homotopy theory of simplicial
Abelian groups is equivalent to the homology theory of chain com-
complexes.
Theorem 22.1 has the following corollary comparing the
homology and homotopy groups of simplicial Abelian groups.
COROLLARY 22.5: Let G be an Abelian group complex. Then
there exists a natural epimorphism of groups j: Hn(G) -> nn(G) such
that j oh is the identity map of rrn(G), where ft is the Hurewicz
homomorphism. In particular, ft is a monomorphism.
Proof: Using 0 as base point, Cn{G) is the free Abelian
group generated by the non-zero elements of Gn. Thus there is a
natural epimorphism C(G) -> A(G), and this induces
Clearly / o h = i, i: wn(G)==» H n(A(G)). Letting ] = i~l-f, the
result follows.
We will later use the corollary above to prove that every
Abelian group complex has the homotopy type of a product of
Eilenberg-MacLane complexes.

&Ch
f
ii
11
98
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
COROLLARY 22.6: Let K be a complex. Regarding C(K) as an
Abelian group complex, we have nn(C(K)) = Hn(K) and
nn(C(K)) = ffn(/f).
This corollary states that in some sense homology is a spe-
special case of homotopy.
§23. Eilenberg-MacLane complexes
Recall that an Eilenberg-MacLane complex of type (n, n) is
a Kan complex K such that nn(K, $) = n and n,(K,<f>) = 0, i A n.
Such a complex K will be called a K(tt, n) if it is minimal. We
will prove the existence and essential uniqueness of K(n,n)'s.
LEMMA 23.1: Let n be a group. Define a group complex K by
Kn = it, n>_Q, and by letting each d} and s; on K n be the
identity map of n. Then K is a K(tt, 0) and any other K(ji, 0)
L is naturally isomorphic to K.
Proof: Clearly K is a #G7,0). Now K0 = L0 = n and
the identity map Ko -» Lo induces a unique simplicial monomor-
phism f:K->L. Suppose /: Kn -* L n is an epimorphism, n >_0,
and let yeLn+1. If f(xi) = diy, then /((^x,.) = ftfj.,^) implies
x,. = x;- = x, say. Let z = dty, all i. Then (9,-Soz = (9,y for all i,
hence since 77n+1(L) = 0, soz ~ y. By minimality soz =y, and
thus /(sox) = y-
Now observe that if G is an Abelian group complex, then
if we regard Wn(G) as the group Gn_1 x . .. x Go, W(G) becomes
an Abelian group complex. Define W"(G) = W(Wn~l(G)) for all
n > 1, W°(G) = G. The next theorem follows inductively from
Lemma 21.4 and Proposition 21.5.
THEOREM 23.2: Let n be a group. Then W(K(n,0) is a K(?7,1).
¦**¦
t.
K(n, n)'s AND POSTNIKOV SYSTEMS
99
If n is Abelian, then the Abelian group complex W"(K(tt, 0)) is a
KGr.n).
Before proving uniqueness, we need some preliminaries.
DEFINITIONS 23.3: Let (K, <f>) be a Kan pair, where 9 is the
only vertex of K. Define a complex P(K) by letting
Ln+1
be an isomorphism of sets and by defining dt =
'1
1. Define p:Pn(K)^Kn by
1 and
1 and let
= p~a@). It is easily verified that p is a Kan fibration and
therefore P(K) and
are Kan complexes. Let
y*
Then d0S is the identity map of P(K), d^ix) = A.(so<?) if
x e P0(K). and <9;+1 S = Sdj on Pn(K), n > 1. As in the proof of
Proposition 21.5, it follows that P(K) is contractible.
LEMMA 23.4: Let (K,<?) be a Kan pair, where <f> is the only
vertex of K. Then p: P(K) -> K is a PTCP of type (W) if L(K)
is a group complex.
Proof: Define a = As0: Kn ^ Pn(K). Then pa is the iden-
identity map of K and we may identify Pn(K) with LJK) x o(Kn). An
easy calculation proves that we may define an operation from the
right of L(K) on P(K) by (?lfo(*))?j = (?, ?2,o(*)). It follows
that p: P(K) -> K may be regarded as a principal fibration with
group L(K). Proposition 18.7 now implies that P(K) is a PTCP.
By definition d0S is the identity of P(K) and Sd0 is clearly the
identity on en x oiKn). Therefore p: P(K) -»L is of type
REMARK 23.5: If K is a group complex, then so is L(K), and
therefore K is isomorphic to W(L(K)). This implies the existence
of a group complex structure on W(L(K)). However, unless K is

100
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
Abelian, this product will not be that obtained by regarding
Wn(L(K)) as Ln_1(K)x...xL0(K).
THEOREM 23.6: Let K and L be K(n, n)'s. Then there exists
an isomorphism K -* L. Further, if n is Abelian, then any K(n,ri)
has a structure of Abelian group complex.
Proof: It suffices to obtain an isomorphism K -> L, where
L = W"(K(tt, 0)). We proceed by induction on n and assume the re-
result for n' < n, n > 0. Since K is a minimal complex, it is easy
to see that p: P(K) -» K is a minimal fibration. Therefore L(K) is
a K(n, n— 1). By the induction hypothesis, L(K) is isomorphic to
Y/n~l{K(ir, 0)). But then L(K) is a group complex and, by the lemma
above and by Theorem 21.7, it follows that K is isomorphic to
(n, 0)).
REMARKS 23.7: Observe that if n is an Abelian group and we de-
define a chain complex C{n, n) by Cn(n,n) = n and Cq(n, n) = 0 for
q J n, then TCin, n) is a K(n-, n). This gives an alternative proof
of the existence of K(jt, n)'s. Now, knowing that any K(n,n) is an
Abelian group complex, it follows that any K(n, n) is equal to
TC(jt, n). This is true since by Proposition 17.5 (recalling that
what is there denoted G is the chain complex N(G)), we must have
NK(tt, n) = C(n, n), and since FNKin, n) = K(n, n) by Theorem 22.4.
As a corollary of these remarks, we have:
PROPOSITION 23.8: Let I: n -* n' be a homomorphism of Abelian
groups. Then there exists a unique simplicial homomorphism
0: K(n, n) ->K(n',n) such that 0 = /: K(n,n)n -> K(n',n)n.
When n is an Abelian group, there is an alternative, and
more usual, description of K(n,n). Let C*(A.[q], 77) denote the
normalized cochain complex and C*(A[q]) the normalized chain
K(n, n)'s AND POSTNIKOV SYSTEMS
101
complex (previously denoted CN(A[q]), so that
C*(A[q],77) = Hom2 (C*(A[qj), „).
Fix an integer n and let L(.n,n + l)q = Cn(A[q], „), q > .0. Define
a structure of Abelian group complex on L(ji, n + 1) by
<9/u(x) = uE1.x), u«LG7;n + l)q+i, x « Cn(A[q]),
and
s,-v(y) = v(a;.y), v e L(n,n + l)ql y f Cn(A[q + l]),
where 5,.: C(A[g]) -, C(A[q + l]) and a,: C(A[q + l]) - C(A[q]) are
induced from the simplicial maps St: A[q] -» A[q + 1] and
at: A[q + 1] ^ A[q] defined in Definitions 5.4. Let
the subgroup of L(n, n + l)q consisting of all cocycles. Observe
that K(jt, n) is a subcomplex of L{n, n +1).
THEOREM 23.9: The Abelian group complex K(n,n) defined
above is in fact a K(n, n).
Proof: First observe that K(n,n)q = 0 if q < n because
every element of A[q] in dimension greater than q is degenerate.
Next, K(n, n)n = n since Z"(A[n], n) = Homz (Cn(A[n]), 77) and
Cn(A[n]) is the free Abelian group generated by An. By Remarks
23.7, it suffices to prove that the chain complex NK{n,n) is
C(tt, n). Clearly NnK(n,n) = K(n, n)n = 77, and it remains to prove
NqK{TT,n) = 0, g>n. Thus let fi(NqK(n,n), q>n, so that
(9,/x = 0, 0 < i < g- 1. Suppose for a contradiction that /x(x) ^ 0,
where x = (a0, ..., an), 0<ao<...<an<g is some basis ele-
element for Cn(A[q]). Then x^80y, since fi(80y) = 0, so a0 = 0
(otherwise x = 50(a0-1, . .., an-1)). Similarly x^5ay implies
aj = 1 and, inductively, a,. = i, 0 < i < n. But /n is a cocycle,
and this implies:

111
rill
102
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
n+1
.,n + l)= 2 (-1)'"
1=0
. ,n +
= 1 (-D'
+ (-
j=o voo = (-i)n+VGo = °.
where x~ denotes x = @,1, . . . ,n) regarded as an element of
A[q—l]n. Thus /x(x) = 0 and /x = 0, as was to be proven.
From now on, the symbol K(ji,ri) will denote the explicit
complex K(n,ri) described above. Now since Hn(A[q], tt) = 0 ,
S(Cn(A[q], tt)) = Zn+1(A[q], tt). Thus the coboundaries 8 define a
simplicial homomorphism 8: L(n, n + 1) -» KG7, n + 1). Further, addi-
addition defines an operation of K(tt,ti) on L(n-, n + 1) and of course
#G7, n) = Ker 8. It follows that 8 is a principal fibration with
group K(n,n). Define a: K(n, n + l)q -* L(n, n + l)q by
(i) o(/zXao> •••> an) = ft@,ao,. ••, an).
n+1
Then 5o(/xXa0,..., an+a) = ^ (-l)'/z@,a0,. . -, a,-.1,ai+1,...,an+1)
)) + /x(a0, • • -, an+1)
Thus 5ct is the identity of K(n, n +1), and it is easily verified
that a is a pseudo-cross section of 8, the corresponding twisting
function t being explicitly defined by:
(ii) K/xXao< • • •; an) = /x(O,ao+l, • • • ,an+l)-/x(l,a0+l, • • -an+l).
Observing that 1^ f Image (ct) if and only if i4,a0, . . ., an) = 0 when-
whenever a0 = 0, we define S: L(rr, n + l)q -» o{K(jt, n + l))q+i by
(iii) S(/x)(a0,..., an) = /x(ao-l, • • • ,an-l) if a0 > 0 and 0
if ao = O.
Then d0S is the identity on L{n, n + 1) and Sd0 is the identity
on o(K(n,n + l)). Thus 8 is a PTCP of type (W). Identifying
KGr,n) with Wn(K(n,0)) and letting Wn+1(K(n, 0)) = W(Wn(K(rr,0)),
Kf77, nj's yl/VD POSTNIKOV SYSTEMS
103
Theorem 21.7 implies:
THEOREM 23.10: The PTCP 8: L(n, n +1) ^ K(n, n +1) with
group #G7, n) is naturally isomorphic to the universal PTCP
0)) of /f(ir, n).
§24. KG7, n)'s and cohomology operations
We develop here certain fundamental properties of K(n,n)'s.
These properties will lead to a proof that every Abelian group com-
complex has the homotopy type of a product of K(n, n)'s and to the
characterization of cohomology operations. In the next section,
these properties will be used to define the /^-invariants of Postni-
kov systems.
All chains and cochains mentioned in this section and the
next are normalized.
Recall that L(jr, n + l)n is n regarded as the group of
homomorphisms of the free Abelian group generated by An into n.
Define an n-cochain u e C"(L(tt, n + 1), it) by u(/) = /(An),
/ f Homz(F(An), 77). u is called the fundamental cochain of
L(n, n + 1).
LEMMA 24.1: Let x e L(n, n + l)q and consider the simplicial
map x: A[q] - L(ir, n+1). If x*: Cn(L(ir, n + l),ir) - Cn(A[q],7r) is
induced by F, then x*(u) = x.
Proof: Since Cn(A[q], n) = L(n, n + l)q, the formula x*(u)=x *
makes sense. From the definition of L(ji, n + 1) as a complex and
the requirement that x be a simplicial map, we derive
x(y) = y*(x) e Cn(A[p], n) for y e A[g]p, where y: A[P] - A[q]. In
particular, if. ye A[q]n__is nonrde generate, then we must have:

