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Kenneth S.. Brown


Cohomology of Groups


With 4 Figures






.


Springer...Verlag
New Y ork Heidelberg Berlin








:.:::;,
. ,<:'.
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'-
Kenneth S. Brown
Department of Mathematics
White Han
CorneH U niversity
Ithaca. NY 14853
V.S.A.
Editoriel Board
P. R. Halmos
Managing Ediror
Department of Mathematìcs
Indiana University
Bloomington. IN 47401
US-A.
F. W. Gehring
Department of Mathematics
U niversity of Michigan
Ann Arbor, MI 48109
U.S_A.
C. C. Moore
Department of Mathematics
U niversity of California
BerkeJey, CA 94720
V.S.A.
Library or Cçngt;A\C!"oging in Publication Data
BrowB;Kèimeth S:-- ". -- '. - .
.  . ,
0ihòmo]ogy or groups. .
(l?= tet i mathematie; '87)
Blbh phy.; p.'
Includes index.
L Grpups. Theory of. 2. Homology theory.
I. Title. ft:-Set';es:' .. .
QA171.B876 512'.22 82-733
AACR2
@ 1982 by Springer-Verlag New York Ine.
AlI rights reserved. No part ofthis book ma)' be translated or
reproduced in any Corm without written perrnission from Sprjnger-Verlag.
175 Fifth Avenue New York New York 10010, V.S.A.
Printed in the United Stafes or America
. .
.
#t-  .
9 8 7 6 5 4 3 2 l
.:...._1
. ... f:
" .
..
. .
. . _.....-
"
ISBN 0-38790688-6 Springer-Verlag New York Heidelberg Berlin
ISBN 3-540-90688-6 Springer-Verlag Berlin Heidelberg New Y ork
.
.
Preface
This book is based on a course given at Cornell University and intended
primarily for second-year graduate students. The purpose of the course was
to introduce students who knew a little algebra and topology 10 a subject in
which there is a very rich interplay between the two. Thus I take neither a
purely algebraic nor a purely topological approach. but rather I use both
algebraic and topological techniques as they seem appropriate.
The first six chapters contain what I con sider to be the basics of the subject.
The remaining four chapters are somewhat more specialized and reflect my
own research interests. F or the most part, the only prerequisites for reading
the notes are the elements of algebra (groups rings, and modules, including
tensor products over non-commutative rings) and the elements of algebraic
topology (fundamental group, covering spaces, simplicial and CW-com-
plexes, and homology). There are however, a few theorems especially in
the later chapters, whose proofs use slightly more topology (such as the
Hurewicz theorem or Poincaré duality). The reader who does not have the
required background in topology can simply take these theorems on faith.
There are a number of exercises, some of which contain results which are
referred to in the text. A few or the exerdses afe marked with an asterisk to
warn the reader that they are more difficult than the others or that they require
more background.
I am very gratefu1 to R. Bieri, l-P. Serre, U. Stammbach, R. Strebel, and
C. T. C. Wall for helpful comments on a preliminary version or this book.

Notational Conventions
.....
All rings (including graded rings) are assumed to be associative and to ha ve an
identity. The latter is required to be preserved by ring homomorphisrns.
Modules are understood to be left modules, unless the contrary is explicitly
stated. Similarly, group actions are general1y understood to be left actions.
If a group G acts on a set Xt I wiIl usual1y write XJG instead of G \X for the
orbit set, even if G is acting OD the left. One exception 10 this concerns the
notation for the set of cosets of a subgroup H in a group G. Here we are talking
about the left or right trans1ation action or H on G, and I wilJ always be
careful to put the H on the appropriate side. Thus GfH = {gH:g E G} and
H\G = {Hg:g E G}.
A symbol such as
L f(g)
gE GIH
indicates that f is a function on G such that f(g) depends only on the coset gH
of g; the sum is then taken over a set of coset representatives.
Finally I use the Utopologists notation"
ln = 7l.fn71.;
in particular, lLp denotes the integers mod p, not the p-adìc integers.
.
",
>'
-r-
....
.:
.
VI
Contents
lntroduction
CHAPTER I
Some Homological Algebra
O. Review of Chain Comp]exes
l. Free Reso]utions
2. Group Rings
3. G-Modu]es .
4. Resotutions of 7L OVer?LO via Topology
5. Tbe Standard Resolution
6. Periodic Resolutions via F ree Actions on Spberes
7. Uniqueness of Resolutions
8. Projective Modules
Appendix. Review of Regular Coverings
CHAPTER II
The Homology or a Group
I. Generalities
2. Co-invariants
3. Tbe Definition of H.G
4. T opol ogical Interpretation
5. Hopfs Theorerns
6. Functoriality
7. The HomoJogy of Amalgamated Free Products
Appendix. Trees and Amalgamations
CHAPTER III
Homology and Cohomology wì th CoeffÌcìents
O. Prdiminaries on (8)G and Horna
l. DefÌnition of H.(G, M) and H+(G. M)
i
l
4
4
IO
12
13
14
)8
20
2i
26
]]

33
33
34
35
36
41
48
49
52
55
55
56
..
Vll

-
Contents
.
,
"
...
VIII
2. Tor and Ext
3. Ex(ension and Co-extension of Scalars
4. Injective Modules
5. Induced and Co-induced Modules
6. H. and H* as Functors ofthe Coefficient Module
7. Dimension Shifting
8. H. and H-' as Functors ofTwo Variables
9. The Transfer Map
lO. App]ications of the Transfer
60
62
65
61
71
74
78
80
83
,

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CHAPTER IV
Low Dimensional Cohomology and Group Extensions
I. Introduction
2. Split Extensions
3. The Classification or Extensions with Abelian Kernel
4. Application: p-Groups with a Cyclic Subgroup of Index p
5. Crossed Modu]es and H3 (Sketch)
6. Extensions With Non-AbeHan Kerne] (Sketch)
..
...'
. .
86
86
87
91
97
102
104
! .
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......
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.::..
'",,.
l
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(":"
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1.:...
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CHAPTER V
Products
L The Tensor Product of Resolutions
2. Cross-products
3. Cup and Cap Products
4_ Composition Products
5. The Pontryagin Product
6. Application: Ca1cutation of the Homology of an Abelian Group
107
107
108
109
114
117
121
,.-
CHAPTER VI
Cohomology Theory or Finite Groups
L lntroduction
2. Relative Homological Aigebra
3. Complete Resolutions
4. Definition of il *
5. Properties of il"
6. Composition Products
7. A Duality Theorem
8. CohomologicaUy Trivial Moduies
9. Groups with Periodic Cohomology
128
128
]29
131
-.
134 ::
136 "
.!
.-'
142 ...-
:
144 :.: :
.:,
148
153
CHAPTER VII
Equivariant Homology and Spectral Sequences
L Introduction
2. The Spectral Sequence of a FìJtered Complex
,r,.
)i;-
!,.;,:J!'
: '.
161
16]
161
Conteots
3. Double Complexes
4. Example: The Homology of a Union
5. Homology of a Group with Coefficients in a Cbain Complex
6. Example: The Hochschìld-Serre Specrral Sequence
7. Equivariant Homo]ogy
8. Computation of dI
9. Example: Amalgamations
lO. Equivariant Tate Cohomology
CHAPTER VIII
Finiteness Conditions
L lntroduction
2. CohomologicaJ Dimension
3. Serres Theorem
4. Resolutions of Finite Type
5. Groups of Type FPn
6. Groups ofType FP and FL
7. T opological InterpTetalion
8. Further Topologicai Resu]ts
9. Further Examples
IO. Duality Groups
1 t. Virtual Notions
-
:-----.....
. r. k-.. M A 'jr.. '4
".. ,I-. ,.. - . Ci 1 1>0
)' ..... . .---.---....... 'I,;,
. 'èt-;Q: ,. 
,( :
04:-:----- ------ 'f(..
IL<'EAIT A' t: .
CHAPTER IX
Euler Characteristics
1. Ranks or Projective Modules: lntroduction
2. The Hattori-StaJJings Rank
3. Ranks Over Commutative Rings
4. Ranks Over Group Rings; Swan's Theorem
5. Consequences or Swan's Theorem
6. Eu]er Characteristics of Groups: The T orsion- Free Case
7. Extension to Groups with T orsion
8. Euler Characteristics and Number Theory
9. lntegrahty Properties of X(f)
lO. Proof of Theorem 9.3; Finite Group Actions
Il. The Fractional Part of X(r)
12. Acyclic Covers; Proof or Lemma 11.2
13. The p-Fractional Part of X(r)
14. A Formula for Xr(..W")
CHAPTER X
Farrell Cohomology Theory
t. lntroduction
2. Complete Reso]utions
3.--Definition and Properties of n+(r)
.
IX
'j
164
166
168
171
172
175
178
180
]83
183
184
190
191
197
199
205
210
213
219
225
230
230
231
235
239
242
246
249
253
257
258
261
265
266
270
273
273
273
277

x


I,
,
,
f


'""


Contents


I


4. Equivariant Farrell Cohomology
5. Cohomologically Trivial Modules
6. Groups wirh Periodic Cohomology
1. Jì.(r) and che Orde re<! Set of Finite Subgroups or r


,
.
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28]
287'
288
29]


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References 295
Notation Index 301
Index 303


.


,i


-
i


Introduction


Tbe cohomology theory of groups afose (rom both topological and a1gebraic
sources. The starting point for the topological aspect of the theory was the
work of Hurewicz [1936] OD "aspherical spaces.

 About a year earlier,
Hurewicz had introduced the higher homotopy groups nlrX of a space X
(n > 2). He now singled cut for study those path.connected spaces X whose
higher homotopy groups are ali trivial) bu t w bose fundamental group
7t = 1t l X need not be trivia1. Such spaces afe called aspherical.
Hurewicz proved, among other things, that tbe homotopy type of an
aspherical space X is completely determined by its fUDdamental group re. In
particular, the homology groups of X depend only on x; it is therefore reason-
able to t hink of them as homology groups oJ 7L [This terminology, however, was
not introduced until the 1940's.] Throughout the remainder of this introduc-
tion, then
 we will write H * 1l for tbe homology of any aspherical space with
fundamental group n. (Similarly, we could define homology and cohomology
groups of 7t witb arbitrary coefficients.) As a simple example, note that
H 2(Z E9 Z) = 7L. [T ake X to be the torus.] Although H urewicz considered
only tbe uniqueness and not the existence of aspheriçal spaces, there does in
fact exist an aspherical space with any given group as fundamen tal group-
Thus our topological detìnition of group homology applies to ali groups
F or any group 7t we obviously have H o 1t = 7L and H (1t = 1tgb, the latter
being tbe abelianization of x, i.e., 7t modulo its commutator subgroup. Far
n :2:: 2, however, it is by no means clear how to describe HI/n algebraical1y.
The first progress in this direction was made by Hopf [1942]t who expressed
H 21t in purely algebraic terrns. and who gave further evìdence or iiS impor-
tapce in topology by proving the following theorem: for any path-connected
space X with fundamental group n. there is an exact sequence


(0.1)


11: 2 X -+- H 2 X --+ H 2 1t --+ O.


l




2
IotroducUoo
-'
[To put this result in perspective one should recall that Hurewicz had
introduced homomorphisrns hft: 7tJlX -)o H"X (n 2: 2) and had shown that
h" is an isomorphism if '!tI X = O for i < n. In particular h2 is an isomorphism
if 1t = '1r1X = O. When n is non-trivìaI however, h2 is in generaI neither in-
jective nor surjective, and HopCs theorem gives a precise description. in
terms of 1t of the extent to which surjectivity fails.]
Hopf's description of H 2 n. inddentally. went as follows: Choose a presen-
tation of 1l as F I R., where F is free and R <J F; then
J"'
. .. 
, I..:'.
; .:::.{-:-..
..l.
.......,
.?'.
";.".r,,'
:.
,.
(0.2)
H21t = R n [F, F]/[R. F],
where [A, B] for A, B ç F denotes the subgroup generated by the com-
mutators [a, b] = aba-1b-1 (aeA. beB).
Following Hopf's paper there was a rapid deveIoprnent of the subject by
Eckmann. Eilenberg- MacLane, Freudenthal, and Hopf. (See MacLane
[1978] for some comments about this development.) In particular one had
by tbe mid-1940's a purely algebraic definition of group homology and
cohomology, from which it became clear that tbe subject was or interest to
algebraists as well as topologists. Indeed. the low-dimensional cohornology
groups were seen to coincide with groups which had been introdnced much
earlier in connection with various algebraic problems. H\ Cor instance,
consists of eq uivalence classes of ., crossed homomorphisrns" or ,. deri V8-
tions." And H2 consists of equivalence classes of "factor sets" tbe study of
which goes back to Schur [1904], Schreier [1926], and Brauer [1926]. Even
H3 had appeared in an algebraic context (Teichmiiller [1940]). These are
the algebraic sources of group cohomology referred to at the beginning of
tbis introduction. (Of course, there had been nothing in this a1gebraic work
to suggest that there was an underlying ,. homology theory'; this had to wait
for tbe impetus from topology.)
It is not surprising, in view of this history, that the subject of group co-
homology offers possibilities for a great deal of interaction between algebra
and topology. For instance a 'transrer map." motivated by a classical
group-theoretic construction due to Schur [1902], was introd uced into group
cohomology (Eckmann [1953], Artin- Tate [unpublished]) and from there
into topology, where it has become an important tool Another exarnple is
the theory of Euler characteristics of groups. This theory was rnotivated by
topology. but it has applications to group theory and number theory.
Our approach to the subject will be as follows: We begin in Chapters I
and I l by defining H * 1t from the point of view of "homological algebra.u
This is the point of view which had evolved by the end of the 1940s. The
topological motìvation, however will always be kept in sight, and we will
immediately obtain tbe topological interpretation of H * n in terms of
aspherical spaces. In particular. we will prove 0.1 and 0.2.
Chapter 111 contains more homological algebra, involving homology and
cohomology with coefficien ts. These arise natural1y in applications, both in
algebra and topology. They are also an important technical tool, since they
,-:.o,"
..-
lotroduçtion
3
 j
,
make it possible to prove theorerns by .. dimension-shifting." In Chapter IV
we treat the theory of group extensions which involves the crossed homo-
morphisms and factor sets mentioned above.
Cbapter V introduces cup and cap products (motivated by topology), and
these afe then used in Chapter VI in the study of the cohomology of finite
groups. Much ofthe material in Chapter VI (such as the '"Tate cohomology
theory") was originally motivated by algebra (c1ass field theory), but it
turns out to be related to topological questions as well, such as the study of
groups acting freely on spheres.
In Chapter VII we introduce spectral sequence techniques, which are
used extensively in tbe remaining chapters_ The reader is not expected to
bave previously seen spectral sequences; I give a reasonably self-contained
treatment, omitting only some routine (buI tedious) verifications.
Beginning with Chapter VIII the emphasis is on infinite groups, with the
most interesting examples being groups of integraI matrices. In Chapter VIII
we discuss various finiteness conditions which can be imposed on sucb a
group to guarantee that the homology has nice pro perti es. Chapter IX
treats Euler characteristics. which can .be defined under suitable finiteness
hypotheses. This theory, as we mentioned above. has interesting connections
with number theory. Finally, Chapter X develops the "Farrell cohomology
theory," which is a generalization to infinite groups ofthe Tate cohomology
theory for finite groups.

CHAPTER I
Some Homological Algebra


o Review of Chain Complexes


We
ollect bere for ease of reference some terminology and results concerning
cham complexes. Much ofthis will be weJl-known to anyone who has studied
algebraic topology. Tbe reader is advised to skip this section (or skim it
lightly) an
 refer back to it as necessary. We will ornit some of the proofs;
these are elther easy or else can be found in standard texts. such as Dold
(1972], Spanier [1966], or MacLane (1963].
Let R be an arbitrary ring. By a graded R-module we mean a sequence
C = (C/t)I1EZ of R-modules. If x E C"' then we say x has degree ti and we write
deg x = 1'!. A map of degree p from a graded R-module C to a graded R-
module Cf is a family f = (f.t: C" -+ C
 + p)ne Z of R-module homomorphisms;
thus deg
I(x» = degf + deg x. A chain complex over R is a pair (C, d)
where C IS a graded R-moduIe and d: C - C is a map of degree
 1 such that
d2 = O. The map d is called tbe differentilll or boundary operator of C. We
often suppress d from the notation and simply say that C is a chain complex.
We define tbe cycles Z(C)
 boundaries B(C), and homology R(C) by Z(C) =
ker dt B(C) = im dt and H(C) = Z(C)jB(C). These are alI graded rnodules.
One often comes across graded modules C with an endomorphism d of
square zero such that d has degree + I instead of - 1. In this case it is custoro-
ary to use superscripts instead or subscripts to denote the grading, so that
C = (C")lId and d = (dft: C" -+ cn+ 1). Such a pair (C, d) is called a cochain
complex. There is no essential difference between chain complexes and
cochain complexest since we can always convert one to the other by setting
C" = C-Il. We will therefore confine ourselves, for themost part, todiscussing
chain complexes
 it being understood that everything applies to cochain
complexes by reindexing as above. [Note, however, that there is a difl'erence
when we consider non-negative complexes, i.e., complexes such that C
11


. ,-
'-'-.
_.11
....
. .
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:. '.
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<


0,.


4


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o Revìew or Chain Complexes


5


[or C'1 = O for n < O; if the differentiaJ is thought of as going from lert to
right, then a non-negative chain complex extends indefinitely to the lert,
whereas a non-negative cochain complex extends indefinitely to the right.] In
discussing cochain complexest one often prefixes "co" to much of tbe teT-
minology; tbus d may be called a coboundary operator, and we have co-
cycles Z(C), coboundaries B(C), and cohomology H(C) = (H"(C»"eZ'
If (C, d) and (Cf, df) are chain compJexes, then a chain map from C to C' is a
graded module homomorphism f: C -+ C' of degree O such that dj = Id. A
homotopy h rrom a chain map Ito a chain map 9 is a graded module homo-
morphism h: C -+ C' of degree l such that dfh + hd = f - g. We write
f
 g and say thatfis homotopic to g if there is a homotopy from/to g.
(0.1) Proposition. A chain map f: C -+ C' induces a map H(f): H( C) -+ H( C'),
ami H(f) = H(g) iff
 g. D


Tbe abelian group of homotopy c1asses of chain maps C ---. Cf will be
denoted [C, C']. It is ofien useful to interpret [C, Cf] as the Q-th homology
group of a certain '6function complex" K.o?Jt:R(C, CI), defined as follows:
.)f'
R( C, C')11 is the set of graded module homomorphisms of degree n rrom
C to C' (thus .Jf.o?HR(C, C')" = n4E.l HornR(Cq, C
+II)]' and the boundary
operator D,.: Jt'mnR(C, C')n...... .Jt'
R(C, Cf)"_l is defined by DlI(f) =
d1- (-l)"fd. [rhe sign here makes D2 = O. It is also consÌstent with other
standard sign conventions, cf. exercise 3 below.] Note that the O-cycles are
precisely the chain maps C -+ Cf, and the O-boundaries are the null-homo-
topic chain maps. Thus Ho(.J'fcmR(Ct C'») = [C, C']. More generally. there
is an interpretation of Hn(.)t'omR(C' Cf» in terms of chain maps. Consider
tbe com
lex (1:" C, I:,"d) defined by (1:"C)p = Cp_,p I:,"d = (-l)lId; this
complex 18 called the "-fola suspension of C. [lf n = l. we write1:C instead
of l?C.] Let [C, Cf]1'I = [l''C. C']. Then we have H,,(JltomR(C
 C
)) =
[C, Cf]". The elements of [ . ]n are called homotopy classes of chain maps
01 degree n.
A chain map f: C ..... C' is called a homotopy equivalence if there is a chain
map l': C' -+ C such thatf'f
 idc andf f'
 ide-. And a chain mapfìs caned
a weak equivalerrce if H(f): H(C) -+ H(CI) is an isomorphism.


(O
2) Proposition. An y Iwmotopy equivalence is a weak equivalence. D
A chain complex C is called contractible if it is homotopy equivalent to
the zero complex, or, equivaIently
 if idc
 O. A homotopy from idc to O is
called a contracting homotopy. Any contractible chain complex is acyclic. ie.
H( C) = O. '


(O
3) Proposition. C is contractibiejf and only if it is acyclic and each short
eXQct sequence O -+ Zn+ 1 C. eli + 1
 ZII -+ O splits, where d is induced by d.
PROOF. If h is a contracting homotopy, then (h I z): Z -+ C splits the surjection
à: C -+ Z. Conversely, suppose we have a splitting s: Z -I> C, whence a




6
I Some Homological Algebra
graded module decomposition C = ker a E!J irn s = Z E9 im s. We then get
a contracting homotopy h: C -- C by setting h I Z = s and h I im s = O. D
(0.4) Proposition. A short exaCl sequence O....,. C' --4 C  C"  O oJ chaiPl
c'omplexes gives rise to a long exact sequence in Iwmology:
H (C') H(;) H (C) H(JI) H (C") i! H (C')
. . . -+ Il "'" }-  ---+ 1'1- l -+ - . . .
The . connecting homomorphism" ò is nalUral, in the sense that a commuta-
tive diagram
..t-
',.
1'-
"t.'
"
o
· C'
,e
1
C"
)
...0
1
1
o
E"
,
.0
) E'
)E
with exact fOWS yields a commutative square
H rt( C")
1
H ,JE")
) Hrt-1(C')
1
, Hn-I(E').
D
(0.5) Corollary The inclusion i: C' ---+ C is a weak equ;valence if and only if
CH is acyclic. D
This shows that the cokernel C't of i is the appropriate o bject to consider
if we want to measure the "difference" between R(C) and H(C'). We now
wish to define a "homotopy-theoretic" cokernel for an arbitrary chain map
f: C' --+ C, which plays the same role as the cokemel in the case of an
inclusion: The mapping cone of f : (C', d') -+ (C, d) is defined lo be the complex
(C" d") with C" = C tE> I:C' (as a graded modu1e) and d"(c, c') = (dc + le',
-d'c'). In matrix notation, we have
d" = ( lr)-
See exercise 2 below for the moti vation for this definition.
(0.6) Proposition. Letf: C' -+ C be a chain map with mapping cone C". There
is a long exact homolog y sequence
- - - -+ H(C') H(f) H,,(C) -+ Hn(C") - HII-1(C') --+ .. . _
In particular,fis a weak equivalence if and only ifC" is acyclic.
PROOF. There is a short exact sequence O --+ C -+ C" --+ :EC' ....... O; now apply
(0.4). By checking the definition or the connecting homomorphism H n(I:C')
-4 H" -1 (C), one finds that it equals 811-1(!); H ,,-I (C') -+ H,.-; l (C). D
o Review or Chain Complexes
7
Tbe mapping cone is alsa userul for studying homotopy equivalences, not
just weak equivalences:
(0.1) Proposition. A chain map f: C' -+ C is a homotopy equivalence if arul only
if its mapping cone C" is contractible.
PROOF. A straightforward computational proof can be found in tbe standard
references (or can be supplied by the reader). F or the sake or variety, we will
sketch a conceptua] proof. Suppose first that C" is contractible_ One then
checks easily that tbe function complex.Jff CJnR(D, C") is contractible for any
complex D; in particular, it is acyclic. One also checks that.Jff'umR(D. C") is
isomorphic to the mapping cone of:K DmR(D,f) : jf' ()mR(D, C') -+.J't omR(D, C).
It therefore follows from (0.6) thatjf' mnR(Df) is a weak equivalence. Looking
at Ho, we deduce thatfinduces an isomorphism [D, C'] --+ [D, C] for any D;
hence f is a homotopy equivalence by a standard argument. Conversely,
suppose f is a homotopy equi valence. Then one shows easily that jf' mn R(D,f):
Jf' bmR( D, C') -+.K .tWtR(D. C) is a homotopy equivalence, so its mapping cone
:Jf£onlR(D, C") is acyclic by 0.6. In particular, [D, C"] = O for any D, and this
implies that C" is contractible. D
Finally, we recall briefly the Kiinneth and universa) coefficient t heorerns.
If (C, d) Cresp- (C', d'» is a chain complex of right (resp. left) R -modules, then
we deftne their tensor product C Q9R C' by (C <8IR C')n = EB,,+q=n C" Q9[{ C.
with differential D given by D(e C8> c') = dc @ c' + ( - 1 )del £c Q9 d'c' for c E C,
C'E C'. The sign here can be remembered by means of the following sign
convention: When something of degree p is moved past something of degree q,
the sign (-1)J1q is introduced- [In the present case, the differential, whicb is of
degree -1, is moved past c, so we get the sign (-1) -de&c = (l)dCl!:c.] Note
that C @ R C' is simply a complex or abelian groups for generai R but it is a
complex of R -modules jf R is commutative.
(0.8) Proposition (Kiinneth Formula). Ler R be il principal ideai domain and
let C and C' be chain complexes such that C is dimensio n-wise free. There afe
naturalexacrsequences
0-+ EB Hp(C) Q9R HII_p(C') -+ H"(C Q9R C')
pl.
-+ EB Torf(H "cC). H,.-p-I(C'» -+ O
peZ
and
o -+ n Ext1(Hp(C). Hp+rt+ l(C'» --+ Hn(:/fUTnR(C, C')
p".Z
-+ o HomR(H,,(C), Hp+,,(C'» -+ O.
PEZ
aruJ. rhese sequences spUto
o

.
>.
8
I Some Homologica1 Algebra
o Review or Chaio Complexes
9
We will not recaIl the definitions ofTor and Ext at this point since we will be
defining them in much greater generality in I11.2.
An important special case or 0.8 is that where C' consists or a single
module M. regarded as a complex concentrated in dimension O (i.e. C = M,
C = O for n =F O). In this case 0.8 is caUed the universal coeffic ient theorem,
and the exact sequences take tbc following Corro:
O -Hn(C) @R M ----io H,,(C @R M) ----io Torf(Hn_t(C), M) --+ O
,>
'l' "

.
:...
3. Given J,.I EJt' t"mR{ C, C)p and x E C'I' set <Il. x> = u(x) E CT q - Let d, d', and D be the
differentials in C. C. andomR(C. C). respectively. Verify that d'<u.x) = (Du, x) +
(-IY(u dx). [In fact. this is nothing but a restatement ofthe definition of D, but it
is a convenient Corro in which to remember that definition.] In otber words. the
evaluarion mapJf'bfflR,(C, C) @ C  C. given by u @ X I-t> (u. x> is a chain map.
i" . )
i .
, ,
, 'i:
::,:'
and
O -t> Extl(H,,_ I(C), M) --+ H"(R(C, M» ----io HomR(H,,(C) M) --+ O.
[Here e are following standard conventions ìn regarding .Jf'mnR(C, M)
as a cocham complex, with .}f',mnR{ C, M)lJ = Jf'omR( C, M) _ Il = HomR( C"' M);
the last equality comes from the fact that the only non-zero component or a
graded map f: C --+ M of degree - n is .In: Cn --+ M.]
4. Given v E R(C, C)1Il and U E Jf'm-nR(C" C"}p, their composite U <J V is in
.Jf'.o-mR(C. C")p+qo
Veriry that D(ti c v) = Du o v + (-l)pu o Dv. where D denotes the differential
in any or the three JIf pm complexes  il1 other words. composition or graded maps
defines a chain map.H'R(C" C") @Jt"lWJlR(C, C) --+ 3f'.omR(C, C"). [Hint: This and
the remaining exercises are most convenient]y done b}' taking the detinition of D
in the form given in exercise 3. From this point of view. one starts with the equation
(u,-, v, x) = (u. (v. x» (x E C), which is tbe definition of composition and one
applies d" to botb sides. Applying exercise 3 several tirnes, one obtains
EXERCISES
(D(u c v). x) + (-l)p+q(u G v, dx) = (Du. (v, x» + (-l)P(u. (Dv. x»
+ (-l)P+Q(u. <v. dx»
L Let T: (R.rnoduJes)'-" (S-moduJes) be a covarianr funetor which takes the ro
module to te zero module (or, equivalently. which takes zero maps to zero maps).
For any cham complex (C. tI) over R, there is then a chain complex (TC, Td) over S_
If T is an exact functor, show that H(TC)  TH(C). [Recall that T is e:wct if il
ces, exact sequences to exact sequences. It follows that T preserves injections,
SUfJectIons. kernels. cokemels. etc.]
'.'
which simplifies to
<D(u" r.,1. x) = <Du" v. x) + ( -lY(u c Dv. x),
-.
. -;.
,...
. .
,
Y..
as requiced.]
2. (Motivation for the definition or the mapping cone) Given a chain map f: C' _ C,
tbe .. homotopy theoretic cokernel" of f should 6t into a diagram C' .4 C -4 CI with
gf  O. Thus C" must receive a chain map g [oE degree O] from C and a bomotopy
h [f degree 1] from C'; this suggests setting C" = C €t) I:C as a graded module. and
takmg 9 and h to be incIusioDS, Show that the definjtion of the boundary operator d"
is now forc:ed on us by the requirement that 9 be a chain map and that h be a homotopy
from gfto O. [Using matrix notation set
..'"':c'"
N. .
I..
, .
. ..

;.-.:
.'.
.'0:
5. If C and C' are chain complexes over a commutative ring R. there is an isomorphism
of graded modules C @R C  C' @R C given by c @ c' 1---+ (_t)del!f'dlgc'c' @ c.
Prove that this isomorphism is a chain map,
-'
6- LeI C be a complex of righI R-modules. C' a complex or lert Rmodules, and C'a
complex of li -modules.
(a) There is a canonica] isomorphism or graded abelian groups
cp: .K01X£{C @R c, C")  Jf'09nR(C, Kmnz(C', e")),
given by < <q>(). c), c') = <u. c (8) c') for u E:l(C <2$JR C. C"), C E C, c' € c.
[Sketch of prooL An element of )f'.omz(C @R C'. C')n is a family or Z-module maps
Cp Q9R C - C;+q+lI' In view of the universal mapping property of the tensor
product. this is the same as a family of R-module maps Cp - HOm.l(Cf C;+q+n). Le..
as a graded R -modu]e map C - £'OMZ( C'. C") or degree n. Note here that .¥&m.zf,.C'. C")
is indeed a complex of right R-modules via the lert action of R on C' and the con-
travariance of Hom in the first variab]e: (urXc') = u(rc') for u Ez(C'. C"), ,. E R.
C'E C'.] Show that ({) is a chain map. [Hint: let d. d', and d" be the differentiats in C.
C, and C", and let D be the differential in any oftheJtbm complexes under considera-
tion_ Apply d" to both sides or thc equation defining qJ; using exercise 3 several times.
you should obtain
'.
dIr = (; )
and note that
. ;'.
g = () and h = ().
Now write oul the matrix equations dN g = gd and d"h + hd' = gf and salve for
IX, fJ. i" «5.]
Tbe remaining exercises are designed lo illustrate various compatibility properties
ofour sign conventions in Jf'm:n(-. -) and - @ -. Few readers wiIJ bave tbe patìence
to do aH of them. but you should at least do enough to convince yourself that alI reaSOD-
abIe identjties involving Kmn and (81 are true. provjded one foJJows the sign conventìon
stated above.
. .
..
, ,
:.i, .
< (D(.-p(u). c). c') + (-lY< (tp{). dc), c') + (_1)J1"'4( <tp(u), c). d'c')
,
 
on the Jen. where p = deg u and q = deg c. and
<Du. c @ c') + (-I)"<u. dc (8) c') + (-ly+q(u, c @ d'c')
..
. .
..
"
, 
<

IO
I Some Homolosìçal Algebra
'.
.-
.
.
;
; .
'. '.
OD the right. Remembering the definition of f{J. you can conclude that
< (D(rp{u)), c). c') = (Du. c Q9 c')
and hence that < (D(4p(u», c). c') = < < (,O(Du). c). c'>.]
(b) uce.from (a) [or check directly] that a map u: C (8)R C' --.. C" of degree O js a
cham map dI' the corresponding map C -io :1l'omz(C'. C") is a chain map.
.
. . 
.'
.'
......'..;.
ti.:
..}t
.:......
"_-!. :"
r -2:'..
- .'
:i'-;
'.
,',
.
',

7. let C and C' (resp. E and E') be compJexes or right (re.sp. left) R-modules. Given
U E Jf".onlR(C. C') and v e.Jfl'MnR(E, E'). their tensor producl
u @ vE.Jf'mnl:(C (8)M E. C' 0R E')
is denned by
(u (8) v. c 0 e) = (-l)de:8".dI!<'<u, c) @ (ti, e> for C'E C. e E E.
(a) Let D be thc dHferential in any ofthe three-*mn complexes under consideration.
Verify that D(u <E> v) = Du @ v + (_1)de8I1u (8) Dv. In other words, the operation
'.tensor product of graded map.s" defines a chain map
£mnR(C, C) @JF.omR(E, E') -lo Jf'omz{C @RE. C' 0R E').
[Hint: Once again. you are advised to do this by differentiating both sides of the
equation defining U @ v.]
(b) Deduce from (a) (or chcck direcUy) that jf u: C  C' is a chain map (of degree O),
then (here is a chain map mR( E. E') -+ jf"l»J1!.z( C @ R E, C' (8) RE') given by v I--t
u@v.
(c) Deduce from (a) (or check directJy) that the tensor producI of chain maps is
copatible with homotopy, in the following sense: Given chain map.s and homo-
toples Uo ::= (lI: C - C' and Vo ...... Vl: E -+ E'. one has!lo @ Vo  u1 @ v] : C (8J1I E -+
C' @R E'. [Hint: Uv and UI are bomologous O-cycIes in R(C. C'), and similarJy
for Vo and VI; 11 now foIlows from the boundary formula in (a) that 1.1.0 (8) Vo and
u 1 @ VI are bomologous O-cycJes in £onrz( C @ R E. C @R E').]
1.