hi
w
i
104
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
as was to be proven.
Let K be a complex and let Hom(K, L{n, n + 1)) denote
the set of simplicial maps K -> L(n, n + 1). The addition in LGr,n+1)
induces a structure of Abelian group on this set.
LEMMA 24.2: Let K be a complex. Define
0: Hom(K, L(ir, n +1)) - Cn(K, n)
by 0@ — f (u). Then 0 is an isomorphism of groups with inverse
iff defined by ^(y)(x) = x*(y) for y(Cn(K,n), xeKql where
x*:C"(K,7r)->C"(A[q],7r).
Proof: We must first prove that i//(y) is in fact a simpli-
simplicial map. However, since dj = Sj and sf = at on L(n, n + 1), we
find:
and similarly for the degeneracy operators.
0 and ip are obviously homomorphisms, and it remains to prove
that they are inverse to each other. If x e Kn is non-degenerate:
@ °<WKx)=^(y)*(u)(x)=u(iKy)(x))=u(i*(y»=x*(y)(An)=y(x).
Finally, if x t Kq, then using the first lemma we find:
LEMMA 24.3: 0 and i/f define an isomorphism between the co-
cycles Z"(K,7r) and the maps Hom(K, K(n,n)).
Proof: If y e Zn(K, n), we must prove that <A(yXx) t K(n, n)
for all x e Kq, that is, ip(J){x) is a cocycle. But
8(tfi(y)(x)) = S(x*(y» = x*E(y» = o.
Conversely, suppose that i/r(y)(x) is a cocycle for all x e Kq.
Then if x e Kn+1 is non-degenerate, we find that:
sy(x) = 5y(x(An+1))=5(x*(y)XAn+1)=5(<A(yXx)XAn+1)=o,
K(n, n/s AND POSTNIKOV SYSTEMS
105
= /*(u),
and y is a cocycle.
cf>: Hom(K,K(n,n)) -> Zn(K,n) is defined by
where u is regarded as an element of Zn(K(n,n),n). Here u is
called the fundamental cocycle of K(n,n) and its cohomology class
is called the fundamental class. We note that the following diagram
commutes:
Hom(K,L(ir,n +
C"(K,tt)
[5
Gr, n + 1)) -^- Z
Further, we have the following theorem.
n+1
(K,7r).
THEOREM 24.4: Let /,gfHom(K,K(i7,n)). Then f^g if and
only if <f>(f) is cohomologous to ${?). Thus 0 induces an iso-
isomorphism of n(K,K(n,ri)), the group of homotopy classes of maps
K-> K(w, n), onto Hn(K,n).
Proof: If / =* g, then 0(/) = f*(u) = g*(u) = 0(g) on the
cohomology level, hence the cocycles / (u) and g (u) are coho-
cohomologous. Conversely, suppose cf>(f) = (f>(g) + 8(a),
a e C"-\K,7t). Let i0: K - K x @) and iV K.Kx A) be the
simplicial maps identifying K with the cited subcomplexes of
K x /• It suffices to find Y e Zn(K xl, n) such that
and /t(y) = y|K x A) = <?(?), for then if/(y): K x / -. A'(w,n) will be
a homotopy from / to g. Let p: K x I -> K be the projection onto
K and let yo = p\<j>(f))eZn(KxI,n). Clearly /t(>/o) = 'S(>/o) = 0(O-
Further, regarding a as a cochain defined on i\(K), we may choose
a cochain /3 f Cn~'(K x /, ^) which extends a and vanishes on
/0(K). Thus JoQ3) = 0 and i*(fi) = a. Let y = y0 - S(j8). Then
and /f(y)'= (f>(f)-8(a) = cf>(g), as desired.

106
SIMPUCIAL OBJECTS IN ALGEBRAIC TOPOLOGY
Before studying cohomology operations, we prove:
THEOREM 24.5: Let G be a connected Abelian group complex,
and let nn = nn(G). Then G has the homotopy type of the infinite
Cartesian product x^i K(nn,ri).
Proof: By Corollary 22.5, there exists an epimorphism
;: Hn(G) -¦ nn(G) such that ;' oh: nn -* nn is the identity map,
n> 1. Let ynfZ"(G,7rn) satisfy y"(x) = /ix|, xtCn(G), where
|x) denotes the homology class of the non-bounding cycle x. By
Lemma 24.3, y" determines a map /": G -» K(nn,n). If g e Gn re-
regarded as A(G)n, then f"(g) = ~g*(yn) and
/"(g)(An) = yn(g) = j\g\ = j oh[g] = [gl
Therefore /?: nn(G) ?* rrn(K(nn, n)). Let '=x~=1/n: G^x~=1KGTn,n).
Clearly / is a weak homotopy equivalence, hence by Theorem 12.5
a homotopy equivalence.
Now we define and characterize cohomology operations.
DEFINITION 24.6: Let n and it' be Abelian groups, n and ri
integers. Regard H"(K,n) and H" (K,n') as defining functors
from the category of simplicial sets to that of sets- Then a coho-
cohomology operation of type (n, n', tt,tt') is a natural transformation
of functors C: Hn( ;n) -. Ha\ ;ir').
K(n, n/s AND POSTNIKOV SYSTEMS
107
Let H"(n,n,n') denote the cohomology group Hn(K(jr,n);7r').
Then we have:
THEOREM 24.7: The cohomology operations of type (n, n', tt,'tt')
are in natural one-to-one correspondence with the elements of
Hn'(jr,n,n').
Proof: If C is a cohomology operation of type (n, n', it, n'),
define a(C) = C(u) e H" (n,n,n'), where u denotes the fundamental
class of Hn(n,n,n). Conversely, suppose c e H" (n, n, it') and
y e H"(K,7t), where K is some complex. By Theorem 24.4, y de-
defines a homotopy class of maps </f(y): K -* K(ir,ri). Define a coho-
cohomology operation /3(c) by /3(cXy) = ^(y)*(c). Then since ip(u) is
the class of the identity map K(jr,ri) -»KGr,n), we find
(a o 0) (c) = /S(c) (u) = 0(o)*(c) = c.
Finally, if y eHn(K,Tr) for some complex K, then using the natu-
rality of cohomology operations and the fact that ifr(y) (u) = y
find
8 (C)()()**u))=c(y).
we
§25. The ^-invariants of Postnikov systems
Let (K,cj)) be a connected Kan pair. We wish to obtain in-
variants which characterize the homotopy type of K. Choosing a
minimal subcomplex if necessary, we may assume that K is mini-
mal (by Theorem 9.5). Let X = (X@), ..., XM,...) denote the
natural Postnikov system of K (Definition 8.5). Then by Lemma
12.1, each A^n) is a minimal fibre space. Letting nn denote
nn{K,4>), the fibre F(n+1> of ^(n) is isomorphic to K(nn+l, n + 1)
(by Lemma 10.2 and Theorem 23.6). Observe that K@) = cf>, so
that KA)==! K(iri, 1). Now by Theorem 11.11, each XM is a fibre

108
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
hi
if ir
?#
bundle, n >_ 1. We will show that, under assumptions on tti, the
group of the bundle Xin) can be reduced to K(TTn+1, n + l), n >^ 1.
Then by Theorem 19.4, it will follow that X1-"' can be given a
structure of PTCP. By Lemma 21.9, there will result a map
fn+2: KM ^W(K(TTn+1,n+l)) = K(nn+l,n+2) which will induce the
PTCP Xin). By Lemma 24.3, fn+2 will determine a cocycle
kn+2 e Zn+2(K<-n\nn+l)- A sequence kn+2, 1 < n, of cocycles so
chosen is called a set of /c-invariants of K, and such a sequence
certainly determines the homotopy type of K.
Of course a similar argument holds for fibre spaces. Thus
let (F, i/r) -i-> (E,i//) -?+ (B,(/)) be a fibre sequence of connected
Kan pairs. By Corollary 10.11, we may assume that (E, p, B) is a
minimal fibre space. Let ? = (?@),.. . , ?(n), . . . \ denote the nat-
natural Postnikov system of (E, p, B) (Definition 8.10). Lemmas 10.3
and 12.6 imply that each ?( is a minimal fibre space. Letting
nn denote nn(F, if/), the fibre of ?(n) is isomorphic to K(jrn+l,n+l),
n >_ 0. Under assumptions on nl3 tti(E), and tt^B), we will show
that the group of each of the fibre bundles ?(n) can be reduced to
K(nn+I,n + 1). Then each ?(n) will have a structure of PTCP,
there will be a map fn+2: E(n) -> W(K(nn+1, n + 1)) = K(nn+l, n+2)
which will induce the PTCP ?^n), and there will result a cocycle
kn+2 { Zn+2 (?(n)( ^^^ A sequence J^n+2^ q < ^ q{ cocycles SQ
chosen is called a set of ^-invariants of p, and such a sequence
determines the strong homotopy type of (E, p, B).
To validate the arguments above, we must study fibre bun-
bundles with fibre K(n, ri). The group of any such bundle is contained
in A(K(n, n)), the group complex consisting of invertible elements
of the monoid complex M(n, ri) = K(w,n)KGr'n). We determine the
structure of M(n,ri) and A(K(n, n)) in the next two propositions.
K(n, tO's AND POSTNIKOV SYSTEMS
109
We will need the following lemma.
LEMMA 25.1: The group of simplicial maps Hom(K(n, n),K(n', n))
is isomorphic to the group of homomorphisms HomzGr, it'), and
every element of Hom(K(ff, n), K(n', n)) is a homomorphism.
Proof: We observe first that every element
/ e Horn (K(n,n),K(n',n))
is a homomorphism. This is true since / induces a homomorphism
n -» v' on K{n, n)n = nn(KGT, n)) = n and since if x,y t K(jr,n)q, q>n,
then, by induction on q, f(x +y) and /(x) + f(y) have the same
faces, hence are homotopic and therefore equal. Now / determines
a homomorphism n -> n' and the converse was proven in Proposition
23.8.
PROPOSITION 25.2: As a complex, M(n,n) is isomorphic to
End {n, it) x K(n, n), where End (n, tt) is the complex defined by
End (n, n)q = Horn (K(n, ri), K(n, n)) for all q, with each dj and s;
the identity map.
Proof: M(it, n)q = Horn (K(n, n) x A[q], K(n, n)). Define
Lq = Zn(K(n, n) x A[q], tt). Then cf> of Lemma 24.3 defines an iso-
isomorphism M(n,n)q -¦ Lq. Anticipating the Eilenberg-Zilber theorem,
to be proven later, we know that the chain complexes
and C*(K(jt, n)) ® C*(A[q]) are chain homotopic. Further, since
K(n, n)q = 0, q < n, and 0 is a degenerate simplex, q > 0, we find:
(i) Cn(K(n,n)xA[q])^Cn(KGT,n))®C0(&[q])®C0(KGT,n))®Cn(A[q]),
(ii) Bn(KGT,n)x A[q\) ^Bn(
Now suppose y e L q, so that 8(Y) = 0. From the second term on

110
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
the right of (ii), we find y(x, y) = Y(x, z), where x e Cn(K(n, n)) and
y, z e A[q]0- Then we see that if Y = Y \ + Y2, where
y1 = y|Cn(KGT,n))®C0(A[q])
and y2 = y | C0(K(n, n)) <E> Cn(A[q]), we may regard Y i as an ele-
element of Zn(K(n,n),n) and Y2 as an element of Zn(A[q],n)=KGT,n)q.
Translating back to M(n,n), we obtain the result.
The result above determines the structure of M(n, n) as an
Abelian group complex, but we must still determine the monoid
structure and identify A(K(n, n)) C M(n, ri).
PROPOSITION 25.3: The product in M(n,ri) is given by
(i) (f,x)(g,y) = (fg,f(y) + x), f,geEndGT,7r)q, x,y eK(n,n)q.
Let AutGr,7r)q c EndGr,7r)q denote the group of automorphisms
K(n,n) ^K(n,n). Then (/,x) e A(K(n,n)) if and only if ft AuKtt.tt),
and in that case:
(ii) (/,x)-1=(/-1,-/-1(x)), / e Aut(n,n)q, xeK(n,n)q.
Finally, the embedding v: K(ji, n) -> A(K(n,n)) of K(n,n) as a
subgroup of operators on K{n,n) is given by:
(iii) v(x) = (l,x), xeK(ir,n)q, where 1 denotes the identity
map of K(n, n).
Proof: If (/, x) and (?, y) are as in (i) and z eK{n,n)r,
w e A[q]r, then using Lemma 25.1 we find:
(f,x)-(g,y)[(z,w)] = (f,x)[(g(z) +y(w),w)] = / • g(z) +f<j(w))+-x(w),
where x,y~: A[q] -> K(n,n) and f(y(w)) = f(y)(w).
This implies (i) and (i) implies (ii). If w = ?Aq, then:
v(x)(z,w) = z +E(x) = z +~x(w) = (l,x)[(z, w)].
This proves (iii). —- —
In the next two theorems, we obtain conditions on a bundle
K(n, n)'s AND POSTNIKOV SYSTEMS
111
with fibre K(n,n) which guarantee that the group of the bundle can
be reduced to K(n, n) and show by how much two K(n, n)-bundles
so obtained can differ.
THEOREM 25.4: Let p: E -> B be a fibre bundle with fibre K(jr,n)
and assume B is a connected Kan complex such that there exist
no non-trivial homomorphisms of 7r1(fi,^)) into the group of auto-
automorphisms of n. Then the group of the bundle can be reduced to
K(n,n).
Proof: Let \a(b)} bean A(K(n, n))-atlas of p with twist-
twisting function r(b) = (f(b),x(b)). Let c e B2 • Then from
dor(c) = r(doc)-lT(dic)
and di t(c) - tC2 c) we derive f(d2 c) = /(c) = f(d0 c) f(di c).
Considering / as a function Bt -> AutGr, n), we find that b ~ b'
implies f(b) = fib') and that / induces a homomorphism
7Ti(B,(pi) -* AutG7-, IT)
for any base point (f>. Therefore f(b) = 1 for be By
J3F) = p(b)(g(b), 0), where g(b) =1 for b e Bo and
g(b) = r\b)r\dob).. . rHd^b) for b e B
The relations doT(b) = T(dob)-1T(dl b) and
Define
, q
> 0.
o)(l ) and dtr(
together with the condition f(b) = 1 for be B
imply that g(b) = g(dtb) for / > 0. Then:
(i) dt J3F) = d, fi(b)(g(b),0) = P(dt bXgtft b),0) =
and
(ii)
= T(di+lb), i > 0,
are easily seen to
b), i > 0,
Thus \a(b)\ is A(K(n, n))-equivalent to the K(;r,n)-atlas (a(b)\,
and the result is proven.

112
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
THEOREM 25.5: Let p: E -> B be a fibre bundle with fibre K(n,n),
and suppose the group of bundle can be reduced to K(n, n). Then if
Bo has just one element, the resulting K(n, n)-bundle is unique up
to an automorphism of n (in a sense to be made precise in the
proof)-
Proof: Let \a(b)\ and ]a(b)\ be two K(n, n)-atlases for p
with twisting functions r(b) = (l,xF)) and T(b) = (l,y(b)). \a(b)\
and \a(b)\ are necessarily A(K(n, n))-equivalent, say
/3F) = /8F)(*F),2F)).
Then:
(i) d0P(b) = P(dob)(llX(b))(g(b),d0z(b))
b) [(g(d0 b), z(dob))]~l A, xF)) (g(b), d0 z(b))
It follows that gidob)-1 g(b)=l, g(b) = g(dob). Since Bo has just
one element, we find inductively that gib) is the same element g
for every b. In this sense, any two K(n, n)-atlases of p differ at
most by an automorphism of n. In fact, replacing {a(b)\ by the
K(n-,n)-equivalent atlas \d(b)\ where J3(b) = ]3(b)(I, -g~\z(b))), we
have:
(ii) ?F) = P(b)(g, 2F)) A, -g~l BF))) = j8F)fe, 0) .
The resulting twisting functions may be inequivalent, but
(g-1,0)r(b)(g,0)=T(b)
or, considering r and r to take values in v~l(l,K(n,n)),
REMARKS 25.6: Let r: B -*K(n,ri) be a twisting function and let
/: B -> W(K(n, n)) = K(n, n + 1) be the corresponding map. Then
4>{f)eZ"+\B,rr) is given explicitly by <j>(f )(b) = r(bX& n), b t B n+1 .
This follows from a calculation showing that t(K(tt,n))(i//j>)({) = r
K(n, n)'s AND POSTNIKOV SYSTEMS
113
when <?(/) is so defined (where T(K(n,n)) is the twisting function
for W(K(n, n)) defined by formula (ii) on patfe IOZ).
Therefore, if k, k e Zn+1(B, jf) arise from twisting functions t,T
such that 7 = g ¦ t, where g is an automorphism of K(n, n), then
k = g k, where g is regarded as an automorphism of n. Now we
extend the concept of a map of PTCP's with fixed fibre and base
to include g-equivariant maps, or ?-maps, g an automorphism of
the group. Then two cocycles k and k in Zn+1(B,n) determine
?-isomorphic PTCP's with fibre K(n, n) if and only if g*\k\ = \k\,
where g*: Hn+1(B,n) -» H"+l(B,n) is induced by the automorphism
g Of 71.
Now we have proven the following theorems:
THEOREM 25.7: Let (K,j>) be a connected minimal Kan pair and
let nn denote irn(K, j>). Suppose there exist no non-trivial homo-
morphisms n-j -» AutGrn, 7rn), n > 1. Then each of the fibre bundles
X(n), n > 1, of the natural Postnikov system of K is a
K(nn+i, n + l)-bundle, unique up to ^-isomorphism,
g e Ant(nn+1,7Tn+i).
Xin) is determined by kn+2 = 0(/n+2) e Zn+2(KM,nn+l), where
fn+2. ^(n) -+K(nn+l, n+2) is defined by any twisting function of
Xin), Another cocycle kn+2 also induces X(n) if and only if
\~kn+2\ = g*\kn+2\ for some g e Aut(n-n+lf nn+l).
THEOREM 25.8: Let (F,4i) -» {E,ijj) -?-+(B,j>) be a fibre sequence
of connected Kan pairs, where (E,p, B) is a minimal fibre space.
Let nn denote wn(F,t/r) and assume ttx is Abelian. Suppose
there exist no non-trivial homomorphisms n^E,^) -> Aut (nn, wn) ,
n >^ 1. Then each of the fibre bundles ?(n), n > 0, of the natural
Postnikov system of p is a K(nn+i> n + l)-bundle, unique up to

114
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
w
II
i
^-isomorphism, g e Aut(nn+l,nn+i)- ?(n) is determined by
*n+2=#/n+2)eZn+2(E(n)f,rn+1),
where /n+2: E^n) -» K(jrn+i, n+2) is defined by any twisting func-
function of ?(n). Another cocycle kn+2 also induces E(n) if and
only if \kn+2\ = g*\kn+2\ for some g e Aut(irn+i,irn+1).
REMARKS 25.9: There are various comments to be made on the
arguments above:
A) The ^-invariants could be defined and should be inter-
interpreted as obstructions to the cross-sectioning of the relevant bun-
bundles.
B) Had we developed the theory of the operation of n^K^
on nn(K,j>) (and of n^B,^) on nn(F,i//)), then the assumption
as to non-existence of homomorphisms could have been replaced by
the assumption of n-simplicity for all n. Further, the appropriate
generalization to the case where n-simplicity fails could be devel-
developed by use of the results on the structure of A(K(n,n))'s.
C) If (K,cf)) is a Kan pair and n is an Abelian group
such that Aut(n-, 7r) is an Abelian group, then the statement that
there exist no non-trivial homomorphisms n^K, ^>) -¦ Aut Or, n) is
precisely the statement that H^K, AutGr, n)) = 0. This follows
from the universal coefficient theorem and the fact that
H,(K; Z) S ir,(Kf 0)/MK, 0), n^K, 0)].
D) There is an obvious inverse procedure for constructing
any (simple) homotopy type: Let there be given a sequence of
groups (jti, . .., nn, ¦ ¦ ¦), nn Abelian if n > 1. Define
K(n, n)'s AND POSTNIKOV SYSTEMS
115
and let B be a connected minimal complex. Define E^ * = B.
Choose k2 e Z2(E@), n{) and let EA) be induced from
Choose k3 t Z3(KA), n2), and construct KB) as the PTCP induced
from x/f(k3): K<-1) -»K(ir2,3). Choose ?4 e Z4(KB), tt3) and con-
construct K^3' from i//(k4), etc Similarly, suppose 77x is Abelian
^)K(irlf2).
Choose k3 e Z3(E(-1\tt2), and construct EB) from i/>(k3), etc.
E) The precise statement of the topological results implied
by the arguments above should be clear. Starting with a connected
Cff-complex X or a Serre fibration p: E -» B of connected CW-
scomplexes, we derive (via the functor S, minimalization, and the
1 functor T) a sequence of principal fibrations (in the topological
sense) which are determined by ^-invariants and which determine
"the homotopy type of X or of (E, p, B). The additional assumption
that all homotopy groups in sight are countable must here be made
unless one takes the precaution of working in the category of com-
.pactly generated spaces.

116 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
Bibliographical Notes on Chapter V
The result that nn(N(G)) = H n(A(G)) and the resulting com-
comparison of the homotopy and homology of Abelian group complexes
is due to Moore [52], but our proof is essentially that of Cartan [6].
The equivalence of the categories of simplicial Abelian groups and
of chain complexes is due to Dold [8] and Kan [27]. Our construc-
construction of the functor V follows MacLane [41]. The relation
nn(C(K)) = Hn(K)
is the simplicial analog of a result of Dold and Thorn [9] on the in-
infinite "symmetric power" of a space.
The material on K(n,n)'s is mainly due to Eilenberg and
MacLane [13]. The proof of uniqueness of K(n,n)'s is due to
Moore [5,52]. Our presentation of the results of [13] comparing
maps into K(n, n)'s and cohomology groups follows Douady [7].
The relationship between K(n, n)'s and cohomology operations
was first observed by Serre [58].
The approach to the /c-invariants of Postnikov systems via
the study of simplicial fibre bundles with fibre K(n,n) is new. An
obstruction theoretic approach in the semi-simplicial case is taken
by Heller [21] and Moore [53].
CHAPTER VI
LOOP GROUPS, ACYCLIC MODELS,
AND TWISTED TENSOR PRODUCTS
The main object of this chapter is the construction of the
Serre spectral sequence by means of twisted tensor products. The
basic tool in this development, which is completed in section 32,
is Brown's theorem, which gives a natural equivalence between
CN(F >V B) and C^fi) <8>t C^F) on the category of twisted Car-
Cartesian products, where t is a twisting cochain determined by the
twisting function r. In order to define twisting cochains, the theory
of cup, Pontryagin, and cap products is developed in section 30.
Of course, the definition of those products depends on the Eilen-
berg-Zilber theorem, and this is proven in section 29. The proofs
of both Brown's theorem and the Eilenberg-Zilber theorem rely on
the method of acyclic models, which is described in section 28.
The models for Brown's theorem are defined in terms of functors
which assign to a (reduced) simplicial set K a simplicial group
G(K) and a PTCP G(K) xT K such that T(E(r)) is contractible.
G(K) is called a loop group of K. G(K) and G(K) *T K are de-
defined in section 26. In section 27, it is shown that G and W are
adjoint functors, the suspension E(K) of a complex K is defined,
and miscellaneous results about the functors G, W, and E ate ob-
obtained.
117

116 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
Bibliographical Notes on Chapter V
The result that nn(N(G)) = H n(A(G)) and the resulting com-
comparison of the homotopy and homology of Abelian group complexes
is due to Moore [52], but our proof is essentially that of Cartan [6].
The equivalence of the categories of simplicial Abelian groups and
of chain complexes is due to Dold [8] and Kan [27]. Our construc-
construction of the functor F follows MacLane [41]. The relation
nn(C(K)) = Hn(K)
is the simplicial analog of a result of Dold and Thorn [9] on the in-
infinite "symmetric power" of a space.
The material on K(n,n)'s is mainly due to Eilenberg and
MacLane [13]. The proof of uniqueness of K(n-,n)'s is due to
Moore [5,52]. Our presentation of the results of [13] comparing
maps into Kin, n)'s and cohomology groups follows Douady [7].
The relationship between K(n, n)'s and cohomology operations
was first observed by Serre [58].
The approach to the /c-invariants of Postnikov systems via
the study of simplicial fibre bundles with fibre K(n,ri) is new. An
obstruction theoretic approach in the semi-simplicial case is taken
by Heller [21] and Moore [53].
CHAPTER VI
LOOP GROUPS, ACYCLIC MODELS,
AND TWISTED TENSOR PRODUCTS
The main object of this chapter is the construction of the
Serre spectral sequence by means of twisted tensor products. The
basic tool in this development, which is completed in section 32,
is Brown's theorem, which gives a natural equivalence between
CN(JF xT B) and C^(B) <8>, C^F) on the category of twisted Car-
Cartesian products, where t is a twisting cochain determined by the
twisting function r. In order to define twisting cochains, the theory
of cup, Pontryagin, and cap products is developed in section 30.
Of course, the definition of those products depends on the Eilen-
berg-Zilber theorem, and this is proven in section 29. The proofs
of both Brown's theorem and the Eilenberg-Zilber theorem rely on
the method of acyclic models, which is described in section 28.
The models for Brown's theorem are defined in terms of functors
which assign to a (reduced) simplicial set K a simplicial group
G(K) and a PTCP G(K) xT K such that T(E(r)) is contractible.
G(K) is called a loop group of K. G(K) and G(K) xr K ate de-
defined in section 26. In section 27, it is shown that G and W ate
adjoint functors, the suspension E(K) of a complex K is defined,
and miscellaneous results about the functors G, W, and E ate ob-
obtained.
117