"=.Js.
...
.:.t
...
.S
','
"
".
. ":
;:-
.....
.;
__o:
f.'
.'1. :
" ,
. ,
-, .
. 
"." ..:
- -.
.'
.'.
:!
8. Pro \le that thc tensor product operation or thc previous exercise is compatible with
composition of graded maps. i.e" that
(u @ r..-)" (u' @ li') = (_l)deKZJ-del!lI'(u c u') @ (v o v')
whenever the cOmpOSl[eS are dcfined.
1 Free Resolutions
Let R be a ring (associative, with identity) and M a (Iert) R-module. A
resolution ol M is an exact sequence or R-modules
. . . - F 2  F l  F o -!. M -+ O.
If each Fi is free, then this is called afree resolution. Free resolutions exist
for any modu)e M and can be constructed by an obvious step-by-step
1 Free Resolutions
1]
procedure: Choose a surjection t;: F o --+ M with : o. ree, then choose a sur-
jection F 1 --+ ker e with F l free, etc. Note that thc IDlttal segment F l --+ F o -
M --+ O of a free resolution is simply a presentation of M by generators and
relations.
There are two useful ways to interpret a reso1ution in terrns of chain
complexes.1 Tbe first is to regard the resolution itself as an acyclic chain
complex with M in degree - 1. We wiU rerer to this as the augmented chain
complex associated to tbe resolutioD. Thc second way is to consider the non-
negative chain complex F = (Fi, ÒJi2: o and to view E: F - M as a chain map,
where M is regarded as a chain complex concentrated in dimension O. The
exactness hypothesis, from this point of view, simply says that 8 is a weak
equivalence.
In case there is an integer n such that Fi = O for i > n, wc will say that the
resolution has length < n. In this case we will simply write
0- FII -+. -. -+ Fo - M - O,
it being understood that the resolution continues to the Iert with zeroes.
EXAMPLES
1. A free module F admits the free resolution
'd
O-+FF-O
or length O.
2. If R = Z (or any principal ideai domain) then submodules of a free
module are free, hence any module M admits a free reso1ution
O-F1-+Fo-+M--+O
or length < 1. For example, the Z-module 7L2 = Z/27L adrnits the resolution
O - 7L. ..; 7L - "£2 - o.
3. Let M again be 7L 2' regarded now as a module over the polynomial ring
R = Z[T], with T acting as O (i.e.,j(T) acts as multiplication by j(O». Then
M can be regarded as the quotient of R by the ideai generated by T and 2.
Using unique factorization in R and the ract that T and 2 are relatively
prime, one easily o btains tbe free resolution of length 2
0-+ R  R ffi R 01 R  7L2 - O
where 8(/) = f(O) mod 2, and al and Ò2 are gìven by the matrices (T 2) and
(- i), respectively.
4. Let R = Z[ T]j(T2 - l) and let t be the image or T in R. Let M be the
R-module Rj(t - 1). (Equivalently, M = li.. with t acting as the identity.)
t   for termino1ogy and basic facts (;(l[]cerning chain complexes.

12
I Some Homological Algebra
Since T2 - l = (T - J)(T + l), it isclear that an element ofR is annihilated
by t-l (resp. t + l) if and only if it is divisible by , + l (resp. t - l). One
therefore has a free resolution
. . . R t - l) R l + l. R t-I R _ M -+ O.
Note that, unlike the previous examples, this resolution is oI infinite length.
We will see, in fact, that M admits no free resolution offinite length, cf. II.3,
exercise 2.
In this book we will be primarily interested in the case where R i8 a
group ring, so we digress now to recall what a group ring is.
2 Group Rings
Let G be a group, written multiplicatively. Let ZG (or Z[G]) be the free
Z-module generated by the elements of G. Thus an element of ZG is uniqueIy
expressible in the form LI1E1G a(g)g, where a(g) E Z and a(g) = O for almçst
alI g. The multiplication in G extends uniquely to a l-bilinear product
ZG x 7l.G - ZG; this makes ?LG a ring, called the integrai group ring of G.
Note that G is a subgroup of the group (ZG)* of units of lG and that we
bave the following universal mapping property:
(2.1) Given a ring R and a group homomorphismj: G - R*. there is a unique
extension of f to a ring homomorphism ?LG -+ R. Thus we bave the "ad-
junction formula .,
Hom(rinplZG, R) :=:::: Hom(lJroUPS)(G, R*).
EXAMPLES
1. Let G be cyc1ic of order ti and let t be a generatoT. Then the powers ti
(O < i < n - I) Corro a l-basis for lG. and one bas tn = 1. It follows that
ZG  Z[T]j(TIt - l).
2. If G is infinite cyclic with generator t, then lLG bas a Z-basis {ti}iEZ'
Hence lLG can he identified with the ring (usually denoted .l[ t, t - ]]) of
Laurent poIynomials LiEZ uitj (ai E l, ai = O for almost alI i).
EXERCISES
1. For any group G we define the augmentarion map to be the ring homomorphism
e: 1:.G - l such thal e(g) = l for ali g E G. The kemel of li is catled the augmentation
ideai of ZG al1d is denoted l or IG.
(a) Show that the elemel1ts g - t (g E G. g #= I) forro a basis for l as a Z-moduJe.
(b) If S is a set of generators for G. show that tbe elements s - 1 (s E S) generate l as
a left ideaI.
"
l. .":".
.',
.'. ....!
, .
.rr\;
.....
",.-
-!".
.:....
..
 .
.c
:...:
.'1."'7.10
-;. f
"
-'
"l:"
"-:
't..
':....IIC'
II.. 
..' ....
 ::; .
.,

<
3 G-modules
13
(c) Conversdy. if S is a set or elements of G such that the elernents s - I (s e S)
generate 1 as a lefi ideai, show Ihai S generates G. [Hint: The h.ypothesis on S implies
tbat everyelement of l is a sum of elements ofthe form g  g'. where 9 = g'S:tl for
some SE S. Writing 9 - 1 in this way. where 9 E G is arbitrary. deduce that there is a
sequence (/1. gl..... Ori such that gJ = O. g" = 1. and gj = gi+lSj:tl Cor some SjES.
See execcise 2 of3 be]ow for an a]temative proof.]
(d) Show thai G is a finitely generated group if and onl}' if l is finitely generated as a
left ideai.
2 Let G be cydic with generaror t and let M be the O-module lL. with l acting as the
identity. (Equivalently, M = lLG/l = ZG/(t - 1).) Write down a free resolution or
M. [See l. exarnple 4. for the case where IGI = 2.]
3 G-modules
A (left) ZG-module. also called a Gmodule. consists oC an abelian group A
together with a hornomorphism from ZG to the ring of endomorphisms of A.
In view or 2.1, such a ring homomorphism corresponds to a group homo-
morphism from G to the group of automorphisrns of A. Thus a G-module is
simply an abelian group A together with an açtion of G OD A. For example,
one has for any A the trivial module structure, with ga = a for g E G, a E A.
(Thus ra = e{r) - Q Cor r E lG, w here f: l G -+ .l is the augmentation map
discussed in 2, exercise I.)
One way of constructing G-modules is by linearizing pennutation repre-
sentations. More precisely, if X is a G-set (i.e., a set with G-action), then one
forrns the free abelian group 7l.X (also denoted Z[X) generated by X and
one extends the action of G on X to a 7L -linear action of G on ZX. The re-
sulting G-module is called a perrnutation module. In particular, one has a
permutation module Z[G/H] far every subgroup H or G, where G/H is the
set or cosets gH and G acts on G/H by Ieft translatioo.
The operation or disjoint union in the category of G-sets corresponds to
the direct sum operation in the category or G-modules:
llU X;] = E9ZXi.
rt follows that every permutation module ZX admits a decomposition
.lX R;: E9Z[GjGx],
where x ranges over a set or representatives for the G-orbits in X and Gx is
the isotropy subgroup of G at x. In particular, if X is afree G-set (Le., if ali
isotropy groups are trivial), then GJGx = G. and we obtain:
(3.1) Proposjtion. Lec X be afree G-set a"d let E be a sel ofrepresentativesfor
the G-orbits in X. Then ZX is a/ree ZG-module with basis E.

14
I Some Homolo@ical Algebra
Remark. If k is an arbitrary commutative ring. then one can form the group
algebra kG or G over k. This is tbe free k-module generated by G, with the
unique k-bilinear product extending the group multiplication on G. Every-
thing we have done in this section generaIizes in an obvious way to this
situation. F or example, a kG-module is simply a k-module A together with
a (k-linear) action of G on A.
._1':
.r_
EXERCISES
L Let H be a subgroup of G and 1et E be a set or representatìves for the right eosets Hg.
Show that ZG, regarded as a lert-module over ìts subrjug ZH. is free wìth basis E.
2. Use permutatÌon modules to gìve a non-eomputational so]ution of exercìse le or 92.
[Let H be the subgroup of G generated by S and consider Z[GfH]. It has an element
fixed by H and henee annihilated by I. But then the element is fixed by G.]
.
4 Resolutions of 7L over 7l..G via T opology
In this section we will consider 7L as a G-module with trivial G-action, and we
will see that free resolutions of this module arise from free actions or G on
contractible complexes.
By a G-complex we will mean a C W -complex X together with an action
of G OD X which permutes the cells. Thus we ha ve for each 9 E G a homeo-
morphism x .-.. gx of X such that the image g(J of any celi (J of X is again a
celi. For example, if X is a simplicial complex on which G acts simpliciaJlYJ
then X is a G-complex.
If X is a G-complex then the action of G on X induces an action of G OD
the cellular chain complex C .(Xh w1ùch thereby becomes a chain complex of
G-modules. Moreover, the canonical augmentation E Co(X) -+ 7L (defined
by E( v) = l for every O-cell v of X) is a map of G-modules.
We will say that X is afree G-complex if the action or G freely permutes
the cells of X (i.e., gu #= (J for all (1 if 9 #- 1). In this case each chain moduJe
Cn(X) has a l-basis which is freely pennuted by G hence by 3_1 C,,(X) is a
free ZG-module with one basis element for every G-orbit of cells. (Note that
to obtain a specific basis we mUSI choose a representative celi from each orbit
and we must choose an orientation of each such representative.)
Finally, if X is contractible. then H.(X)  H...{pt_}; in other word the
sequence
-. .
-.1:"
...  Crz(X} 1'. Cn-1(X} -a. ... -+ Co(X)  7L -+ O
is exact. We bave, therefore:
."
. .;..;,
.......
:40:-
,. .::
(4. t) Proposition. LeI X. be a contractiblefree G-complex. Then the augmented
cellular chain complex oJ X is a free resolution of 7L over ZG.
.... '..
.: . . 
,
,
.1-
4 Resolutions of Z over ZG via Topology
15
Tbe reader who has studied covering spaces has or course seen many
examples of free G-complexes. Indeed suppoe p: Y ---+ Y is a regula coverin.g
map with G as group of deck transformatl?ns. (See the appendlx o. thls
chapter for a review of regular covers.) If Y 15 a CW --complex, then.lt IS an
elementary roct that Y inherits a CW-structure such that tbe G-actlon pe
mutes the cell cf. Schubert [1968], 111.6.9. Explicitly, the open cells or y
lying over an open celi u of Yare simply the connected components of p - lU;
these ceJ1s are permuted rreely and transitively by G and each is map
homeomorphically onto (J under p. Thus :P is a free G-complex and C. Y
is a complex or free ZG-modules with one basis element for each celi or Y.
. In view of 4.1, it is natura! now to consider CW-complexes Y satisfying
the following three conditions:
(i) Y is connected.
(ii) 1t] Y = G.
(iii) The uni versai cover X of Y is contractible.
Such a Y is called an Eilenberg-MacLarle compie x oftype (G l) or simply a
K(G, l)-complex. Condition (ii) above is to be interpreted as meaning that
we are given a basepoint Y E Y and a specific isomorplllsm 1t1(Y, y)  G by
which we identify 1tl(Y' y) with G.
h is sometimes convenient to note that condition (iii) above can be
replaced by:
(iiY) HiX = O for i > 2.
Indeed, we c1early have (iii)  (iii'). Conversely, (iin implies that X is acyclic.
i.e., that H.X  H *(pt.); and 1t is shown in e1ementary homotopy theory
that simply-connected acyclic CW-complexes are contractible. [fhe reader
who is not familiar with this fact can sim ply take (i), (ii), and (iii') as the
definition of a K( G  1 }-complex, since we will never make serious use of the
fact that X is contractible. Note, for instance, that 4.1 remains true if" con-
tractible n is replaced by "acydic."]
It also follows from elementary homotopy theory that (iii) can be replaced
by
(iii") ni Y = O for i > 2.
Thus the Eilenberg-MacLane complexes are precisely the aspherical com-
plexes studied by Hurewicz (cf_ Introduction). Once agaif4 we will not need
to use this fact.
If Y isa K(G, l)then the universal cover p: X -+ Y is aregularcover whose
group is isomorphic to 11:1 Y = G- [Th.is depends on a choice of basepoint in
X, cf. Appendix.] We therefore obtain from 4.1:
(4.2) Proposition.lf Y is a K(G, 1) then the augmented cellular chain complex
ofthe universal cover of Y is af.,.ee resolution oflL over ZG.
"

16
I Some Homologica1 Algebra
..
.-
..;
<,
We now give one example. There will be many other examples in Chapter
II and later in tbe book.
'</...
!::
.
'-
(4.3) EXAMPLE. Let G = F(S), the free group generated by a set S. Let Y =
VJES S:.. a bouquet of circles S: in l-l correspondence with S. Thus Y is a
l-dimensional CW-complex with exactly one vertex and with one l-celi for
each element or S. One knows that 11:1 Y  F(S}. Moreover.. Y is a K(F(S), l)
since condition (iii') above holds for trivial reasons. [X is l-dimensiona1.J
As basepoint in X we take a vertex Xo; this then represents tbe unique G-orbit
of vertices of X and hence generates the free ZG-madule Co(X). As basis for
C l (X) we take, for each S E S, an oriented l-celi es of X which lies over S:
(traversed in tbe positive sense). Replacing es by ges for suitable g E G, we
can assume that tbe initia] vertex of eli is the basepoint xo; the endpoint or es
is then sXo (cf. Appendix, Al) so that iJes = sXo - Xo = (s - l)xo. We
obtain, tberefore, the free resolution
(4.4) o --t> ZG(S)  7lG  7L - o
where ZG(S) is a free ZG-module with basis (e,s)seS' òelj = S - l and £(g) = l
for ali 9 E G.
'1
t.
,
.1_"
:....
Remarks
l. Note that t coincides with the natural augmentation or the group ring
7L.G, as defined in 2, exercise I. The exactness of 4.4 says, therefore, that the
augmentation ideai or Z[F(S)] is a free lert l[F(S}]-module with basis
(s - l).ses. [We williater indicate a purely algebraic proof of this fact, cf.
IV.2,. exercise 3b.] The reader who is not used to working with modules
aver non-commutative rings may find it surprising that l.G, the free module
of rank 1,. can contain a submodule which is free of rank > I; this 80rt of thing
cannot happen over a commutative ring R, since then any two elements
a, bER are linearly dependent: (b). a + a. b = O.
2. If S contains a single element t (i.e.,. G is infinite cyclic) then (4.4) reduces
to tbe 'obvious" resolution which the reader probably wrote down in 2,
exercise 2:
.-
(4.5)
o -+ 7l.G r - l) ?LG  7L - O.
Note that X, in this case, is simply IRl  with l acting by x  x + l:
:>

o
:
o
t-2XO
t-1xo
t1xo
Xo
txo
3. The contracti ble free F(S)-complex X, and hence the resolution 4.4
can easily be constructed directly, without appeal to the theory of covering
complexes, or to the fact that 'Tr1(V Si) ::::: F(S). Namely X can be detlned
as the l-dimensional simplicial complex whose vertices are tbe elements of
G = F(S) and whose l-simplices are the pairs {g, gs} (g E G, s e S). The action
:(0..
'.:

i
4 Resolurions of Z over ZG via Topology
17
of G on itself by leCt translation induces a simplicial action of G on X, which
is easily seen to be free. Finally 9 one can construct an explicit contracting
homotopy which contracts X to the vertex l along paths of the form
o
c
o
'J
. . .
g = gli
g]
90 = 1
gll-l
where 9 = Sl. . . s:" is a reduced word or length ti (Si E S, l>i = + l l3i = Gi+ J
if Si = Sl+ tJ and g = si] . .. sii (O < i < n). In case S is a two-element set
{s. t}  for example, X is the tree pictured below:
(2 t2s
Lo
f'
..
t (S
s- tts SI
l
-2 - l l S $2
S S
sr- t
...
4 ,- 1 _'!il
-l
,
C'l
EXERCISES
L ]f X is an arbitrary G-complex, is Cr:(X) necessarily a permutation modu]e?
*2. Let X be a free G-complex where G is an arbitrary group. Show that every point
of X has a neighborhood V such that gU n U = 0 for alI 9 *' 1 in G. Deduce that
K - X jG is a regular covering map with G as group of covering transformations.
In particuJar. if X is contractìble then X IG is a K(G. 1) and X ìs its universal cover.

18
I Some HomologicaJ AJgebra
[Hinl: A C W -compJex X. with explicitly given characterisrSc maps (B". S" - J) -+
(lI, èa) for its cells, admits a canonical open cover {U a} indexed by the cells, such tbat
V ti (ì V'[" = 0 if Cf and 1" are distinct ceUs or the same dimension. If X is sirnplicial.
for example. we can take U" to be the open star of the barycenter or CT in the bary-
centric subdivision of X.]
.,-
.,.
. :.i.
,,'
';1 "
:1#
'i.<,
.I
3. Let G be a free abelian group of rank 2. Use the methods or this section to construct
a free resolution or Z over IG-
*4. (This exercise requires some homotopy theory_)
(a) For any group G, show that there exists a K(G, 1). [Start with a connected 2-
complex with 1t1 = G; then attach cells to kiH the hìgher homotopy.]
(b) Show that the construction in (a) can be made functorial in G. [Given the
(n - l)-skeleton y"-l  attach an n-celi for every possible map sn-l _ y"- 1.]
5 The Standard Resolution
Let X be a contractible simplicial complex on which a group G acts simplici-
ally. lt may happen that the G-action is free 00 the vertices but not on the
higher-dimensional simplices. [Note: This cannot happen if G is torsion-
free.] In tbis case we can stiU obtain a (ree resolution of Z over lG by using.
instead orthe usual cbain complex C.(X), tbe ordered chain complex C(X)
(cf. Spanier [1966], ch. 4, 3)- Thus C(X) has a Z-basis consisting of tbe
ordered (n + 1 )-tu ples (vo, . . . , VII) of vertices of X such that {vo, . . . , VII} is a
simplex of X (of dimension < n if the v[ afe not all distinct). Since G c1early
acts freely on these (n + I)-tuples, we obtain a free resolution of li. over lG.
The most o bvious example of such an X is the . simplex" spanned by G;
ie., tbe vertices or X are the elements of G (wìth G acting by left translation),
and every finite subset of G is a simplex of X. [If G is finite, this really is a
simplex; otherwise it is an infinite dimensional analogue. In any case it is
contractible by a straight-line contracting homotopy.] The corresponding
free resolution F!tI = C(X) is called tbe standard resolution of 7L over lLG.
Exp)icitly F Il is the free Z -module generated by the (n + 1 }-tuples <Do. - . . . gJ
of elements of G, with the G-action given by 9 . (90' - . . , gli) = (gg, . - . , gg,J
The boundary opera tor ò: F n  F 11_ l is gìven by a = Li- o ( - l )dB where

(5.1) d/.9o, ..., gli) = (go. .... Di' . .., g,,).
The augmentation 6: F 0-- 7L is given by r.(go) = I. Note that tbe acyclicity
(ie., exactness) or this augmented complex has been deduced from the
contractibili ty or X, but one can also verìfy it directly by writing down a
contracting homotopy h: F Il """"'t> F n + 1 for tbe underlying augmented complex
Or 7/..-modules (i.e., h will not be a map of G-modules). Namely, we detìne h
by h(go... ., g,J = (1,90' ..., g") if n  O and h(1} = (l) if n = -1.
::.
"y,;,"(
i-.-
-'
"i
J ,
.: .. :1";:
. '.....
.. ....
.' ...
.
,. J:
 .
5 The Standard ResoJution
19
As basis for the free ZG-module F" we may take tbe (n + l}-tuples whose
first element is l, since these represent the G-orbits or (n + l)-tuples. It is
ofteo convenient to write such an (11 + l)-tuple in the form (l, gl' 9192"'. 
9192.' . g,,) and to introduce the "bar notation.'
[gllg21. - -19,,] = (1, gl' glg2, .. - . 9192 . . . gli)'
(If n = O there is only one such basis element, denoted [ ]; if we identify
F o witb ZG in the obvious way, then [ ] = I.) li is eay to compute ò: F 1r 
FII-J in terrns of this ZG-basis {[gll- - .lglI]}; one finds Ò = Li=o (-I)idj,
where di is the ZG-homomorphism given by
91[g21. . 'lg,J
(5.2) di[gll. --lgl'I] = [gJ I. ..19[- dgjgi+llgi+21... fg.J
[g 11. . .1911-1]
i = O
O<i<n
l = n.
This standard resolution F. is often called the bar resolution. In low di-
mensions it has the forro
F 2  F 1  7LG  Z -+ O,
where ò2([gjh]) = g[h] - [gh] + [g], òt([g]) = g[ ] - [ ] = 9 - 1,
and e(l) = 1.
FinaIly, we mention tbe normalized standard (or bar) resolution F. =
F../D., where D. the "degenerate" subcomplex or F., is generated by the
elements (go, . . . , g,.) such that 9i = (li + I for some i. In terms of the bar
notation, D 1ft can be described as the G-subcomplex of F. generated hy tbe
elements [gll-. 'Ig,a such that gi = 1 for some i. Thus F,. is a free 7LG-
module with one basis element (stili denoted [g 11 . . . 191'1]) for every n-tuple of
non-trivial elements of G. The fact that F. is acyclic over lL, and heoce a
reso lution , follows from generaI facts abou t normalization (cf. MacLane
[1963], VIIL6); alternatively, one can simply observe that the contracting
homotopy h defined above carries D * into itself and hence induces a COD-
tracting homotopy or the quotient F * .
EXERCISFS
I. Write down the homotopy operamr h in terrns ofthe 7L-basis g[gll.. .Ign] for E".
2. Write out the normalized bar resolution in case G is the group of order 2; compare
it with the resolutìon given in I. exampJe 4.
.'
,
.3. (a) Show that the standard resolution is the ceJlular chain complex of a free con-
traclible G-complex X. Note thal this reproves the result of exercises 4a and 4b
of. [For each (n + l)-tuple a = (g(). .... gli)' let A(J be a copy of the standard
n-simplex with vertices V{I. . . + . Vn. Let di (J = (go.. . + . {j i. _ . . . g..) and let Di: l1.t.a  
be the linear embedding which sends vQ, ..., V,,-l to vo. .... Vi' .... Vn. Frm the
disjoint union Xo = UrJ tJ... and obtain X by identifying &.t,a with its image Under.si

:1.
,
"
.
':
20
1 Some Homological Algebra
:.,
"
"
,,'
for a11 (J'and aH i. Use the Quotient map C(Xo) -+ C(X) lo compute tbe boundary
operator on C(X) and see lhat C(X) ::::::; F. To prove X contractible, consider
110" = (1,90' . ... glI) for each 0", and lise the straight-line homotopy between tSo:
à.,. - "... and the constant map 11fT ---+ A"(1 at vo.]
(b) Do thc same for the normalized standard resolution. [U se the same method as in
(a). but make further identifications in X o lo coHapse degenerate simplices. Explicitly.
[or each (J = (go' . . . glI) lec St(J = (go, . . .. li,. g.. . . . .9,,) and collapse ASJG lO lia via
the )inear map which sends Vo. .... Vn+ 1 to Vo. . .., VI> l'h . .., V".]
.j. "'-.
i. .
...r.:.';'::
.......
6 Periodic Resolutions via Free Actions on Spheres
"
'.
l-
Cf"
-f
Let X be a free G-complex which is homeomorphic to an odd dimensional
sphere S2J:-l. (G is then necessarily finite since X is compact.) By an easy
special case of tbe Lefschetz fixed-point theorem (cf. exercise 1 below), G
acts (rivially on H 2k-l X = l. Writing C. = C*(X) we have then an exact
sequence of G-modules
(6.1)
o --io 7L. -\ C2lt.-l --io . . . --io Cl - Co 4 7L. -+ Ot
"
 .,.
where each Ci is free. Splicing together an infinite number of copies or 6.1,
we obtain a free resolution of 7L over lLG which is periodic of period 2k:
(6.2) ... - CI -+ Co  C2k-t -+... - CI --+ Co  7L - O.
.!". ..
----..
.......
<;..:: .
._..t... .
'-
, ,
EXAMPLE. Let G be a finite cyclic group of order n with generator l. Then G
acts freely as a group or rotations of SI, regarded as a CW-complex with n
vertices and ti l-cells:
.."':.":
':j:
'-i::
li..
v!,,-
:
)..
le
.'1-.
.'
v
r!'-le
..
j
-,
'"
..j

'..
'.o:
-
, ,
, .
::...::......
Note that H 1St is generated by the cyc1e e + te + t2e + ... + f-1e = Ne
where N is the "norro e1ement  I + t + . . . + t"- l of lLG, so 6.1 tak.es the
form
"
.
[,
o -7L .!l. 1:.G t-l. ZG 4 Z -+ O
..
"
i
7 Uniqueness of Resolutions
21
where £(1) = I and ,,(1) = N. We therefore obtain the Collowing periodico
resolution of period 2, which the reader has probably already seen (2..
exercise 2):
(6.3)
lLG l - l 7l. N r - 1 7JG l
. . . -+ ) G --io 7l..G ) IL --io 7L --+ O.
We will see other examples or groups with periodic resolutions in Chapter
VI.
ExERCISES
1. Prove the following spccial case or the Lefschetz fixed-point theorem: Let X be a
finite CW -comp]ex and f: X --lo X a map such that. for every celi u
f(o) ç U T.
"'a
dm:<;diDia
(ln particular, f has no fixed-point.s.) Then the Lefschetz number L ( _1)i Irace h
is zero, where li is the endomorphism induced by f on the free abelian grOlip
Hj X jtorsion. [Hint: The Lefschetz number can be computed on the chain level, where
the matrix of f has zeroes on the diagonal.] If H .(X)  H .(S211. - 1). deduce that f. :
H 2,n-1 X -+ H 2n - l X must be thc identity.
2. Prove that the group of order 2 is the only non-trivial group which can aet freely on
an evcn-dimensional sphere.
7 Uniqueness of Resolutions
We return to the generality or  l, i.e., we work over an arbitrary ring R. It is
obvious that a given R-module M adrnits many different (ree resolutions;
the purpose or this section is to show that alI such resolutions are homotopy
equivalent. In the course of proving this we will be faced with mapping
problerns which can be put in the form
(11)
.
#
.
'i1.
,
.
p
i. 
..
M'
M
:tMIf
J '
i
where the solid arrows represent given maps (with jqJ = O) and the dotted
arrow represents a map we would like to construct. A solution to this pr")blem,
theot consists or a map t/1  p -+ M' such that i.p = ({l. A module P is called
projeClive if a solution exists for every mapping problem 7.1 in which the
row is exact. More concisely, P is projective if the functor HornR(P, -) is
exact.
.,
,

'.;:
."
.\
.-
22
I Some Homologìcal Algebra
,
'.
For our present purposes, the main interest in projectivity is provided by:
(72) Lemma. Free modules are projective.
PROOF. Let F be free with basis (eli)' and consider a mapping problem
F
../ 1. 
I,.
":r
,.
,".!'
.1:.....:
..;.
i.:.;i
, .-
ç..
.,
)M
M"
)
j
:i.
..
M'
i
with exact row. Then tp(eJ E ker j = im i, so we can choose Xa E M' with
i(xa) = <p( t\.). Now let I/J be the unique R -module map with .;,( eJ = Xa' D
The next lemma treats the particular mapping problems which one has
to solve in trying to construct chain maps and homotopies inducti vely:
(7.3) Lemma (a) Suppose given a diagram
".' .
P d Q
, l,
,
,
.
(/,
,
,
....
M' ..M M"
111 112 -+ ·
,'".
 .
.-'
",,,:"..
-=.:
..:-<-
  .
!;::".
l'''
:-. ...
) .;- -
.::: .:.
':).
where d2 fd = O and it is desired to find a g such that di 9 = Id. II p is projecttve
und the bottom row is exact, then such a g exists.
(b) Suppose given a diagram (not necessarily commutative)
p d. Q
.../ ,l /
-.<"
..':
.1"
I ..,
<.
.1' ,.
M, M
d1 )
M"
4.2 ,
. :. .
"",-,"
where d2hd = d2 f and it is desired lo finti a k such that dlk + hd = f.lf P is
projective and the botlom row is exact. then such a k exists.
PROOF. (a) is obvious, since the given mapping problem is of the form 7.1
with rp = fd. Similarly, (b) is a problem of the forro 7.1 with tp = f - 1Id. D
We can now prove the 'fundamentallemma or homological algebra,"
which says, roughly speaking, that it is easy to construct chain maps and
homotopies from a projective complex to an acyclic one:
(74) Lemma Let (C, ò) anti (C', il') be chain complexes and let r be un integer.
Let (fr.: Ci  C)ir be afam11y ofmaps such that iJiii =!,-tÒi!or i  r.ll
Ci is projective for i > r and Hi(C') = O for i  r, then (fJ, r extends to a
chain mapf: C -+ C', andfis unique up lO homotopy. More precisely, any two
extensions are homotopic by  homotopy h such that h. = Ofor i .o:;: r.
::!..
,pt'o
. ;:..
....., ...
.::,....:,'
',.. ...
-""'"'i
." ,
f

1
7 Uniqueness of Resolutions
23
PROOF. Assume inductively that li has been defined for i < 11, where n > r;
and that oh = h-tOi for i < n. We then bave a mapping problem
Cn+1 è ) C,.  )C
,,-1
. l" l'H
.
,
JPJ + I:
.
.
.;..
C+J 4 c · C'
C' i,. !'I-h
where aj"Ò = f" - 1 00 = O. The desired In + l therefore exists by 7.3a.
Suppose now that 9 is a second extension of (1); < r' We wish to find a
homotopy h between f and g. Assume inductively that hi: Ci - c+ l has
been defined for i < n, where n > r, and that a'hi + Ili-lO = h - gi' (Note
that we can start the induction by setting h; = O for i < r.) Setting Ti = h - Yt,
we then have a mapping problem
Cn+ l
lJ )C  .c
" Il-I
..;.'.....,...1 y 1" /"-.
C+2 D' J C+ l
C,

,}' 1'1
with
iYh,. a = (1"" - h"-1 ò)iJ
= filÒ
=éYTn+1
by tbe inductive hypothesis
since Ò2 = o
since 1: is a chain map.
The desired h" + l with iJ'hll+l + h"o = Tn+l therefore exists by 7.3b. D
Remark The proof of 7.4 should have looked familiar to anyone who has seen
the method of acyclic models in algebraic topology. We will explain tbis
'6similarityU in exercise 3 below.
Now let t: F -+ M and f/: F' -+ M be two free (or projective) resolutions
of a module M. We can then form tbe augmented chain complexes with M
in dimension - 1 and apply 7.4 with r = -1:
. . .
l F1
) Fo l: M -O
) - - .
, 1 M 1
.
.
.
,
.
,
'"
) Fo ' M o t.. . - .
.
I
I
...
.. . - Fl
We condude that there is a chain map f: F -+ F' which is augmentatiorJ-
preserving, ie., which satisfies e'I = E. Moreover,jis unique up to homotopy.
(Note that the homotopy h given by 7.4 on the level of augmented chain

24
I Some HomoJogical Algebra
complexes yields a homotopy F -t F't because h_l = O.) Similarly, there is
an augmentation-preserving map l': F' --+ F, and we bave /1':::1. idp and
1f'  idF" again by the uniqueness part or 7.4. This proves:
,

(7S) Theorem9 Given projective resolutions F and F' oJ a module M there is an
au.gmentation-preserving chain map f: F -+ F', unique up lo homotopy. and f
is a homotopy equivalence.
,
-
,
-
,
"
,
. ,
.< '-
,
-
"
We wiU express this informally by saying tbat projective resolutions are
"unique up to canonicai homotopy equivalence.' At the moment, of course,
we are mainly interested in free resolutions, but later (beginning in Chapter
VIII) we will need to consider projecti ve resolutions which afe not known to
be free.
F or future reference we record two special cases of 7.5 :
,
- ;..:;
'.
.
.,
...

,
,
,
..
.-
: . ..
.',
.,
.