118
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
§26. Loop groups
In this section, we will prove the existence of loop groups
of reduced complexes, that is, of complexes having just one vertex.
We will also give a reinterpretation of the Hurewicz homomorphism.
DEFINITION 26.1: A group complex G is said to be a loop group
of the complex K if there exists a PTCP E(j) = G *r K such that
T(E(r)) is a contractible space.
EXAMPLES 26.2: K(n,ri) is of course a loop group of K(n, n + 1).
By Lemma 23.4, if K is a Kan complex, then L(K) is a loop group
of K provided that L(K) admits a structure of simplicial group.
If K is a reduced complex and K0 = k0, we will let kn
denote sJAQ. We now define a loop group of such a complex.
DEFINITION 26.3: Let K be a reduced complex. Define Gn(K)
to be the free group generated by the elements of Kn+l modulo the
relations sox = en for xEKn. Gn(K) is of course a free group.
If x ?Kn+i, let r(x) denote the class of x in Gn(K). Define
face and degeneracy operators on the generators of G(K) by:
r(dox) dor(x) = <dix)
(T) df(x) = r(di+1x) if i > 0
s/(x) = r(s,+1x) if i > 0.
The dj and s,- extend uniquely to homomorphisms Gn(K)->Gn.,(K)
and Gn(K) -> Gn+l(K). G(K) so defined is easily verified to be a
group complex, and t: K -» G(K) is clearly a twisting function. We
let E(r) = G(K) xr K.
We must prove that T(E(r)) is contractible, and it suffices
to prove n-iCrCECr))) = 0 and ~tfn(E(r)) = 6, n > 0.
LEMMA 26.4:
= 0.
LOOP GROUPS AftD TWISTED TENSOR PRODUCTS 119
Proof: Recall that ni(T(E(r))) can be considered as a
group having one generator for each 1-cell not in a maximal tree
and one relation for each 2-cell. We regard non-degenerate simpli-
ces (g: x) as denoting the corresponding cells. The 1-cells
(sog, x), xEK, non-degenerate and g E G0(K), form a maximal
tree. This holds since dQ(sog, x) = (j(x)g, k0) and di(sog,x)=(?,k0),
which implies that any two 0-cells can be connected in one and
only one way by 1-cells of the cited form. We must show that every
1-cell (g, x), g non-degenerate, is homotopic to the product of 1-
cells in the maximal tree and their inverses (reverses). The 2-cell
(Sig^gx) shows that (g, x) is homotopic to (s0dog, x)(g, kt). If
g = Hy)~lg', the 2-cell (sQg,y) shows that (g, dxy) is homotopic
to (g',d0y)(s0dlg,d2y). If g = i<y)g', the 2-cell (sog',y) shows
that (?, doy) is homotopic to (g', dxy) (sQdig\ d2y)~1- Combining
these relations, (?, x) is homotopic to the product of (?', k^ with
1-cells of the maximal tree or their reverses, where the group theo-
theoretic length of g' is strictly less than that of g. Inductively,
since e: = s0e0, the result is proven.
LEMMA 26.5: Hn(E(r)) =0, n > 0.
Proof: Consider Cn(E(r)), where (e0, k0) is taken as base
point. For g e GJK) and x ? Kn+l, define [g,x\ e Cn(E(r)) by:
(i) [g, x] = (r(x)g, dox) - (g, kn).
Observe that [g, kn+1\ = 0 and define B = [[g, x]|x ^ fcn+i!. Suppose
for the moment that we know that B is a basis for the free Abelian
group Cn(E(r)), and define 5 : Cn(E(r)) ^ Cn+1(E(r)) by
(ii) S[g, x] = . 1 (-II [s,g, (soJ'+'WJ'x] •
Using the easily verified relations

120
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
(iii)
and
a simple calculation proves that dS +Sd= 1 on C(?(r)). It remains
to prove that B is a basis for Cn(E(r)). We show first that the sub-
subgroup Z(B) of Cn(E(r)) generated by B is all of Cn(E(r)). Let
(g, x) be an element of the natural basis for Cn(E(r)). Since
(g,x) = (g,kn) + [g,sox], (g,x) = (g,kn) mod Z(B). Since
(iv)
(v)
and
= (g',kn) + [g,y],
and g can be written in one of the forms g = r(y)~1g' or g = r(y)g',
it follows that (g,x) = (g',kn) mod Z(B), where g' is an element
of Gn(K) with length strictly less than that of g. Inductively, we
find Z(B) = C „(?(»). We must still prove that the elements of B
are linearly independent. Let q be an integer and define B q to be
the union of the following sets:
\[g,soy] \y ?kn, length (g) < q\
(vi) \[g,x] | length (r(x)g) = length (g) + 1 < q\
ItKx)^*] I length (r(x)^) = length (g) + 1 < q\
Since B = U B q, it suffices to prove that the elements of Bq are
linearly independent for each q. B0 = \[en,soy]\ = \(en,y)\, so the
result is true for q = 0. Now we need only show that the elements
of Bq — Bq_l are linearly independent mod Z(Bq_l), q >_ 1. But
Bq — Bq_j mod Z(B q_l) is the union of the following sets:
\(g,y)-(g,kn) | length (g) = q\
(vii) i(r(x)?,dox) I length (r(x)g) = length (^) + 1 = q!
KKxr^.^n) I length (Kxr1^) = length (g) + 1 = q\.
Clearly these elements are linearly independent, and this completes
the proof.
LOOP GROUPS AND TWISTED TENSOR PRODUCTS
Thus we have proven the following theorem.
121
THEOREM 26.6: Let K be a reduced complex. Then T(E(j)) is
contractible, hence G{K) is a loop group of K.
COROLLARY 26.7: The homotopy groups nn(K) = nn(ST(K)) of a
reduced complex K are calculable as nn_l(G(K)), n >_ 1.
REMARKS 26.8: In principle, the corollary defines a procedure for
a purely group theoretic calculation of the homotopy groups of any
reduced complex. The construction of G can actually be carried
out for an arbitrary connected complex. The details of this general-
generalization may be found in Kan [30].
There is an interesting interpretation of the Hurewicz homo-
morphism in terms of the functor Q. Let A(K)=G(K)/[G(K),G(K)].
Then the projection p : G(K) -» A(K) can be verified to be a Kan fi-
bration, whose fibre we denote by F(K). We will show that
p*: ?7-n_1(G(K)) -> nn_1(A(K)) is essentially the Hurewicz homomor-
phism h:nn(K)^Hn(K).
THEOREM 26.9: nn(A(K)) is naturally isomorphic to Hn+l(K),
n > 0, and the following diagram commutes
_ fl/\ h rr
nn(G(K)) -?*-+ nn(A(K)),
where the vertical arrows are the natural isomorphisms.
Proof: By Theorem 22.1, if we regard A(K) as a chain com-
complex with differential d= 2(-l)'d., then nn(A(K)) = Hn(A(K)). Ob-
Observe that An(K) = Cn+1(K)/s0Cn(K). Thus it suffices to prove that
Hn+i(c(K))=Hnn(c(K)/s0C(K)). Now if A is any Abelian group
complex, then r)s« +.<?„/)- <*.A ™ a ^ •- 1 i

122
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
dso(sox) + sO(9(sox) = sox for x E An_1. Therefore s0A is stable
under d and is null-homotopic. Since the commutativity of the
cited diagram follows from the definitions, this proves the result.
REMARKS 26.10: We have the following commutative diagram,
where the rows are exact and the vertical arrows are isomorphisms:
X, K)^ ...
The first exact sequence arises from p: G(K) -* A(K), the second
from the PTCP A(K) >V K. The outer vertical arrows result from
the fibration G(K) >V K -> A(K) xr K with fibre F(K).
§27. The functors G, W, and E
Here we prove that G and W are adjoint functors. From
this will follow the essential uniqueness of loop groups. We then
define the suspension E(K) of a complex K and make some re-
remarks about the functor E.
THEOREM 27.1: Let § denote the category of simplicial groups,
S that of reduced complexes. Then the functor G: S-» § is an ad-
adjoint of W: § -> S.
Proof: Define if/: Homg(G(K),4) -> Hom§(K,JF(/4)) as fol-
follows. Given a simplicial homomorphism Y: G(K) -» A, there exists
a unique y-equivariant map 6: G(K) xr K -* A such that
d(eq xK,)Ce,x Wq(A), q > 0 (by Theorem 21.7). Define
if/(y): K -* W(A) by pd = tfj(y)p. As in A) of Lemma 21.9, t/,(y) is
given explicitly by:
A)
LOOP GROUPS AND TWISTED TENSOR PRODUCTS
Define 0: Hom$,(K,W(A)) -» Hom§(G(K),A) by
B) 0(/) (r(x)) = r(A) (/(x)), x?K,
123
This means that if /(x) = [an_1, . . . ,a0], then 0(/) (r(x)) = an_j .
0(/) is extended to G(K) by the requirement that it be a simplicial
homomorphism. It is easily verified that 0 o^ and if/ o0 are the
respective identities.
COROLLARY 27.2: There is a natural one-to-one correspondence
between twisting functions K -> A and simplicial homomorphisms
G(K) -> A.
Proof: Each twisting function t: K -> A is induced by a
unique simplicial map /(?): K -> W(A) by the rule t = r(/4)o/(r).
Notice that
C)
= r(A)tG) (x) = 7(x) ,
COROLLARY 27.3: Any PTCP with base K is determined by a
unique simplicial homomorphism of G(K) into the group of the
PTCP.
COROLLARY 27.4: Let A be a loop complex for K. Then there
exists a simplicial homomorphism G(K) -> A which is a homotopy
equivalence.
Proof: There exists a twisting function t: K -* A such that
T(A x- K) is contractible. If /G): K -> W(A) induces A Xp K ,
then clearly /(r) is a weak homotopy equivalence. Consider
0(/(r)): G(K) -> A. By definition of if/, the 0(/(r))-equivariant map
G(K) >^ K -> ff'(^) of Theorem 21.7 covers (iff o<f>XKf)y=f(f): K^W(A).
Therefore 0(^(T)) is a weak homotopy equivalence, hence a homo-
homotopy equivalence.

124
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
REMARKS 27.5: Free groups play a role in the category § similar
to that played by Kan complexes in the category S. To see this,
we note first that the natural concept of homotopy in § is that of
loop homotopy: Two simplicial homomorphisms i,g: G -> H are loop
homotopic if there exists a homotopy F: i ~ g such that
F(xy,v) = F(x,v)F(y,v) for all x,yz:Gn and vE/n. It can be
shown, by appropriately redefining H° in the category § and
proving that Hc is a Kan complex if G is a free group, that loop
homotopy is an equivalence relation on homomorphisms defined on
free groups. Similarly, a weak homotopy equivalence of free groups
is a loop homotopy equivalence. Further, 0 and i// of Theorem
27.1 induce one-to-one correspondences between the homotopy
classes of simplicial maps K -> W(A) and the loop homotopy classes
of simplicial homomorphisms G(K) -> A. Let 0: GW -> lg and
*P: 1§ -> WG correspond to 0 and if/ (as in Lemma 15.2) so that
FO o W : W -> W and OG o GW : G-> G are the respective identity
natural transformations. In analogy with Theorem 16.6, we can
prove that ^(K): K -> WG(K) is a weak homotopy equivalence (and
is a homotopy equivalence if K is a Kan complex) and that
0(/4): GW(A) -> A is a weak homotopy equivalence (and is a loop
homotopy equivalence if A is a free group complex). Finally,
G(F) : G(/) =^ G(g) is a loop homotopy in § if F: f c*g in S, and
W(F) : W(f) c± W(g) in S if F : t ~ g is a loop homotopy in § .
Proofs of these results may be found in Kan [31].
Now we define the suspension of a complex.
DEFINITION 27.6: Let K be a complex with base point k0. De-
Define the suspension E(K) as follows. EQ(K) = b0; En(K) consists
of all Symbols (i,x), i >_ 1, xEKn_j, modulo the identification
(i,kn) = s"+'?0 = bn+j, where kn = s^kQ . Define face and degen-
LOOP GROUPS AND TWISTED TENSOR PRODUCTS 125
eracy operators on E(K) by:
(i) s'0(j,x) = (i+j,x)
(ii) s;+1(l,x) = (l,s,x)
(iii) (90(l,x) = bn, xeKn
(iv) d1(l,x) = b0, xeK0
(v) <9;+1(l,x) = (l,a,x), xEKn, n>0.
Sj(j,x) for i: > 1, ; > 2 and dt(j,x) for i > 0, / ^ 2 are deter-
determined by the operations above and the requirement that E(K) be a
complex. We note that (i' + l,x) is often denoted by s'QEx in the
literature.
Recall that if (AT,x0) is a topological pair, then the (re-
(reduced) suspension E(X) of X is the identification space obtained
from X x / by collapsing {.X x /) U (x0 x I) to a point.
LEMMA 27.7: TE(K) is canonically homeomorphic to ET(K).
Proof: Define /: ET(K) - TE(K) by the formula:
(i) f(|x,S|,l-0= |(l,x),5@|, where x E Kn and where
if 8 = (to>...,tn), then S(t) = (l-t,tt0,... ,tta).
i is the required homeomorphism.
Writing r(l,x) = x, we find that GE(K) is just the free sim-
simplicial group generated by K, subject only to the relations kn=en.
It follows that AE(K) = C(K), so that nn(AE(K)) = H JK). The
inclusion map K -> GE(K) induces a: nn(K)^nn(GE(K))^nn+1(EK).
a is called the suspension map of the homotopy groups. Using
Theorem 26.9, we obtain the commutative diagram:
*„(*)
nn(GE(K)) -=
lP*
nn(AE(K)) _=
nn+1(E(K))