.
.;
"
,
I
-'
':j
:1
'1"
;.,
..'
::_t
, .
'.,fI.
- .
'. ,
;
"
...
,
.
..
:
.'
!
!
(7.6) CoroUary. Let £: F --+ Z be a free resolution of li as a 7L -module. T hen e
is a homotopy equivalence.
PROOF. idz: 7l. - 7L is also a free resolution, and f: F -+ Z can be viewed as an
augmentation-preserving chain map from ane resolution to another. D
(7.7) CoroUary. 1/ F is a non-negative acyclic chain complex oJ projective
nwdules (over an arbitrary ring R), then F is contractible.
PROOF. F and the zero complex are projective resolutions of O, so they are
homotopy equivalent. See also exercise 3 of  for an alternati ve proor. O
Remarks
1. In practicet a (ree resolution usual1y comes equipped with an explicit
basis in each dimension, and one usually proves that it is a resolution by
exhibiting a contracting homotopy for the underlying augmented chain
complex or lL-modules. In this case, tbe proof of 7.5 yields a specifiç rnap
f: F -+ F'. Namely, ir k is a contracting homotopy for the augmented
complex associated to F', then f is determined inductively by fn+ l(ea) =
knhoea, where (e!X) is a basis for F",+ l.
2. The uniqueness theorem 7.5 can be thought of as the algebraic analogue
of Hurewicz's tbeorem, which we quoted in the introduction, asserting the
uniqueness up to homotopy or an aspherical space with a given rundamental
group.
"::'I..
L- ..:\
".
,.
" -
. lo
:.i;\
. ..- 
.11I .. ;:
..
... I_t-
-.
. .:r
. .:
.'
"
. 
f.
,..
.-
EXERCISES
l. Let G be a finite cydic group. Let F be the rree resolution of Z over ZG given ìn 6.3
and let F' be the bar resolution. Write down an augmentation pceserving chain map
f:F -+ F'.
- 
7 Uniqueness or Resolutions
25
2. The method of proof of 7.4 applies to many situations. some of which wìll arise later
in thìs book. It is therefore useful to axiomatize 7.4, as follows, By an additive CQlegory
we mean a category .rd in which Hom(A. B) is endowed with an abe)ìan group stcuv
(ure for any two objects A, B, in such a wa}' that the composìtioD law Hom(B, C) x
Hom(A. B) -+ Hom{A. C) is Z-bilinear. In particular. since wc have a zero element
O E Hom(A, B) it is cIear what we mean by a chairl complex in d. The usual definitions
of chain map and homotopy also go through with no change. [Note that additivity is
needed to define .. homotopy."] Suppose now that we are given a class 8 of sequences
M' ...... M -+ M" in d called exact sequences. We then say that an object P or..91 is
projeclive relative to " if every mapping problem 7.1 whose row is in ti adrnits a
solution. Verify that the proof of 7.4 is valid in this situation and leads to an analogue
of7.4 for chain complexes in.9l. [Note: In the statement ofthis analogue. the "acy-
dicity" hypothesis on C' should be replaced by the assumption that c+ 1 -+ c -+
Ci- J is in 8 for i > t'.] Deduce an anaJogue of 7.5.
?.
3. Let <ti be an arbitrar}' category and let .d be the additive category whose objects are
covariant functors  ---+ (abe1ian groups) and whose maps are natura] transforma-
tions of functors. Fix a subclass At or the class of objects of. A seq uence T' - T -+ T"
in d will be called ..$I --exact jf the resultìng sequence T'(M) -+ T(M) -+ T"(M) of
abelian groups is exact for alI M eJ(. The purpose of ihis exercise is to show that
when exercise 2 is applied to.91 (with ti equal to the class or ....N-exact sequenccs). the
resuJt ìs the acyclic models tbeorem as given, for instance. in Spanier [1966], Theorem
4.2.8. or Dold [1972], Proposition Vl.l1. 7. Tbe crucial step is to prove an analogue in
od of Lemma 7.2; this is done in (a) and (b) below.
(a) (Y oneda's lemma) Let A be an object of Cjf and let hA:  .--)- (sets) be the covariant
functor represented by A. i.e.. hA. = Hornv(A. -). Let UA, e hA.(A) be the identity
map A -+ A. Let T: (j......,. (sets) be an arbitrary covariant functor, For any v E T(A),
show that there ìs a unjque natura} transformation qJ: hA - T such that tp(UA,) = v.
Thus Hom,.(h,4, T)  T(A). where :F is the category of functors f{ --+ (sets). [This
can be thought of as saying that hA is "freely generated .. by uA' The proof js strajght-
forward definition-checking. Tv prove unìqueness. for example, note that we must
have tp(f) = T(/Xv) for any f E hA(B) = Hom'j(A. B). in view of the diagram
hiA) hAll) I h..t(B)
.1 1.
T(A) Tifi  T(B).

To prove existence. take tbis equation as a definitioo and cbeck that it works.]
(b) Let ZhA: <tf..... (abelian groups) be the composite or h with the functoc (sets)-
(abelian groups) which associates to a set the free abelian group it generates. A functor
F: rj -+ (abelian groups) wilI be caIJed .lt1ree if it is isornorphic to a direct sum
Ef)a Zh.. . where Aa E .AI for alI :1. Deduce from (a) that ..K-free functors are projective
(Il
relative to the class of J( -exact sequences.
(c) Using (b) and exercise 2, state a theorem about chain maps in d from ...If.free
complexes to .. A( -acyclic" complexes. [Note: This theorem IS precisely the acyclic
models theorem cited above.]

"
26
J Some Homological Algebra
4. Another important example or exercise 2 is obtained by taking d lo be the d'4al ofthe
category of R-modules. Thus si has one object MO for every R-module M and one
map fO: M ......... Mi. for every R -module map f: M 2 -+ M l' Composition is defined
by fOgO = (JIfT. As exact sequences in si we tate those sequences M'I _ M - M'j
such that the corresponding sequence M 3 -+ M 2 - MI of R.modules is exact.
Applying exercise 2. we get an analogue 0[7.4 for chain complexes in d. which can
obviously be restated as a theorem about cochain complexes of R-modules. Explicitly
state this theorem. as well as rhe theorem corresponding to 1.5. [Note: An R-module
Q is caUed injective if QO is projective in d. or. equivalent1y. if the fundor HomR( _. Q)
is exact. Y our theorem should therefore be stated in tenns of maps of an acyclic
cochain complex into a cochain complex of injectives. T ogive this theorem substance.
of course. we should have an analogue or 7.2. so that wc will have examples of in-
jectives. We will pro vide such an analogue ]ater. in 9111.4.]
'.
".
,.
.1"'\:
, \.
, -
..;: \.
.l0. "=....
'f.',.
;1
t;",.
/f :
1-';.::-:
"
J.
. -
.:
:r-
r ....::
. <.
.: 
i--.,
.-'&t;
,:;
.:e-..."SI..-
':';'r
:,,':h;
.f
.=-¥-
,Ù:
::.:
i. : .
 ,-..
:: ::..-.!
:.' {.
.....
1-....
8 Projective Modules
Tbe reader may be curious at this point to know more about projective
modules other than the fact that (ree modules are projective. We give in this
section therefore, some miscellaneous results and examples conceming
projective modu1es and complexes. We will not make serious use or these
results (except as theyappJy to free modules) until Chaptee VIII. One may
therefore skip this section now and return to it later.
The first observation is that to prove a module P is projective one need
onlyconsidermapping problems 7.1 in which M" = O; for 7.1 can be replaced
by
.'.
. ,.
....
. 'iL
. ,.
;" . ,.-}'"
.. .....
- }
:. .
.. :','.
-j;
--.."-:.-
-'".. 1;".
",1..'.'.0.:.:
.:.,;.:
", i' , _
.:. :';\.
. :: . -;:
V".'
.l",: j\
or.:. l',.
:';::'.
. .t.
. "." . :'.
ir-r'.
;."1'. j-
',', :.-:=
.. 1:
-'>
,'-' -'.
;"";a.......
., -.
, ,
.
p
. l
.
.


,
,
l<:
M'  ker j ) O.
Thus we ha ve:
(8.1) Propositioo. P is projective il and only iffor every surjeclion n: M -+ Al
and every map cp: P -+ M rhere is a map t/J: P -+ M such that tp = mjJ:
-
"
-".
-..,:
.:
'..
.I":-
,
, .
......-:;:
,. .
. :  . -'
p
 l.
¥I .' .
,
.
.
..
M · M.
re
., .
. ".
,
"-
8 Projectjve Modules
27
One also has the following characterization of projectivity:
(8.2) Propositiou. The following conditions on an R-module P ore equivalent:
(i) p is projective.
(ii) Every exact sequence O -+ M' -+ M -+ P -+ O splits.
(iii) P is a direct summand oJ a free module.
(iv) There are elements fi E P and li E HornR(P, R) (where i ranges aver some
index set 1) such that fo, ever y x E P  Jl(x) = O fQr almosl all i and x =
Ed h{x)ei'
PROOF. Il P is projective and we are given an exact sequence as in (ii), then we
can split the sequence by lifting id: P -+ P to a map P -+ M. Hence (i) "* (ii).
Choosing sucb an exact sequence with M free, we see that (ii) => (iii). It is
immediate from tbe definitions that any direct summand of a projective is
projective; so (iii)  (i). Finally, (iv) is simply a restatement or (iii). D
EXA M PLES
1. Let e ERbe idempotent, i.e. e2 = e. Then right multiplication by e is a
projection operator of R onto tbe direct summand Re. So the left ideaI Re is
projective.
2. Let R be a (commutative) integraI domain and I an invertible ideaI.
[This means that tbere is an R-submodule J 01 the field or rractions of R
such that l J = R, where l J is thc set of finite sums L aibi, ai E l, bi E J.] Then
J is a projective modu.e. For ir we write l = L eih where ei E l,fr. E J, then
tbe criterion or 8.4(iv) is satisfied. [Eacb/; gives rise, by multiplication, to a
homomorphism l -+ R which plays tbe Tole of the Ji. in 8.4(iv).J On tbe other
hand, l is free only if it is a principal ideai, since any two elements a, b E l
are linearly dependent.
3. Let R = Z[CJ. where' is a primitive twenty-third root of unity. It is
known from algebraic number theory that R has an ideai l which is not
principal and that every non-zero ideai in R is invertible. Hence l is projective
but not free.
4. Let G be a cyclic group of prime oeder p. There is a theorem due to
Rim which relates projective modules over lLG to projective modules over
Z[{], where , is a primitive p-th root of unity  c[ Milnor [1971], 3. In particu-
lar, if p = 23t we deduce from example 3 that Z G ha.s non-free projectives.
5. If R is the rational group algebra OG of a finite group G. then one can
show that every R-module is projective. cf. exercise 5 below. As an illustration
we will prove that C, with trivial G-action, is projecti ve. Note first that tbe
functor HomQG(Q. -) is simply the "invariants" functor M MG where
MG = {meM:gm = mforallgeG}.

28
I Some Hornological Algebra
Thus we need to show that any surjection M -+ 1J or OG-modules gives
rise 10 a surjection MG -+ MG. This is shown by averagi,zg: if m E MG, lift
m to m E M; then (1/ I G I) LEG gm is also a lifting of m and is in MG.
/
r "."
.,
.:: ..:::
We turn now to the duality rheor y for finitely generated projectives.
analogous to that for finite dimensional vector spaces over a field. F or any
left R-module M. let M* = HornR(M, R). Here R is regarded as a left R-
module in forming Hom, but it also has a right R-module structure which
we can use to make M* a right R-modu1e; namely, we sel (ur)(m) = u(m)r for
U E M*, rE R, m E M. Similarly, we can define the dual or a right module,
and it is a left module.
The main facts about duality afe gi ven in the following proposition.
Parts (b) and (c) afe tbe most importanl ones for Qur purposes; they allow
one to use duality to convert Rom to (8) and vice-versa.
. .
.'. .
. :.........
-.:t":
.' i-o
! . .:.:
. .,
'. .'J.
:"  . .....;
',.:. ':.
::': ".....
, ,-
!"':'
....; .
";.......
(',".:
-:::::..
t'" :
"o, :
: ,::-.
".
0<=.
,-
.:c.
". ......
"'
="
'. ,"-
.....-
.  "".r
"':.
(8.3) Proposition. Let P be afinitely generated projective left R-nwdule.
(a) P* is a fin itely generated projective right R-nwdule.
(b) for any lrt R-module M, there is an isomorphism
qJ:P* 0RM HomR(P' M)
DJ abelial1 groups, given by q>(u (8) m)(x) = u(x). mlor U E P*, m E M, x E P.
(c) For any right R-module M, there is an isomorphism
ql: M 0R P  HotnR(P*, M),
given by qJ'(m @ xXu) = m. u(x)for m E M, x E P, U E P*.
(d) There is an isomorphism
,"
'-':::Ii
.'-
-"...
.:;;".
:!.I' .
,.
I:.:
. ".-
"
". ......
..
:&:..."....-
,-.......
;.....:...:
 .:
....... .

.-
:,
,
, " -
. . I
-.;.....
I...:.';t'
: ',:......
.......
:-....)
 .:-
- "
-" 'Il;:"
, .-
'. )
ql': P  prtl*
.'
: ::;
. J:"- 
'o'
olleft R-modules, given by q>"(x)(u) = u(x)for x E p U E P*.
,. .
';I.'.
.?:"
.-. .
i'"':
.....
"
(In connection with (b) and (c), the reader should recall that one can rorm
M 0 R N whenevef M is a right module and N is a left rnodule; the tensor
products written down above therefore make sense.)
PROOF. It is clear from the proof or 8.2 that P can be written as a direct sum-
mand of a finitely generated free module F. By additivity, then, it suffices
to prove the proposition for F. In more detail: If F = P €a Q then
F* = p. EB Q*, F. (8)R M = (p. (8)R M) $ (Q* @R M). etc., and the mapS
rp., q/, and <p'T preserve these decompositions. So (a)-(d) for P will £ollow
from (a)-(d) for F. [In order to use tbis argument, of course., one must first
check that cp and ql are well-defined.] By additivity again. it suffices to con-
sider the case where F is free of rank l, i.e., we may assume F = R. In this
case R*  R. whence (a), and it is easy to verify (b)-(d). To prove (b) for
instance, one need only check that qJ is tbe composite of tbe canonical
isomorphisrns R* <8IR M .4 M  HomR(R, M). D
..:"
",:,:
,:
..-
..
1h"
"-:
K
, .
,-.
._'V,",:
'<
8 Projective Modula
29
Next we give some properties of chain complexes or projectives.
(8.4) Theorem. lf f: p' - P is a weak equivalence between non-negative
cQmplexes oJ project ives, then f is a homotopy equivalence.
PROOf. The mapping cone or f is non-negative, projective, and acyclic
{cf. 0.6). It is therefore contractible by 7.7, soJis a homotopy equivalence by
ro. D
Using similar methods, we will prove a c10sely related mapping property
of projective complexes., from which we could bave deduced 8.4:
(8.:5) TbeoreDl. Let f: C' -+ C be a weak equivalence between arbitrary com-
plexes. Il P i.s a non-negative complex DJ projectives then
:Jft"omR(P,f)::KomR(P, C') -+:KomR(P, C)
is a weak equivalence. In particular, the map [P., C] -+ [P, C'] irnluced by f is
an isomorphism.
PR.OOF. Let C" be tbe mapping cone of f It is acyclic, and the mapping cone
of .}ff()of1lR(P,!) is Jft'omR(P, C"); so it suffices to show that .tfh.mR(P, C") is
acyc1ic, i.e., that [P, C"]n = O for ali n E 71.. Now [P, CIi]n = [P. I; -nc,,]., and
the latter is zero by the uniqueness part of the fundamentallemma 7.4, since
any map on P is zero in negati ve dimensions. D
Finally t we prove an analogue of 8.5 for tensor products. Projectivity is
unnecessarily strong ror this purpose and can be replaced by a "ftatness"
hypothesis. Recall that a (left) R -module F is flat if the functor - @ R F is
exact. Free modules are ftat, for example, and hence projectives are flat by
8.2(iii).
(8.6) Theorem. Let f: C' ..... C be a weak equivalence between complexes DJ
right R-modules. Il P is a non-negative complex o[ flat le!t R-modutes, then
f@R P: C' <8> R P -+ C (8) R P is a weak equivalence.
PROOF. Let C" be tbe mapping cone or f It is acyclic, and C" 0 R P is the
mapping cone or f @ P; sO it suffices to sbow that C" (8) R P is acyclic. Let
p<n) be tbe n-skeleton of P., i.e., the truncation (Pi)ilI. We will show induc-
tively that C" 0 R P<-) is acyclic. Note first that C" 0 R F is acyclic for any
complex F consisting of a ftat module concentrated in a single dimension,
since the exact sequences C'+J - Ci' - C'-l remai n exact when tensored
with F. But pf.1I)jP(n-l) i5 such a complex F. So if we assume inductively
that C" <8IR p<n - 1) is acyclic, it follows from the exact sequence 0-+
C" i8>R p(n-U --t' C" C8>R p(n} -+ C" C8>R (p(n)/p(n-l})  O that C" <8>R p(") is

30
I Some Homo.ogical Algebra
acyclic. Finany C" @R P is the increasing unìon or the acyclic complexes
C" (8)R P(II), hence it is acyclic. o
.'
-.
( .
:..
.
. .;:/
. "1:......
, :..
I.,,J
.
. ..:4
:f'_"
'::1:
-'1-
f...
EXERCISES
1. For what groups G is 7L a projective ZG-modu]e? [Hint: When does t: lLG - Z split
as a map of G-modules?]
2. If P is a projective ZG-module, sbow that P is aiso projective as ZH-module Cor any
H c G. [Use criterion 8.2(iii).]
3. (a) Use 8.2 10 give another proof of 7.7. [Acwrding to 0.3, il suffices to show that
the surjection òn: Pn - Zn-! il1duced by alt splìts for ali n. Assuming inductively that
all_ 1 splilS, Zn - 1 = ker è If- 1 is a direct surnmand or P n- h hence it - is projective.
Therefore ò" splits.]
(b) Use the same method to show that the non-negativity hypothesis in 7.7 can be
dropped for certain rings R. e.g., for principal idea] domai ns. [If R is a principal ideai
dornain. then subrnodules or a projective module are projective (in fact. free). So
Zn_l above is automatically projective and we do not need to use induction.]
-,
. ..'
4. lf G is a group and X is a Gset such that aH isotropy groups G x are finite, show that
the pennut.ation module QX is a projective QG-mooule.
.' . .,:
":
. :
,
...
':l''
.,
. ';,1"':
.,-
-I..
5. If G is finite and k is a field or characteristic zero. show that every kGmodule is pro-
jective. [Given an exact sequence as in 8.2(ii), choose a splining f : p - M Cor the
underlying sequence of k-vector spaces. Then x...... (1{1 G I) 2.,e-G g(g-l x) is a
kG-splitting.]
.., .
-:;:-:
',.
. o!'.'
./..
"'..;" ,
}
',:

6. Prove the following converse or 8.3b: If P ìs a rnodule such that ({J: P* @R P -
Hom(P. P) is surjective. then P is finitely gellerated and projective. [Write idp =
rp('Lh. 0 eJ and show that 8.2(iv) is satisfied.]
. .
: l.
:'.>,.:
_ '-,J.
 '". .
f';;
'1 ".
". -.
. .
7. Let P be finiteJy generated and projective. For any Z E p. @,R P and any module M
there is a map r/I-z: HomR(P. M)  p. @lt M, dcfincd as foJlows: '-/I-z(!) is the image of
2 under p. @f:p. @R P -+ P* 0R A-f. Show that the inverse of thc canonical
isomorphism tp: p. @R M=' HomR(P, M) or 8.3b is a map of the fonn I/Iz for some
fixed z (independent of M). (Method 1: Vìew <P - 1 as a natura] transformation
HornR( P, -) - p. @,R -. By Y oneda's lemma (exercise 3a or 7). li' - 1 is uniquely
determined by what it does to idp. Moreover. the proof of Yoneda's lemma tells j'ou
pow to descrìbe Cf' - 1 in terrns of z = tp - l{id,.). and this description says preciseIy that
qJ-l = I/Iz. Method 2: Choose (e,) and (h) as in 8.2(i\l) and set z = I/;  ej. By
direçtly checking definitions. verify that VJ-z :) tp = id andJor tbat cp " "'.z = id.]
..
,.
! '
:.i
.'
, .

- .'
8. If P is finiteIy presented and flat. show that P LS projective. [Take a finite presentation
F 1 -+ F o -+ P -+ O with F o and F l free or finite rank. This gives an exact sequence
O -+ p. -+ F - Ft of right R-modules. .Tensor with P and deduce lhat  @I/: P 
Hom.R(P' P). Now appIy exercise 6.]
,
.\
Appendix. ReYÌew or Regular Coverings
31
Appendix. Review of Regular Coverings
The material summarized in this appendix can be found in many algebraic
topology texts, such as Massey [1967] or Spanier [1966].
Let p: R --+ X be a covering map of connected. locally path-connected
spaces. A deck transformation of p is a homeomorphism g: R - X such that
pg = p. The group G of all deck transformations actsfreely on g, in the sense
that gi +- x far ali XEX and 9 -+ l in G.
The cover p is said to be regular if it satisfies the following conditions
which are eq uivalent:
(i) G acts transitively on p-Ix for ali x eX. [Hence X ::t: r/G.]
(ii) The image of 1r1X --+ 'lr1X is normal in XIX for some (and hence every)
choice of basepoints.
(iii) For any dosed loop ro in X, if one Iift of Q) to R is closed then alllifts
of w are c1osed.
In this case we have G  11:1 X/x l g. In particular. if X is simply connected
(so that p is the universal cover of X), tben G  1tIX.
To get an explicit isomorphism G  1t1X/1tlg above. we must choose
basepoints x E X and i E p-l x. We then have a homomorphism (/J: 1tl(X, x)
-+ G, deft.ned as follows: Let w: [0, 1] ---+ X represent [ro] € 7t1(X, x) and let
w: [O. 1] -+ X be the lift of (Q with cò(O) = x. Then W(l) e p-1x, and fP([w]) is
defined to be the unique element of G sucb that
(Al)
cp([w])i = W(l).
- .
One verifies that cp is a homomorphism is surjective, and has kernel1t1 x: so lt
induces the required isomorpbism.
Finally, we mention a sIightly different point of view which is sometimes
useful. [For our purposes this will be needed only. in exercise 2 or II.7.] Fìx
an abstract group G and a connected, locally path-connected space X. By a
regular G-cover of X (also called a principal G-bundle over X) we mean a
covering map p: g - X  where X is not necessarily connected, together with
a free G-action OD R satisfying condition (i) above.
In case g is connected, such a p is simply a regular cover in the usual
sense, together with an isomorphism or G with the grou p of deçk trans-
formations.
We will assume tbat a basepoint x E X has been chosen and that alI
covering spaces X come equipped with a basepoint xep-lx. We can then
define a homomorphism cp: 1ttX --. G exactly as in Al abovet the only
difference being that qJ wiU not be surjective if g is disconnected. In fact, ODe
checks easily that G/im lp  noX (isomorphism of G-sets).
Tbe main theorem OD regular G-covers (with basepoint) says that they are
completely classitìed by rp E Hom(1tlX G):

32
I Some Hor.noloJical AJgebra
.
.
,
(A2) Tbeorem. Let f{/ G(X) be the seI oJ isomorphm classes of pointed, regular
G-covers 01 X. The assignment olcp lo p gives a bijection
<€G(X)  Hom(7t1X, G).
-,.,'
. :.'
. J:
} .. 
o,' r
. ."'.!
; . :
.' ',
(Note: Isomorphisms are required lo preserve basepoints, commute with
the G aCtlon, and commute with the projection ooto X.)
SKETCH OF PROOF. Using tbe usuaJ classification of connected covering
spaces in terrns of su bgroups of n l X, one easily sees that connected, poin t ed,
regular G-covers correspond to surjections cp: '1t1X ..... G. The study of
disconnected covers is easily reduced to the connccted case by considering
tbe connected oomponents of X. D
". .:.:"
": r.
.,
. .
.' .:."":"
-:\"':.,
'ìi:-::-::
l ':
. t:
:':t; .
".'- "<
1f.";....":-,..
,.,
. III(;
:....r.
......
,-
..
'.
. ...
;
-
. ..
:i"
"

. :
.
..
.
"-
,;
,'."1:
. .. .
': "-:'
.
.-
. :f'
:."  ;
;.: ti.
,".: 1-
. -.,,0
..
I,
,
.
"
. .O:i
. .;; 
". -,
,. ..r:
.- "i.
. ..
'-
. ,.
1'\
.f
- -:
'1':.._
<o,
,_ 44
;:
..-
,
.
:'i--
,.r;-
- .: ,
.. -"_'$
"
,
-i
CHAPTER II
The Homology of a Group
1 Generalities
In homological algebra one constructs homological invariants or algebraic
o bjects by the fol1owing process, or some variant or it:
Let R be a ring and T a covariant additive functor from R-modules to
abelian groups. Thus the map HomR(M, N) -+ Homz(TM TN) defined by T
is a homomorphism of abelian groups for alI Rmodules M, N. For any R-
module M, choose a free (or projective) resolution t: F -+ M and consider the
chain complex TE oi abelian groups obtained by applyjng T to F termwise.
Now T, being additive, preserves chain homotopies; so we can apply the
uniqueness theorem for resolutions (1.7.5) to deduce that tbe complex TF is
independent, up to canonical bomotopy equivalence, of the choice of resolu-
tion. Passing to homology, we obtain groups H n(T F) which depend only OD
T and M (up to canonical isomorphism).
This construction is of no interes4 of course, if T is an exact functor; for
then tbe augmented complex
... -+ TF] -+ TFo -+ TM -+ O
is acyclic, so that H,.(TF) = O for n > O and Ho(TF} = TM. Thus we can
regard the groups H n(T F) in tbe generai case as a measure or the failure of T
to be exact.
In this chapter we will apply this construction with R = lLG, M = Z, and
T equal to the 'co-mvariants" f\,lfictor which we will describe in 2 below.
This particular choice or R, M and T is not arbitrary, as we wiJl see, but
rather it is a reflection o£the topology which motivates the homology theory
of groups.
33

34
..
,
"
Il The Homology of a Group
.,.
, .
2 Co...invariants
r..
If G is a group and M is a G-module, then tbe group or co-invariants of M,
denoted MG, is defined to be the quotient or M by the additive subgroup
generated by the elements or the form gm - m (g E G, m E M). Thus M G is
obtained from M by "dividing out  by the G-action. (The name . oo-ln-
variants , comes from the fact that M Q is the largest quotient of M on which G
acts triviaIly, whereas MO, the group ofinvariants, is the largest submodule or
M on which G acts trivially.) In view of exercise la ofI.2. we can also describe
MG as M/1M, where l is the augmentation ideai or 7LG and 1M denotes the
set of ali finite sums L aibi (ai E l, bi E M).
Stili another description of MG is given by:
(2.1) MG  Z @ZG M.
Here. in order for the tensor product to make sense, we regard li.. as a right
ZG-module (with trivial G-action). To prove 2.1, note that in l0zG M we
bave the identity 1 @ gm = I. g @ m = 1 @ m; hence there is a map Ma-i'
7L 0zG M given by m f--+ I @ m. wbere m denotes tbe image in M G of an
elernent m E M. On tbe other hand, using the universal property of the tensor
product, we can define a map 7l. @ZG M ..... Ma by Q @ m H amo These two
maps are inverses of one another.
In view of 2.1 and standard properties of the tensor product we im-
mediately obtain the following two properties of the co-invariants functor;
(2.2) Right-exactness: Given an exact sequence M' --+ M -+ M" --+ O of
G-modules, the induced sequence M -+ M G -+ Me. -+ O is exact.
(2.3) If F is a free ZG.module with basis (ei1 then F G is a free Z-module with
basis (ia.
Finally, we note that tbe co.invariants functor arises naturally in the
topologicaI setting or I.4:
'.
"
+':" .!
:". . .....,
:. .
)- _"'V,
. "
; ,. f,"
- A "1-"
'.
".:..'
, ,-:
''-.l'\:.
.. -,
... "I!  !
;-.'.:?"-
.: ,I,:
. . ,
: .:';:':,
. .,
'. '.
... . .. 
. ."
. :;CI
'v
.: .:'1:
."'';
.' , '-"'.
": ._t:.'
. :."';.
. "'..
-:':
"
. .;
. -,
- ,.
.,
... . l'"
.  .: :...
 .-
. .'"
),.: .!.
: \ ::.'.!'
:l?';..
jo':.'..\'
.;. i'.:
'-'..1.
.' :!.:
(2.4) Proposition. Lel X be a free G-complex and let Y be the orbit complex
X 1(;. Then C .(Y)  C.(X)G'
PROOF. The projection C.(X) --io C.(Y) induces by passage to the quotient
a map ({J: C.(X)G -+ CIt.JY). Now C.(X)G bas (by 2.3 and the observations in
I.4) a Z -basis with one basis element for each G-orbit of cells or x. But C.( Y)
also has a Z-basis with one element for each G-orbit or ceIls of x, and jt is
clear that ({J maps a basis elernent of C.(X)G to tbe corresponding basis
element or C.(Y) hence rp is an isornorphism. D
I. \
.it
-).';
. .:...
.........:.
-t'-'-:-
': .
.l -
,
"".
,- .
.(1-
,-
-:'.
--.
 J:.
.
" ,
. .:-!-.
EXERCISES
l. If S is al1 arbitrary G-set. show that (ZS)G A::: Z[SjG].
2. The freeness hypothesis in 2.4 ìs unnecessarily strong. Find a weaker hypothesis under
which the conc1usion remains true. [Hini: U.se exercise 1.]
 ".,:..
,-
3 The De6nition of H*G
35
3. Let H be a normal subgroup or G and let M be a. G-modu]e.
(a) Show t hat the actiol1 of G 011 M induces an actiol1 of G /H on M 11-
(b) Show that MG  (MH)G/H.
(c) Show tf1al M li  Z[GjHJ @ZG M as GjH-modules. (Here the right-tral1s1ation
action or G 011 Z[G/H] is used to form tf1e tensor product. and Ihe left-translation
action ofG/H on .l[G/H] is used to give the tensor product a G/H-module structure.)
3 Thc Definition or H *G
Let G be a group and E;: F --io 7L a projective resolution or 7L over ZG. We define
tbe homolog y groups of G by
HlG = HI..FQ)'
As we explained in 1 the right.hand side is independent of the choice of
resolution, up to canonica1 isomorphism. For example, suppose G is a finite
cyclic group of order n. Using the resolution
... t-l) 7LG  7LG r-t 7LG -+ 7L --fo O
of 1.6.3, we obtain for Fa the complex
o " o
...-+7L7L-+7L
Thus
li. i=O
HiG  7L" i odd
O i even, i > O.
The reader is invited to similarly compute the homology of a free group (cf.
1.4.3) or a free abelian group or rank 2 (cf. I.4, exercise 3). We will treat these
examples topologically in the next section.
For any group G we can always take F to be the standard resolution
(1.5) in which case we write C.(G) for the chain complex FG' Using 2.3 and
the formula 1.5.1, we can describe C.(G) explicitly, as foIlows: Define an
equivalence relation on the (n + l)-tuples (go, . . . , gn) (gi E G) by setting
(go ... , gli) -- (ggo, .. . , gg,.) for alI g E G and let [go,...  gli] denote the
equivalence class of <90. . .'. g,.). Then CII(G) has a Z-basis consisting or the
equivalence classes (go,..., gJ, and ò: C,.(G) -+ CI'I-l (G) is given by
a = Li=o (-l)idi, where
dl[go, . . . , g,J = [go, ...  Di. . .., gn].
C.(G), when described in this way, ìs often called the horoogeneous chain
complex of G, because of the analogy with homogeneous coordinates for
projective space.
(3.1 )

36
Il Thc HomoJogy of a Group
Tbc non-homogeneous description or C.( G) is obtained by using the bar
notation and the fonnula 1.5.2. From this point or view C,,(G) has a Z-basis
consisting of n-tuples [gli.. .1911], and o = Li o (_1)fdj, where
;
'.
[g21... JgJ
dt[gll.. .19lJ] = [gli.. .IOigl+d.. .Ig.a
[g1 J. . .1911-1]
i=O
O<i<n
1= n.
..
.oI
"'."
,"
,
:-i.
l
.. .'
Note that the symbol [911.. . I gli], which previously was used to denote a
typicaI ZG-basis elernent of F Ir  now denotes thc image or that basis element in
(F Jo = C,,(G). This abuse ofnotation which is standard, might occasionally
cause confusion. The.reader is also warned that some authors write [g 1, . . .  gli]
or (gl' .. . ,9,,) instead of [glI.. 'lgJ.
In low dimensions C.(G) has tbe form
Cz(G)  Cl (G) .i Z,
where 8[glh] = [h] - (gh] + (g]. Consequently, HoG = 7L and BiG is
isomorphie to the abelianization Gab = G/[G, G). (Explicitly, ifwe denote by
g the homology class or the cycle [g], the reader can easi1y check that there is
an isomorphism H 1 G --io G szb such that 9 1--+ g rnod [G, G].)
.:"{
"
.
':,
"
 "
-'
, ";'l':
1.,:':"
>
:.....