126 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
The pair (GE(K),K) gives rise to the exact sequence:
REMARKS 27.8: The construction GE could be modified by using
free monoids instead of free groups. We denote the resulting free
monoid complex by G+E(K). The interest in G+ECK) arises from the
fact that if K is a countable complex, then TG+E(K) is canoni-
cally homeomorphic to the reduced product (James [23]) of T(K).
However, a homological argument shows that the inclusion of G*~E(K)
in GE(K) induces an isomorphism on homotopy groups when K is
connected.
REMARKS 27.9: Let S denote the category of simplicial sets,
S+ that of reduced complexes. One would like G: § -> S and
E: S -¦ S+ to be adjoint functors, since in the topological case we
have Homg-(AT,flF) = Uomj(E(X),Y), where Q(F) denotes the loop
space of F. This conclusion is false: It is easy to construct a
natural transformation i/r: Hornc+ (E(K),K') -> Hornc(K,G(K')), but
if/ has no inverse. If G is replaced by L of Definition 23.3 and
S is taken to be the category of Kan complexes, then ft is still
defined and now does have an inverse. Thus the functor L is more
closely analogous to Q. than is G. However, L(K) is only defin-
defined if K is a Kan complex, and L(K) may admit no structure of
group complex.
LOOP GROUPS AND TWISTED TENSOR PRODUCTS
127
§28. Acyclic models
a opnpral nrocedure for the construction of
chain maps and chain homotopies. The method applies when a cat-
category has certain distinguished objects, called models, which are
acyclic with respect to a homology theory defined in terms of one
functor from the given category to the category of chain complexes
and which, in a suitable sense, represent another such functor.
Let S denote a fixed category. Let DH denote a set of ob-
objects of S, called the models. ? will denote the category of chain
complexes, 2) that of Abelian groups. Let A: S -> jj be a covariant
functor. We must define the concept of representability of A.
Assume (or arrange by use of the free Abelian group functor) that
S is an additive category and A is an additive functor. Define A(K) =
©itf«3TC (Horns (M,K) ® A(M)), K an object of S. For maps a: K-^L
and /i: M —> K in S and a e A(M), define A(a)(n,a) = (a o n,a).
Then A: o -> % is an ii-litine. functor. Define a natural transforma-
transformation of functors A: A -> A by A(K)(/z,a) = A(^Xa), n'.M^K, aeA(M).
DEFINITION 28.1: A: S -»IB is said to be representable if there
exists a natural transformation of functors ?: A -> A such that
A^: A -> A is the identity natural transformation. ? is then said to
be a representation of A.
Theorem 28.2: Let A and B be suUitive functors S -> ? and
let n >_ 0 be an integer. Suppose given natural transformations
/,•: A^Bj such that </,/,- = /,_!</,., 0 < i < n (where A^B^. S -> %
are determined by A^K), B{K), K an object of S). If each Aq:h-3>
is representable for q > n and each Hq(B(M)) = 0 for q > n and
M t M, then there exists a natural transformation /: A -» B such that
l\Ai = ti, 0< i < n.
Proof: We construct lq: Aq^> Bq by induction on q, start-
starting with the given f,•, i < n. Suppose /9_j has been defined on

128
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
Aq-i, g> n + 1. If aeAq(M), M eX then tq_
because by the induction hypothesis we have
_,B(M)
Since Hq_1(B(M)) = 0, we may choose b e B q(M) such that
dq(b) = fq_i dq(a). Define a natural transformation 3.: Aq -> B q by
«{K)(/i,a) = Bq(ii)(b), n-.M+K. Then we find:
Thus dq^.= fq_ldqk. Now let
Aq and define fq = %?: Aq
B
q.
A q -> A q be a representation of
Then we have
as desired.
THEOREM 28.3: Let A and B be additive functors S -> ? and
let n >_ 0 be an integer. Let / and g be natural transformations
A -> B and suppose given natural transformations s,-: A{ -> B/+1
such that dj+1Sj + si_idl = il— gt, 0 < / < n— 1. If each /4q:S-*fB
is representable for g > n and each H q(B(M)) = 0 for g >. n and
W f 5I, then there exists a natural chain homotopy s: I ^ g such
that s|/4, = Sj, 0<_i< n-1.
Proof: The argument is similar to that above. We construct
sq: Aq -> Bq+1 by induction on q, starting with the given s,-,
i <_ n— 1, or with s_j = 0 if n = 0. Suppose sq_: has been de-
defined on Aq_!, g> n. If a(Aq(M), M e %, then we find that
(gq— fq— sq_ j cfq) (a) f ZqB(M) by use of the induction hypothesis.
We then choose c f Bq+1(W) such that rf9+1(c) = (gq-/q-sq_j<fq)(a)
and define a natural transformation rf.
B
q+1
by
LOOP GROUPS AND TWISTED TENSOR PRODUCTS 129
An easy calculation proves that dq+1rj = (gq-fq-sq_1d )A.. Let-
ting
and then
A be a representation of Aq, we define
s9 =
= gq- iq- sq_xdq, as desired.
REMARKS 28.4: In the applications, one often encounters the sit-
situation where B(M) is an augmented chain complex satisfying
Hq(B(M)) =0 for g > 0 and E: H0(B(M))^* Z. In this case, we
say that M is acyclic with respect to B. Here, if n = 0 in Theo-
Theorem 28.2, we must assume that /(^(/KM)) CKer ? for all M t %.,
while if n = 0 in Theorem 28.3 we must assume that
Co-go) W))CKer E.
The proofs then go through with the obvious trivial modifications.
REMARKS 28.5: If each model M has a canonical contracting
homotopy with respect to B, then there are canonical choices for
the chains b and c in the preceding proofs. Therefore explicit
formulas for the constructed chain maps and homotopies are then
obtainable.
§29. The Eilenberg-Zilber theorem
Let Q denote the category of simplicial Abelian groups.
If K and L are objects of u , it is necessary to modify the defi-
definition of KxL by letting (K xL)n = Kn®Ln in order for K x L
to be an object of Q ¦ Now we have two functors A and B: QxQ^>?,
defined as follows. A(K, L) is A(K*L), that is, KxL regarded
as a chain complex, with dn = 2JL0(-l)'<?i. B(K,L) is A(K&A(L),
so that Bn(K,L) = 2i+j=n AffL) ® AfL) and d = d<8> 1 + 1 <g> d
(with the sign convention: A ® cf)O ® y) = (-l)dee x* ® d(y)). We
will obtain a natural chain homotopy equivalence of functors /: /4->B
h

130
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
with inverse g: B -> A by use of the method of acyclic models.
Since A0(K,L) = Ko ® L0 = B0(K,L), we define fQ:A0->B0
and gQ: Bo -> AQ to be the identity natural transformations. We will
obtain models in ft x Q which represent both A and B and are
acyclic with respect to both A and B. Thus let Mq denote the
free simplicial Abelian group generated by A[g] and define the
models JR in ft x ft to be the pairs (Mp,Mq).
LEMMA 29.1: Both An and Bn are representable, n > 0.
Proof: Define ?: A „ -> A „ by
A) &K.L) (*n ® O = (*„ xTn, An ® An).
Then
\(K,L)?(K,L)(kn ® ?n) = i4(*n x !n)(An ® An) =
*n(An) <g> Tn(An) = *n ® ?n.
Similarly, define ^: B„ -> Bn by
B) &K,L)(kp ® ?,) = (Fp x T,, Ap ® A,), p + g = n.
Again we find A^ = 1, as desired.
LEMMA 29.2: The models are acyclic with respect to both A and
B.
Proof: Mp is augmented with augmentation ?: Mp -> Z de-
defined by ?(x) = 0 if xfWp, n>0, and ?(x)=l if xeA[p]0. Define
?: AWjM'1) -> Z by ?(x ® y) = ?(x)?(y), and similarly for
B(Mp,Mq). We must prove that Hn(A(Mp,Mq)) = 0, n>0, and
?: HQ(A(Mp,Mq))=* Z, and similarly for B. Define s: M^ -» W^+J by
s(ao,...,an) = @,ao,...,an), 0 < ao < . .. < an < p. Then
A) dos=l; diS = @) on A[p]0; di+1s = sdt on A[p]n, n > 0.
Therefore ds + sd = 1-oE, where a: Z -> Wp is defined by
LOOP GROUPS AND TWISTED TENSOR PRODUCTS
131
o(l) = @). Now define s: An(Wp,Mq) -» An+l(Mp,Mq) by
B) s(x ® y) = s(x) ® s(y), x e MP, y f Mq .
n n
Then ds + sd = 1 <g> 1 - ff? ® CT? on A(WP Wq). Similarly define
s: Bn(Mp,M")^Bn+1(Mp,Mq) by
C) s(x ® y) = s(x) ® y + (-l)va?(x) ® s(y), x f W^, y f Mqy .
u + v = n.
Then ds + sd = 1 <g> 1 - oE % oE on B{Mp,Mq). This proves the
result.
Now the following theorem, due to Eilenberg and Zilber
[16], follows immediately from Theorem 28.2 (with n = 0) and Theo-
Theorem 28.3 (with n = 1 and s0 = 0), coupled with Remarks 28.4.
THEOREM 29.3: Let A, B: B x Q -> ? be defined by
A(K,L) = A(K XL)
and B(K,L) = A(K) <g> 4(L). Then there exist natural transforma-
transformations /: A -* B and g: B -» /I lying over the respective identity
transformations /0: Ao -> Bo and g0: Bo ^ /40. Any two such /
are naturally chain homotopic, as are any two such g. Further, for
any such / and g, f o g: B -* B and g o /: A -> A are naturally
chain homotopic to the respective identity transformations.
COROLLARY 29.4: Let AN, BN: ft x & -> C be defined by
AW(K,L) = /l#x L) and BN(K,L) = AN(K) ® AW(L). Then the
theorem remains true if we replace A, B by AN, B N.
Proof: The models of (I x Q are acyclic with respect to
AN and BN by Corollary 22.3. It is easy to check directly that
AN and BN are representable, but since it follows from Theorem
22.2 and Corollary 22.3 that AN(K,L) and BN(K,L) are direct
summands of A(K,L) and B(K,L), this result is a consequence of
t|);S
:. i !'
1 I

132
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
LEMMA 29.5: Let A, A': S -> C be sJ.«litivefunctors. Let a:A-*A'
and {$'¦ A' -> A be natural transformations such that j8oct=l: A -> A.
Then if A' is representable, so is A.
Proof: The following diagram is commutative:
JA'
Here a(K) Qi,a) = (^,a(W) (a)), p. K -, A# , a f A(Af), and
/800(/*,a') = (^.jS(W)(a')), a' e M'flf). Let ?': A' -, A' be a repre-
representation of A' and define ?: A ^> A by ?= jS^'a. Then:
= jSA'^'a = jSa = 1, so ? is a representation of A.
COROLLARY 29.6: Let K and L be simplicial sets. Then there
are natural chain homotopy equivalences /: C(KxL)->C(K) ® C(L)
and g: C(K) ® C(L) -> C(K x L), and similarly with C replaced by
Proof: We need only observe that C(K*L) = A(F(K),F(L))
and C(K) ® C(L) = B(F(K),F(L)), where F(K) and F(L) are the
free simplicial Abelian groups generated by K and L.
As stated in Remarks 28.5, the contracting homotopies on
the models determine explicit choices for / and g. It is simpler
in this case to construct explicit transformations ad hoc. For con-
convenience, if K is a simplicial set and x e Kn, we let dx = dnx and
in general d"~'x = di+1 ¦ ¦ ¦ dnx.
DEFINITIONS 29.7: Let K and L be simplicial Abelian groups.
Define /: A(K x L) -> A(K) ® A(L) by
0)
?„)=
LOOP GROUPS AND TWISTED TENSOR PRODUCTS 133
/ is called the Alexander-Whitney map.
Define g: A(K) ® A(L) -, A(K x L) by
where the sum is taken over all (p,q)-shuffles ([i,v)
(Definition 6.5) and where o(/i) = 2/=1 [/z,—(/'— 1)] is
the signature of the corresponding permutation.
We will call g the Eilenberg-MacLane map, since these authors in-
introduced and studied it in [13].
PROPOSITION 29.8: The Alexander-Whitney map / is a natural
transformation A(K x L) -> A(K) ® A(L) lying over the identity
AQ(K x L) -» A0(K) <S> i40(L)- It is associative in the sense that
the following diagram commutes:
A(KxLx MX'
A (K)
/ induces a natural transformation fN:AN(KxL)-*AN(K) ® AN(L).
Proof: Clearly /0 is the identity and / is natural. To prove
that df = Id and A <8> /)/=(/ <8> 1)^, we need only demonstrate
these equations on An ® An and on An <S> An <8> An. Here these
results follow from explicit calculations, which will be omitted.
Next, we have an exact sequence (with the obvious maps)
(a) D{K) ® A(L) © A(K) ® D(L) -> 4(K) ® A(L) -> i4w(K) ® /4
Now if x <8> y t D(KxL), say x ® y = sAx' (8> sAy', then if / < k,
d sky' is degenerate and if / > k, dn~'skx' is degenerate. Thus
one factor in each term of /(x (g) y) is degenerate, and therefore,
by (a), / does induce fN.