:,
-'
..
,
,
1
, '-
,
.>
, .
., ;

,
:
.
,
.;
.'
,
/f
'i"
EXERCISES
1. Let gl' . . . . gJl be n elements of G which pairwise commute. and Iet
Z = L (-lY'-'Hr(gu(l}l. . '!9tJ1nJ
..
in Cn(G). where (J ranges over aH permutations of ft.... 11}. (If n = 2. for example.
z = (g 1192] - [92 ]gJ.) Verify that 2 is a cyc1e. [Such cyc1es playan ìmportant role in
the homology theory of abeIian groups. as we wiJ[ see in Chapter V.]
.>
.'
. ," . -.
2. If G is a non-trivia1 finite cycJic group. show that 7!.. does not ad mi t a projective resolu-
tion of finite lengrh over ZG.
.-'."':::
. -..,
: . ,
3. If G has torsion (Le.. non-trivia] elements of finite order), show that Z dOe5 noI adrnit a
projectìve resolution of finite Jength over lLG. [Hint: This follows from exercise 2.]
4 TopologicalInterpretation
..
}.
If Y is a K(G. l)-complex with universal cover X tben we know (1.4.2) that
C.(X) is a free resolution of 7L over 71.G. Since C * (X)G  C *( Y) by 2.4 above,
we obtain:
-.;;
;:-'-.
<':i
.. . i
, n'
-..,
'..'
, - 
(4..1) Proposition. IfY is a K(G. l}comjJlex then H.G :::::: H. Y.
,.
"
'"','
::c
;
4 T opologjcal InterpretatioD
37
[In some treatments of the homology theory of groups, this result is taken
as the definition of H * G. as wc indicated in the introduction.]
EXAMPLES
I. Let Y be a bouquet or circles, indexed by a set S. Then, as we saw in
1.4.3, Y is a K (£(S), l). Hence
7L i=O
HJ.F(S» = Hi Y = .lS(=F(S)ab) i = 1
O i> I
2. Letgbean integer  1 and let G = (ah" . , Qg,bt, ..., bg; Of-l [ah ha),
i.e., G is the grou p with generators a l. . . . , u9. b 1. . . .  b g and the single defining
relation nf= I [a; ba = L Thus G is the fundamental group of the. dosed
orientable sunace Y of genus 9 (cf. Massey [1967], eh. 4, 5), and I clalm thal
Y is a K(G, 1). To see this, we need only note that the universal cover X or Y
is a non-compact surface, since G is infinite. Consequently, one knows from
the homology theory or m anifolds (cf. Dold [t 972], eh. VIII, 3) that H i X = O
for i > 2, so Y is a K(G. 1). [Alternatively, onecan explicitly exhibit X as the
hyperbolic pIane tiled by 4g-sided polygons (cf. Siegel [1971], eh. 3, 9), hence
X is contractible.] Thus
li
H.G = H. Y = 7f.2(J
I J
i=0,2
i = 1
i> 2.
o
The interesterl reader can similarly treat the group <CI"'" c,,; n = 1 Cf>
(k > 2), which is the fundamental group of the non-orientable c10sed surface
with k crosscaps. Finall y, we remark that non-compact surfaces and surfaces
with boundary are also easily seen to be K(G lfs. but we obtain no new
examples in this way since the fundamental groups are free.
3. The surface groups just considered are examples or one-relator groups.
Suppose now that G = (5; r) is an arbitrary one-relator group, i.e.. G is the
quotient or a free group £(S) by the nonnal closure of a single eternent r. Let Y
be the 2-complex (V!JeS s!) Ur e2 obtained from the bouQuet of circles Vs:
by attaching a 2-cell via the map SI -+ VS: corresponding to r. Then
1[1 Y = G (cf. Massey [1967], ch. 7, 2). If r is not a power r.l (n > I) in F(S).
then a deep theorem or Lyndon implies that Y is a K(G, I); proofs can be
found in Lyndon [1950], Dyer- Vasquez [1973] and Lyndon-Schupp [1977]
(eh. III, 9-11). (Ir r is a power t on the other hand then one can show that G
has torsion, so that there cannot exist a finite-dimensional K(G, l») cf.
exercise 3 of3 above.) The chain complex C* Y is easily seen to ha ve the forro
j} o
1L. -fo lLS --io Zt

r
-
f
38
II The Homology or a G.-oup
..
".. I.
..:" . I
..
:( ,.,'1.
.: .
where ò(1) is thc image of r in lLS = F(Slw' Hence HoG = 7l., H1G = Gab,
H 2 G = {7L if rE [F(S), F(S))
O otherwise.
and HiG = O for i > 2.
4. If G = lL". the free abelian group or rank n, then the n-dimensional tOTUS
y = SI X ... X SI (n factors) is a K(G, l) since its uni versaI cover II is
contractible. Hence H j G is a free abelian group of rank (1) = n !fi!(n - i)!.
Exmple 4 can be described in terms of the em bedding of the group G = ZII
as a dIscrete subgroup of the Lie group L = ". Indeed, the K(G, l) Y =
SI X ... X Si is simply the quotient L/G. Our remaining examples wiU
furth.er illustrate this method of constructing K(G. 1)'s. These examples will
reqUlre somewhat more effort to read than the previous ones, and they will
Dot be referred to again until Chapter VIII; the reader may therefore want to
simply glance at them now and read them more careruHy later.
5. Let G be the n x n striet upper triangular group over l, i.e., G is the
group of n x n integrai matrices with I s OD the diagonal and O's below the
diagonal. Let L be tbe n x n strict upper triangular group over IR. Then G is a
discrete subgroup or L, and we can form tbe coset space L/G. Since G is
discrete, the projection L --+ L/G is a covering map. Indeed9 let U be a neigh-
borhood of l in L such that U lì G = {l} 9 and let V be a neighborhood of 1
such that V - l V c: U. Then for an y l E L, the neighborhood W = IV of l has
the property that its transforms W g (g E G) are disjoint; our assertion follows
at once. Finally, L is obviously homeomorphic to Euc1idean space 
d = .n(n - 1)/2 so the manifold LIG is a K(G, 1) and H. G  H*(L/G).
[Stnctly speakmg, we should verify that LjG admits a C W -structure, since
4.1 was proved only for K(G, l}complexes. This is in fact true by Whitehead9g
triangulation theorem for smooth manifolds (cf. Munkres [1966], eh. II), bul
an easier way to deal with the pro blem is to simply o bserve that the proof of
4.1 goes trough with no difficulty in the context of singular homology theory;
see exerCIse 1 below.]
6. Let L be the Lie group GLII(R). In contrast to tbe Lie groups considered
in examples 4 and 5, L is not contractible. Nevertheless, there is a contraetible
anifold X assoçiated to L wbich can be used to study the homology or
discrete subgroups of L. In order to describe X, we need to recall some ele-
rnentary linear algebra.
RecaI! that a quadratic form on " is a funetion Q: IR" --+ IR of the forro
..' "j.
\.....
"
.. !1
...r.;i.
,....
. '1,"
, .,..
 ':"
.;\"-
.....'
:-.\:
:r:
:.:..
": .:
i;'
'"
} i'
or.
'.':"
v.
, ,
:. .
, f',
;. "li
 '
:-. ..
,(1\.
',- .
,I. .:
.....;-
..
. .
" -
"I"
;...
:.' ......
.. .
".
-,
,o
,
. o
"
,.
: ,..:
",
"
.-i
. -
. ".
,
Il
Q(xh ...  XII) = L aljxjxj.
t.l"" 1
The mtrix A = (aij) can be taken to be symmetric, and it is then uniquely
determmed by Q. We will orten identify Q with A. The forro Q and the matrix
A are called positive definire if.Q(x) > O for ali x ::J:. O in R". The set or positive
..
4 TopologicalInterpretation
39
definite symmetric matrices is a convex open subset or the space or alI
symmetric n x n matrices.
We now define X to be the space of positive definite quadratic forrns on ",
topologized as a subspace of the space of symmetric n x n matrices. It is c1ear
from what we ha ve said above that X is a contractible manifold of dimension
d = n(n + 1)/2. [In fact, X  .] There is a right action or L = GL..,(R) on IW
by right matrix multiplication (where an element of [R" i5 thought of as a row
vector), and this induces a left action of L on X, said to be given by 'change of
variable ":
(gQ)(x) = Q(xg)
for Q EX, 9 E L, x E ". [In terms or symmetric matrices, this action takes the
form
9 . A = gAg1,
where gr is tbe transpose of g.] ]t is welI-known that any Q E X is equivalent
under change or variable to the standard form Qo = L x , so the action of L
00 X is transitive. Moreover, the isotropy group LQo is the orthogonal group
K = O,,(IR). We therefore bave a bijection
X  L/K
of left L-spaces, which can be shown to be a homeomorphism.
Since K is eompact it follows that the action of L on X is proper, i.e., that
the following condition is satisfied: F or every compact set C c X, {g E L:
gC lì C '1= 0} is a compact subset of L. [This says, roughly speaking, that the
transforms gC of any compact set C tend to 00 in X as g --+ 00 in L.]
Suppose now that G is a discrete subgroup of L. Then for any compact
C c X, {g E G: gC lì C '# 0} is finite. One deduces easily that the isotropy
grou p G Jt or any x E X is finite and that x has a neigh borhood U such that
gU lì V = 0 forgEG - Gx.
FinallY9 suppose further that G is torsion-free. Then the finite isotropy
groups G x must be trivial, and it follows at once that the projection X -+ X jG
is a regular covering map with group G. Thus X /G is a K(G l) and H * G 
H.(X/G).
This discussion does not apply to GL,,(lL), the most obvious discrete sub-
group of GLII(IR) because it is not torsion-free. But G L,{L) does bave tOIsion-
free subgroups G of finite index (cf. exercise 3 belo w), and our discussion
applies to them. U nfortunately, it is extremely difficult in practìce to actually
compute H .(X /G).
7. Consider now the speciallinear grou p SL,JIR) = {g E G LlI(R): det g = l}.
We can then replace tbe space X of example 6 by a contractible manifold X o
on which SLR(R) acts properly, with dim X o = dim X - L Namely, we take
X o to be the quotient space of X obtained by identifying two quadratic forrns
which are (positive) scalar multiples of one another. One can verify that
Xo  SL,,(Iii)/SOIl(IR).


40
Il The Homolol)' of a Group
so tt the SL(IR)-action OD X o is proper, SOn(lR) being compact. Moreover,
?C o l a contracible manifld; in fact, the interested reader can verify that X o
IS dlffeomorphlc to Euclidean space of dimension n( n + 1)/2 - 1. As in
example 6, then. we have H.G  H.(XoIG) ror any discrete, torsion-free
sbgroup G c SLlI(IR). We note that if n = 2 tbe space X o can be identified
wlth tbc upper half pIane .JF c IC, with S L 2( IR) acting by linear fractional
transformations:
;.
. . : 
,
, '.'
..\
(a b)z = az + b.
c d cz + d
Indeed, ne checks that this defines a transitive action or SL2(1R) OD.JIf and
that the lsotropy group at z = i is S02(1R); this leads easily to the desired
homeomorphism
.  . .:1.1.
-'
. _.
. ".!;...
,..:.;"
," ,t"
'r....
. ,. 
)'
.i'
o
;t'  SL2(IR)jS02(R)  X o.
8. Finally, we state the generaI facts which were iIlustrated in the previous
examples. Let L be a Lie group witb only finitely many connected components.
Then L has a maximal compact subgroup K (unique up to conjugacy), and
the homogeneous space X = LI K is diffeomorphic to . d = dim L - dim K.
(Proofs can he found, for example, in Hochschild [1965], eh. XV.) Con-
sequently, if G c L is a discrete, torsion-free subgroup then tbe quotient
manifold X/G is a K(G, 1) and H.G  HiX/G).
" '.
. :.5-
. .,
,.
:.
...
'..
,.
.'-
.,.
 .
..'
'r-
EXERCISES
'.
.
l. Let Y.be a path-connected space. If Y has a contractibJe. regular covering space X with
coerlDg group G, show that H.. Y ::::; H * G. [Hint: The singular chain complex
C"X) provides a feee resolution or Z over ZG- and c:nX)G  GQI!(Y).]
2. This exercise requires some elementary homotopy theory.) Let Y be a K(G, J )-space.
I.e., a path-conneçted space with 1! l Y = G and lli Y = O for i > L Prove that H.. Y 
H.G. [Hint: This is clear if Y is a CW-complex; in Ihe generai case, replace Y bya
CW-complex which is weakly l1omotopy equivalent to Y. cf. Spanier [1966], 7.8.
Ahernatively, one can directly construct a chain homolopy equivalence C:nll( Y) 
C. G. cf. EiIenberg-MacLane [1945].] Note that exercise l is a special case of the
present exercise.
..
"
.,
. . ..
.
.r
:";"
,.
t
3. Fìx an integer 1'! > L For any integer N > 2.let r(N) be the kemel of the canonical
map GL..(Z) -+ GL..(ZfNl). i.e., r(N) = {g E GL..(Z): g = 1 mod N}. where I denotes
the identity matrix. The group r(N) is caIJed the prirlcipal cOflgruetlCe subgroup 01
GLnfZ) of level N. Note that r(N) has finite index in GL,,(Z). since GLtI(Z/NZ) is
finite. The purpose of this exercise is to prove that r(N) is torsion-free for N > 3.
(a) Lei p be a fixed prime and let A be an n x 11 matrix of integers such that A =
l mod p. If A #:- l, then there is a uniq ue positive integer d == d(A) such that
A = l modpd and A;f:. 1 mod ,1+1.
-i-
. 'e.
:
,.
i
S Hopf's Theorems
41
Show that d(A Il) = deA) ror any prime q  p. If p is odd or d(A)  2. show that
d(AP) = deA) + l. [Hint: Write A = l + pdB with B ¥ O mod P. and look at tbe
binomial expansion or (l + rfB)I, l = p or q.]
(b) Deduce that r(N) 15 torsion.free for N  3 and that r(2) has on]y 2-torsioo.
(Hint: Suppose A E r(N) has prime order and apply (a) with p a prime divisor of N.]
5 Hopfs Theorerns
The purpose of this sectiOD is to prove the results stated as 0.1 and 0.2 in the
introduction. We will need to use the Hurewicl theorem (cf. Spanier [1966],
eh. 7, 5), which says that if 7tiX = O for i < n (where n  2), then H,X = O
for O < i < n and the Hurewicz map h: XliX -+ HjtX is an isomorphism. (In
fact, an examination of our proofs will show that we only need the surje<::tivity
or h, which is considerably easier to prove; indeed, it foIlows directly from
Spanier's Thm. 7.4.8.)
We begin by observing that for any group G and integer none can COID-
pute H i G for i < n + l from a parlial projective resolution of length n:
(5.1) Lemma LeI F" -+ . . . -+ F o ....... Z ...... O be ari exact sequence oJ ZG-
modules where each F i is projective. Then Hf G :::; H t..F G)for i < 11. and there is
an exact sequence
0-+ HJI+ ](G) - (RjtF)G -+ HiF G) --io H,.(G) --+ O.
PROOF. Extend F to a Culi resolution F + by choosing a projective module
F+ 1 mapping onto HnF, etc.:
. . . ...... F + 2 ...... F + l  F Il -+ F Il - 1 --+ . . . -- F o  li.. -+ o.
\. J
HnF
It is easy to see (either by direct inspection or by considering the homology
exact sequence assoçiated to the exact sequence O .... F G -+ F --+ FZ / F G -+ O
of chain complexes) that HI..F:) = Hi(F d for i < n and that there is an
exact sequence
0-+ Hn+1(F'ri) - A - H,,(FG) -+ H.,(Fri) -+ O,
where A = coker {(F+ :Z)G --+ (F+ l)G}' By 2.2 this cokemel caD be identified
with (H"F)G' whence the lemma. D
(5.2) Tbeorem. For any connected CW-complex Y there is a canonical map
tJ1: H. Y -+ H.n: (7t = 1t1 Y). /f Xi Y = O for 1 < i < n (for some n;2 2)
then '" is an isomorphism H i Y  H i 7t for i < n, and the sequence
,. '"
x.. Y --io H" Y --io HJl7t --+ O
is exact.

42
II The Homology of a Group
(Note that the hypothesis always holds with n = 2, so 5.2 yields the result
0.1 stated in the introductìon.)
PROOF. Let X be the uni versai cover of Yand let F be a projective resolution of
l over 1l:1r:. Since C.(X) is a complex offree Z1!-modules augmented over lL,
the fundamentallemma (1.7.4) gives us a chain map (aver 7L.rc) C$(X) ..... F.
well-defined up to homotopy. Taking co-invariants we obtain a map C.(Y)
..... F'II;' which induces the desired map 1/1; H. Y  H.. n. One knows that
1tiX  Xi Yfor i > l (cC. Spanier [1966], 7.3.7), so oue hypothesis implies that
1tiX = O for i < n. Hence HiK = O for O < i < n and the Hurewicz map
h: 1t1l X  HnX is an isomorphism. We therefore ha ve a parti al free resolution
Cn(X) --+ . . . --+ Co(X) -+ 7L. -+ O.
whose n-th homology group is the group ZII X of n-cycIes of X. Lemma 5.1
DOW implies that Hi Y  Hi1t for i < n and that there is aD exact sequence
-
'41
ZIIX -+ ZII Y  HJl1t - O.
The map  which arises here is easi1y seen to be the composite ZII y _
H,. Y .!.. Hn1t. Consequent1y the sequence
HIIK  Brt y  HlIll -+ O
'.
..
.,
.
"
;'
,
'
"i.'
is exact, and the desired exact sequence
h .p
X" Y  Hn Y --+ H,. n - O
.'>
"
'.
,
now follows from the diagram
nrlX ,. H X
;;;; 1'1
l 1
1tIlY Il ) HII Y
.
!'
. . . I"
. ';;)1
D
.:' -,.;;,,,
. ..
-1
.':
We turn now to formula 0.2 of the introduction.
"
. ;:
.4;'
. '. '.'
(5..3) Theorem.lfG = F/R where F isfree, then H2G  R n [F, F]/[F, R].
PROOF. Let F = F(S), let Y be a bouq uet or circles indexed by S, and let r be
the connected reguIar covering space of Y corresponding to the normaJ sub-
group R of F(S) = 1t J Y. Cboosing a basepoint li in Y lying over the vertex of
Y, we identify G = F / R with the group of covering transformations of r as in
Chapter I, Appendix, Al. For any f E F we regard f as a combinatorial path
in Y and we denote by Ì the lifting of f to Y starting at v. [By a combinatorial
path in a CW-complex we mean a sequence e 1  . . ., e,. of oriented l--cells 8uch
that the initial vertex of ei+ 1 is equal to the final vertex or ei for i = l,...,
n - l.] This path /. then, ends at the vertex Iv. where J is the image or fin G.
S Hopf's Theorems
43
The complex C. Y is a complex of free ZG-modules. and it provides a
partial resolution CI Y ..... Co y  7L ...." O. We can therefore apply 5.1 to
obtain H 2 G  ker{ (H l 9)6 --i' H l Y}. Now H l r  (1t1 Y)ab  Rab7 and I
claim that the composite isomorphisII! H l r  RQb is an isomorphism of G-
modules where the G-action on Rab is induced by the conjugation action or F
on R. [This makes sense because the conjugation action or R OD itself in?ces
the trivial action or R OD R4Jb.] To verify the claim, one checks tbe definlhons
and finds that the isomorphism RfIb  H 1 r is induced by a map d: R --+ H I V
de:fined as follows; For any r E R the lifting r is a c10sed path in Y; taking the
sum or the orien tOO l-cells whicb occur in i, we obtain a I-cycle in r hence an
element of H1 Y, and this element is by delìnition dr. Now if f eF and reR,
the lifting of f r f - 1 is the path
l
jF
/-1
Q
',)
-o
--o .
v
IV
v
IV
hence d(f r f - 1) = J dr. This shows that the isomorphism RQb =+- H l Y is an
isomorphism of G-modules, as claimed.
We therefore bave a diagram
(H} y)o  (Rab)G = R/[F, R]
1 l 1
H1 y  FQb = Fj[F, FJ
where the second and third vertical arrows are ind uced by the indusion R 4 F.
HenceH2G  ker{R/[F R] - F/[F7F]} = R n [F, FJ/[F, R]. D
(The G-module Rab which occurred in the above proof is called the
relation module associated to the presentation of G as F / R.)
Note that the chain complex C Y can be described explicitly, exactly as in
1.4.3. We therefore obtain, as a crollary of the above proof, the rollowing
resuIt:
(5.4) Proposition.lfG = F(S)/R then there is a11 eXQct sequence
O -+ Rab !!.. ZG(S)  7LG  7L  O
ofGmodules, where ZG(S) isfree with basis (e.s)a:eS, and òe& = s - 1.
(Here 5, as usual, denotes the image of 5 in G.)
Tbe map () which occurs here has been described explicitly in the proof or
5.3 in terms ofpath lifting. We will see in exercise 3d below that 6 can also be
described purely algebraically, in terms or the .free dìfferentÌal calculus_'

".
44
Il The Homology or a Group
Remal'ks

1. It is possible to give an algebraic proof of 5.4 (see exercise 4d or IV.2)
and then deduce Hopf's formula 5.3 directly from 5.4.
2. Hopf's formula suggests that, rougbly Speaking H G consists of
1. f1 ' 2
commu tator re atlOns [ai. bJ = 1 in G, modulo those relations that hold
trivialIy. See C. Miller [1952] for a precise formulation and proof of tbis
statement.
EXERCISES
1. In the situation of 5.2. suppose in addition that Y is ri-dimensional; prove that there is
an exact sequence
0--+ Hl(+ J1t' --+ (1t1l Y)1I: -+ Hn Y --+ HII1[ --+ O.
Hee (1t1f Y)1r makes sense because 1t acts on 1t''1 Y. cf. Spanier [1966]. eh. 7. sec. 3; this
aclIon corrsponds t? the obvius action of 7t on HIIX (X . universal cover of Y)
undr he lsomorphism 1[" Y - :IT"X  HJjX, [Remark: W!thout any dimension
restnctJOns on Y one can prove that there is an exact sequence
H...... Y --+ HII+ 111: --+ (1t1l Y)" -+ H" Y -+ HII7r. --+ O.
..:;.
This will be proved in exercise 6 of SVII.7 by a spectral sequence argument; the
reader may want to try to give a direct proof now.]
2. Let G = < S  r 1. r 2. . . .). Le.. G = F(S)f R where R is the Donnal c10sure in F(S) of
"1. r2. -...
(a) Show that the relation mooule Roto is generated (as G-module) by the images of
rl.r2.... .
(b) Show that these elementsfreely genen1te R/lb as ZG-module ifand only ifthe 2-
complex associated to the given presentation of G is a K(G. 1). (By the 2...complex
associated to the presentation we mean tbe complex (y SI) U e2 U e2 u .. .
2 . ES J rl" r] 2 ·
where the ?-ceJl ei IS attached to YS; by the map Sl --+ VS: corresponding to
,.. E 11: l (V Sa) = F(S).) Thus. for example. L yndon's theorem about onerelator
groups  = < S; r) which we stated in example 3 of  can be interpreted as saying
that Rllb 15 free]y generated by the image of r. provided r is not a power. (11 is in this
formo in fact. that Lyndon stated and proved his Iheorem.)
*(c) Let G = (S;,.> be an arbitra,.y one-relator group and write r = un in F(S)
here n > 1. is maximal. One can show (d. Lyndon -Schupp [1977]. I V.5.2) that th;
Jmage t of Il ]D G has order exact1y u. and we denote by C the cyclic group of order n
gencrated by t. If n > 1 the:n R.. is not freely generaled by r mod [R. RJ since thìs
genen!lor is c1early fìxed by C. But Lyndon [1950] proved that no other relations
hold. ì.e.. tbat the obvious surjeçtion Z[G/C] - Rllb is an isomorphism. Show that
thìs result can be interpreted topolosicaJly, as follows. Lei Y, r. and v be as in the
,
. ..
S Hopf's ThCOteros
45
proof of 5.3 and consider the lìfting r of thc Ioop r in Y. Since r = u" and ù ends at
rv, r is the composite path pktured schematically as foUQws:
v
...
t"-1ù'
.
.
.
.
'.
,
-,
Thus the map Si ......,. Y corresponding to r is compatible with the action oftbe cydic
group C. where C acts on Si as a group of rotations as in *1.6. Deduce that we can
form a 2-dimensional G-complex X by attaching 2-cells lo Y along thc loops gr.
where g ranges over a sei of representatives for the cosets G/C; if u is the 2-cell
attached along r. then the sotropy group Cc is equal to C. wìth C acting on (t as a
group of rotations. Show that L yndon's theorem about RCIb stated above is equivalent
to the statement that X is contractibJe. [Note: In the terminology ofLyndon -Schupp
[1977], ch. III. X is the Cayley complex associated to the presentation (S; r).]
,
,
"ò
!
'.
.:
,
,-
3. In this exercise you will construct the Fox "free derivatives" by using the ideas
introduced in the proof of 5.3. See the exercises in gIV.2 for a purely algebrak treat-
ment or the same resuJts. If G is a group and M a G-module. then a derivation (or
crossed homomorphism) from G to M is a function d: G - M such that d(gh) =
dg + gdh for aH g. h € G.
(a) Let the notation be as in 5.3 and its proof. Show that the definition of dr given in
that proof can be used (almost verbatim) to define a function d F -+ Cl Y which
satisfies d(fl 12) = d/l + Il djl for alI /1. /2 E F. Thus if we regard the G-module Cl Y
as an F-module via the canonical homomorphism F -+ G. then d: F --+ Cl Y is a
derivation.
:
.'
y
"
"
(b) F or any free group F = F(S), show that there is an F -module g which admits a
derivation d: F - Q such that (} is a free 7l.F -module with basìs (ds)  s. [Hint : Apply
(a) with R = {l}.] We ca]) dI for f E F the forai (free) derivat ive or f. The coefficient
of ds when dI is expressed in terrns of the basis (ds) is called the pa rtin I deri va rive of f
with respect to s and is denoted òI/iJs; thus dI = L! E S (òf jiJs )ds. where òj /òs E ZF.
Show that ò /f1.ro: F - llF is a derivation and satisfies al !ÒS = (j.!. r (l E S). These
properties completely characterize il/os. (For example. if S = {s. t}. you should be
able to use these properties to compute ò{lS- tts2)/òs.)
(c) With F as in (b). show that c:my derivation d: F - M. where}.tl is an F-moduJe.
satisfies di = L  S (òl jos)ds.
(d) Show that the map o: RDI! -}> ZGIS) in 5.4 is induced by the map R --+ lLG(S) given
by r........ O!!S (orjòs)es where orjiJs is the image or òr/i)s under the canonical rnap

46
II The Hornology or a Group
?LE - ZG. If R is the nonna! closureofa subset Ts F, deduce that there is a partìal
free resolution
,.
,
"
ZG(TJ :; 7LGlsl  lLG  7L -+ O
"
'",-
. !
such that the matrix or a 2 is the .. Jacobian matrix' (at/òs ),.. T. S s.
"
"
",'."
,
:-:
.4. Let G = F I R as in 5.3. The purpose or this exercise is to estab1ìsh exphcit fonnuJas for
the isomorphism r.p: H 2 G .; R n [F, F}/[F. R] and its inverse. where we view H l. G
as the second homology group of the standard chain complex C.( G).
(a) Exhibit a specìfic chain map in dimensions ::5:2 from the bar resolution to the
partia1 resolution
,; "."
'. ...
 . '!
. :'''."
. '.'
: ".::
":: '.'.
 , .
.
. :'.
(*)
:"/1
: \{.
ìX
, ..
ZG(R) - C 1 Y - Co f - z -. O,
where r is as in the proof or 5.3. (Here 7l.G(R) is a free module wìth bas.is (e),...R; it
maps onto R/lb = H l Y c: C l Y in the obvìous way.) Deduce that lp can be computed
as foUows: Choose for each g E G an eJement f(q) E F such that f (g) = g. Given
g, h E G, wrìte f(g)f(h) = f(gh)r(g. h) whece r{g. h) E R. Then there is an abelian
group homomorphism C2(G) -+ Rub given by [g Ih] H r(g, h) mod [R. R]. and this
induces the isornorphism tp: H 2 G -4 R n [F. F]/[F. R] by passage to subquotìents.
(b) Exhibit a cha..in rnap from (.) to the bar resoJution. and deduce that tp - · can be
computed as follows: Let D: F -+ C2(G) be the unique derivation such that Ds =
[11 S] for each free generator S E S, where the group C 2( G} is regarded as an F -module
by f. (glh] = [fglh]. (ExpJicitly, Df = Lse;s [oflè.sls]. where the symboJ [,1-] is
understood to be Z-bilinear.) Then (DIR): R - Cz(G) is a homomorphism which
induces tp - l by passage to subquotiems.
(c) le! a.. . u. Qg, blt ..., bt; be elernents of F such that the element r = nf"'1 [aj, bl]
is in R. Then (j) -l(r mod [F, R]) is represented by the cycJe
..
"
. ,
.,
" :r,
. ,,-,
"
'. ...
-.. . -i.
, .t
.,
,. .
..
:';:
;'
.'
','

 
... J,
,
"
!I
L ([li-dà.] + [/j-lodE;] - [li-lGiDiOj-ljiia - [l,rEa),
.=1
. ,
where lì = [Gl,1)1]'" [aj, [ja. [Hìnt: Il suffices lo prove this in the universal
examplewhereFisthefreegrouponal,.... ag.h]. ..., bjlandR isthenonnalclosure
of T. In this case. li' - I(r moo [F. RJ) is easily computed by the fonnula or (b).]
;.
.'
.. 
. .:!
.:':'
5. (a) Let G be a group which adrnits a presentatìon with n genemtors and m relations.
Let r = rkz(Go") = dimQ(Q @ Gob). Prove Ihat the abelian group H2G can be
generated by m - rz + r e]ements. [Hint: Let Y be the 2-complex associated to the
presentatìon. Computing the Euler characteristic l( Y) in two different ways. one
tìndsl - n + m = l - rk(H1 Y) + rk(H2 Y) = 1 - r + rk(H2 Y),whencerk(H2 Y)
= m-Il + r. Now H 2 Y is a free abelian group. beìng a subgroup of the group of
ceUular 2-chains of Y, and we ha ve a surjection H 2 Y -» H l G.]
Parts (b) and Cc) be]ow illustrate typical applìcations of (a).
(b) Let G be a perfect group (i.e., a group such Chal GBb = O) which admits a finite
presentation with the same number of generators as relatjons. Prove that H 2 G = O.
(c) Let G be a perfect group such that H 2 G = 71.2, $ Z2' Show tha. any n-generdtor
presentation of G must involve at least n + 2 relations.
- -
.
""1;
"l'."
"
}'
, ..
', :,
"" .H
. .
.
,-
S Hopf's Theorems
47
6. (a) By a group exrensiOtJ we mean a shoTt exacl sequence ] - N --+ G - Q --+ 1 or
groups.. Deduce from Hopf"s fOnTIula that such an extension gives rise to a 5-tenn
exact seq uence
CI (J )' "
H2G - HzQ -+ (H.N)Q -+ H1G - H1Q - O,
where the Qactìon on H I N = N.:.b is induçed by tbe conjugation action of G on N,
and y and  are induced by the maps N <+ G -» Q. [Hint: Write G = F/R and Q =
F /S. where R c S c: F. The desired sequence is then
R rì [F, F]
--
[F,R]
S lì [F, F] -1> N _ G _ G -+ O.
[F, S] [G. N] [G, G] N. [G. G]
whìch is easily proved to be exact.] Remark  We wiJ] give another derivation or this
5-term exact sequence. and a generalization of it, in Vn.6. It has ìnteresting applica-
tions to the study ofthe lower centrai series.due independendy to Stallings [1965a]
and Stammbach [J966]. See alsoStammbach [1973], Chapter IVforfurtherdeveJop-
ments aJong these lines.
(b) Conversely, show that Hopf"s formula can be deduced from the 5-term exact
sequence. [Apply (a) to l -+ R --+ F --+ G - 1 and reC'"dIl that H2F = O.]
.7. (a) RecaI] that the 3-sphere S3 has a group .structure. [It is the muJtiplicative group or
q uatemions or norm l.] If G is a finite su bgroup of S. deduce from the results or Ihìs
section that H 1 G = O. [Him: S3/G ìs a dosed. orientable 3-manifold with finite
fundamemaJ group, hence H 2(S3/G) = O.] Remark = A lisi of the groups G to wbich
this applies is given in W olf [1974] and recaJJed briefly in exampJe 2 or VI. 9.2 below.
We will also be able to give there a more elementary solutìon or the present exercìse.
cf. exercise 3 of VI.9.
(b) One or the most intereslìng groups to which (a) applies is the "binary icosahcdral
group.' This is a group G of order 120 which rnaps omo As, the alternating grop on
:5 letters, with centrai kernel or order 2. Moreover. G is perfect. Using these facts, the
resuJt of(a), and the 5-term exact sequence, deduce that H2(As) = 1:2,
(c)(R. Strebel) It ìs known that As admits the presentation (x, Y. z; x2 = y3 = ZS =
xyz = l). Considernowtheabstractgroup G = (x, y, 2; x2 = y3 = z:5 = xyz),and
Jet C be the cycJic su bgroup of G generated by the centrai element );;2 = y3 = z =
xyz of G. Thus GjC = A5' Determine the order or C and hence that of G. [Hìnt;
From the given presentation of G you can show that G is perfect, hence H 2 G = o by
exercise 5b above. The 5-term exact sequence now yields H 2( A s) ..!. H 1 (C) = C. so C
has order 2 and hence G has order ]20. In fact. with a Jittle more work you can show
that G ìs the binary icosahedral group.] Similar methods can be used to al1alyze
other abstract groups defined by presentations dose]}' relate<! to presentations of
known groups. You mighllook, for ìnstance. at some or the examples ìn Coxeter-
Moscr [1980] in connection with Miller's generalization or the polyhedral groups;
the treatrnent there can be simplified by the use of the method of the present exercise.
Remark.1t is clear from this exerc:ìse that H 2 is dosely related to the theocy of centrai
exlensions. It would be possible lO develop this connection systematicaUy on the
basis of Hopf's formula and the S-term exact sequence. but we will instead deduce it
from the generaI theory of group extensions, to be dìscussed in Chapter IV. See
exercise 7 in IV.3.

48
Il The Homology or 8 Group
6 Functoriality
Tbe standard chain complex C *( G) is cIear ly functorial in G, hence H.. G is a
{covariant} functor or G. This functoria1ity can a1so be described in terms or
arbitrary resolutions, as follows: Given a homomorphism IX: G -+ G' and
projective resolutions F and F' or Z aver ZG and ZG', respectively, we can
regard F' as a complex of G-modu1es via . Then F' is acyclic (aIthough Dot
projective, in generaI, over ZG), so tbe fundamentallemma (1.7.4) gives us an
augmentation-preserving G-chain map 1": F -io F', well-defined up to homo-
topy. The condition that t be a G-map is expressed by tbe formula
.,
..
,
.,'
,.
'.'
(6.1 )
t"(gx) = a(g}t(x)
.
for 9 E G, x E F. Clear1y T induces a map F G -+ Fé;" well-defined up to homo-
topy, hence we obtain a weU-defined map alli: H.G -+ H.G'.
-
"
..!
(6.2) Proposition Fix 90 E G and let 0:: G -+ G be given by rJ{g) = goggo 1.
Then IX.: H. G -+ H * G is the identity.
PROOF. Let F be a projective resolution of 7L over lLG and define t": F -i' F by
T(X) = go x. Then 't commutes with the boundary operator and satisfies 6.1 so
'f can be used to compute 0:.. But clearly'f induces the identity map OD F G,
whence the proposition. D
.
r
<,
"-)
r
-.
,.
"
:;.
I
.,
l.
,
": 
"
(63) CoroU8ry lf G is a group and N is a norma l subgroup, then the conjugation
action oJ G on N induces an action 01 GIN on H. N.
,
EXERCISES
1. Gi ven N -<J G as in 6.3.1et F be a projective resolution of 71. over 1:G and consider th
complex F /li' Then F /li is a complex of GIN-modu1es (2. exercise 3a), hence H.(FN)
inherits an action of GIN. Sbow that H .(F N) = H. N and tbat the resuJting GIN-
actioD on H. N agrees with that defined in 6.3. [Hint: Given g e G, the action or 9 on
H.., N can be compllted via the map 't: F -+ F given by t(x) = gx.]
,-
.,-
 "..::.
,.
.t
'>
2. F or any finite set A let I:( A) be the group or permu tations of A. If I A I :::s; I B I (where 1.1
denoles cardinality). cboose an injection i: A 4 B and consider the injection 1:(A) 4
1:.(B) obtained by extending a permutation of A to be the identity on B - iA. Sl10w
that the induced map H.I:(A) - H..(B) is independent of the choice or i. In
partìcular, if I A I = I B I. then H... I:(A) is canonicaUy isomorphic to H .E(B).
. ):
.!
, .'  .
". ":-:-
.
-. -
 '::'..
, l,.:
'. I
..: :
3. (a) Under thc isomotphisrn H l( )  ( )"'b of 3. show that H l (:x): H t (G) - H l (G')
corresponds to the map Gah - G obtained from a: by passage to the quotient; in other
words., the isomorphism H l( ) :::::: ( )U(I is natura!. In particutar. the action or GIN on
H l (N) in 6.3 aOO\le agrees with that dened in exercise 6 of 5_ [Hint: Use the bar
resolution to compute H . (ex).]
'.
,
7 Tbe Hornology of Amalgamated Free Pròducts
49
(b) Prove the fol1owing natuntlity propeny of HopCs isomorpbism 5.3: Suppose
G = FJR andG' = F'jR' withFandF'free,and suppose IX: G - G'liftsto&: F -+ F'.
Then the diagram
,

H2(G) 
H'l
H 2(G')  R' n [F', F']j[F'. R']
R lì [F, F]I[F R]
l
commutes. where the right-hand vertical arrow is induced by à. [Hint: Let Yand Y be
associated to the presentation G = F I R as in the proof of 5.3. and similarly let Y' and
Y' correspond to G' = F' jR'. Then a yields a rnap Y -+ Y', which yields a map C...( Y)-+
C.( :V'). which can be extended to a map 't or resolutions, which can be used to compute
H 2(!X)']
Remark. This exercise allows one to interpret in terms ofthe functoriaHty or H1 and H2
three of the four maps which occur in the 5-term exact sequence of exercise 6 of 5.
7 The Homology or Amalgamated Free Products
As an i1Iustration of the topological interpretation of group homology we
will derive in this section a Mayer- Vietoris sequence for computing the
homology or an amalgamated free product. We begin by reviewing the
necessary group theory.
Suppose wc are given groups Gb G2, and A and homomorphisrns 1X1:
A --. G1 and (i2: A --'--* G2. EventualIy we will assume further that 0;1 and 2 are
injective, so that A can be viewed as a common subgroup of G1 and G2,
but for the moment we do not make tbis assumption. By the amalgamated
free product (or amalgamaced sum, or amalgam) of G 1 and G2 along A we mean
a group G which fits into a commutative square
A .:J: G
). 2
',1 lp,
(7.1 )
01  G
fJ,
with the following universal mapping property: Given a group H and homo-
morphisms 'Vi: Gf -+ H (i = 1 2) with YICX. = J'2Cl:2, there is a unique map
(f): G -i' H such that lpfJj = Yi' We write G = G1 *A G2, and we say that the
square 7.1 is an amatgarnation diagram,
The universal property above shows that amalgamation is the group-
theoretic analogue of pasting two topological spaces together along a com-
mon subspace. Tbe Seifert-van Kampen theorem. which the reader has
probably seen in some form, makes the analogy precise via the 7tl-functor. We
will need tbe following simple versìon of that theorem:

50
Il Tbc Homology or a Group
(7.2) Theorem. Let X be a CW-complex which is the union DJ two connected
subcomplexes Xl and X l whose intersection Y is connected ami non-empt y.
Then the square
1tl Y ) Xl X 2
l l
'-
.,
,::i".
,.
,
'.
. .
 ':
XIX 1 )o nIX
is an amalgarnation diagram, where alI fundamental groups Qre compuled al a
fixed vertex Y E Y and ali maps are induced by inclusions. Thus
1tlX = ntX1 .1]'1tlX2'
"
.
;,
,
..
.',
.':
This is an easy consequence or the usual combinatorial description of the
fundamental group of a C W-complex. Details can be found in Schubert
[1968],1115.8, or Cohen [1978], Il.2.3. See also exercise 2 below for a proof
based on covering space theory.
We can express this theorem more concisely by saying that the functor
Xl: (connected, pointed complexes) -+ (groups)
preserves amalgamations. In order to study tbe homology of amalgamations
of groups, we would like to have a result going in the other direction, saying
that the UfunctorU K( -. I): (groups) -+ (complexes) preserves amalgama-
tions. This turns out to be true as long as the maps IX 1 and CXl or7.1 afe injective:
...
.'
>
'-
(7.3) Theorem (Whitehead). Any amalgamatiofl diagram 7.1 with !Xl and al
injective can be realized by a diagram
y'-- )X2
I [
XI"" ...x
01 K(1t, I )-complexes such thar X = X I V X 2 and Y = X l n X 2 .
The proof will req uire three elementary lemmas:
(7.4) Lemma.1f a l and a2 are injective then so are P 1 and f32 . Thus Gl' G 2' and
A can be regarded as subgroups ofG.
This is part of the  normal form theorem" for amalgamations. See, far
instance Serre [1977a], Lyndon-Schupp [1977], or Cohen [1978].
(7.5) Lemms4 Let X'  X be an inclusion 01 connected CW....complexes such
that the induced map x' ...., n offurrdamental groups is injective. Let p: X -+ X be
the universal cover oJ X. Then each connected componen! oJ p-I XI is simply
.:'.
7 The Homology or Amalgamated Free Products
51
connected (hence it is a cop y oJ the universal cover oJ X'). M oreover, these
compo1Jents are permuted transitively by the action DJ 1t on g, and 7C is the
isotropy graup oJ one oJ them; in other words, 1ro(P - 1 X')  n/n'.
PROOF. For any basepoint in p-l X we have a diagram
n.(p - l X') -
) 1rlX
[ I
. ('
n: ) 1t
J
where the vertical maps afe induced by p and the horizontal maps by in-
clusions. Since 7r1 g = {l}, the first assertion rolIows at once. The second
assertion, which wc wilJ no t make serious use of, is lert as an exercise for the
interested reader. D
(7.6) Lemma. Any diagram G1 - A -+ Gz 01 groups catl be realized by a
diagram X l  Y c.. X 2 01 K(n:, 1 )-complexes.
PROOF. According 10 exercise 4 of I.4 or exercise 3 of 1.5, K(n, 1 }complexes
can be constructed functorially. We can therefore realize the group homo-
morphisrns by cellular maps X l .- Y ...., X 2 of K(n, 1)'8. Taking mapping
cyJinders if necessary (cf. Spanier [1966]. 1.4), we can make these maps
inclusions. D
PROOF OF 7.3. 8tart with X l  Y c.. X 2 as in 7.6 and rorm the adjunction
space X = Xl Uy X 2, i.e.. X is obtained from the disjoint union Xl li X 2 by
identifying the two copies or Y. Then 11; l X = G l * A G 2 = G by 7.2, so we need
ani y show thatthe universa) cover X satisfies Hi.r = O for i > 1. Let g h X 2;'
and Y be the inverse images of X l' X 2, and Y in X. Since X l' X 2 t and Y have
acyclic universa] covers, i t folJows from 7.4 and 7.5 that X l, X 2, and Y have
trivial homology in positive dimensions. The Mayer-Vietoris sequence
associated to the square
y ('
[
. -
Xl c....
--J.X
I:"-.<CA MAr£'.A
t;ç\} - ---- .--
I / , .
'. /. ./' -
, V"4: -- .-<0 ,;'
, ::TA' D\'
D
-
)X2
I
therefore shows that HiX = O for i > 1.
(7.7) Corollary. Given G = G1 *A G2 where A c.. G1 and A  G2 there Is a
. Ma yer- Vietoris  sequence
... -+ HnA ........ HreGl G'J HnGl ...., HrrG -+ HII-1A -+ ....
This is immediate from 7.3.

52
Il Tbc: Homology or a Oroup
Remark.. As a bi-product or tbe proof of 7.3 we obtain an exact sequence of
permutation modules
(7.8) 0-+ Z[GjA] -+ Z[G/Ga E9 Z[GjG2J ...... 7L -+ O.
Indeed, tbis is just tbe low-dimensional part of the Mayer- Vietoris sequence
which we used in the proof of 7.3. (The second part of 7. 5 allows one to identify
H o( :V), H o(X 1), and H o(g 2) witb permutation modules.) We will give another
derivation of 7.8, again based on topological ideas, in an appendix to this
chapter.lt isalsopossible to prove 7.8 algebraically, cf. Bieri [1976], Prop. 2.8,
or Swan [1969], Lemma 2.1. Moreover, it is possible to give a purely alge-
braic proof or 7.7, using 7.8 aS the starting point. We will explain this in tbc
exercise or IlI.6 below.
.0
.'0:
. :
EXERCISES
I. Let G = G. *."1 G2 with A -+ G. al1d A -+ G1 not necessarily injective. Let Ch G2 and
A be the images or G i' Gz. and A in G. Show that G = G. '" A C2. Thus any amalgam
as defined at the beginning or this section is isomorphic to one in whicb the maps
A - G1 and A -io G2 are injective.
.'0
,o
.
.'
*2. Give a proof of7.2 based on the classification or regular covering spaces as stated in
the appendix toChapterIA1. (Hint: ForanygroupH,apointed regular H-covering
X or x is specified by giving pointed regular H...coverings g 1 or X l and g 2 of X 2
such that the induced coverings or Y are isomorphk (as poimed regular H-cover-
ings).]
.,.
.'
3. It is a classicaJ fact that SL2(Z)  Z4 'l'z Zt,. (See Serre [1977a] for an indication or
an easy proof ofthis; see also example 3 or VIII.9 below.) Use the Mayer-Vjetoris
sequel1ce to ca]culate H.(SL2(Z». [Suggestion: You ca.n save a lo. of work by.
considering separately the- 2-torsion and the 3-torsion in the Mayer-Vietoris
sequence. As far as 2-torsion is concerned. SL2(Z) behaves like Z4 *.z 7Ll = Z4t
whereas it behaves like {l} *(1} Z3 = ZJ with respect to 3-torsiol1.]
.
,-
.1

'_o
...
. . 
. '.,
,",'4- :
.....
-,
Appendix.. Trees and Amalgamations
. :(
The results of 7 were based on a topological interpretation of amalgamated
free products in terms or "amalgamations n oftopological spaces. The pUlJ.'ose
of this appendix is to describe a different topological interpretation of am-
algamated free products, due to Serre [1977a]. In particular, we will obtain
anotber proof of 7.8.
Recall that a graph is a l-dimensional CW-complex and that a parh in a
graph is a sequence e h . . . , en of oriented edges such that tbe final vertex or ei
equals tbe initial vertex of e, + I for l. < i < n. The path is a loop if the finai
vertex of eJ'l equals the initial vertex or et. We allow the case n = O, in which
"
"
. .'
," .t
.0
I
,
Appendix. Trees and Amalgamation'S
53
.0
.
case the path is called trivial. Finally, the path is called reduced if el+ 1 ;/; ei
for O .s i < n, where e, is the same geometric edge as ei but with tbe opposite
orientation.
The graph X is called a tree if it satisfies the following conditions, which are
easily seen to be eq uivalent:
(i) X is contracti ble.
(ii) X is simply connected.
(iii) X is acydic.
(iv) X is connected and contains no non-trivial reduced loops.
Suppose a group G acts as a group of automorphisms or a tree X, and let e
be an edge of X with vertices v and w. We will say that e is afundamental
domai1'l for the G-action if every edge of X is equivalent to e mod G and every
vertex or X is equivalent to either v or w but not both. In other words, we
require that tbe subgraph
..
.<
V
Q
w
o
e
map isomorphically onto tbe orbit graph X/G. Note, in this case, that the
isotropy groups Ge, Gv' and Gw satisfy
Ge = GvnG...,.
Indeed, we have G,. :;:) Gv (ì Gw since e is the only edge with vertices v and
w [X is a tree]; and the opposite inc1usion holds since no element of G can
interchange v and w [they afe inequivalent mod G].
The theorem of Serre that we wish to state says that actions of G or this
type (i.e., with an edge as fundamental domain) are essentially the same as
decompositions of G as an amalgamated free product :
(Al) Theorem. Le! a group G acl 01'1 a tree X. Suppose e is an edge with vertices v
and w such that e is a fundamental domain for the action. Then G = Gr] *G.. Gw.
Conversely,givenanamalgamationG = G1 *..t G2(whereA c. GlandA C. G2),
there exists a tree on which G acts as above, wirh Gl' G2, and A as the isotropy
groups G!:. Gw and Ce.
SKETCH OF PROOF. Note tbat two edges gle and g2 e of X Wi E G) bave a
vertex in common if and only if gllg2 E Gv or Gw. This allows ODe to relate
reduced paths in X to reduced words ("normal forms") in Gv *G. G'OV.
Consider, for example, a reduced path of length 4 starting with e, oriented so
that v is its first vertex. Such a path must ha ve the form
l7
gli
ghw
ghkv
w
O--
.
o

.
oltke
.
ghe
e
ge
where g, k E G", and h E Gl" Moreover, {l, h, k f; Ge. Thus the path gives rise to
the reduced word ghk in G" *a.. Gw. Since X has no loops, we must have

54
Il The Homology of a. Group
."
yhk =I- l in G. Generalizing this argument, one sees that the canonical map
cp: G(1 *G. G", -+ G is injective_ Similarly, it follows from the connectivity of X
that q> is suIjective. [Given g E G9 consider a path ending at gv, with e as its
first edge.] This proves the fust part of the theorem.
Conversely, given G = G1 "'A G2., we have no choice as to how to con-
struct X. Namely, we must take G/GJ il G/G2 aS the set ofvertices of X and
GjA as the sel of oo.ges. There are canonical maps a: G/A -+ GIG1 and
fJ: GjA --io GjGz, by which we attach the edges to the vertices, i.e., an edge
e E GjA joins the vertices rt(e) and fJ(e). In this way we obtain a graph X on
which G acts witb an edge as fundamental domain and witb Gh G2, and A as
the isotropy groups. Since G = G1 *.A. G2, tbe first part ofthe proofshows that
X is connected and has no non-trivial reduced loops., so that it is a tree. D
.
:i
.,
.:
,.
).
0. :.
..
".
t
j
It is apparent from this sketch of the proof that the theorem is not partic.
ularly deep; indeed, tbe existence of tbe tree X associated 10 G1 "'A G2 is little
more than a reformulation ofthe normal form theorem for amalgamated free
products. Nevertheless, the tree is a very convenient tool for keeping track of
tbe combinatorics of normai forms. It is often considerabIy easier to prove
things about G by using X than it is to work directly with nonnal forrns.
For example, we obtain the promised proof or 1.8 by notjng that tbe
sequence in 7.8 is simply the augmented chain coplex of X. Its exactness
therefore follows from tbe acyclicity of X.
Here are two other applications of Al:
..
"
!
.'
',".
1:-
,.
i
.-
.,
':':
,.
(Al) Corollary. Let F c: G1 *.A. G2 be a subgroup which iPltersects every con-
jugate ofG1 and G2 trivially. Then F is afree group.
PROOF. The hypothesis implies that F acts freely on X. The latter being simpIy .
connected, it rollows that X is the universal cover ofthe orbit graph X/E and
F  1tl(X/F). But it is well-known tOOt the fundamentai group of a graph is
. D
.
,-
'.
"
l
'r

,
..
,
. ,:&.
-.
,
,
(A3) CoroUary.LetH c G1 *A G2beafil1itesubgroup- ThenH iscol1jugatetoa
subgroup of G l or G 2'
PROOf. We must sbow that H fixes some vertex of X. But this follows from tbe
elementaTY fact that every finite group or automorphisms of a tree has a fixed
pomt. See Serre [1977a], 1.4.3, for more details. D
<

.
CHAPTER III
Homology and Cohomology with
Coefficients
o Preliminaries on (g)G and HornG
Recai) that the tensor product M @R N is defined whenever M 1s a right
R-module and N is a left R-module. It is the quotient of M @z N (which we
denote M @ N) o btained by introducing the relations mr @ n = m @ rn
(m E M, r E R, n E N).
In case R is a group ring 7LG, we can avoid having to consider both Ieft and
right mod1Ùes by using tbe anti-automorphism 9 1-+ g-1 of G. Thus we can
regard any left Gmodule M as a right G-module by setting mg = g - 1m
(m E M. 9 E G), and in this way we can make sense Qut of the tensor product
M @ZG N (also denoted !vI &r N) of two left G-modules.
Note that M  N is obtained from M @ N by introducing the relations
9 - t m @ n = m @ gn. If we replace m by gm, these relations take the form
m @ n = gn @ gn, and we see that
(0.1)
M &r N = (M @ N)G,
where G acts ..diagQnally' on M @ N: g. (m @ rz) = gm @ gn_ In particuIar,
this shows that - @G - Is oommutative:
M <8>G N  N &r M.
Warnil1g. Tbe passage between left and right modules as above is convenìent,
but it can sometimes be confusing, for instance if M naturally adrnits both
a leCt and a right G-action. (e.g., M = ZG'. where G c G'). In such cases we
will revert to the standard ntation M (8)ZG N if we want to indicate that the
tensor product is to be fonned with respect to tbe given right action of G OD
M rather than tbe right action obtained from the left action_
çç

56
111 HomoJogy and Cohomology with Coefficicnts
The diagonal G-action used above is quite generai and can be used when-
ever a functor of one or severaI abelian groups is applied to G-modules.
Consider, for example, the functoe Hom( , ) = Hornz( , ). If M and N
are G-modules, then the action of G on M and N induces by functoriality a
"diagonal" action of G OD Hom(M. N), given by
(gu)(m) = g. U(g-Im)
for g E G, U E Hom(M, N). m E M. [The use of 9 -I here is needed because of
the contravariance of Hom in the first variable. In effect, we compensate for
the contravariance by converting M to a right module.]
Note that gu = u if and only if u commutes with the action of g. Thus
(0.2) HomG(M, N) = Hom(M, N)G.
This observation has already occurred implicitly in exercise 5 of I.8, where
we used an averaging procedureto convert an arbitraryelement ofHom(M, N)
to an element of Hom(M, N)G = Homo(M, N).
EXERCISES
1. Let F be a flat lG-module and M a G-module which is l-torsìon-free (i.e., Z-ftat).
Show that F @ M (with diagonal G-:actìon) is ZG-flat. [Hint: (F (8) M) @G - =
(F @ M 0 -)G = F Q9G (M <8J -). which is an exact functor.]
2. Let F be a projective ZG-module and M a l-free G-rnodule. Show that F (8) M (with
diagona] G-action) is projective. [H.im: HomG(F 0 M, -) = Hom(F @ M. _)G =
Hom(F, Hom{M. - )G = HomG(F, Hom(M, - ). which is an exact functof. See
exercise 3 of5 below for an alternative proof.]
1 Definition of H.(G,M) and H*(G,M)
Let F be a projective resolution of 7L over?LG and let M be a G-module. We
define the homology DJ G with coefficients in M by
(1.1)
H.(G, M) = H.(F @c; M).
Here F (8)G M can be thought of as tbe complex obtained from F by appl ying
tbe functor - (8)G M. Thus 1.1 is a natural generalizalion or the definition
of H * G in Chapter II; indeed, we recover the lattee by taking M = 7f..:
(1.2)
H.(G, Z) = H*G.
As in Chapter II, H...( G, M) is well-defined up to canonical isomorphism.
The complex F @o M can a180 be thought of as the tensor product of
chain complexes (cf. I.O), w here M is regarded as a chain complex con-
centrated in dimension O. From this point ofview there is a certain asymmetry
.!
"'
'.
I
,.
,
''''.
t
.J.
.
".J:
'.,
.-','
.:
'I.'
"" 
,'.-
"
,.'
....;
....
, l
-j
:',
'.
'.

.;
.
.
..
:':
.
.
"
.
.
"
..
j
J

l Definìdon of H.(G, M) and Jr(G. M)
57
in 1.1 which did not appear in the context or Chapter Il. A more symmetric
looking definition is obtained by choosing projective resolutions of both
7L and M, say f: F -+ Z and ,.,: P -+ M, and setting
(1.3)
H.{G, M) = H.(F @GP),
F o rtunately , 1.3 is consistent with 1.1; for f] ind uces a weak equi valence
F (81 '1= F @G P -+ F @G M by 1.8.6.
Note that 18.6 also gives us a weak equivalence E @ P: F &7 p -+ 7l. @G P.
Thus
,.
(1.4)
H.(G, M) = H.(PG).
Clearl y, then, we ha ve considerable flexibility in the choice or a chain complex
from which to compute H .(G. M). For the moment we content ourselves with
a trivial example or such a computation:
( 1.5)
Ho(G, M)  MG'
This follows from the right exactness of the tensor product. [A pply - @a !-vI
to F l -+ F o -+ 7L -+ O and use 1.1, or apply ( )G to P 1 -+ p o -+- M -- O
and use 1.4.]
We turn now to cohomology with coeffidents, which is defined via.J'fNn
rather than (8). Choose a projective resolution F -+ 7L as above and consider
the complex (F, M), where M is again regarded as a chain complex
concentrated in dimension O. Checking the definition of .J'fcm in I.O, we see
that Jf't:mtdF, M)n = HornQ(F -li' M). It is therefore reasonable to regard
.1'lbmG{F, M) as a cochain complex by the usual indexing conventions, i.e.,
by setting
(F, M)" = :1l'omG(F, M)-n = HomG(FIi. M).
It is then a non-negative cochain compie x with coboundary operator lJ
given by
(ou)(x) = ( _l)n+ lu(8x)
for u e KomG(F, M)", x E Fn+ l' This formula is most easj)y remembered in
the forro
(1.6) (iju, x) + ( _1)deR*,(u, Bx) = O,
cf. exercìse 3 of 1.0.
We now define
(1.7)
H.(G, A-I) = H*(:Kl»n(;(F, M»).
Remar". Because of the sign in the definition of lJ above, .Jft'omG(F, M) is
not the same as the complex HomG(F, M) obtained from F by applying the
contravariant functor HornG( -, M) dimension-wise. This somewhat annoy-
ing fact oI nature is not serious for changing the sign of a coboundary
4

':!.
..,:
S8
111 Homology and Cohomo)ogy with Coefficients
..
.
operator does not change cocycles, coboundaries, or cohomology. Qur
convention or using OM' instead or Rom is dictated by our desire to be able
to use the properties of.Ttbm developed systematically in the exercises of I.O.
Tbe reader is warned. however, that H*(G. M) is ofteo defined in the litera-
ture by means of Hom instead or JFmn.
'.
. "
'.
.,
'.' .;.
'. f-.
7-
:':'
Note that the exact sequence F 1 --+ F 0-+ 7L --to O yields an exact sequence
0-+ HOIDQ(Z, M) -+ HOffiG(Fo, M) -. HomG(Fl, M). Since HomG(.l, M) =
MG, this gives

.r-
. ...1,:
. . .
. .'
.
'. i-::
.;...:.....
,
.'
. '':/::;1
-; .
". :(
(1.8)
HO(G, M) = MG.
. "'"
. ...:
"" .
':-:"::
There exist analogues for H*( G, -) of 1.3 and 1.4, but these involve the
notion of injective resolution, which we bave only mentioned briefly (cf.
exercise 4 or 17). Tbe interested reader can find these analogues in exercise
4b below, which will not be needed elsewhere in this book.
: -
'.  I.
(
....:f

;'..
:'
EXAMPLES
1. Suppose G is infinite cyc1ic with generator t. Then we have a resolution
(1.4.5)
'..i
,
.',
.t...
o'i'
. "1
",i
. ::'.
,
"
r-l
0-.... lLG ) lLG -+ 7l. -+ O,
hence H.(G, M) is the homology of
f- 1
...-+Q-.M- M
. -
-;
1 ':'."!
J
-'
:\
and H*( G M) is the cohomology of
, I
M M-+O..._
.'"
Thus Ho(G,M) = Hl(G, M) = Me, H1(G, M) = HO(G, M) = MG, and
HG, M) = H[(G, M) = O for i > l.
2. Suppose G is cyclic of finite order n with generator t. Then we ha ve a.
resolution (1.6.3)
N ,-1 N f-l
. . - -. ZG .. lLG --4 lLG ). lLG -- 7L - O,
where N = If-Jti. Hence H*(G. M) Is the homology of
N f- I N t- J
----+M ,M-+M .M
.-
-
..' .'i
.' ..
.,'
....!.
- .'1 .
 .:
: ""'.
., -
, '.
.:'tr
  .
,'. "lo
. .
" }.!i
and H*(G. M) is the cohomology of
. ..
.:.!,
.'
l-l N t- 1 N
M M-+M .M-+.--.
Notethat N:M-M satisfies Ngm=Nm (g E G. mE M) and that NMcMG;
thus N induces a rnap N: M G -+ MG, called the norm map. [This is in fact,
true for any finite group G, where N = LtoG g.] We can no w read oD' from
the above that
Ht.G, M) = HHl(G9M) = M6jNM = coker N
F.....:..
",!-'I
-.....
. ;'
:1:-
.. .
'. ..'.
. -;
.-
I Definition of H.(G. M) and W(o. M)
59
Cor i odd (i > 1) and that
HJG M) = Hi-1(G, M) = ker N
for i even (i > 2).
3. For any group G we can always take our resolution F to be the bar
resolution. In this case we write C.(G, M) for F <29G M [=M @G F] and
C*(Gt M) for .Komv(F. M). Thus an element of CIt(G M) can be uniquely
expressed as a finite SUffi of elements of the form m @ [g 11 . . . 191'1]' i.e., as a
formallinear combination with coefficients in M ofthe symbols [gli.. .Ig,,].
Tbe boundary operator ò: Cn( G M) -+ Cre _ l (G, M) is given by
O{m @ [gll-' 'lgJ) = mgl @ [g21-' . 1 gli]
- m @ [glg21. "IOn] +... + (-l)"m @ [gli.. .lgR-l].
Similarly an element of C"(G, M) can be regarded as a functionJ: Gli -+ M,
i.e.. as a function of n variables from G to M. The coboundary operator
lJ: CII-I(G. M)  Cn(G. M) is given, up to sìgn. by
(bf)(gb . .. ,9,,) = 91!(g2... . , glI)
- f(g]U2,' ". gl!) + ... + (  1)'1(g17"" 9,,- t).
(Note: If n = O then Grt is, by con vention a one-element set" so that
CO(G, M)  M.) We could also use the normalized bar resolution here; the
resulting normalized cochain complex C(G, M) ç C*(G. M) is tbe sub-
complex consisting or those cochains f 8uch that f (g l' . . . , gn) = o whenever
some 9i = 1.
Finally, we remark that H .(G, M) and H*(G, M) have a topological
interpretation analogous to that of H * G:
H.(G, M) = H*(K(G, 1); AI)
H*(G, M) = H*(K(Gt 1).; ..H),
where .AI is tbe local coefficient system OD K (G. I) associated to the G-module
M. (See Eilenberg [1947], Chapter V, for the relevant facts about local
coecient systerns.) We will not make use ofthese isomorphisms. so we leave
the dctails to the interested reader. [Hint: Let F be the chain complex ofthe
uni versai caver or a K( G, l) Y. Then C*( Y; .AI) = F @G M and C*( Y; A') =
(F, M).]
EXERCISES
l. Let G be a finite group. M a G.module, and N: M G - Mri the norm map defined in
ExampJe 2 above.
(a) Show that ker N and coker N are annihilated by I G I. [Hint: Consider the obvìous
map MG -+ MG']

60
111 Homology and Cohomology with Coefficients
(b) Suppose M is a module ofthe fonn M = ZG @ A. whereA isan abeliangroup and
G acts by g . (r  a) = (}r (g) a. [Such a module M is said to be induce d.] Show that
N: M(i - MG is an isomorphism.
(c) Show that N is an isomorphism if M is a projective ZG-mod ule.
2. Use the standard cochain compJex C*(G. M) to sho\V tbar Hl(G, M) is isornorphic
to the group or derivations from G to M (IL5. exercise 3) modulo tbe subgroup of
principal derivations. C A prindpal derivation is one of the fOnTI dg = ghJ - m, where
m is a tixed element or M.) In particu]ar. if G acts triviaJly on M then HJ(G. M) 
Hom(G. M) = Hom(Gub' M) = Hom(H1G, M).
3. Let A be an abelian group with trivial G-action. Show that F @G A  F fJ @ A and
that .Jf'.o?iJ(ì(F. A) .;::;: Jf'MJ'l(F G. A). Deduce universal coefficient sequences
0-+ Hn(G) 0 A - Hn(G, A) -+ Torf(Hn_ I(G), A) - O
and
o -+ Ext(H"_ l (G), A) .... HR{G. A) -- Hom(Hn(G), A) _ o.
*4. (a) State and prove an analogue of 1.8.5 for maps loto a non-negati ve cochain
compJex or injectives.
(b) Let {: = F -+ 7l. be a projective reso]ution and Jet t] : M -+ Q be an injecti\ie resoJurion.
Prove that € and " induce weak equivalences
.JFOMG(F. M} -+ G(F, Q)+- (l. Q).
so any ofthese three complexes can be used to compute H.CC. AI). In particular.
H.C G, M)  H.(QG).
2 Tor and Ext
There are obvious generalizations of H.(G, -) and H"(G. -), called
T or( -, ) and Ext( -, -). obtained by removing the restriction in 1.1
and 1.7 that F be a reso]ution of the particular module 7L.. For Tor, we take
projective resolutions F --). M and P -+ N of two ar bitrary G-modules M and
N and set
Tor(M, N) = H..cF @(ì N) = H.(F @GP) = H*(M @GP),
We recover H*(G, -) as Tor(.l. -).
Similarly. if F -+ M is again a projective resolution, then \Ve set
Ext(M. N) = H*((F. N».
-:'
.
l:
2 Tor and Exr
61
We recover H*(G, -) as Ext(Z, -). The reader who has done exercjse 4
of I will note that we could also take an injeçtive resolution N -+ Q and
write
..
<
Ext(M, N) = H*{Jff'o.m(;(F, Q» = H*(.JIfo-mc(M, Q)).
"
.,
We do not intend io systematically develop the properties ofTor and Ext,
bu t it will be useful to record a few results about thern for future reference.
Tbe first shows that one has considera bly more flexibility in the choice of
resolutions for computing Tor than is apparent from the definition.
.!
-
,
(2.1) Proposition. Let E: F...,. M and '1: P  N be resolutions, not necessarily
projecrive. /f either F or P is a complex o[ ftat modules. then Tor:(M, N) 
H.(F  P).
PROOF. Suppose for instance, that F is flat, and let il: P  N be a projective
resolution. By the fundamental lemma 1.7.4, there is an augmentation-
preservingchain mapf: P ---t P. Two appJications ofl.8.6now yield
 -  - G
H.(F @G P)+-H.(F @G P) - H.(M @G P) = Tor.(M. N),
where the first ma p is induced by f and the second by E.
o
ODe way to construct a resolution or a module M is to start with a resolu-
tion f: F -+ 1.. and tensor it with M to obtain 8 @ M: F @ M .... 7L @ M = M.
This will be a resolution under mild hypotheses on F and/or M. Suppose
for instance, that F is 7L -free; then E is a homotopy equivalence if we ignore
tbe G-actioD (cf. 1.7.6), so the same is true of 8 @ M. [A I ternativeJy, use the
universaJ coeffic-ient theorem to compute that
H 'feCE (8) M) = H .(F) (8) M = M.]
This method will be used now to show tha t T or and Ext ean often bt;
computed in terrns of H..(G. -) and H*(G, _).
(2.2) Proposition Ler M and N be G-modt4les. lf M is 7L -lorsion-free then
Tor2(M, N)  H*(G, M <29 N),
where G acts diagon.ally on M (8) N. lf M is 7L.-free, the,r
Ex[(M. N)  H*(G, Hom(M, N),
where G acts diagonaliy on Hom(M, N).
PROOF. Let E: F -+ 7l. be a projecti ve resolution, and consìder the resolution
8 <29 M: F @ M-M. This is a flat resolution jf M is Z-torsion-free and a