134
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
PROPOSITION 29.9: The Eilenberg-MacLane map g is a natural
transformation A(K) <8> A{L) -* A(K x L) lying over the identity
A0(K) <8> AQ(L) -> A0(K x L). It is associative in the sense that
the following diagram commutes:
A(K) <g> A{L)
g induces a natural transformation gN'-AN(K)(8) AJV(L)->/4JV(KxL).
Proof: Clearly gQ is the identity and g is natural. To
prove that dg = gd and g(g <8> 1) = g(l <g) g), it suffices to obtain
these results on Ap <8> Aq and on Ap <8> Aq <8> Ar. Here these
equations follow from lengthy computations, which will be omitted.
Finally, if either x e Kp or y e Lq is degenerate, then so is each
term of g(x <g) y), and this implies that g induces gN.
Corollary 29.10:
^i, fog~i,
^i, and
Proof: The existence of the cited chain homotopies follows
from Theorem 29.3 and Corollary 29.4. The fact that fN°gN = 1
(with no chain homotopy required) is verified by explicit computa-
computation.
Finally, / and g are homotopy commutative in the sense of
the following lemma.
LEMMA 29.11: Let K and L be simplicial Abelian groups. De-
Define t: KxL ->LxK by t(x <g) y) = y <g) x. Then the following
diagrams are chain homotopy commutative:
J—A(LxK) A(K)~® A(L)-L* A(L) ® A(K)
[f and \g
. A tT \ rfO A tlS\ A fT/ \s T \ '
A(K\ E?) Ad A
A(LxK)
LOOP GROUPS AND TWISTED TENSOR PRODUCTS
where T(x ® y) = (-l)defi * dee yy <g> x.
135
1
\
Proof: We note first that the sign in the definition of T
makes it a chain map. Since ft = Tf on A0(KxL) and gT = tg on
A0(K) (g) A0(L), the result follows from Theorem 28.3 (with n = 1
and s0 = 0), using the same models as those used in Theorem29.3.
§30. Cup, Pontryagin, and cap products; twisting cochains
Here we define cup products, Pontryagin products, cap pro-
products, and twisting cochains. Twisting cochains define twisted
tensor products, which are related to tensor products in a way ana-
analogous to the way twisted Cartesian products are related to Carte-
Cartesian products.
We first define algebras, coalgebras, and Hopf algebras.
DEFINITIONS 30.1: Let X be a graded A-module, A a commuta-
commutative ring. X is said to be a A-algebra if there exist morphisms of
A-modules 0: X <S>\ X -> X (the product), -q: A -> X (the unit; rfX)
is the identity), and ?: X -* A (the augmentation) such that the
following diagrams commute (all tensor products are over A in this
definition):
x,
A
X is said to be commutative if the following diagram commutes:

136
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
x®xx *
Reversing all the arrows, we obtain the definition of a A-coalgebra
(with coproduct, counit, and coaugmentation), and of cocommutati-
vity of a coalgebra. If X is an algebra with product 0, X <g) X is
an algebra with product @ <8> 0)A <8> T <8> 1). If X is a coalgebra
with coproduct vfr, X ® X is a coalgebra with coproduct
A ® T <g> 1)(^ ® ^).
If X is both an algebra and a coalgebra, and if the unit is the co-
coaugmentation, the augmentation is the counit, and the product is a
morphism of coalgebras, then X is said to be a Hopf algebra. The
last condition is that ^0 = @ <g) 0)A <g> T ® 1)(^ <g) ^) on X <g> X.
If X is both an algebra and a differential A-module and if d(f> = (f>d
(d=d®\+l®d on X <8> X), then X is said to be a differen-
differential algebra. If, further, Y is both a (left) X-module and a differ-
differential A-module, then Y is said to be a differential X-module if
da = ad, where a: X <8> Y -> Y defines the X-module structure on
Y. Differential coalgebras and comodules are defined similarly.
In what follows, if the base ring A is not mentioned it is
assumed to be Z, and ® and Horn without subscripts are taken
over Z. We will here work with unnormalized chain and cochain
complexes, but by Corollary 22.3 (and Lemma 29.5 in acyclic model
arguments) we could equally well use normalized complexes.
Let (B,b0) be a simplicial pair. Define A: B -> B x B by
A(b) = (b,b). Let / be the Alexander-Whitney map and define
A) D = / o C(A): C(B) -» C(B) <g> C(B) .
Define ?: C(B) - Z by EF) =1 if 6 f BQ and 6F) = 0 if b e B n,
n > 0. Define 77: Z -> C(fi) by r^l) = b0. Since (A x 1)A = A x A)A
LOOP GROUPS AND TWISTED TENSOR PRODUCTS 137
and (f <g> 1)/ = A <g> f)^, we find (D ® 1)D = A ® D)D. Clearly
(e ® 1)D = 1 ® 1 = A ® ?)D and Dr, = G7 ® 77)/}. Finally, <fD =Dd.
Therefore C(B) is a differential coalgebra with coproduct D, co-
counit 6, and coaugmentation 77. (The counit is, of course, usually
called the augmentation).
Now let X be a differential A-module with product n, A a
commutative ring. Consider Horn (C(B),X) = C*(B;X). heCn(B;X)
is a sequence of homomorphisms hq: C q(B) -> X q_n. The A-module
structure on X induces such a structure on C (B;X). Define the
differential 8 on C*(B;X) by
B) 8(h)(b) = dh(b) + (-l)deg h+1 hd(b).
Then define the cup product C*(B;X) ®A C*(B;X) -. C*(B;X) by
C) (A U h')(b) = ff[(A ® hy>(b)].
Define ?*: A -> C*(B;X) by 6*A)F) = B(b)e, e the identity of X,
and define 77*: C*(B;X) -» A by iy*(A) = A(iy(l)). Then C*(B;X) is
a differential A-algebra with product U, unit 6*, and augmentation
77*. The relation SU = US may be written out as:
D) 5(/i U h') = 8(h) Uh' + (-l)de* h/i U 8(h').
Together with Lemma 29.11, the constructions above imply
PROPOSITION 30.2: Let B be a simplicial set. Then //*(B) is
a cocommutative coalgebra. If X is a differential A-module, then
H*(B;H(X)) is a A-algebra, which is commutative if H(X) is com-
commutative. (For the first statement, assume that H^(B) is A-flat.)
EXAMPLES 30.3: The most common instance of the proposition is
X = A with zero differential. We will have occasion to use the
cases X = C(G), where G is a simplicial group, and X = C*(F;A),
where F is a simplicial set. In the latter case, we must take
X_n = Cn(F;A) to make sense of the grading, and the cun

138
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
is here given explicitly (with n the product in A) by:
E) (h U h') (b) @ = ir[((A ® A ')D D))D (/)].
Next, let G be a simplicial group with product n, and sup-
suppose G operates from the left on a simplicial set F via a: GxF^>F.
Let g denote the Eilenberg-MacLane map and define the Pontryagin
product by:
F) a = C(a)g: C(G) ® C(F) ¦* C(F) .
With F = G and a = n, we find that C(G) is a differential algebra
with product n, unit 77 (base point e0) and augmentation 6. Then
C(F) is clearly a differential C(G)-module, and we have:
PROPOSITION 30.4: Let G be a simplicial group which operates
from the left on a simplicial set F. Then tf*(G) is a Hopf algebra,
which is commutative if G is commutative, and //*(F) is a left
H*(G)-module. //*(F) is in fact a left module coalgebra over//*(G),
in the sense that the module product is a morphism of coalgebras.
(For both statements, assume that H%(G) is A-flat.)
Proof: We need only prove that the chain maps Do and
(ff®a)(l®r®lXD®D):C(G)®C(F)^C(F)®C(F) are chain homo-
topic (taking F=G and a=W, this will imply that ff*(G) is a Hopf al-
algebra). Now if K and L are any simplicial Abelian groups, then
by Theorem 28.3 (with n = 1, s0 = 0, and the models of the previous
section) the following diagram is chain homotopy commutative
i
A(K) <g> A(L)
®D) J
A(K) ® A(L) <g> A(K) ® A(L)
A(K x L)
A(K xL)(g) A(K x L)
By the naturality of D, Do* = (a*(8>o*)D. Therefore Do*g and
(a* <8> a*)(g <8> gXl ® r ® 1X0 ® D) are chain homotopic, as was to
be proven.
LOOP GROUPS AND TWISTED TENSOR PRODUCTS 139
For the remainder of this section, B and F will denote sim-
simplicial sets and G will denote a simplicial group with product n
which operates via 0 on F. Define the cap product:
D: C*(B;C(G)) <g> [C(B) ® C(F)] -» C(B) <8> C(F)
by
G)
PROPOSITION 30.5: The cap product gives C(B)®C(F) a struc-
structure of left differential C*(B;C(G))-module.
Proof: We must prove the following two formulae:
(8) d(fnF®/))
(9) (tuon(i®
Define ^(f): C(B) ® C(F) -> C(B) ® C(G) ® C(F) by
/i@ = (l® f®l)(D® 1),
t e Cn(B;C(G)). To prove (8), it suffices to show that
But:
where we have used formula B) and the relation dD = Dd. It re-
remains to prove formula (9). Here we find:
(f U O D = (l<g>a)(l®F(f<g>OD® 1XD® 1)
= r n(f' D), as desired.

140
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
The cap product is natural in two senses, as shown in the
following lemma.
LEMMA 30.6: Let /3: B -> B' be a simplicial map, let Y: G -> G'
be a simplicial homomorphism, and let a: F -* F' be a y-equivariant
map. If re C*(B;C(G)) and t' e C*(B';C(G')) satisfy y*(Q = j3*(O.
that is, if C(Y) ot = t'o C(/3), then:
A0) (|3*®a*)[tn (i®/)]=('n [/8*D)®
On the other hand, if A e C*(B';C(G)), then:
A1) (l®a*)[A n(j8*(A)®/)]= (jS*®l)[y^
The proof is straightforward, and will be omitted.
REMARKS 30.7: If F = G is the trivial simplicial group and we
replace C(G) by CN(G) = Z, then the cap product reduces to a map
C*(B) ® C*(B) -. C*(B). Explicitly, if f e CP(B) and 6 f Cp+q(B),
then:
A2) tn4 = (l®0D(*) = (-l)p*^PA • KdqQb)fCq(B).
Now let A be a commutative ring. We can and will make
the identification:
A3) C*(B;C*(F;A)) = Horn (C(B) ® C(F),A) .
Then the dual of the cap product, which we shall also call a cup
product, defines a structure of right differential C (B;C(G))-module
on C*(B;C*(F;\)). Explicitly, if h e Horn (C(B)<g> C(F),A), we de-
define:
A4) (A U 0 (b ® /) = hit D ib ® /)), and then
A5) 8(h U 0 = 5(A) U f + (-l)deg hh U 5@ and
A6) hUitU t') = ihU t)U t'.
Here in deriving A5) we have used the special case of B):
A7) 8(h) ib ® I) = (-l)deg h+1 AdD<gi f).
LOOP GROUPS AND TWISTED TENSOR PRODUCTS ¦ 141
At this point we have developed all the requisite machinery
to define twisting cochains. ¦ -' *
Definition 30.8: Let t e Cl(B;C(G)), so that fq:Cq(B)_C ^(G)
Define dt: CiB) ® dF) -» CiB) ® CiF) by:
Using (8) and (9), we find d^ib ® /) = E@ +(Ut)nD®/). t is
said to be a twisting cochain if 5@ + t U t = 0, that is, if
A9) dtn+tn_1d + 'l tiUfn_,. = 0, n>l.
and if etx = 0 (so that (e® e)d, = F® 6)(rD) = 0). Then d, is
called the differential twisted by t and CiB)®CiF) furnished with
this differential is denoted by CiB) <8>, CiF) and is called a twisted
tensor product. Dually, Horn(C(B) ®f C(F),A) is given the differ-
differential 8t defined by A7) with 8t and dt replacing 8 and d, or:
B0) 8tih) = 8ih) +i-l)degh+1hU t.
§31. Brown's theorem
Brown's theorem states essentially that there is a natural
way to assign to every twisting function r a twisting cochain t in
such a manner that CiF xr B) is chain homotopy equivalent to
CiB) ®, CiF). In the last section, this result will be used to con-
construct the Serre spectral sequence.
Unless otherwise specified, the symbols C and C* will de-
denote the normalized chain and cochain functors in this section and
the next, and the symbol (n) will refer to formula (n) of section 30.
We will use the method of acyclic models, and we must first
define a category, the objects of which are all twisted Cartesian
products.
DEFINITION 31.1: Let F xr B and F'x7'B' be TCP's with
'--¦