62
111 HomoJogy and Cohomology with Coefficients
i
,.
'.
projective resolution if M is 7l. -free. by exercises 1 and 2 of . So i£ M is
Z-torsion-£ree, then wc have
.
,.
Tor(M N)  H.«F C8J M) @o N)
= H.«F @ M @ N)G)
= H *(F (8)G(M C8J N»
= H.(G, M @ N).
by 2.1
"
,
,'.
.
-\
And if M is 7L -free, then
Ext(M, N) = H*(4fb.mc;(F C8J M N» by definition of Ext
= H*(ftum(F @ M, N)G)
= H*(Jfom(F, Hom(M, N»G)
= H*(JfbmG(F, Hom(M, N»
= H*(G, Hom(M, N). D
"
. .
",
.."'!
"
','
/.
A
We will outline alternative proofs of this proposition in exercise 2 or
7 below.
Finally, we remark that T or and Ext can be defined over an arbitrary ring
R, using the same definitions as we gave for G-modules. One need only be
careful about which modules are left modules and which are right ones.
Thus Tor:(M, N) is defined when M is a right module and N is a left module,
whereas Ext(M. N) is defined when M and N are either both left modules or
both right modules.
"5
J
,-
''\
':
,
"
j.
.
EXERCISE
"
r
Show that Tor(M. N)  Tor(N, M). (Hìnt: See exercise 5 ofI.O.]
'J
,
::
:r
3 Extension and Co-extension of Scalars
Before proceeding further with tbe study or H.(G, M) and H*(G, M), we
digress to discuss, in this and the next two sections, some module-theoretic
eonstructions which pIa y afundamental role in the homology andcohornology
theory of groups. Let CL: R -+ S be a ring homomorphism. Then any S-module
can be regarded as an R-module via 0:, and we obtain in this way a funetor
from S-modules to R -m od ules, called restriction oJ scalars. The purpose of
this section is to study two constructions which go in the opposite direction,
from R-modules to S-modules.
For any (left) R-module M. consider the tensor product S @R M where S
is regarded as a right R-module by s. r = 50:(r). Since the naturalleft action
of S OD itsel£ commutes with this right action of R OD S, we can make S @R M
a (left) S-module by setting s. (s' @ m) = 58' @ m. This S-module is said to
be obtained from M by etension oJ scalars from R to S.
,
.  .;.
" ;,.;/'
'.
'.
J:

3 E'J{tension and Co-extension or Scalars
63
Note that there is a natural map i: M -+ S @R M given by i(m) = 1 (81 m.
Since 1 @ rm = (X(r) @ m = lX(r). (1 @ m) for rE R, wc have
(3.1)
i(rm) = a(r)i(m);
in other words, i is an R-module map, where the S-module S @R M is re-
garded as an R-module by restriction of scalars. Moreover, the following
universal mapping property holds:
(3.2) Given an S-module N and an R-module map f : M -+ N,
there is a unique S-module map g: S @R M -+ N such
that gi = f:
'.
M
Il
i
J S@RM
,
,
,
-
..
f 9
o'
f
ti
N
[Thus wc have
. (3.3)
HOffis(S @R M, N)  Ho(M, N),
showing that tbe extension of scalars functor (R-modules) -+ (S-modules)
is left adjoint to the restriction of scalars functor (S-modules) -+ (R -modules ).]
Heuristically, 3.2 says that S @R M is the smallest S-module which receives
an R-roodule map from M. To prove 3.2 we need only note that (J, ifit exists
must satisfy g(s <8J m) = sg(1 @ m) = sg(i(m» = sf(m). This proves unique-
ness and tells us how to define g in order to prove existence; the remaining
details are left to the reader.
Note that 3.2 can be applied, in particuiar. with M = N (regarded as an
R -m odu le by restriction of scalars) and f = 1dN. We o btain, then, for any
S-module N, a canonical S-module map
(3.4)
S @R N -+ N,
given by s C8J n 1--+ sn. This map is surjective; moreover, as an R-module map
it is 'a split surjection.
We now consider a duai çonstruction, which uses Hom instead of @.
Given a (le£t) R-module M, consider the abelian group HornR{S, M). where S
is regarded as a Ieft R-module by r. s = a(r)s. Since the natural right action
of S on itself commutes with this left action or R OD S, we can make HomR(S. M)
a left S-module by setting (sf)(s') = f(8'S) for f E HOffiR(S, M). [Note: It is
because of the contravariance of Hom in the fust variable that the righe
action of S OD itselfioduces a left actioo or S on HornR(S, M).] This S-module
is said to be obtained from M by co-exlension oJ scalars from R to S.
There is a natural map x: Horn,R(S, M)....... M, given by n(f) = [(I).
Note that n(r:x(r)f) = (o:(r)f}(l) = f(rx(r» = if(1) = rn(f), so 1t is an

64
111 Homology and C()homo]ogy with Coefficients
R-module map ifthe S-module HomR(S, M) is regarded as an R-module by
restriction ofscalars. Moreover, we have:
(3.5) Given an S-module N and aD R-modu)e map f: N -+ M, there
is a unique S-mod ule ma p 9 : N --+ HomR( S, M) such that ng =f:
HomR(S, A-/)
l.
';
"
'.
:..
111
9 .-
,
,
,
.
.
"
t
.<
N
tM
I
[fhus
(3.6)
Horns(N, HornR(S, \1»  HomR(N, M),
.,
'.'
:....
so that eo-extension of scalars is right adjoint to restriction or scalars.]
Heuristically., 3.5 says that HomR(S, M) is. the smallest S-module which maps
to M by an R -module map. T o prove 3.5, note that g must satisfy sg(n) = g(sn)
far SE S. n E N; evaluating both sides at l, we find g(n)(s) = g(sn)(l) =
:rc(g(sn» = f(sn); existence and uniqueness of g folIow easi1y.
Taking M = N (regarded as an R-module) and I = idN, we obtain
from 3.5 a canonicaJ S-module rnap
,
.,
,
.,;.
:.
... .
,
.
.
.
-
::.:
(3.7)
.......
.1".'
, .
,.

N -+ HOffiR(S, N),
given by n 1--+ (5 I--t sn). This map is injective; moreover, as aD R-module map
it is a split injection.
Examples or extension and co-extension of scalars ha ve already occurred
in these notes. Namely, if C( is the augmentation map t: 7LG -+ li., then the
"extension of scalars" functor (ZG-modules) -+ (l:. -mod ules) is simply
M t-+ }.,tI G' and the ., ço-extension of scalars" functor is M t----f> MG. More
generally. exercise 3 of 1I.2 treated the extension or scalars relative to a
surjeçtion IG -+ 7L[G/H], where H <::J G. What we are prirnari1y interested
in, however, is the case where ex is an inc1usion ofthe farro ZH 4 l:.G, where
H c G. This case will be discussed in detail in 5 beJow.
,.
,.-
,s;-
i'
..
..
"
,-
.
.:

. .
, '
. ..'
EXERCISES
. :.i
L Show that extension of sca]ars takes projective R-moduJes to projective S-modules.
[Hint: If P is a projective R-module. 3.3 shows that HomS<S @R p. -) is an exact
functor.]
. ""'.c-
. ,
.'
"
2. RecaJ1 rrom exercìse 4 of I. 7 that an R -module Q is called injectit>e if the functor
HomR( -. Q) is exact. Show that co-extension of scalars takes in jective R -mod ules to
ìnjectivc S-modules. [Hint: If Q is an injective R-moduJe. 3.6 shows that
Homs(-. HOIDR(S, Q)) is an exact functor.]
3. If S i.s flat as a right R-module (Le., if S @R. - is an exact functor).. show that restriction
of scalars takes jnjective S-rnodules to injectìve R-modules. [Simì1ar hint.]
.:
4 Injcctive Modu]es
65
4_ If S is projective as a left R.module. show that restriction of scalars takes projective
S-modules to projective R-modules. [SimiJar hìnt.]
The reader familiar with adjoint functors may wish lo fonnulate a generaI state-
ment or which 1-4 are specia] cases. [Noie: Exercises 1 and 4 can also be done in a
more concrete way. using the fact that a projective module is a direct summand of a
Cree module. A specjaJ case or exercise 4 was a]ready done in this way in exercise 2 of
I.8,]
4 Injective Modules
Recall from exercise 4 or I.7 that an R-module Q is called injective if it
satisfies the foIlowing equivalent conditions, dua1 to those by which we
defined  projecti ve" :
(i) HomR(-, Q) is exact.
(ii) Every mapping problem
M' M ) M"
 l ,

.
,
.

,
lIf
Q
with exact row can be solved.
(iii) For any inclusion M' c.. M, every map M' --+ Q can be extended to a
map M -+ Q:
M'r 1M
l
,
,
.
,
,
.
,
"
Q.
The purpose of this section is to prove that every module can be embedded
in an injective module; this is analogous to the (obvious) faet that every
module is a quotient of a projective module.
(4.1) Propositiooe An R-module Q is injeclive if and only if every map l -+ Q,
where l is a left ideai 01 R, extends io a map R --+ Q.
PROOF. The H only if" part is trivial. T O prove the " if" part. suppose that every
map l -+ Q extends to R for ali ideals l c: R. It follows at once that if C' c C,
where C is a cyclic R-module, then every map Cf  Q extends to a map
C -+ Q. Now let M' c: M be aD arbitrary ìnclusion. Givenf: M' -+ Q, there
exists by Zorn's lemma a maximal extension F: M" -+ Q of f, where M' 
M" cM. If M" :F M tben there is a cyclic module C s M such that C $. M".

.'
66
..
.
.r
111 Homolùgy and Cohomology with Coe:fficients
But we know that F I c n M" extends to C so F extends to M" + C contra-
dicting the maximality of F. Thus M'I = M and Q is injective. D
.'.
.
.,:
. .
,.
"
. .:.
<.
, '
.:;!;
(4.2) COl'ollal'Y. IfR is a prindpal ideai domain chen an R-module Q is injective
if and only if it is divisible, i.e. if arul only if r M = M for every r =1= O in R.
.... .{
I, ,
.. ", t....
: \:.::
:.: ..,.:-.
 .,
. :rl_
. ;:.r
The proof is immediate.
As an example or the corollary, the Z-module Q 18 divisible and hence
injective. Sirnilarly, O/Z is injective. The lauer module is particular1y
interesting. because of the following result:
'
..)."
"
.,=j
.' .
..,
.,
(4.3) Proposition. If A is a l-module and O ¥ a E A, then there is a map
f: A -+ O/lL such thatf(a) ::F O.
PRODE Since a/lL contains (l/n )Z/Z  lL/nlL £or every n  O, it is clear that
there is a map /0: 7l.a -+ O/lL such that foCa) =F O. But O/Z is injective, so /0
extends to a map f : A -+ O/Z. O
7
 r
\-.-
'-'::-
. ,.
.,
. .
.
.'
..
,.
,
..'
:.r:-

:,r
This shows that O/lL plays a role in abelian group theory dual to that played
by 7L; for the latter admits non-zero maps to every non-zero abelian group.
Returning now to an arbitrary ring R, let R' be the R module Hornz< R. a/l)
o btained from Q/lL by co-extension of scalars with respect to the ring homo-
morphism Z - R. It is injective byexercise 2 of 3. As we will see, it plays the
role in the theory of injectives that R = R @z 7L plays in the theory or pro-
jectives. By a co{ree module we mean an arbitrary direct product of copies
of R'. The following result is the analogue of 17.2;
.-
.
,
"
. ,.
:.-
.
,-
..
:
(4.4) Lemma. Co-free modules are injective.
PROOF. It is immediate from the definition of .'injective' that an arbitrary
direct product or injectives is injective. D
.-
.:t
..
\)
:f.
RecaIl now tbe standard proof that every module M is a quotien t of a free
module F; one takes F = EB R, where there is one summand ror each element
of M, ie. for each R-module map R -+ M. Dually we can consider Q =
O/e,- R', where !F = HomR(M R'). There is an obvious map i: M -+ Qt
whose f -companent for f E:ffi is the map f.
. .. 
... ;
.. .."
". ';''
.'
,.
"
...
o . 
.':-
. ".1
(4.5) Theorem.. The map i is a monomorphism. Consequently any Rmodulecan
be embedded in an injective module.
PROOF. If O ;f; In E M  there is by 4.3 a 7L -module map /0: M -+ O/1l. witb
lo{m} ;f; O. By 3.5. lo factors through an R-module map f:M - R'. It follows
that f(m) =F Ot and hence that i(m) '#- O. D
!l'.
',;
I ...
, ,
,
". l.c
.,
, ,..
.J_
. ..
..':.,:
". ;.
-: :.
l. 0"1
..
See Cartan-Eilenberg [1956], I.3.3 for a different proof of the second
assertion or 4.5. based directly on 4.1. -
S Induced and Co-induced Modules
67
.
EXERCISE
Let R = 7/../1171.. Show tha. R is an ìnjective R-moduJe. Deduce:
(a) If A Is an abelìan group such that nA = O. and C c: A is a cydic subgroup of order n.
then C is a direct summand or A.
(b) U A is as in (a). then A is a direct SUffi,Or cyc1ic groups.
5 Induced and Co-induced Modules
In this section we apply the constructions of 3 to ring homomorphisms or
the form 7LH c., 1..G, where H c G. In this case extension of scalars (resp.
co-extension of scalars) is called induction (resp. co-induction) from H to G.
We will often write
ZG (8iZH M = Ind M
and
HomzIA.lG, M) = CoindM
for an H-module M. Note that a module of tbe form Indg} M is precisely what
we called an induced module in exercise lb of l above_ Similarly. a rnodule of
the forro CoindY.} M will be called co-induced. _.
Since the right translation action of H on G is free lLG IS a free nght
ZH -module; as basis we can take an y set E o£ representatives far the left
cosets gH. It follows that 7L.G <8IzH M. as abelian group, adrnits a decompo
sition
7LG <8IZH M = Eeg <81 M,
fieli
where 9 @ M = {g (8) m: m E M} and g (8) 1\1  M via () (8i m +-+ m- In
particular, since we can take l as the representative or its coset, it follows that
the canonical H-map i: M - lLG @ZH M defined in 3 maps M isomorphically
onto its image 1 @ M. We can therefore lise i to regard M as an H-subrnodule
of Ind M. Moreover, the summand g (8) M which occurs above is simply the
transform or this submodule under the action of gt since g. (1 @ m) =
9 @ m. We bave there£ore established:
(5.1) Proposition. The Gmodule Ind M contains M as an H-submodule and
is the direct sum oJ the transforms gM, where g ranges aver any set oJ representa-
tivesfor the left cosets oJ H in G-
More briefly, the second assertion of the proposition says:
(5.2) IndZM = E9 gM.
f/IEG/H

68
111 Homology and Cohomology with Coefficients
..
.'
This makes sense because M is mapped onta itself by tbe action of H, so thal
the subgroup gM ofIndZ.M depends only on tbe class of gin GjR.
The description 5.2 completely charactecizes G-modules of the form
Ind M. More precisely, suppose N is a G-module whose underlying abelian
group is a direct sum EBi e l M i' Assume that the G-action transitively
permutes the summands in the sense that there is a transitive action ofG on I
such that glWi = Mgi for alI g E G and i E 1. Then we have:
"
'.
.
..
f
,
"
. .
o'.
(5.3) Proposition. Lei N be a G-module as above. let M be one oJ the summands
Mj, and let H ç; G he the isotropy group oJ i. Then M is an H-module and
N  Ind M.
PROOF. It is obvìous that M is an H-submodule of N, and 3.2 implies that the
inclusion M 4 N extends to a G-map Ind M  N. Clearly tp maps the
summand gM of Indg M isomorphically onto the corresponding summand
M gl of N, so f{J is an isomorphism. D
).
 :.
6.
:f
..'
:t
';
i.
,,
,
'.
.;
(5.4) Corollary. Let N be a G-module whose underlying abelian group is ofthe
form EBi'lOl Mj. Assume that the G-action permutes the summands according lo
some action oJ G on I. Let G i he the isotropy group oJ i and let E be a ser oJ
representatives for l mod G. Then Mi is a Gi-module and there is a G-iso-
nwrphism N  EBleE Ind8; Mi'
:pROOF. We have I = Il ieE Gi, so N = EBi ",E EBjeGi Mj; now apply 5.3 to the
mner SUID. . D
.
I:,
o''
"
"

--,
<
!

"
<,
.!
<
,
i
(5.5) EXAMPLFS. (a) The permutation module Z[GjH] is isomorphic to
lnd Z, with H acting trivially OD Z. This can be seen directly from the
detinition of Ind* Z 0£9 alternatively, by writing Z[G/HJ as a direct sum of
copies of 7l. and applying 5.3.
(b) Let X be a G-CW-complex and consider the G-module CII(X). This is a
direct sum of copits of 7L. ODe for each n-celi of X, and the summands are
permuted by the G-action. Hence 5.4 gives
Cft(X)  E9 Ind8., lL(J)
d" f< In
>
.<
'.
o:
,.-
.,
where 1:" is a set of representati ves for the G-or bits of rz.cells,
G(/ = {g E G: gu = a}
and 7L is the "orientation module'" associated to a, i.e., 7L.(f is an infinite
cyclic group whose two generators correspond to the two orientations of (1.
(Thus g E G acts OD "La as + I if 9 preserves the orientation of (1 and -1
otherwise.) Note that if G acts freely on X, then the isomorphism above
simply reduces to our observation in. I.4 that CJ..X) is a free ZG-module
with one basis element for each a E EII_
. .-
. .!.
:.. "i
. -:
.'
S Induced and Co-induced Modules
69
Before stating our next result we remark that the summand gM of Indg M
is dosed under the action of gHg- \ hence gA-l is a gHg-1-module. Note
that the action or g-1 gives a bijection f: gM ..; M such that f(kn) =
g-lkg. f(n) for k E gHg-1, n E gM. Hence gM can be identified with the
g Hg - l-module obtained from the H -module M by . restriction of scalars .
via the conjugation isomorphism gHg-l  H.
For any G-module N we denote by Res N the H-module obtaìned by
restriction of scalars from G to H_
(566) Proposition. (a) Let N be a G-module. Then
Ind Res N  Z[GjH] Q?) N,
where G acts diagono.lly on the tensor product.
(b) LeI H o.nd K be subgroups ofG and let E be a set ofrepresentativesfor
the double cosets KgH. For any H-module M, there is a K -isomorphism
Res Ind M  EB Ind""8HfI-1 ResIHI1g1 gM.
8eE
l n particular, if H -<J G, then there is an H -isomorphism
Res* Ind M  EB gM.
g'lOGIH
Remark6 In view of the observations preceding the statement of the proposi-
tion, we can identify the module Rest:H;" - 1 gM which occurs above with
the module ReS(ìI7Hg-1 M, where the latter restriction is with respect to the
conjugationmap K (ìgHg-l -t> H, kl-+g-lkg.
PROOF OF 5.6. (a) will be left to the reader; it is the special case M = 7L of
exercise 2a below. Far (b) we note that the summands of Ind M =
ffi"e-G/H gM are permuted by the action of K according to the natural action
of K on GjH. Since the K -orbits in G IH correspond to double cosets KgH
and the isotropy group in K of the coset 9 H is K n gHg - l. (b) fol1ows from
SA D
We will have severa l occasions to use 5.6a in the special case H = {l}.
Tbe content or the result in this case is:
(5.7) Corollary. Let M be a G-module and let Mo be its underlying abelian
group. Then 7LG @ M (with diagonal G-action) is canonically isomorphic
lO the induced module llG @ Mo. In particular, ZG C8> M is afree ZG-module
if M isfree ll5 a Z-module.
The interested reader can veriry that the proof of 5.6a yields the specific
isomorphism 7LG @ M o -+ lLG @ M given by g @ m  g @ gm. with inverse
9 @ ml--+g@ g-lm.

70
111 Homology and COhomolo.8Y with Coefficients
.;
-.
Co-induction has properties analogous to the properties of induction
given above, but with E.B replaced by n- The precise statements, however,
are somewhat more awkward and require some notation: If 1[1: N ....,. !vI J
and 1t2: N""""* M2 are surjections of abelian groups, then we write 1tl ....... 71:2
if there is an isomorphism h: M l .; M 2, such that hn: l = 1t 2, , or, eq uivalently,
ir ker n 1 = ker 1['2' If 1t: N -++ M is a surjection and N has a G-module
structure, then we denote by ng for g E G the surjection N -# M defined by
(ng)(x) = n(gx)- Fina1Jy, by a direct product decomposition of N we mean a
family of surjections (1ri: N...,.. Maid such that the corresponding map
N --+ nie[ Mi is an isomorphism. It is now easy to state and prove analogues
for co-induction of 5.1, 5.3, 5.4, and 5.6. For example, the analogue of 5-1
is that the underlying abelian group or CoindZ M admits a direct product
decomposition (ng)geH\G, where n: Coind M -+ M is the canonical
H-module map defined in 3. (Note that n:h -- 1r for h E H so tbe equivalence
class or tbe surjection 1tg depends only on the cIass of g in H\ G.) Similarly,
the analogue of 5.3 is:
(5.8) Proposition. Let N be a G-module which, as an abelian group, admits a
direct product decomposition (1ti: N --* Mi.)iI' Assume that tllere is a transi-
tive right actiolt oJ G on l such that 7fig '" 7till for all i E l and 9 E G. Let
n: N --+ M be one ofthe TCi and let H c: G be the isotropy group ofi. Then M
inherits an H-module structure from N, and N  CoindZ M_
.,
,
"I:
.-f
., ::
,
.,,
. '1:
-,
,,':
' I
'0.
"
.,.
.',
.
.\
'.'
..
.. ,
,..
o 
O"i
.;:
,
,.

J
<
r
..
,
, I.
',o
,.
.-
,'o
.t",
"I."
:i:
The reader can similarly formulate the analogues of 5.4 and 5.6.
Finally, since a direct product indexed by a finite set can be identified with
the corresponding direct sum, the following resu)t is not surprising:

.,
:!'
',:
..
(59) Proposition If(G; H) < 00 then Ind M  Coind* M.
PROOF. There is an H-map ifJa: M - HomIl(.&::G, M) given by
{gm geR
q>o(mXg) = O gtlH.
This extends to a G-map qJ: lLG @ZH M -+HOIDH(ZG, M) by 3.2. Choosing
coset representatives and expressing ZG @ZH M (resp. Hom.zG, M» as a
direct sum (resp. direct product) of copies or M, one sees that f/J can be
identified with the canonical inclusion of the sum into the product. Con-
sequentJy, q> is an isomorphism if (G: H) < 00. Altematively, one can set
rjJ(f} = LeG1Hg Q9 f(g-I) and vecify that  = 'P-t. D
"
-'
.t
..
...
t
",l'
. ..
..
...
. . ,
. .
, ,
. .
EXE.R.CISF.S
. 
l. ",.
. ......
. .' 
1. Pro\'e that induction is "invariant under conjugation;' in the sense that IndgM 
lodIIg-!gM for any H-module M. Deduce that the K-module
Indnlill,-l ResJ.:...- IgM
which occurs in 5.6b depends (up to iSOrriorphism) onJy on the c1ass of gin K \ GIR.
. .-
. -
-:=-
,

6 H. and  as Functors oftbe Coeffident Module
71
,.
'.
2. (a) For any H-module M and G-module N prove that
N @ IndM  Ind(Rcst N 181 M),
where the tensor product on the left has tbe diagonal G-action and thal on the rìght
has the diagonai H -action. [Hint: Write the Ieft side as EB (N @ gM) and appIy 5.3.]
(b) State and prove simdar resuIts for Hom(lnd M. N) and Hom(N, Coind M).
3. Use 5.7 lo give a new proof of the result or exercise 2 of 9(1
4. (a) If(G: H) = 00, prove tbat (Ind*M}G = O for any H-module M.
*(b) If G is finitely generated and (G: H) = crJ, prove that (Coiò M)G = O .for
any H-modu]e M. [See Strebel [1977] for an algebraie proof of thIS. A !opologlcal
proof can be given as folJows_ We may assume that G is a f!ee group_ln tIS case let Y
bc a finite. connected l-complex with 1t" l Y = G and let Y be the covenng comlex
corresponding to H. Then M (resp. CoindgM) can be regarded as a local.coefficlent
system on Y (resp. Y). and it is easy to see that (CoindM)G = Ho(Y. Comd M) =
Ha:;I(y. M). where Hr.<:J) de[)otes homology based on infinite chains. The reult now
follows from the following elernentary faet: If X is an infinite. Iocally fimte, eon-
nected complex:, then HòW)(X. M) = O for any loeal eoefficient system ""1.]
*(c) Given H c: G. prove that the following conditions are equìvalent:
(i) There is a finitely generated su bgroup G' ç G such that (G' : Gr fì gH 9 - 1) = Xi
ror ali 9 E G. [If H <J G, for example. tl1is just says that the group GjH
contaìns a finitely generated infinite subgroup. i.e., GjH is not locally finite.]
(ii) (Coind* M)G = O for alI H-modules M. .
(iiì) The element or (CoìndZ .l)G represented by the augmentatIon rnap E; E
HotnH(ZG. Z) = CoìndZZ is zero. [Hint: Use (b) and the double-çoset
formula.]
.5. If G is finite and k is a fidd. show that a kG-module is projective iff it is injective.
[Hint: Start by showing that the free module kG is injeetive; thìs follows from the
analogue over k or 5.9. whicb impUes that HomIcG{-, kG)  Homl«-' k).]
6 H and H* as Functors of the Coefficient Module
.
Since F @G - and .1t'omdF, -} are covariant functors, it is dear that
H.(G, -) and H*(G -) are covarjan funetors.of the coefficient module.
The Collowing proposition gives the baslc propertles or these functors_
(61) Proposition. (i) There is a natural isomorphism Ho(G, M)  MG'
(i') There is a natural isomorphism H.°(G, -¥)  MG_
(ii) F or any exact sequence O -+ M'  M  M" -+ O of G-modules amI arzy
integer n there is a natural map ò: H"(G, M") -+ HlI-l(G M') such that the
sequenct!
... -+ Ht{G, M) -Jo Ht(G. M").i Ho(G, M') -+ Ho(G, M) --+ Ho(G, M") -+ O
is eXQCt. (The unlabelled atrows here represepu the maps induced by i andj.)

'.
"
72
111 Homology aod Cohomdogy with Coefficients
(ii') Por any exact sequence as in (ii) and any integer'l there is a natura I map
D: H"(G, M") -+ HIf+ l(G M') such that the sequence
O -+ HO(G M') -+ HO(G, M) ..... HD(G, M")  Hl(G M') -+ H1(G, M)  . . .
is exact.
(Hi) Il P is a projective lLG-nwdule then HJ..G, P) = Ofor n > O.
(iii') lfQ is an injective llG-module then. H"(G, Q} = Ofor n > O.
I
.':
.,
"
d.
"
f
,

Note: The naturality assertion in (ii) means that for any commutative
diagram
. .
"'I,;"
'.
I
i:
o
M'

-M
M"

)0
:..:
1
1
l
"!
o
) N'
N
10 N"
) O
.'
"
.,
!,
with exact rows, the square
:'
H"(G, M")
l
ò
) Hn-l(G, M')
1
B _ H"_l(G, N')
,-
:."
",
HII(G, N")
.-;
.,
J
is commutative.
..0
PR.OOF OF 6_1_ (i) and (i') were proved in 1_ Given an exact sequence as in (ii)
and a projective resolution of 7L over ZG. we have an exact sequence or chain
complexes
i
,
.
,
.
o ---+ M' @G F -+ M @o F - M" @G F -+ o
since projectives are flat; the corresponding long exact homology sequence
yields (ii). Similarly, (ii') follows from the sequence of cochain complexes
O --t ftb.mG(F, M') - (F, M) -+ JIt'£Hncj(F, M") --Jo O,
which is exact by tbe definition of "projective." Final1y. (iii) and (iii') are
immediate consequences or the detìnitions of H $(G, -) and H*(G, -) and
the exactness of - @G P and HomG( -, Q)- D
l
.
,
.
--,
__o
.,
. '-
'j
-,
'.1"
,I
"
-,
Properties (iii) and (Hi') are usualIy expressed by saying that projective
modules are H * -ac yclic and injective modules are H*-ac ydic.
A func[or T (say from R-modules to abelian groups) is said to be eJJace-
able if every module M is a quotient or a module M such that T{M) = O.
Clearly T is effaceable if T(P) = O for every projective P_ Conversely, if T
is effaceable then T(P} = O for every projective P. For let n: P -+ p be a
surjection with T(P) = O; since P is projective 1t rnust spIit, hence T(n): T(P)
-+ T(P) is a split surjection and T(P) = O. Thus the coment of (iii) above is
"
.'
..

,-
i'
'.
.
"
...
,
-j
6 H. and IP as FUDClors cr the Coefficient Module
73
that H n{ G, - ) is effaceable for n > O. Similarly. (Hi') says that H"( G, -) is co-
ej[aceable far n > Oe., that every module M can be embedded in a rnodule
M such that HIf( G. M) = 0_
(6.2) Proposition ("Shapiro's lemma "). 11 H c G and M is an H-module, then
H.(H, M)  H.(G, Ind M)
and
H*(H M)  H*(G, Coind* M).
[N ote : After we bave discussed the fW1ctoriality or H * and H* with respect
to group bomomorphisms, we will be able to make 6_2 more precise by show-
ing that tbe isomorphisrns are induced by the inc1usion H  G and the
canonical H-maps M -- IndZ M and Coind M -+ M. See exercise 2 of 8.]
PIt.OOF. Let F be a projective resolution or 7L over 7LG. Tben F can also be
regarded as a projective resolution or Z over ZH, so
H *(H, M)  H $(F @ZH M).
But F @ZH M  F @ZG (ZG @ZH M)  F @c, (Indg M), wbence the first
isornorphism. The second isomorphism follows from the universal property
of coinduction, wbich implies KomH(F, M)  .JIt'mnG(F. COind* M) cf 3.6.
D
Taldng M = Z, for example, we conclude from 6.2 that
(6.3)
H$(H)  H.(G, Z[G/H]).
(This shows that homology with coefficients is or interest even if one is
primarily interested in ordinary integraI homology). In case (G: H) < 00,
CoindZ M  Ind* M by 5.9, so we also have
(6.4)
H*(H, 7L)  H*(G Z[GjH]).
Similarly, stilI assuming (G: H) < 00, we find
(6.5) H*(H, ZH)  H*(G, 7LG),
since ZG @ZH 7LH :==: lLG. (It is also true, of course, that H *(H, lLH) 
H .(G, ZG1 butthis isofno interest in viewof6.1(iii).) Final1y. we remark that
Shapiro's lemma can be applied with H = {l} to yield:
(6.6) Corollary.lnduced modules ZG @ A are H *-acyclic. Co-.-induced modules
Hom(ZG, A) are H*-acyc/ic.
Note that 6.6 yields another proof that H J.. G -) is effaceable and Hh( G -)
is co-effaceable for n > O; for we ha ve seen (3.4 and 3.7) that every module is a
quotient or an induced module and that every module can be embedded in a
co-induced module.

74
In Homology and CohomoJogy with Coeflìcients
EXERCISE
(a) Show that the Mayer- Vietoris sequence 11.7.7 can be deduced from the shon exact
sequence Il.7.8 ofpermutation modules. [Hìnt: Use 6.1(ìi) and 6.3.]
(b) More generalJy. derive tn this way a Mayer-Vìetoris sequence for homology and
cohomo[ogy with coefficients. [AppJy - @ ll.1 andHom(-, 114) to II.7.8; use 5.6a and
6.2.]
7 Dimension Shifting
We saw in  that HJ..G, -) is effaceabJe and H"(G, -) is co-effaceable for
n > o. The significance or this is that it aUows us to use the following di-
mension-shifting technique: Given a module M, choose an H.-acyclic
modu]e A-f which maps onta M (e.g., take M = ZG @ M), and let
K = ker{A-f -+ M}.
Then the long exact sequençe of 6.1(ii) yields
(7.1 )
H,.(G. M)  {Hn-I(G, K) _
ker{Ho(G, K) -+ Ho(G, M)}
n> l
n = L
Thus a questjon about Hn can, in principle. be reduced to a question about
H,,-l, provided we are wiUing to change the coefficient module. Iterating this
procedu we are uJtimateJy reduced to Ho. Similarly, embedding M in an
- -
H*-acyclic module M (e.g. M = Hom(ZG. M) and letting
......
C = coker{M --+ M},
we find
(7.2)
HJr(G lYf) __ {W-1(G, C)
, - coker{HO(G. M)  HO(G. C)}
n> 1
n = L
This argument shows. at least heuristically, that a 06 homology theory'
H * having properties analogous to (ii) and (iii) of6.1 is completely determined
by Ho. Similarly, a "cohomology theory" H* satisfying (ii') and (iii') is
determined by HO. The rest of this section will be devoted to a precise
formulation and proof or these assertions.
Lei R be an arbitrary ring and T = ()'uz a farnily of çovariant runctors
from R-modules to abelian groups. Assume that we afe given "connecting
homomorphisrns" iJ: 7;,(MN) -+ 1;.-1(M') for every short exact sequence
O  M' -+ M -+ M"  O of R-modules. We require that ò be narural (in the
,
,
,
':
.:
.. .'"::
,;
..'
. .(
. :
'.
..
.,
.,
,
:,"
,
-
.i
..
1
;.--
;;.
, .
;
,.
;.
.
.,.

-'.
-
.
..
:
o
.,
.'
,.
..
7 DimensÌon Shifting
75
.J
sense explained after the statement of Proposition 5.1) and that alI com-
posi tes be zero in the sequence
(t) ... -+ 1"..+ I(M").!. 7".t(M') -+ (M) -+ T,,(M")  _I(M') -+ -".
Wewill thensay that T (or, more precisely.(T, il» is aò-junclOr.lf. in addition,
1:. = O for 1'1 < O and (t) is exact for every short exact sequence
0"-+ M' -+ M -+ MIf -+ O, then we will say that T is a homological fUl'lctor.
Thus. for example, the content or 6.1 (ii) is that H.(G. -) is a homological
runctor on the category of G-modules.
If S and T are o-functors, then a map from S to T is a ramily li' or natural
transforrnations rp,.: SII -+ 1;. such that tbe square
y.
..0

S'I( MJf)
...[M") 1
TJ.M")
 ,SIl_I(M)
1...,(M")
ò ) 4-t(M')
commutes for every short exact sequence o -+ M' -+ M -+ M" ...... o and
every n.
(7.3) Theorem. Let H be a homologicalfunctor such that H", is effaceableJor
n > O.lfT i5 an arbitrary ò-functor and CPo: To -+ Ho is a naturallrasforma-
tion. rhen rpo extends uniquely to a map lp: T -+ H oJ ò-functors. Th,S map lp
is an isonwrphism if and only ifthefollowing three conditions hold:
(i) lpo is an isomorphism.
Ci i) T ls homological.
(i i i) 1',. is effaceable [or n > O.
Thus a homological functor H which is effaceable in positive dimensions is
UDiquely detennined (up to canonical isomorphism or o-functors) hy Ho.
In particular, it follows that H iG. -) is uniquely deterrnined by the proper-
ties (i)-(Hi) of 6.1.
PR.OOF OF 7.3. We wish to construct lpn: 4. -+ HJt so that
T,,(M") v.. 1',.- l (M')
( · ) .,(M11 l... ,[M')
Hn(M") lJ ) H,.-t(M')
commutcsforeveryshortexactsequenceO -+ M' -+ M -+ M"  O. Forn < O
there is no choice and ali diagrams (*) commute automatically. Asswne,
then. that n > O and that CPn-l has been defined. For any module M choose a

76
nl Homology and Cohomology with Coefficients
short exact sequence O -+ K -+ P - M -J> O with P projecti ve, and consider
the diagram
T,.(M) a ) T.r- l (K)  1;.- l (P)
(..) . 1 ¥.- ,(K) 1..- ,(l')
q;>.(M} I
.
I
.....
O ) H,,(M) è ) H"-i(K) H"-I(P)
where the dotted arrow represents tbe map we want to define. Note that the
composite OD the top row is zero, that the botto m row is exact (because
H n( P) = O), and that the right-hand square çommutes (by naturality of ({J" - 1)'
Consequently" there is a unique map tp"(M) which makes (_$o) commute.
Since this definition of €p,. was forced OD us by (-), the uniqueness assertion of
the theorem is c1ear. It is also clear from (**) that if conditions (i)-(iii) hold
then we can prove inductively that qJ" is an isomorphism. It remains, therefore,
to complete the proof of the existence of (jln by proving that qJn is weU-detìned
and natural and that ali squares of tbe form (*) commute. We will need the
following lemma.
(7 .4) Lemma. Let cp,,(M) be defined as above b y means oJ a short exact sequence
0--+ K - P - M-O. For any diagram
M
II
o
 Nt
'J-N
N"
)
.;.0
with exact row, the diagram
T,,(M) TJ.f) J T.a(N")
¥Al
H,,(M) R.(f)) H,,(N")
il J -I(N')
l .- ,(N")
il +Hn-l(N)
commutes.
PROOF. Since P is projecti ve9 the given diagram can be completed to a
commutative diagram
o
o
M
,K
J-P
.1
l
Il
o
) N'
}N
 N"
) O.
.',
O'l
.
',.
;
-
..
.""
$'
I.
,
--
:",:
.r
A
. . ".'
.
.
".=-
,
'J.
1__
.....
)':
.c
.,
"
!
!
"
! -
"
"
,i,

. -'
.,
,.
"
. "
. ,
:...... :
7 Dìmension Shifting
77
Consider now thc diagram
T,.(M)
M)
Hn(M)
TAl) H.(f) 1
Hrc{N")
il
.
 T..-t(K)
aD -
è ) H"_l(K)
(2) 1 H. - ,C.I Q) 1;- ,I<)
e ) Hn-1(N')
H
'1;.- dN'),
1',,( N")
iJ
where the maps labelled ò in eD afe the . ò-maps'9 associated to O --+ K --+
p -+ M -+ O. Then eD commutes by (**)9 (2) commutes because H is a ò-
functof9 and Q) commutes by naturality of (/)11-1' Moreover, the outer square
çommutes because T is a ò-functor. The lemma follows at once. D
Returning now to the proof of the theorem, let f: MI --+ M 2 be an arbitIary
map of R-modules. Choose exact sequences 0-+ KI --+ Pi --+ Mi -+ O with Pj
projective (i = l, 2), and use them to detìne 'Pn(Mi): T,,(M,J -+ H,,(Mj) as in
(* *). A pplying the lemma to the diagram
Mi
Il
O tK2 +P2 J-M2 'o
,
we conclude that the outer rectangle is commutative in the diagram
1;.(M1) T1.f)  T,,( M 2) è t T,,-1(K2)
AM.)l 1...M,) 1..-,(K,)
HII(M l) HJf) ..H ,,(M 2) è t H"-l (K 2).
Now the right-hand square js commutative by (**), and
ò: H,,(M2) --lo Hn-1(K2)
is injective; hence the left-hand square commutes. This shows that ({J,.
is natural. It also shows that lp(M) is well-defined, independent of the
choice of exact sequence O -to K -+ P --.-. M -+ o. [Take f = idM.] Finally,

 :
78
III Homology and Cohomology with Coefficients
if O --+ MI ---., M --+ M" ---., O is an arbitrary exact sequence. then we can apply
the lemma to thc diagram
M"
"
'i,:
. .
"I::
o
::..
.,:..
 :.
i-.:
- .
:. :;
:..:
.MF
.M
1 M"
o
.,
. ..
'!,'::
to conclude that (*) commutes.
.' 
,
D
Similarly one can define tbe notations or o-functor and cohomological
funcror in the obvious way. and one can prove:
(7.5) Theorem. Let H be a cohomo logica l functor such that HII is co-effaceable
for n > o. lf T is an arbitrary b-Junctor and cpo: HO -t' TO is a natura l trans-
formation then epo extends uniquely lO a map rp; H -+ T of o-functors. This
map <p is an isomorphism if and only if the following three conditions ho1J:
(i) rpD is an isomorphism.
(iì) T is cohomological.
(iii) TI! is co-effaceable for n > O.

.I:..
 .;
-,
'''-.
;,
>
. --
.:. 
.
.,
:::.
,
--'
EXERCISES
1-_ J:
..
1. (a) Prove the ro]]owing generaJizatìon of 1.4: Let. .. ......... Cl .E.. CI}  M ---., O be exact.
where each C i is H. -acycHc; then H .( G, M)  H.( C G). [Hint  Apply the dimension-
shifting argument using the exact sequences O - ker E: - C o -+ M -+ O. O - ker a -+
Cl - ker e -+ O. etc.]
(b) Similarly. let O - M -+ CO - Cl -+ ... he exact. where each Ci is H.-acyclic.
Prove that H*(G, M)  H*(CG).
;

,.
2. Use the resuJts of this section to give a new proof of Proposition 2.2. [Method I  rr M
is E-torsion-free. then Tor(M. -) and H .(G, M (8ì -) are homological functors
which agree in dimension O and are effaceable in positive dimensions. A similar
argument applies to Ext(M. -) and H.(G, Hom(A-l, -». Method 2: Compute
Tor(M. N) in tenns ofa projective resolution of M. Expres.s this computation in the
form Tor(M. N) = H",(Cc.) and apply exercise la. Similarly, deduce the result on
Ext from lb.]
"::,
,
3. In the discussion at the beginning or this section. show that the surjection M "* M and
the injection M Cj. K1 can aJways be taken to be Z-split (Hint: Use 3.4 and 3.7.]
.....:
.......,:
.::;' t:
i:r..,
..../'
8 H. and H* as Functors ofTwo Variables
. "i:
.... ....
.... .:
Let Cjj be the following category: An object ofttl is a pair (G, M), where G is a
roup and M is a G-module; a rnap in  from (G, M) to (G'  M') is a pair
(CL. f). where a:: G -+ G' is a map or groups andf: M -t' M' is a map or abelian
J:
8 H. and W as Funccors or Two Variab]es
79
..,
,
.'
groups such thatf(gm) = a.(g)f(m) for 9 E G, m E M. (In other words.fis a
G-module homomorphism if M' ìs regarded as a G-module via a..) Given
(a: f), let F (resp. F') be a projective resolution of Z over ZG (resp. ZG'),
and let 1": F -+ F' be a chaìn map compatible with a as in II.6. Then there is a
chain map t <81 f: F @G M -t' F' <8IQ. M', and L @ f induces a well-defined
map (a., f).: H.(G, M) -+ H.(G', M'). In this way H. becomes a covariant
functor on fjj. In case M = M' and f = idM., we will simpIy write a. for
(a, f).: H ..(G, M') -+ H .(G', M'). Note that the generaI map (ex, f). can
always be written as the composite
H*(G, M) H.(G,f)J H.(G, M')  H.(G', M').
where H .(G, f) makes sense because f is a map of G-modules.
As an example of this functoriality we consider the maps induced by
conjugation. Given H c G, a Gmodule M. and an element 9 E G, we denote
by c(g): (H, M) -t' (gHg - I, M) the isomorphism in CC given by
(h I---i' ghg-l, m  gm).
Tv compute c(g).: H .(H  M) -+ H .(g H 9 - 1, M) OD the chain leve!, we choose
a projectìve resolution F of 1. over lG and use F to compute the homology
of H and gHg - 1, We can take t': F -+ F to be multiplication by 9 as in the
proof of II.6.2 and we then find that c(g)* is induced by the chain map
F @H kl -+ F @gHg-1 M given by x @ m........ gx @ gm. Note that this is just the
map obtained from the usual diagonal action of 9 on F @ M by passage to
q uotients.
Far Z E H*(H, M) we set gz = c(g).z EH .(gHg-1, M). In view of the
above description of c(g)* on the chain level. we ha ve:

(8.1) Proposition. II h E H then hz = z for ali Z E H .(H, M).
(8.2) CoroUary. 11 H -<J G and M is a G-module. then the conjugation action oJ
G on (H M) induces an actior1 ofGjH on H.(H M).
The situation for cohomology is similar. Let g} be the category with the
same objects as t'(f hut where a map (G. M) - (G' M') is now a pair
(a G -+ G' f: M' --+ M);
as before we require that f be a G-module map i.e., thatf(a(g)m') = gf(m')
for 9 E G) m' E M'. If F and F' are resolutions far G and G' and 't: F --+ F' is
a chain map compatible with Cl, then tbere is a chain map
Kum(t" f)&mG.(F'. M')..... .J¥'NnG(F, M).
which induces (C(, Il''': H*(G', M') -+ H.(G, M). Thus H. is a contravariant
functor on f!}. In case M = MI and f = idM" we will write a. for (ex, f).; note
that the generaI map (a, f). is the composite
H*(G', M')  H$(G. M') H.(G./),. H.(G M).

80
III Homology and CohomoJogy with Coefficients
Let qj o be the su bcategory of tJt' consisting of ali the objects (G, M) and of
those maps (CX9 f) such that f is bijective. Then  o is isomorphic to a sub-
category of 9 by (a. f) p-+ (a. f - I), so H* can re regarded as a contravariant
functor on cç o. In particular, the conjugation maps c(g) discussed above are in
fio 9 so we have an isomorphism c(g)*: H*(gHg - l, M) -- H*(H, M) for any
9 E G, H c: G. and G.module M. If Z E H*(H, M) then we set
gz = (c(g)*)- I(Z) E H*(gHg-., M).
In terms of a projective resolution F or 7L over lLG, tbe map z ........ gz is induced
by the cochain mapH(F, M) - .Jf'c.JnfHg- t(F, M) given by
f H [x 1-+ gf(g-lX)J.
Note that this isjust the usual diagonal action of (j onK.o.m(F, M)9 restricted
to tbe subcomplex F, M). In case 9 E H we clearly obtain tbe identity
map, hence:
(8.3) Proposition. Il h E H then hz = z for ali Z E H*(H, M).
(8.4) Corollary. If H <J G and M is a G-module, then the conjugation actiofl
ofG on (H. M) inducesanaction ofG/H onH*{H, M).
EXERCISES
1. If H is centraI in G and M is an abelian group with triviaJ G-action, show thal G I H acts
trivialI)' on H.{H. M) and H.(H, M).
2. Let ex: H c. G be an inclusion. Jet M be an H-module, and let i: M - IndM and n:
CoindgM -+ M be the canonical H-maps. Show that the isomorphisrns ofShapiro's
lemma (6.2) are given by (IX, i)_: H..(H, M) -+ H.(G, IndgA-l) and (o:, n).:
R*(G. CoindgM) -- H-(H. M). [Hint: You can take t to be the identìty map.]
,.' ,.
9 The Transfer Map
Given an inclusion a.: H c.;. G and a G-module M, we have seen that tbere are
maps a*: H*(G, M) -- H*(H, M) and IX.: H .(H, M) -+ H.(G. M). We wiJ)
often write a* = resZ and cali it a restriction map. SimilarIy. we wnte ex. =
corZ and cali it a corestrictton map. The purpose or this section is to show that
if (G: lI) < 00 then there are maps going in the other direction, called transfer
maps. This extremely useful observation is due to Eckmann [1953] and
Artin- Tate [unpublished]. The existence of the transfer maps is somewhat
more subtIe than that of a. and (X*; in particular, they are not induced by
maps in the categories  and f!À of. [Tbe name LO transfer map .9 comes from
the fact that on H l it coincides with tbe transfer map of classical group
heory, due to Schur [1902]; see exercise 2 below.]
" .
"
. ".':
. ..
. ,-
.,
. ..
-' ':t
. '.
.'
.....
",.-.
"(
. .
:.
',-
,.'
.,'
.,
.
...':
.
"
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.' ,
.'
..
. ..
"'ti
'-

9 The Transfer Map
81
We will explain tbe transfer maps from five different points or view, al] or
which are useful:
(A) For any G-module M and any H c G we have a canonical surjection
or G-modules
.,
"
(9.1)
lLG@ZHM -+ M
and a canonica.l injection
(9.2)
M -+ Hom.lH(ZG, M),
r
.:.
cf. 3.4 and 3.7. Applying H.JG, -) to 9.1 and using Shapiro9s lemma, we
obtain a map H.(H, M) --t H.(G, M); this is easily seen to coincide with IX*.
[V se exercise 2 ol 8.] Sìmilarly, applying H*( G, -) to 9.2 and using Sbapiro9s
lemma we obtain a*; H*(G, M) -+ H.(H, M). If we assume now that
(G: H) < 00, then ZG <&zH M  HomZH(ZG, M) by 5.9. Consequently we
can apply H*(G -) to 9.1 and H.(G, -) lo 9.2 to obtain tbe transfer maps
goìng in the other direction. These maps are often denoted
cor: H*(H, M) - H*(G. M)
and
res: H*(G, M) - H.(H, M)
or simply trg. if it is clear from tbe context whether we are talking about H *
or H..
(B) For any H c G we can regard both H *(G, -) and H*(H, -) as
homological funçtors on the category of G-mod ules, and both are easily seen
to be effaceable in positive dimensions. Assuming now that (G: lI) < 00,
we define tr: M G -+ M H by tr(m) = I" e H\G gm, where m (resp. in) denotes the
image of m in M G (resp. M H)' (Tbe sum makes sense because gm depends only
on the class of g in H\G.) We now defioe res: H.(G, -) -+ H ,<H -) to
be the uniq ue extension of tr to a map or homological funçtors, cf. Theorem
7.3. To see that this agrees with the map defined in (A), one need only verify
that the latter is compatible with ò and equals tr in dimension zero. Details
are left to the reader. Similarly, eor: H*(H, -) -+ H*(G, -) can be defined
as the unique map of cohornological functors which in dimension zero is the
map tr: MB -+ MG defined by tr(m) = LG1H gm. [fo verify co.efface-
ability of W(H9 -) on the category or G-modules, use exercise 3 of 3;
altematively, note that a co-induced G-module is also ço-induced as an
H-module.]
(C) Let F be a projective resolution of Z over ZG. Then F @n M =
(F @ M)lI and F @GM = (F @ M)G. We can now define res: HiG9 AI) -+
H*(H, M)to bethemapinduced bythechainmaptr: (F @ M)G -+ (F @ M)H'
where tr is as in (B). This detines a map or homological runctors which equals

82
III HomoJogy ftl1d Cohomology wìth Coefficients
tr on Ho hence it agrees witb definition (B). SimilarlYt .JfhmG{F. M) =
Jf'mn(F. M)<i and .)f',çmH(F. M) = :Kt»n(F. M-,nt where G acts diagonally on
Ho-NI (cf. ). Hence there is a cochain map tr: .Jf'ont(F. M)H --+ .Kom{F, M)G
whicb induces cor: H*(H. 'M) -+ H*(G, M).
(D) If we wish to compute the transfer maps of (C) using one resolution
F(G) ror G and a different one F(H) for H, then we need an (augmentation-
preserving) H -chain map t: F( G) ..... F( H) in order to realize the canonical
isomorphism H.(F(G) @H/t.-f) -=+ H.(F(H) @fI M). The transfer maps will
then be induced by the composites
(9.3) F(G) @G M  F(G) @H M T0M F(H) @H M
and
(9.4) .1'tb-mH(F{H)t M) Jt<,,,,tT.M'. mH(F(G). M)  Jtb.»t(;(F(G). M).
Taking F(G) and F(H) to be the standard resolutions, for example, we can
easily write down such a map 'l' as follows. Choose a set or representatives
for tbe right cosets Hg. Then there is a unique map p: G --+ H or left H-sets
which sends every coset representative to 1; explicitJy. il 9 denotes the
representative of Hg, then pg = gg-l. Using tbe Hhomogeneous" de-
scription or the standard resolutions, we can now define t: F(G) -+ F(H)
by t(go,. - . , g,.) = (PBo, - . ,  pg,.). The interested reader can translate this
into the bar notation and can write out the corresponding transfer maps
C.(G, M) ..... C..(H, M) and C.(H. M) --+ C*(G. M) given by 9.3 and 9.4.
(E) If :r --+ Y is a covering map of finite degree, then there are transfer
maps H.( M) --+ H.(i'. M) and H*(Y, M) - H*(J': M) for any coefficient
system M on Y. In terrns of cellular chain complexest tbe homology transfer
is induced hy u @ m  La ti @ m, where (] is an oriented celi or Y and ét
ranges over the oriented cells of Y lying over f1. SimiJarly, the cohomology
transfer is induced by the cochain mapf t-+ [a 1-+ La- f(u)].l£ Y is a K(G. 1)
and Y is the covering space corresponding to H c: G, then r is a K{H, 1)
and these transfer maps agree with the transfer maps H .(Gt M) --+ H .(Ht M)
and H.(H, M) --+ H*(G. M), To see this, one need only apply definition (C)
with F equal to tbe chain complex ol the universal cover or 1':
We now give some properties ofthe transfer maps_ Let H(-. -) denote
either H 111 or H..
(9..S) Proposition. (i) Git'enK c H£; Gwith(G: K) < 00.
cor = cor o co and res = res o res.
(ii) Given (G: H) < 00 and Z E H(G. M). cor res z = (G; H)z.
(iii) Given H, K c G with (G: H) < 00 if H(-t -) = H* and (G: K) < 00
ifH(-. -) = H., we have
G G "" K H -L
resI( corHZ = L COrXngHg-t resBUf-1 gz
flf!JE
,
.'
-
J 
L;:,
.
,
:.
'!"
J
,
"
,..;.
.'
,l
>
r. '"
"
r',.
.,
",i'
. "
I:"'
, I_'
,- .
.. . :-
., ..
.
,<
.'-
.
,
- >
-.
".
.... ..
'.
,..
 -'
.:""
.. .
". -
r ".
.<
"
.". .'
-,':t
 :tI".
.:<" .:
-. ..;}
.:
.
.,
:;:
;,;
.'
IO Applications of tbe Transfer
83
" ,.
Ior any Z E H(H, M), where E is Cl set oJ representatives for the double
cosets KgH. In parricular. if H -<J G and (G: H) < 00, lhen resg corg Z =
LEO/H gz = N z. where N is tl,e norm element ofZ[ G/HJ.
(See 8 for tbe definition of gz. Note that in the cohomo]ogy case we can
write resi:H;';g-t gz = resr\gHg- t z, where the restriction map' on the right
is with respect to the conjugation map (k H g  1 kg, m t-+ g - l m)7 regarded as a
map (K n gHg-1, M) - (H, M) in l'ifo.)
PROOF oF9.5: (i) and (ii) follow easily from the definitions and will be lert to
the rcader_ (iii) can be deduced from the double coset formula 5.6b and
definition (A) of the transfert but it is easier to work directly with definition
(C). We wiIl give tbe proof in the case H(-, -) = H*; the homology case is
similar. Let Il E(F, M)H represent z. Then resf cor* z is represented by
L(lIEOGIH gu E.7t'om{F, Mt ç JltbM(F, M)K. Grouping together those terms
gu corresponding to a given K-orbit in GjH, we find
Igu=L I kgu,
tJ e GlH gE E hi K/Ky
where Kg = {k E K: kgH = gH} = K fì gHg-1. Since the inner sum on the
right clearly represents corf reslHf/ - 1 gz, this proves (iii). D
9 -"
EXERCISES
1. Show that res: H(G. M) ---io H(H, M) and cor; H(H, M) _ R(G. M) are "invariant
under conjugation:' in the sense that (J. resw = reS;Hg-'W ror w E H(G, M) and
Ci G
corfHg - 1{JZ = cor HZ for z e R( H, M).
2. The transfer map H 1 (G) - H 1 (H) can be regarded as a map G 1Il> - H ah' Using defini-
tion (D), show that this map is given by
g mod [G. G) Ho n y'gg'g-l mod [H, H],
g'EE
where E is a set of representatives for the right cosets H g and :i ìs the representative of
Hx. This formula is usually taken as thedefinition ofthe transfer map ingroup theory
texts, cf. Hall [1959], formula (14.2.4).
lO Applications of the Transfer
T O simplify tbe notation we will state the results of this sectioo onIy for
cohomology; similar resuJts hold for homology.
(10.1) Proposition. LeI M be a G-module and H c G a subgroup oJ finite index
such that H"(H, M) = Ofor some n. Then H"(G. M) is annihilated by (G: H).
In partit;ular. if(G: H)-is.invertible in M (i.e.t ifmultiplication by (G: H) is an
isomorphsm). then HlI(G. M) = o.

84
111 HomQlogy and Cohomology wìth Coefficients
PROOF. The first assertion follows immediately from 9.5(ii). The second
follows from the tirst, since (G: H) is invertible in H"(G, M) if it is invertible
M O
.. -.
Note that 10.1 applies 10 any n > O if G is finite and H = {l}. Hence;
(10.2) CoroUary .1/ G is finite, then HII{ G, M) is annihilated by J G Ifor all n > O.
II I G I is invertible in M (e.g., if M is a QG-module), then H"'(G, M) = O for
ali n > O.

It follows that H"(G, M) adrnits a primary decomposition
Hh(G, M) = EB H"(G. M)(P)1
l'
where p ranges over the primes dividing I G I and H"( G, M)IP) is the p-primary
component or Hn(G, M). Now fìx a prime p and let H be a p-Sylow subgroup
of G. Since (G: Il) is relatively prime to p, 9.5{ii} implies that cor res is an
isomorphism on H"(G, M){p)' In particular, the restriction map induces a
monomorphism H"(G, M)(p)  H"(H, M). In order to describe the image of
this monomorphism, we nee<! some terminology.
If G is an arbitrary group, H c G is a subgroup, and M is a G-module, we
wiU say that an element Z E H.(H M) is G-invariant if resZ""IIHQ-1. Z =
res"H;tr! gz for ali g E G. In case H -<:J G, thisjust says that Z E H*(H, M)GIH.
Note that li z = resg. w for some w E H*( G, M) then z is G-invariant; for
gz = resH,-1 w by exercise 1 of 9, hence resYi;B-l gz = resL"\BHg-l w =
H
resH f'1 9H9-1 Z.
We can now state:
.
- <
t
,
..
..
,
I.
.,
(10.3) 1beorem. Let G be afinite group and H a p-Sylow subgroup. For any
G-module M and any n > O resg maps Hn(G, M)(p) isomorphically onta the
set ofG-inuariant elements oJ Ir(H, M). In particular, if H -<.:J G then
H' G, M)(p)  Hn(H. M)GIH.
PROOF. We have a1ready shown that res maps Ir(G, M)(I" monomorphically
into the G-invariants. so it remains to show that any invariant z is in tbe image.
Consider the element w = corZ z € Hn(G, M). Since H"(H, M) is annihilated
by IHI, w E H"(G, M)(p)_ We now compute resZ w by the double cosel
formula 9.5(iii); since z is invariant, we obtain
.
. -
G "H gHrr J
resH w = i... cor H f'1 gHg-l resH r.gHfl-1 gz
,,.H\G/H
"H H
L COfH""IIHr,- 1 resHr.9HB-1 z
gftiH\GIH
L (H:HngHg-1)z
t/EiH\GjH
= (G: H)z.
"
lO AppJìcations of tbe Transfer
85
(The last equality is obtained by decomposing the set GIH into H-orbits and
noting that (H: H (\ gHg-l) is the cardinality of the Horbit or gH.) Since
(G: H) is prime to p, it follows that z = res* w', where
W' = w/(G: Il) E H"(G, M)(p)'
o
,-
This proof, which should be thought of as an averaging argument, can be
used in other situations where di vision by (G: H) is possible. F or example, one
pro ves in the same wa y:
'.
(10.4) Proposition. Let G be arbitrary M a G-module, and H c G a subgroup
of finite index. Il (G: H) is invertible in M then resg maps H*(G, M) iso-
morphically onto the set of G-invariants in H*(H, M). In particular, if H..::::] G
then H*(G. M)  H*(H, M)GIH.
EXERCISES
l. Use 10.3 to compute H*(G, l). where G is the symmetric groupon three letters. [This
wjJ] be easier after we have cup products 8V'd.ilable. cf. exercise 5 of V.3, but it is do-
able now.]
2. Let H c G be a subgroup offinite index. For any G-module M and any H-H double
cosetC.defineaG-endomorphismj(C}ofZG@ZHMby l (8) m....... LC!lfg @g-lm.
Using Shapiro's ]emma, we can view the map Hf/I(G.f(C» as an endomorphism
T(C) of H*(H. M).
(a) If C = HgH. show that r(c)z == COrZo"'\9H9-1 resJ:;-lgZ. [Hint: It suffices to
check this in dimension O.]
(b) Hz E H*(H. M)isG-ìnvariant, show that T(C)z = a(C)z, wherea(C) = IC/HI =
(H: H n gHg 1).
(c) In the situation of 10.3 and 10.4. show that the image of resg is the set of z such that
T(C)z = a(C)z for aH double cosets C.
,.
---
----

.'.
i..
",
,
CHAPTER IV
Low- Dimensional Cohomology and
Group Extensions
,
"
<.
".
,
1 Introduction
An extension of a group G by a group N is a short exact sequence of groups
(*) I .... N .... E .... G .... 1.
[Warning: Some people calI this an extension of N by G.] A second extension
l -+ N .... E' .... G -+ 1 of G by N is said to be equivalen.t to (*) if there is a
map E -+ E' making the diagram
1
E
'N/ l G
E'/
q
. <
commute. Note that such a map is necessarily an isomorphism.
The main problem in the subject of group extensions is to c1assify the
extensions of G by N up to equivalence. Roughly speakin& then, we are
trying to understand ali possible ways of building a group E with N as a
normal subgroup and G as the quotien t. This problem turns out to involve
the cohomology functors Hi(G, -) ror i = 1.2.3.
F or a while we will consider only the case w here the kernel N is an abelian
group A (written additively). A special feature ofthis case is that an extension
i Il:
OAE....G-+l
gives rise to an action or G OD A, making A a G-module. F or E acts on A by
conjugation since A is embidded as a normal su bgroup or E; and the conjuga-
tion action or A on itself is trivial so there is an induced action of E/A = G
,
86
n
2 Spii. Extensions
87
on A. Explicitly. given g E G we choose g E E such that 1t(9) = g. and the
action of 9 OD A is thcn characterized by
(1.1)
i(ga) = gi(a)g- l
(a E A).
It is orten convenient to rewrite 1.1 as a commuration rule
(1.2)
gi(a) = i(ga)g.
In particular. this shows that i(A) is centrai in E if and only if tbe G-action
is trivial. In this case the extension is called a centrai extension.
In view of the G-module structure on A, we can refine the classification
pro blem by fixing a G-module A and trying to classify the extensions of G
by A w hich give rise to tbe given action of G on A. As we wiU see in 3, this
problem has a very simple solution: The equivalence c1asses of such extensions
are in l-l correspondence with the elements of H2(G. A).
We begin by reviewing the simplest c1ass of extensions, name1y. the split
extensions.
2 Split Extensions
Fix a G-modu1e A and let
(*)
O i 11
....A-+E......G....I
be an extension which gives rise to the given action of G on A. We say that
( =Il) splits if tbere is a homomorphism s; G .... E such that 1[5 = idG. The
following characterization of split extensions is pro bably well-known to
the reader, but we wilJ write out tbe proof in detail since it motivates the
pr.oof or the main result or the next section.
(2.1) Proposition. T Ire following conditions on the extel'lsion (Il<) are equivalent:
(O ('Il) splits.
(ii).E has a subgroup G which is mapped by 7t isomorphically onto G, i.e.t
which satisfies E = i(A). (; and i(A) n G = {l}.
(iii) E has a subgroup G such that every element e E E is uniquel y expressible
in theJorm e = i(a)g (a E A, g E ti).
(iv) (*) is equivalent to rhe extensiol'l
i' x'
O .... A -+ A ><I G  G .... I,
where A ><I G is the semi-direct producl oJ G and A relative to the given
action. and i' and re' are che canonica/ inclusion and projection maps.
(Tbe definition of A ><I G will be recalled in the course or the proof of
2.1.) ,

88
IV Low-Dimensional Cohornology and Group Extensions
PROOF. Clearly (i) <=> (ii) -<=- (ìii). To prove that these conditions imply (iv),
let s: G -+ E be a splitting and note that we have a set-theoretic bijection
A x G  E, given by (a, g) 1--+ i(a)s(g). There is therefore a unique group
law on the set A x G such that this bijection is an isomorphism. To calculate
this group law, we must express a typical product [i(a).s(g)] . [i(b)s(h)] in E
in the form i( - )s( - ). Using the commutati OD rule 1.2, we find
.,.
. .
.I
..l
u
.';:..'
..
......:.
'.<.
.:, .
: i' .:....;:.{
J li' . .:. .
.. .....
.....:
y
i(a)s(g)i{b)s(h) = i(a)i(gb)s(g)s(h)
= i(a + gb)s(gh).
Thus the grouplawonA x G isgiven by
(a, g). (b, h) = (a + gb, gh).
=;
\:..
:.
,
.
..
,
"r
..
,.
-:..;:
'. <

","
';-r
"
Tbe set A x G with this group law is, by definition, the semi-direct product
A>c:3 G, and (iv) folIows at once. Finally, it is trivial that (iv) =-- (i). D
.f
i:.....
:. ,-
{
;,
J:
.
, .
.
Proposition 2.1 says that there is only one split extension of G by A
(up to equivalence) assodated to the given action of G on A. Nevertheless,
there is an interesting .. c1assification" problem in volving split extensions;
Given that an extension (*) splits, classify ali possible splittings.
In case G acts trivially OD A, for example, so that the group E is iso-
morphic to tbe direct product A x G, then tbe splittings are obviously in
1-1 correspondence with homomorphisms G -+ A. In the generaI case,
I claim that splittings correspond to derivations (also called crossed homo-
nwrphisms). These afe functions d: G -+ A satisfying
.:
-
....
.....
't.
. (
:'
1::
;:-
'.
.
. ..
i
- _'i
. '.."
. ,.
. -
, -
t '"i:
;'
(2.2)
-
'?
d(gh) = dg + g.dh
,
..'
'.-
.,-
for ali g, h E G. To see this we may assume that (*) is tbe canonical split
extension
"
<
. : .:.
O-+A-+A><:IG-+G-+1.
".  ..
. :)'
A function s: G -+ A ><.I G with ns = id then has the form s(g) = (dg, g),
where d is a function G -+ A. We have
.'
.'
. 
. .}"
. '..(....:
. ,'\c,
:'. = .:.
,
s(g)s(h) = (dg + g. dh gh),
. .";:'
. ". ";
so s will be a homomorphism if and only if d Is a derivation. This proves the
claim.
" .
-. . 
..,....
-, ,
. :j,
. ;".1
Remark4 Tbe terro "derivation" seerns more ceasonable if wc think of G
as acring 00 A OD the right by the trivial action, in addition to the given left
action of G on A. The equation 2.2 then takes the familiar forro
d(gh) = dg.h + g.dh,
.- -
.......
-'

-
- ..-::
. .
. ..-:,.
," :
,.
as in the product ruJe Cor derivatives.
"
f
'.
.
"
]
]
2 Split EJttensiODS
89
.
"
:
Two splittings Sh S2 will be said to be A-conjugate if there is an element
a e A such that 81(g) = i(a)s2(g)i(a)-) for alI g E G. Sioce .
(a, lXb, gXa, 1)-1 = (o + b - go g)
in A ><1 G, this conjugacy relation becomes
d 19 = a + d2 9 - ga
in terms of tbe derivations d b d2 corresponding to SI' S1' Thus dI and dz
correspond to A-conjugate splittings if and only if their difference d2  di
is a function G  A of the form 9 I--i> ga - a for some fixed a E A. Such a
function is said to be a principal derivalion.
We can summarize the preceding paragraph by saying that the A-con-
jugacy c1asses of splittings of a split extension or G by A correspond to the
elements of the quotient group Der(G, A)/P(G, A), where Der(G, A) is the
abelian group of decivations G -+ A and P(G A) is tbe group of principal
derivations. On the other hand, a glance at the standard cochain complex
C*(G, A) shows that Der(G. A) is the group of l-cocycles and P(G, A) is
the group of l-.co boundaries (cf. exercise 2 of III.l). We have therefore
established :
(1.3) Proposition. Far any G-module A the A-conjugocy classes oJ splittings
of tlre sptit extension
O-+A-+A><:IG-+G-+I
are in l-l correspondence with the elements of HI(G, A).
EXERCISES
The purpose of these exercises is to develop the theory of derivations and to give alge-
braic proofs of some results that were proved topologically in Chapters I and II.
1. H d is a derivation, show that d( 1) = o.
2. Let l be the augmentation ideaI or lLG and let D: G - l be the derivation defìned by
Dg = (J - 1 for 9 E G. [Regarded as a derivation G - ?La, this is simpIy the principal
derivation corresponding to l eZG.] Show that D is the utliversal derivatioll on G,
in the following sense. Given any G-moduJe A and any derivation d: G  A, there Is a
unique G-module map f: 1- A such that d = fD:
G b l
-I I

/1

ti:
A.