142
SMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
LOOP GROUPS AND TWISTED TENSOR PRODUCTS
groups G and G', and let Y: G -» G' be a simplicial homomorphism.
A y-map B: E(j) -» E(r') is a simplicial map such that
where a: F -> F' is a y-equivariant map, ft: B -> B' is a simpli-
simplicial map, and </r. B -* G' is a function. Clearly p'0 = /3p. We will
write 6 = (a.fi.ifr) . 6 is said to be y-special if Yt = r'ft. The re-
requirement that 6 be a simplicial map is equivalent to the identities:
= x/,(dob)Yr(b)
) if ;>0
>) if i > 0.
(V)
The composite of a y-map 0 and a y'-map 0' is the y'y-map
0'0 = (a'a,)Q')Q,(<A')8)-(y''A))- With the obvious identity maps, we
have defined a category whose objects are all TCP's and whose
maps are all y-maps. If maps are required to be special, we obtain a
subcategory with the same objects, which we shall call %. 9 will
denote the subcategory of $, the objects of which are all PTCP's.
Observe that if 6 = (a,ft,i//) is a y-map of PTCP's, then neces-
necessarily a = y. If base complexes are required to be reduced, we ob-
obtain subcategories !K0 of $ and 90 of 9. The categories %0 and
(Po will be of primary interest. (We implicitly make them additive; see § 28.)
The symbol F xT B will denote ambiguously an object of
%0 or the corresponding total complex. We now define model ob-
objects in the category $0. Let A[n] denote A[/j]/A[n]°, where
A[n]° denotes the zero skeleton of A[q], and define the models
Mp-q of Ro by:
(i) MP-« = (G(A[p] x A[q]) xT(p) Up].
For clarity, we have here denoted the twisting function
5[p] - G(A[p])
143
Y.,, WrV\ C( AT nT\ /-.n^i-o
fikro fitWn\\ ^ AF/Tll
?(?',") = (gg'.u). Clearly
(ii) M'-<S* (G(A[p])xr(p)A[p])xA[q].
Therefore the realization of each Mp>q is contractible.
Using the models, we can assign a twisting cochain to each
twisting function. We first need a definition.
DEFINITION 31.2: A twisting cochain on the category 90 is a
function T which assigns a twisting cochain T(r)eHorn 1(C(B),C(G))
to each twisting function r. B -> G, B a reduced complex, in such
a manner that the following conditions are satisfied:
(T.I) T(rXb) = e0 - T~\b) for all nondegenerate feefij
(T.2) If Kfe) = eq_1 for all beBq and all q < n, then
for all nondegenerate b e B q and all q < n.
(T.3) If 0 = (y,jQ,</r): GxrB^G'vB' is a y-special map of
PTCP's, then the following diagram is commutative:
T(t)
C(B')
THEOREM 31.3: There exists a twisting cochain on the category
9
Proof: Let G xT B be a PTCP, B a reduced complex. De-
Define T(tI by formula (T.I) and define T(rJ by:
(T.4) T(rXb) = -T~\b) ¦ s0T-\d0b) for all non-degenerate beB2.
Clearly E. r(r)j = 0, and dT(jJ + r(r), cf + 7(r)i U T(r)i =0 is
proven by an easy calculation. Condition (T.2) holds for n = 2
since e1 is degenerate and therefore zero in C(G). Suppose in-
inductively that T(r)t has been defined for i < q, q> 2. We require
that
1
1
'I \

144 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
dT(r)q = -ro-)q_,cf - ^i T(r)i U T(r)q_, = Xq, say.
Clearly ctf(q = O. First consider G(A[g]) xr(q) A[g]. Since
ffn(G(A[g])) = 0 for n > 0, there exists m e Cq_i(G(A[g])) such
that d(m) = Xq(Aq). We define r(r(g))(Aq) = m. Next consider
G(K)xr(K)K, t(K): K-> G(K), K any reduced complex, and let
x t Kq be non-degenerate, x: A[g] -> K induces G(x):G(A[g])->G(K)
and we define:
T(t(K)Xx) = G(x)^(n.) t Cq_, (G(K)). Then we have:
dT(r(K))(x) = G(x)*d(m) = G(I)^ q( A q) = X q(x(A q)) = X q(x).
Finally, consider the arbitrary PTCP G xT B and let b e Bq be
nondegenerate. If t is induced by f(j): B -»W(G), define:
T(f){b) = 0(/(r))J^(Kfi)X^)] f Cq_i(G), where
<?(/(r)): G(fi) -» G is as defined in Corollary 27.2. Then again we
find dT(r)(b) = Xq(b). Condition (T.2) holds since for any
c e Bq, <P(((t))(t(B)(c)) = r(c), and since eq_! is degenerate.
Clearly condition (T.3) is satisfied, and this completes the proof.
We can now define the two functors A and BT: 3\0 -»(_ that
we wish to compare by the method of acyclic models. Thus define
A(F xT B) = C(F xT B) and B T(F xT B) = C(B) ®t C(F), where
t = T(t), T being a fixed twisting cochain on the category J 0. Ob-
Observe that if 6 = (a,f3,i//) is a y-special map of TCP's, then (T.3)
and A0) of Lemma 30.6 guarantee that B T(ff) = /3* <8> a* is a chain
map.
LEMMA 31.4: Both An and (??r)n are representable, n > 1.
Proof: By Theorem 22.2, Corollary 22.3, and Lemma 29.5,
it suffices to prove the result when C(F xT B), C(B), and C(F) are
interpreted as unnormalized chain complexes, T(j) being extended
LOOP GROUPS AND TWISTED TENSOR PRODUCTS
145
to C(B) by letting T(r)(b)= 0 if b is degenerate. Thus, for the
purposes of this proof only, we regard A and BT as being defined
in terms of unnormalized chain complexes. Let F xT B be a TCP
with group G, B a reduced complex. If b tBt, define
Kb) = #/(/)) o G(b): G(A[r]) -» G ,
where /(r): ?? -»W(G) induces t and where <f>(f(j)) is as defined
in Corollary 27.2. Explicitly, y(*>X<r)(u)) = r o fc(u), u f A[r], Let
a: G x F -» F define the action of G on F and let e: A[r] -» G be
the map defined by e(u) = em , u e A[r]m . If I e F s, define
d((,b): Mr-S ->F xT B to be the y(fe)-special map
Now define ?: A n -> A n by
(i) ?/,&) = (d(i,b), ((en, An), An)), where / f Fn, i(flB,
and ((en,An),An)f4(A/"'n).
Then clearly A^=l:.4n-»4n. Similarly, define ?: (BT)n -»(fiT)n
by
(ii) ?fc<8>/) = @(/,fc),Ap<g)(eq,Aq)), where i(flpI ( ( F q,
p+q = n and Ap ® (eq, Aq) e B^il/"'").
Again, A? = 1: (BT)n -*(BT)n. and this completes the proof.
LEMMA 31.5: Hn(A(Mp-q)) = 0 and Hn(BT(Mp-q)) = 0, n > 1.
Proof: Since the realization of Mp'q is contractible,
Hn(A(Mp-<r)) = 0 is clear, n > 1. Now A[g] is contractible and
the operation of G(A[p]) on G(A[p]) x A[g] does not depend on
A[g]. It follows that:
(i) BT(Mp>") and fiT(G(A[p]) xr(p) 3[p]) are chain hornotopy
equivalent.
(A more formal demonstration of (i) will be given in Remarks 31.6).
Let ir = inE[p]). Then tf q(G(A"[p])) = 0 if q > 0 and H0(G(A[p])) is

146 SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
LOOP GROUPS AND TWISTED TENSOR PRODUCTS
147
isomorphic (under the Pontryagin product) to the group ring Z(jt) .
Now n has one generator x for each 1-simplex of A[p] and, re-
regarding Z(n) as a chain complex with differential zero, we may
define a chain equivalence h: CG(h[p]) -> Z(n) by /i(x) = 0 if
deg x>0 and /i(r(x)) = [x] if x t A[p]j. Since h is a morphism
of algebras, we have:
(ii) C(A[p]) ®, CG(A[p]) and C(A[p]) ®h o, Z(n) are chain homo-
topy equivalent, where t = T(r(p)).
(Again, a more formal argument will be given in Remarks 31.6). Now
if x e A[p]n is non-degenerate and a e n, then
n
dht(x®a)= 2 (-1)'' diX(g)a + ht f~l (x <g) a), and
h t 0 (x <g) a) = A ® a) A ® h t ® 1) (D ® 1) (x ® a)
i=0
Since t(d *x) = e0 -
l
x) by (T.I), we find
n-l
By Definition 16.4, the last formula is precisely that for the bound-
boundary in the normalized chain complex of the universal covering com-
complex of A[p]. Since all of the higher homology groups of this vanish,
we have proven the result.
Before stating Brown's theorem, we define filtrations of
A(F xT B) and of BT(F xT B). Thus filter C(F xT B) by:
(A) (i,b) eFpCn(F xT B) provided that b = sj ... s^ b',
where 0 < )x < . .. < jq < n, b' is non-degenerate, and
dimfe' = n-q<p. Izj(/i,ii)eFpCn(FxrB) provided
that each (fi,bi)eFpCn(FxTB).
There results a spectral sequence {Er(F xT B)\ which converges
to H*(F xT B). Dualizing, let A be a commutative ring and filter
C*(F xTB;A) by:
(A*) FpC*(F xT B; A) = Hom(C(F xT ByF^'C^F xT B), A).
The resulting spectral sequence \Er(F xT B)\ converges to
H (F xT B;A). The cup product preserves filtration, as is easily
verified from the definition (A) and C)), and it follows that each
Er is a differential A-algebra. (See e.g. Massey [43,44] for the
construction of spectral sequences and of their products).
Filter C(B) ®, C(F), t = T(t), by:
(B) FpC(B) ®( C(F) = _2n C,(fi) ® C(F).
i= 0
Observe that if b ® / e Fp C(B) ®( C(F), then both (d ® 1)F ® /)
and t 0F ®/) are in Fp~lC(B) ®( C(F), the latter fact follow-
following from the definition G) of D and from the form of D(b). There-
Therefore E (F xT B) = Cp(B) ® H q(F) (not necessarily as a complex)
in the resulting spectral sequence {Er(F xT B)\. Dualizing, filter
Horn (C(B) ®rC(F),A) by:
(B *) FpHom(C(B) ®r C(F),A)=Hom(C(fi) <g>, C(F)/FP~ ^(B) ®(C(F),A).
Then Ep>q(F xT B) = CP(B fl q(F ;A)) in the resulting spectral se-
sequence |f r(F xT B)\.
REMARKS 31.6: Suppose in the situation of A0) of Lemma 30.6
that a* is a chain homotopy equivalence, B - B', and /3 = 1, so
that 100* induces an isomorphism E^FxfB) -» E*(F' xT' B) .

148
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
LOOP GROUPS AND TWISTED TENSOR PRODUCTS
149
Then by the comparison theorem for spectral sequences (see e.g.
Moore [5,52]) it follows that l(g>a*:C(B) <g>,C(F)->C(B) ®t'C(F')
is a chain homotopy equivalence. This argument applies to give a
rigorous demonstration of (i) and (ii) in the proof of the lemma above.
We can now obtain Brown's theorem, which may be regarded
as a direct generalization of the Eilenberg-Zilber theorem.
THEOREM 31.7: There exist natural filtration-preserving transfor-
transformations 0: A -» BT and </r: B T -» A lying over the natural trans-
transformations 0O = T: Ao -* (BT)o and </r0 = T: (BT)o -» Ao (where
T(x ® y) = y <8> x). Any two such 0 are naturally chain homotopic
via filtration preserving chain homotopies, as are any two such </r.
Further, for any such 0 and ifj, 0o</r:fiT-»BT and </r o 0: A -» A
are naturally chain homotopic to the respective identity transforma-
transformations via filtration-preserving chain homotopies.
Proof: To prove the existence of 0 and iji, we apply Theo-
Theorem 28.2 with n=l (since H0(BT(Mp>q)) is non-trivial). Thus
define:
1) = b1
and ^
d1b1
Then an easy calculation using (T.I) proves that dr0i = 0od and
(Ji/fj = ipodt, and the existence of 0 and </r follows from Lemmas
31.4 and 31.5. The existence of the cited chain homotopies follows
from Theorem 28.3 (with n = 1 and s0 = 0). It remains to prove that
0, tft, and the chain homotopies are filtration-preserving. Observe
that the maps d(f,b) defined in the proof of Lemma 31.4 satisfy
(i) BT(d(t,b))(BT(Mn'n))cFpC(B) ®tC(F)
if (f,b)eF"Cn(F xTB), and
¦t>3
h
(ii) A(d(f, b)) (A(M «")) C F"C(F xr B)
if b<g)f eFpCq(B)<8>Cr(F), so that q < p.
Here (i) holds since fe*(u) = 0 unless tn<_p, u e A[n]m. (ii) holds
since bt(u) is here obtained from degeneracy operators applied to
a nondegenerate element of dimension < p if u e A[q]m. Now an
inspection of the proofs of Lemma 29.5 and Theorems 28.2 and 28.3
shows that 0, </r, and the chain homotopies obtained above are all
necessarily filtration-preserving.
§32. The Serre spectral sequence
Using Brown's theorem, we can easily develop the proper-
properties of the Serre spectral sequences \Er(F xT B)\ and {?r(^ ><r ^I»
including the products in cohomology and the identification of the
first non-trivial higher order differentials. We note first the follow-
following corollaries of Brown's theorem.
COROLLARY 32.1: cj>:A-*BT and ijj: B T -> A induce natural
inverse isomorphisms 0r: Er -» Er, </rr: Er -> Er and 0r: Er -» Er,
4,r:Er^Er, r> 1.
COROLLARY 32.2: Define a natural transformation:
(i) DT = @<g) 0)D^: BT -> BT<8> BT, where D is the natural
coproduct A -» A <8> 4-
Then DT is filtration-preserving and is chain homotopy coasso-
ciative and cocommutative. Dually, define
(ii) Ur = Hom(Dr,l): Hom(flr,A)g> Hom(?T,A) -* Hom(fiT,A).
Then each Er is a differential A-algebra, and, for r > 1, 0r and
i/fr are isomorphisms of differential A-algebras.
Now \Er(F xT B)\, and therefore also \Er(F xr B)\, con-
converges to H*(F xr fi), and Elp q(F xT B) = Cp(BHHa(F). The

150
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
dual statements hold for cohomology. To complete the development
of the Serre spectral sequence, we must identify the terms E2(FxTB)
and E2(F xT B). We need the following concepts.
DEFINITIONS 32.3: A twisting function t is said to be n-trivial
if r(b) = ej_1 for all b e B it i <_ n. A twisting cochain t is said
to be n-trivial if tt = 0, i < n. By condition (T.2), if r is n-
trivial, then so is T(t).
REMARKS 32.4: If the group of the fibre bundle determined by
F xT B can be reduced to one such that G, = erf for i < n, then
F xT B is isomorphic to F xT' B, where t' is n-trivial. Converse-
Conversely, if t is n-trivial, then the subgroup G of A(F) generated by
|r(fc)l satisfies Gt = e,- for i < n. If w/B) = 0 for i < n, then
replacing F xT B by a minimal fibre space of the same homotopy
type (first applying S oT to a minimal fibration of the same homo-
homotopy type, if necessary, to obtain Kan complexes), we may suppose
that B j has just one simplex, i <_ n, and then t is clearly n-
trivial.
Now let F xT B be a TCP such that t, and therefore
t = T(t), is n-trivial for some n > 1. By the definition G) of A
and the form of D(b), we find that if b® ( e FPC(B) ®r C(F), then
m(fc(g)/)eFp-"-1C(fi) ®rC(F). It follows that d^d®! and
dr = 0, 2 <_ r <_ n, in the spectral sequence Er(F xr B). Therefore
we have:
THEOREM 32.5: If t is n-trivial for some n > 1, then
(i)
(ii)
E2
xT B) = E2pJF xT B) = Hp(B;Hq(F)), and
pJ
Under the hypothesis of n-triviality, we can also identify
ft-
If
W
LOOP GROUPS AND TWISTED TENSOR PRODUCTS 151
dn+1 and Sn+1. Thus if x e E°(F xT B) survives to En+1(F xT B),
then dn+l(x) is clearly induced by t D x and if h t E0(F xT B) sur-
survives to En+1(FxTB), then Sn+1(h) is induced by (-l)deg h+1h U t.
By A9), since tt = 0 for i < n, we have cftn+i = 0 and
dtn+2 + W = 0.
Thus tn+1 e Hom(Cn+i(B),Cn(G)) defines a cocycle
veHom(Cn+1(fl),ffn(G)),
whose cohomology class in H "+1 (B ;H n(G)) we denote by ia Now
we have
(a) dn+1(x)=vDx, xeH*(B;Ht(F)), and
(b) 5n+1(/i) = (-l)deg/l+1/iU,,, heH*(B;H\F;A))
v can be further identified as follows. Since we may assume that
G,- = et for i < n, we can define a cocycle u e C"(G;ffn(G)) by
u(^) = \i\- u is called the fundamental cocycle of G and its co-
cohomology class will be denoted by y.. Since B is reduced, say
Bo = b0, we may consider u to be an element of
Hom(C0(fl) ®,Cn(G),ffn(G)).
and then u U ^ is defined. Now if b e Bn+1, then:
(u U 0F ® e0) = u(t 0F ® e0)) = u(b0 ® <(fe)) = UF)!=v(fe).
Thus y.\Jv=v, hence 5n+1(/i) = (-l)n+V in ?n+1(G xr B) (with
coefficient ring ff*(G)). In other words, v is the transgression of
(—l)n+1[i in this spectral sequence. We have now proven
THEOREM 32.6: Let F xT B be a TCP with group G such that
Gt = ey for i < n. Let /x <r ff "(G;ff n(G)) = E°;"(G xr B) (with co-
coefficients ff#(G)) denote the cohomology class of the fundamental
cocycle of G. Let v e Hn+\B;H JG)) = E"n++\'°{G xT B) denote
the transgression of (-l)nH:V- Then rf"+-1 on En+1(F xT B) and
8n+i on ?n+i(F xr B) (with coefficients A) are defined by formu-

152
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
LOOP GROUPS AND TWISTED TENSOR PRODUCTS
153
lae (a) and (b) above.
It remains only to identify the UT-product induced on
E2(F ><t &)> where t is 1-trivial. Now under the identification
A3), the cup product E) on C*(B;C*(F; A)) becomes:
Horn (D, 1): Horn(C(fi) <g> C(F), A) ® Horn (C(fi) ® C(F), A)
-> Horn(C(fi)®C(F), A),
where D = A ® T ® 1)(D ® D). Of course DT^D in general. An
easy acyclic model proof shows that D and D' are chain homo-
topic (on the category of Cartesian products), where D'=(/r®/f)DgT.
(Here D is the coproduct on C(F x B), and f(/>6) = F,/).) D' and
?>T both define natural transformations E1 -» E1 ® E1 on the sub-
category 0 of fR0. the objects of which are all 1-trivial TCP's
with reduced base complex. We have
THEOREM 32.7: DT and D'tE1-*!1®!1 are naturally chain
homotopic on the category 0. Thus if t is 1-trivial, then
E2(F xT B) = H*(B;H*(F;A)) as a ring, where the product on the
right is the usual cup product.
Proof: Take as models in 0 all induced TCP's FxrjA(p],
where the reduced complex B is the base of a 1-trivial TCP
F xr B and b: A[p] -» fi. Then E „ is re presentable, p > 1,
since if b: F x b A[p] -> F xr fi covers fc, then defining
= F, Ap ® 1/1)
for 6® }/l <rCp(fi)®ff*(F) we find
® 1/1) = 6® {/}. We claim
that if p + 9 > 2, then E* , <8>E* , = 0 on the models of 0. To
see this, we note first that n = ?ri(A[p]) is a free group. In fact, n
is freely generated by {(i, i +1) | 0 < i < p\. Next, if we choose a
minimal subcomplex M of ST(A[/j]), then M is a K(n,l) and is
therefore isomorphic to ^(#G7,0)) by Theorem 23.6. Since
C(W(K(n,0))) is isomorphic to the bar construction of the group ring
Z(n) (see e.g. MacLane [42, p. 114]), it follows that
ff*E[n]) S ff+(r7;Z)
(the cohomology of n with coefficients in Z, or Tor* (Z,Z)).
Since 77 is free, Hp(K[n]) = 0 for p > 1 (see e.g. [42, p. 123])
and ff j(A[/j]) is free Abelian. By the Kunneth theorem, this proves
our claim. Now we will be able to obtain the desired chain homo-
topy by applying Theorem 28.3 with n = 3 and so = Si=s2 = O
(using induction on the base degree in the proof), provided we can
first prove that D' = DT on E ,, p = 0,1,2. It is easily seen
that we may suppose </r = gT on F lC(B) ®, C(F) and 0 = ft on
F1C(FxTB) when we restrict ourselves to the category 0. Thus
D' = DT on Eo » and on Et „. It remains to prove that D' and
DT ate chain homotopic on E2 „. To prove this, let
A[2] = A[2]/A[2]\
where A[2]J denotes the 1-skeleton of A[2], and define new mod-
models in 0 by Mp = (G(A[2]) x A[p]) x A[2]. A proof similar to
that of Lemma 31.4 proves that E is representable for p > 1.
An argument analogous to that given above shows that if p+q > 2,
then Si+;=2 E. ®E. =0 on the models. Now we can apply
Theorem 28.3 with n = 3 to obtain the desired chain homotopy,
provided we first prove that D' ^ DT on E2p, p = 0,1, 2. Here
a lengthy calculation gives the result, noting that on 0 we may
suppose: (b0 = Bo)
(i) 0(/2,62) = fe2®c9j/2 +d2b2®r{b2)d0f2 +bo®f2 ,
s0T~1(b2)s1T(b2)s1 d0f2
(ii)
sii • ¦ ¦
lC(F xT B), where the sum is
taken over all (n,2)-shuffles (fjL,v).

154
SIMPLICIAL OBJECTS IN ALGEBRAIC TOPOLOGY
REMARKS 32.8: There are several comments to be made.
A) Had we developed the theory of the operation of ni(B)
on nn(F), then the assumption of 1-triviality on r would have re-
reduced to the assumption of n-simplicity for all n. However, the
theory does not lend itself to the development of local coefficients.
B) The treatment of products here is quite unsatisfactory.
While it does show that products can be introduced in \Er(F xT B)\,
it would be more desirable to have a procedure for defining a coprod-
uct directly on C(B) <S>, C(F), without having to throw the definition
back to C(F xT B). In case C(??) is cocommutative, this can be
done using a procedure developed in [45].
C) The topological implications are clear. Starting with a
Serre fibration p: E -» B, one can pass to a minimal sub-fibre space
of the Kan fibration S(p): S(E) -»S(B) and apply the theory above.
BIBLIOGRAPHICAL NOTES ON CHAPTER VI
The functor G was introduced by Kan [30, 31]. Our proof
of Theorem 26.6 follows that of Cartan [6]. This construction gen-
generalizes results of Milnor [49], who defined the functors E, GE,
and G+E. The relationship between G and W was investigated
by Kan [31]. Free simplicial groups have been studied by Kan in
[34,35, and 36]. Also Kan [29] has reproved the Hurewicz theorem
by means of a study of the algebraic situation given by the map
p: G(K) -»A(K) and has shown in [36] that the exact sequence of
Remarks 26.10 is essentially that of Whitehead [63].
The method of acyclic models was introduced by Eilenberg
and MacLane in [12], and was used in [16] to prove the Eilenberg-
Zilber theorem. Our treatment follows these sources and MacLane
[42]. Most of the material of the last three sections is contained,
LOOF GROUPS AND TWISTED TENSOR PRODUCTS 155
in rather different form, in Gugenheim [19]. Section 30 relies heav-
heavily on suggestions of J. C. Moore. A systematic study of Hopf al- -
gebras may be found in Milnor and Moore [51]. The proof of Brown's
theorem is parallel to, but simpler than, the topological proof given
in his original paper [3]. An explicit expression for a twisting co-
chain T(t) in terms of the twisting function t has been obtained
by Szczarba [61].
The Serre spectral sequence was, of course, studied in the
classical paper [57], following its introduction in cohomology by
Leray [38, 39]. The approach here shows that the introduction of
cubical singular theory is unnecessary, a fact shown by Gugenheim
and Moore [20] using quite different methods. Brown [3] proved that
the spectral sequence defined here is in fact isomorphic to that de-
defined by Serre.
Szczarba [61] studied the products in the Wang spectral se-
sequence using twisted tensor products. The form of dn+1 and 8n+1
in the case of n-triviality was discovered by Fadell and Hurewicz
[17], but of course the result is there proven by quite different meth-
methods. A generalization of this result is proven by Shih in [59].

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