90
IV Low-DirnensìonaJ CohomoJogy and Group Extensions
Thus Der(G. A)  Hom.zG(I, A). [Hint: A derìvationd:G -+- A extendsto an additive
map a: ZG -- A such that
a(rs) = dr. e(s) + r. ds-
for aH r, SE 1:G. where 1-:; z.G -+- 7L is the augmentation. The restriction of a to l is a
module homomorphism such that i1(g - I) = dg for aH 9 E G.]
3. Let F = F(S) be the rree group generated by a set S.
(a) If A is an F-modure and (a!lu,s is a famil)' of elements of A, show that there is a
unique derivation d: F ..... A such that ds = a for ali SE S. [Hint: Derivations
F -+- A correspond to splittings of O -+- A -+ A :><J F -+- F - l; now me the univer.sal
mapping property of F.]
(b) Deduce from (a) and exercise 2 that thc augmentation ideai of 7lF is a free lLF-
module with basis (Ds)ses- Thus we have reproved [.4.4 as well as exercise 3b or
II.5. As in the latter. we can now define the partial deri va ti ves ò/òs: F -+ 7L.F by
DJ = Lf£s(òl1òs)Ds for f E F.
(c) Reprove. from the present point of view. exercise 3c or II.5. [Hint: The formula to
be proved is true for the uni versai derivation, hence (or any derivatìon.]
4. Let G = FIR. where F == F(S) and R is the norma I closure of a subset T c: F.
(a) Far any G-module A, show that derivatiol1s G ---+ A correspol1d to derivations
d: F -+ A such tha t d( T) = O. [Hìnt: A splitting of O ----Jo A --+ A ><I G -+- G  I can be
defined by constructing a suìtable map F -- A ><I G.]
(b) Using exercise 3. restate (a) as follows: Oerivations G -+- A correspond to families
(a.s)n:s of eleme[Jts of A such that L"s(at/ò.s)as = O for ali t E T. where òt/òs is the
image of òt/iJs under the quotient map 7LF -+- ZG.
(c) Using exercise 2, restate (b) as follows: Let I be the augmentation ideaI of ZG.
Then there is an exact sequence
ZG{T) 5 ZG(S}  J .... O.
where ò.e$ = S - l and iJ2e, = L"s(ot/òs)es. (This reproves the second part of
exercise 3d or 91I.5.)
(d) Make (c) more precise by provmg that there is an exact sequence
(.)
0- Rab.!. ZGIS)  ZG":" Z -----+ O.
where oe = s - 1 and B(,. mod [R. R]) == Les (Dr/òs)es' (This rcproves 1154 and
the first part of exercise 3d ofII.5.) [Hint: We have a free resolutiotl
(u)
o -+- l - ZF  Z - O
of 7L over ZF. where 1 now denotes (he augrnentation ideaI of 7LF. Taking R-coin
variant.s. we obtain a complex whose homoJogy is H... R. Since l  ZF{S\ (hìs yieJds
an e.xact sequence
o -+ H lR .!. ZG(SI - 7LG -.. 7L -+- O. "
"
.
l
}
"
>
,
,
.
"
,
,-
<
. '..
(
.::-=
,
3 Tbc Classification of Extensions with AbeJian Kernel
91
.
;
On the orher hand. we can calculate H.. R from the standard (bar) resolutìon of Z
over 7LR. and tbis is what we used to show H 1R  RQb. Now it is easy to map tbe
standard resolution to (**). by [ ]........ 1 in dimension O. [r] 1-+ r - I = Dr in
dimension l and [r] I. . .1 r n] 1--+ O for " > L Deduce that there is a commutative
diagram
R DIR) l
l l
H1R
<p
)7L G{S)
"
where the vertical arrows are quotient maps. Since the composite F  1 -+- ZG(S)
is a derivation such that s 1-+ e. the exactness of (O$ ) fo)]ows easily.]
-
"
",
3 Tbe Classification or Extensions with
Abelian Kernel
Let A be a fixed G-module. AlI extensions of G by A to be considered in this
section will be assumed to give rise to the given action or G on A_ To analyze
an extension
(3.1 )
i '1'1
o -+ A -+ E -+ G -+ l,
we choose a set-theoretic cross-section or 1t, i.e., a function s: G -+ E such
that ns = idQ. F or simplicity, we will assume that s satisfies the normalization
condition
(3.2)
s(1) = I.
li s is a homomorphism, then the extension splits and we know its structure
by 2.1. In the generai case, however, there is afunctionf: G x G - A which
measures the failure of s to be a ho momorphism. Indeed, for any g, Il E G,
the elements s{gh) and s(g)s(h) of E both map to gh in G so they ditJer by an
elernent of i(A). Thus we can define f by the equation
(3.3)
s(g)s(h) = i(f(g, h»s(gh).
Note that 3.2 implies tt't f is normalized, in the sense that
(3.4)
1(g, l) = 0= 1(1, g)
for alI g E G. The function f is often called the factor set associated tp 3.1 and s.
I claim now that the extension 3.1 can be completely recovered from the
G.module structure OD A and the function f. Indeed. since s(G) is a set of
coset representatives for i{A) in E, we ha ve a bijection A x G -+ E given by
(a. g) t- i(a)s(g). To compute thc group la w OD A x G which makes tbis

92
IV Low-DimensionaJ Cohomology and Group Extensions
bijection an isomorphism of groups, consider two elements (a g) (b, h) in
A x G. Using the commutation rule 1.2, we find
i(a)s(g)i(b)s(h) = i(a)i(gb)s{g)s(h)
= l(a + gb)i(f(g h)s(gh)
= i(a + gb + f(g9 h»s(gh).
Thus the group law on A x G is given by
(3.5)
(a, gXb, h) = (a + gb + f(g9 h), gh).
We will denote by Ef tbe set A x G with the product 3.5. Note that 3.510oks
lite the product in A >cl G, .perturbed.9 by f.
Since i(a) = i(a)s(l) for any a E A9 the composite A  E  EJ is the
canonical inclusion
(3.6)
a f-+ (a I).
And tbe composite E f  E  G is obviousIy tbe canonicaJ projection
(3.7)
(a, g)  g.
Thus tbe originai extension 3.1 is equivalent lo the extension
(3.8)
o  A  EJ -io G -+ I
defined by 3_5, 3.69 and 3.7 entirely in terms of G, A, and f.
It is natural to ask at this point whether we could start with an arbitrary
function f: G x G -io A satisfying 3.4 and define a group E J by me-s or
3.5. Tbe answer is no. lndeed if we define a product by 3.5 and compute the
triple products [(a, g)(b, h)](c, k) and (a, g)[(b, h)(c. k)] we find that 3.5 is
associative if and only if f satisfies the identity
(3.9)
f(g, h) + f(gh, k) = gf(h, k) + l(g, hk)
for alI g  k E G. If 3.9 holds, however, then we do in fact have a group. For
(3.4) implies that (0, l) is a 2sided identity, and it is easy to prove the exis-
tence ofinverses. (Given (a, g) e Eb solve the equations (a, g)(b, g-l) = (091)
and (b',g-1)(a9g) = (O J) for b and b'_ One finds that (ag) has left inverse
(-g-Ia - f(g-l. g), g-I) and right inverse (_g-Ia _ g-lf(g, g-l), g- J).
These two inverses are necessarily equal because of associativity.] Moreover,
one checks easiJy that 3.6 and 3.7 define homomorphisrns making t he resulting
sequence 3.8 exact, that 3.8 gives rise to the given action of G on A, and that
the factor set associated to 3.8 (witb the canonical cross-section g 1-+ (O, g» is
tbe originaI function f.
What we have estabIished then, is essentially al-I correspondence
( extensions 3.1 Witb) (functions G x G -+ A)
\ a norma1ized section +-+ satisfying 3.4 and 3.9 .
,
i
"
.
.... ..

.
a '
.. 
.,


"
,
:
f _
:,:
"
}
,
-.
.
..
.
.
.
I _o.
.'
j:
-
.'
.'
"
,
3 The CJassification of E.xtensiol}s with AbeUan Kernel
93
.1
.'
Now the identity 3-9 can be rewritten in a fonn which should look familiar:
:.
";'::
(3.10) gf(h, k) - f(gh, k) + f(g, hk) - f(g, h) = O.
Indeed, f can he regarded as a 2-cochain of the standard complex C*(G, A)
for computing H*( G, A) (d. III.l), and 3.10 says precisely that J is a cocycle.
Moreover, 3.4 simply says that f is in the normalized cochain compIex
C(G. A) c C*(G, A). Thus
( extensions 3.1 witb )  (normalized 2-cocycles Of)
a normalized section G with coefficients in A
Finally, I claim that changing the choice of the section in 3.1 corresponds
precisely to modifying the cocycle f in C(G, A) by a coboundary. In fact,
an arbitrary nonnalized section of 3.1 is given by
.
,
"
!.:"
..
.
i':"
(3.11)
9  i(c(g»)s{g),
"1
where c: G  A is a function such that c(1) = O, ie_9 c is an arbitrary element
of C(G. A). And we can easily compute the factor Set associated to 3.11;
for we bave
i( c(g) )s(g )i( c(h »s(h) = i( c(g )i(gc(h) )s(g)s(h)
= i(c(g) + gc(h»i(f(g, h»s(gh)
= i(c(g) + gc(h) + f(g, h) - c(gh»i(c(gh»)s(gh).
Thus the new factor set is
(g, h) t--+ c(g) + gc(h) + f(g, h) - c(gh),
which is equal to f + Ct as c1aimed.
We have therefore proved:
(3.12) Theorem. Let A be a G-module and lel 8(G, A) be the set of equivalence
classes oJ extensions 01 G by A giving rise to the given action oJ G 01'1 A. Then
there is a bijection
I(G, A)  H1(G, A).
As a simple application of 3.12 and 2.3 we will prove the following theorem
of group theory:
(3.13) CoroUary. Let E be afmite group o[ order mn9 where m and n are relatively
prime, and suppose thill E coPltains an abelian normal subgroup A DJ order m.
Then E contains subgroups of order n9 and any two such subgroups are con-
jugate.
PaooF. Let G = E/A and consider the extension
O -+ A -+ E  G -+ 1.

94
IV Low-Dimcnsional Cohomology and Group Extcnsions
Since I A I and I Giare relatively prime, H2(G, A) = O (cf. 111.10.2). Thus
l'(G, A) contains a single element and hence tbe extension splits. Tbis proves
tbe existence part ofthe corollary. Tbe uniqueness (up to conjugacy) follows
similarly from 2.3, once one notes that any {j c E or order n must map
isomorphica1ly onto G and hence split the extension. D
Remark. The corollary can be genera)ized lo the case where A is non-abelian,
cf. Zassenbaus [1958], IV.7. Tbe existence part of this generalization, as
weIl as the uniqueness if eitber A or E/A is assumed to be solvable, is proved
by a straightforward inductive argument based on 3.13 and the Sylow
theorems. One then completes tbe uniqueness proof by appealing to tbe
Feit- Thompson theorem, wbich says that every group of odd order is 501-
vable and hence that either A or E/A is solvable.
Further group-theoretic applications of 3-12 wiIJ be given in exercises
3 7 below and in .
EXERCISES
L (Functorial properties of rt(G, A»
(a) Given an extension O ..... A  E - G --+ 1 and a group homomorphism 0:: G' __ G,
show thal there is an extension O  A -+ E' -+ G'  l, characterized up to equi-
valence by the fact that it fits into a commutative diagram
o
'A
)E
I
) G
). I
l'
o
) A
 E'
:t' G'
) lo
Deduce that o: induces a map tf(G, A) -+ 8(G'. A), which corresponds under tht-
bijection of 3.12 to H2(rx, A): H2(G. A) ..... R2(G'. A). [Hint: Take E' to be tbe
fiber -product (or .. puJl.back") E x G G'. which by definition is the set or pairs
(e, g') E E x G' such that e and g' have the same image in G.]
(b) Given an extension O --+ A - E -. G - l and a homomorphism f: A -+ A'
of G-modules, show that there is an extension O  A' -+ E' --+ G - l, characterized
up to equivalence by the fact that it fits into a commutalive diagram
"
. . >"
o
). A
 E
). G
). l
Il 1
o
). A'
, E'
G
t L
Deduce that f induces a map 8(G. A) --+ 8{G. A'), which corresponds under the
bijection of 3.12 to H2(G.f) H2(G. A) --+ H2(G. A'). [Hint: Take E' lo be the
la.rgest quotìent of A' ><lE such tbat the left-hand square above conunutes.]
J'
,
.'
'.
,.
".
"
. .
,
-
"
'i
;
"
..
,-.
'h
8
.,
,
'.
.,
..

, ,
,
.:-
,. -
(
Or":
It..'
,.-
.. .
?
3 The Clagification of Extensions with Abelian Kernel
95
2. (lnterpretation of.:5: H1 -+ HZ)
(a) Let O -+ A' -+ A -. A" - O be a short exact sequence or G-modules. Given a
derivation d; G -+ A". show thal there is an extension O -+ A' --+ E  G _ l,
characterized by tbe Cacl that il fits into a commutative diagram
o
,. A'
I A....
A
 O
f. f'
o
) AI
).G
). E
t I
,
.
with il a derivation_ (The latter makes sense because A is an E-modu]e via E - G.)
[Hint: Take E IO be the set-theoretic pull.back, as in exercise la. regarded as a
subgroup of A:><I G.]
(b) The construction in (a) gives a map Der(GA") -+ 8(G.A'). Show thal the
diagram
Der(G, A")
l
) ,f(G, A')
l-
H1(G, A'/)
.5
) H1{G, A')
commutes. [Hint: A (set-theoretìc) lifting of d to a function G --+ A yie1ds a (set
theore(ic) cross.section of E -+ G. Now check defìnitions.]
3. Let G be a finite group. For any homomorphism G -+ Q/l we can construct a
centraI extens!on of G by o:. by pu]]jng back the canonica] extension O -- Z -. Q --+
Qlo:. -- O:
O
, &:.
'0
r
,. Of&:.
I
,. O
O
 l
 E
)).
,. G
Thus we have a map cp: Hom(G. CI&:.) --+ 8(G. l). where G acts trivially on Z. Prove
that tp is a bijection. (Hint: By exercise 2, ({J can be identified with
15: H1(G, QjZ) _ Hl(G, &:.).]
4. (a) Let E be a group which contains a centraI subgroup C of finite ìndex n. Prove
tha( (here is a homomotphism E - C whose restriction to C is the n-th power map.
[Method l: We have a centrai extension l -- C -+ E --+ G -t l with I G l = 11. Since r1
annihilates H1(G, C). the r1-th power map C -- C induces the zero-map on H2(G, C).
Exercise lb now implies that there is a diagram
]
). C
1
t E
1
). G
 ]
I
,. C
--41.
 C><G
te

96
IV Low-Dimensional Cohomology and Group Extensions
Take the C.component of E -+ C x G. Method 2 Use the transfer map H1E-
H]C = C.]
(b) Deduce that the foUowing two conditions on a finitely generated group E are
equivalent:
O) The center C of E has finite indcx.
(ii) The commutator subgroup [E. E] of E 1S finite.
(Botll of these conditions say. heuristicaUy. that E is "alrnost abelian.") [Hin(:
For (i) => Oì). note that tbe map E -- C of (a) has finite kernel. Conversely, (ii)
impJie.s that every generator or E has ol1]Y finitely many conjugates, hence its centra-
lizer has finite index.]
.
.0
;:-
. 
.'
.,'
.J
.:..
.0
, ,
. ..:
;
"'
,
5. (a) Let E be a group which contains an ìnfinite cydic centrai subgroup of finite
index. Prove that E is isomorphic lo a semi-direct product F><J Z with F finite.
[Hinl: Using either exercìse 3 or exercise 4a, we can find a surjection E ---++ Z with
finite kernel. This surjection must spIit since 7L is free.]
.0
,.
.'.
(b) Deduce the following theorem of group theory: lf E is a torsion-free group
which has an infinite cyclic subgroup of finite index. then E is infinite cyclìc. [Hint:
Let A c E be an infinile cyc]ic norma.1 su bgroup or finite inde.x. Then E has a su b-
group E' ofindex 1 or 2 such Ihat A is centrai in E'. It follows from (a) that E'  Z.
so we ha \le an extension O -+ Z -+ E ----+ G...... I witb I G I < 2. If G acts non-triviaUy on
Z then H2(G. Z) = O by direct computation. But then the extension splils con-
tradictìng the assumption that E is torsion-free. Hence G acts tri vially and E  7L by
(a).]
"
l
:;
.'
!
.';1

+6. Lei E be a finitely generated torsion-free group which contains an abelian subgroup
of finite index. Prove tbat E can be embedded as a discrete, eo-compaet subgroup
of the group 1R":;..q Ol! of isometries of {R" for some n. [Hint: We have an extension
O - lLn -+ E -+ G - l with G finite. Now argue as in the hint to exercise 4a.]
Or
..
7. (Uni versai centra] extensions). In this exercise ali extensions will be a.ssumed tO
be centrdl and alI coefficient modules ror cohomology will be assumed to have
trivia[ G-action. We will assume further that G is perfecl, i.e., that G = [G, G],
or equivalentJy. that H lG = O. A]] (non-abelian) simple groups are perfect, for
ex ampIe.
-,
.,
. -
.

;
'0
(a) Show that there is a universal ooefficient isomorphism
<
,
..r
H2(G, A) :::::; Hom(H2G. A)
for any abelian group A.
(b) Deduce that there is a .. universal" cohomology class U E H2( G, H "2 G) with the
following property: For any abelìan group A and any v € H2(G. A). there is a unique
map f: H2 G - A such rhar v = H2(G.f)u. [Hint: Yoneda's lemma (exercise 3a
or *1.7) tells you how to describe the natural map Hom(H2G,-}  H2(G.-) of
(a) in terms of its effect on idu](] E Hom(H 2 G. H 2. G).]
.
.
O,
..,
"
,-
4 Application: p-groups wilh a. Cyc1ic Subgl'OUp or Index p
97
(c) In view of 3.12 and exercise 1 b, reinterpret (b) as saying that G adrnits a" universal
centrai extension ..
O-H2.G-+E--"'O'G-l,
characterized by the following propert y: Gi ven any abelian group A and any
centrai extension
o ..... A  E' -+ G - 1.
there is a unique map E -+- E' making
E
lG
E/
commute. [Hint: To prove uniqueness, note that two such ma.ps E:::t E' which
induce tOO same map H 2 G - A must differ by a homomorpbism E - A which
factors through G. But G is perfect, so there are no nontri\lial maps G -+ A_]
Remark. In practice ODe often uses (c) to compute H 2 G. Namely, one produces a
centrai extension of G witb the universal property described in (c), and it then
follows that the kernel of this extension is isomorphic to H 2 G. See Milnor [1971]
for an imeresting example of thìs (G = SLn(k» and further information about
unìversal centrai extensjons.
8. Let O -+ A  E -+ G ---io l be a centrai extension with G abelian. The commutalor
pairing associated to the extension is the map c: G x G --+ A defined by c(g. h) =
i - 1([9', ]) = i - I(ghg- 1 h-I). where 9 and li are lifts of {1 and h to E.
(a) Show that c i8 well-defined. l-bilinear and altemating. (The latter means that
c(g. g) = O; this implies, in view or bilinearity, that e(g, h) = -c(h. g).) Thus c can
be viewed as a map !\ 2G -+ A, where 1\ 2G, the second exterior power of G. is the
quotient or G (8) G by thc subgroup generated by the e1ements g @ 9.
(b) lfIis a factor set associated lO the extension, show that c(g, h) = l(g, h) - f(h. g).
(c) Let O: HZ( G. A) - Hom(A 2G. A) be the map which sends the c1ass or a cocyde f
to the alternating map f(g, h) - f(h. g). Deduce from (b) that there is a bijection
ker 8  8Dl>(G, A). where 4ijb(G. A) is the set of equivalence classes of abeJjan
extensions or G by A.
See exercise 5 of V.6 below for further results about O.
4 Application: p-groups with a Cyclic
Subgroup of Index p
As an illustration or the theory of group extensionst we will give in this
section a classification of p-groups which contain a cyclic subgroup of index
p, where p is a prime. (Such a subgroup is necessarily norma}; cf. Hall [1959]

 '
98
IV Low-Djmensional Cohomology and Group Extensions
*
..
g4.2.) We will then use this classification to prove a theorem or Burnside
(Theorem 4.3 below) which will be needed in VI.9.
We begin by Jisting some examples of p-groups with a cyclic subgroup of
index p:
(A) ZfI (q = 1f,11. > 1).
(B) Zq x lLp (q = rf, 11 > 1).
(C) 7Lq ><:I ?Lp (q = p", n  2), where the canonical generator of Z acts OD
7Lq as multiplication by 1 + pII - l. (This makes sense because (1 + ;""- 1:Y 
1 mod p" by the binomial theorem.)
If p = 2, there are three addi tional families:
(D) Dihedral 2-groups: Recall that for any integer m > 2 the dihedral
group D2rn is defined to be llfl Xl7L2, where the generator or 71.2 acts on lm
as multiplication by - 1. If m is a power of 2, D 2m is a 2-group. Note that D 4
is oftype (B) and D8 is oftype(C). For m > 4. however. D2m is not isomorphic
to any of the previous examples; this can be seen by computing abelianiza-
tions.
(E) Generalized quaternion 2-groups: Let Di be the quaternion algebra
IR ffi lRi ffi IRj Ef> k. For any integer m > 2, the generalized quaternion
group Q4m is defined to be the subgroup of tbe multiplicative group u-D*
generated by x = e'd/m and y = j. F or examplet Q 8 is simply the usuai
quatemion group {+ I, +i + j, +kj. Note that x is of order 2m and that
we have the relations y2 = XM and yxy - l = X - I. It follows that the cyclic
subgroup C generated by x is normal and of index 2 and hence that Q41f1
has order 4m. If m is a power of 2, Q4m is a 2-group. For future reference we
record some properties of Q4m: (a) In tbe extension O -+ C -+ Q41n -+ Z2 -+ Ot
the generator of 'll acts as -I on C. (b) Every element of Q4R1 - C is of
order 4. (c) Qa has exactly three cyclic subgroups of index 2; Q4R1 for m > 2
has a unique cyclic subgroup of index 2. [This follows from (b).] (d) Q4m
has a unique element of order 2. (e) The extension O -+ C ---+ Q4m -+ 7L2 -+ O
does not spiit.
(F) 7Lq><1 Z2 (q = 21\ n > 3) where the generator or Zz acts on Zq as
multiplication by -1 + 2"-1.
We now prove that this list is complete:
,
.r
'.'
",
...
. -
;.;
:(
"
1
,
"
,
..
, ..
"'.
,.
'.
,
"
..'
,-
". "..
(4.1) Tbeorem. lf G i5 a p-group with a cyclic subgroup DJ index p, then G is
isomorphic to one ofthe groups listed in (A)-(F) above.
The proof will use the following lemma from elementary number theory:
'.
(4.2) Lemma. Let a be an integer such thar a" == l mod p" for SOmE n > 2.
l! p is odd then a ::: I mod pll- I. Il p = 2 the a == + 1 mod 2"- 1.
PROOF. Assuming. as we may that a -::F t let d(a) be the largest integer d
such that a = 1 mod pd. Note that d(a)  1 since a = a" = l mod p, where
the first congruence is by Fermat's little theorem. If p is odd or dea)  2,
,
4 Application: p-groups with 8 Cyclic Subgroup of Index p
99
then exercise 3a or II.4 js applicable and gives d{a") = d(a) + 1. Thus
dea)  11. - 1 in this case as required. If p = 2 and dea) = l, then a =
-1 mod 4. We can therefore apply the previous argument to -a to conclude
that -a = t mod 2n-1. D
PROOF OF THEOREM 4.1. By hypothesis we bave an extension O -+ ZfI -f' G -+
H -+ l, where q = p" for some 11. > O and I H I = p. If H acts trivially OD 7LiI
then G is abelian, being generated by two commuting elements. It then
follows from the theory or finite abelian groups that G is of type (A) or (B).
Assume now that H acts non-trivially OD lL'l' We wìll use example 2 of
III.I to compute H2(H. 7L'l}' The action of H on 7Lq is given by an embedding
H c. Z*, where Z: is the group of units of the ring ZII = Z/p"7Lt and cIear1y
we mut bave n > 2. Suppose first that p is odd. Then Lemma 4.2 implies
that tbe image of this embedding is {I + b: b E p" - I lLJpn7L}, w hich is a group
of order p. generated by I + p"-I. In particular, H must have a generator
which acts as 1 + p"-I, as in (C). Under this action or H on Zqt we have
Z: = {x E Zq: pll- l X = O} = pZj ptlZ. On the other handt the nOfm operator
LI! e H h on 7Lq is m ultiplication by L (I  b) (b E pll- l 7Lf p"7L), whicb eqals p.
[:L b = O because p is odd.] Thus the lmage of the norm operator IS also
equal to p7L/p'?L. We conclude that H2(H, lq) = O, so tbe extension splits
and G is of type (C). Suppose now that p = 2. By Lemma 4.2 again, tbe
image of H  Z: is a subgroup of {+ l + b: b E 2n- l Z/2nz}. There are
therefore three possibilities for the image a E Z: of the generator or H:
Case 1: a = -1. In this case Z: = 2n-llLj2"Z and the norro operato! is
zero. so H1(H, 7Ltl)  71..2, Thus there are precisely two inequivalent extensions
of H by Z corresponding to tbis action of H on 7Lq. Since we bave already
produced two sucb extensions in (D) and (E)t it rollows that G is either of
type (D) or type (E).
Case 2: a = l + 2"-1. Then Z: = 27L/2117L. and the norro operator is
multiplication by 2 + 2n-1 = 2(1 + 2"-2). We may assume n > 3, since
otherwise we are in case l. Thus 1 + 211- 2 E Z: and the image or the norm
map is 27L./2r17L. We tberefore bave H2(Ht lLq) = O, so the extension splits
and G is or type (C).
Case 3: a = -1 + 2"- l  n > 3. Then Z: = 2n- 17L/2"1.. and the norm
operator is multiplication by 21'1-1. Thus H2(H,lLll) = O. the extension splits,
and G is of type (F). D
Using tbis theorem, we can prove thc following result (cf. Burnside
[1911], 104 and 105):
(4.3) 11aeorem. 1f G is a p-group which has a unique subgroup oJ order p, then
G is either a cyclic group or a gel1eralized quaternion group.
PROOF. Arguing by induction OD IGI we may assume that every pro per
subgroup or G is cyclic or generalized quaternion. Choose H <J G of index p.

100
IV Low-Dimensional Cohomology and Group Exteosions
I
(:...
[Such an H always exists cf. Hall [1959], .2.] If H is cyclic, then we are
done by Theorem 4.1; for the onIy groups in the list (A)-(F) which have a
unique subgroup or order pare the cydic groups and tbe generalized quater-
Dion groups. Suppose then, that H is generalized quaternion (and hence
that p = 2). I claim that H has a cyclic subgroup N of index 2 which is normal
in G. Indeed, if!/ is the set oi cyclic subgroups of H of index 27 then we know
that card(..9') is odd (cf. statement (c) in the discussion of type (E) above);
the conjugation action of the 2-group G on Y must therefore fix some
N E!/, as claimed. Consider now the action of GIN on N. It is given by a
homomorphism G/ N --.. Z:, where q = I N t >4. Composing with the
canonical projection Z: - Z: = { + l} 7 we obtain a surjection G/ N -+ { + l},
whose kernel is a group KjN of order 2. Since the generator of KjN does
not act as -1 OD N, K cannot be generalized quatemion. Thus K is a cyc1ic
subgroup or G of index 2 and we afe done as before. D
..
.:
'.
,
.'
'.
,
<
,
,;
.'
.
[
14: '.
,.
:;
,
.'
Remark.1t isinstructive to look at Burnsides proofofthis theorem. Although
neither cohomology theory nor the classification of group extensions had
yet been developed7 Bumside's proof contains, essential1y, a direct proof
that ,f(G A)  AGINA far G finite cyclic, where N is the norm operator.
[Explicitly, given aD extension O  A -+ E -+ G ...... 1, where G is cyclic of
order n with generator t, Burnside lifts t to t E E and considers a = ?' E A.
Then a is in A G and a mod N A is the element or AGI N A which classifies
tbe extension.]
,.
'-
,.
'.'
.!
Final1y, as one further illustratioo of tbe use or cohomology in group
theory, we wilI determine alI subgroups or a non-dihedral group of type
(C):
(4.4) Proposition. Let G = 7Lq><J Zp (q = p", n > 2) as in (C).lfp = 2, assume
n > 3. Let A c ?L'l be the subgroup pZlpftZ of index p.
(a) A:><J 7L.p = A x 7Lp is a non-cyclic abelian subgroup ofG ofindex p.
(b) Everyelement ofG which is no! in A x ?Lp generates a cyclic subgroup
DJ index p. T here are precisely p distinct such cyc lic subgroups 01 i1'ldex p.
(c) Every proper subgroup H ofG is either a subgroup ofAx Zp or a cyclic
subgroup ofindex p as in (b).ln particular, H is abelian.
PROOF. (a) is obvious7 since lLp acts trivially OD A. Il is possible to prove (b)
by a direct (and tedious) computation, but it is easier to proceed as follows.
First. note that any homomorphism G  7L p must factor through the
quotient ?Lp x ?Lp of G. It follows that the subgroups of G of index p cor-
respond to the subgroups of 7l..p x lLp of index p, and there are precisely
p + 1 or these. [They are the points of the projecti ve line over the field with p
elements.] or these p + l subgroups or G of index p, one is A x Zp and I
claim that the other p are ali cydic. Accepting this for the moment, we can
easily prove (b) by a counting argument. For tbe p cyclic subgroups of
,.
,
"
"
,
4 Application: p-groups Wtth a Cyclic Subgroup of Iodex p
101
index p contain a totai or P(p" - p"-1) generators and none of these gen-
erators can be in A x lLp. But card(G - (A x Zp» = p"+l - Jf' = P(p" -
p"-1)7 so Cb) follows al once. Since (c) is an immediate consequence or (b),
il remains onIy to prove the claim.
In the proof of 4.1 we calculated the norm operator for the actÌon of?Lp
on li.. and it follows from this computation that Hl(Zp, Zq) = O. In view of
4' .
Proposition 2.3, this means that the extenslOn
O -+ lq -+ G - Zl'  O
has a unique splitting (up to conjugacy) and hence thai G contains only two
conjugacy classes or subgroups of order p. Suppose now that H is a non-
. cycIic subgroup of G of index p. Then we ha ve an extension
O -+ A -+ H -+ ?Lp - O
with Zp acting trivially on A (because l + p"- l _ 1 mod I A I). Thus H is
abelian and non-cyc1ic so H contains at Jeast two subgroups of order p.
Since H is normal in G, it follows that H contains both conjugacy classes of
su bgroups of G of order p. In particuIar, O x li..p c H and hence H = A x 7L p'
as claimed. D
EXERCISES
l. Ir G is a genera1ized quaternion group, show that every subgroup of G is either
cyclic or a generalized quatemion group. [This foIlows from Theorem 4.3, of coucse,
but you can also check it directJy.]
2. If G is a dihroral group, show that every subgroup of G is either cyclic or dihedral
[Hint: Every element or G = lq><l 7L2 which is not in 7Lq is or order 2.] If I G I > 8.
show further that the non-cyclic abelian subgroups of G (i.e., the subgroups iso-
morphic to 7L2 x 1'.2 = D4) are nOI normal. Ds, on the other hand, contains two
non-cyc1ic abetian normal subgroups.
3. Let G = 7Lq:><l1l2 (q = 2". n > 3) as in (F). Let A c: Zq be the subgrop of rder 2.
Show that the only nonyclic abetian subgroups of G are A x 1'..1 and lts conJugates.
Show further that A x Zz is not norma] in G. [Hint: If H is a non-cydic proper
subgroup of G. then we have an extension O -t H rì Zq - H - 7..2 - O with the
generator of 1:.1 acting as -1 on H n lq. This can oDly be abelian if H n Zq = A.
Now calcuiate H l (l: 1- . l",) aJ}d use this information as in the proof of 4.4.]
4. If G i a p-group such that every abelian norma] subgroup is cyclic. show that G is of
type (A). (D), (E), or (F). wìth I G I > 16 if G is of type (D). (Hint: Choose a maximal
abeJian normal subgroup or G. and consider the corresponding extension o - Zq -
G _ H - I, q = p". If IHI < p then we are done by Theorem 4.1. sìnce groups of
type (B) and (C) have non-cyclic abelian norma] subgroups. Suppose. then. that
I H I > p2. and consider the norma) subgroups H' c: H of order p. Such an H' cannot
ad tcivially on Zq. since the inverse image G' or H' in G would be an abelian normal

102
IV Low-Dimensionw Cohomology and Group E]!;tensions
subgroup biggec than lLq. And H' cannm act as in CC) unJess p = 2 and n = 2, since
tbe non-cyc1jç subgroup of G' or index p (cf. 4.4) would be norma l ìn G. So the on]y
possibility is thal p = 2 and (hat the non-tri ...iaI element or HI ac(s a:s - 1 or - 1 +
2"-1 (with 11 > 3 in the latter case). But this is absurd. for the comnnsjte H _ 7L*--
. t' Il
{+ l} has a non-tnvla] kernel and we cou]d take H' to be contained in this kerneJ.]
5 Crossed Modules and H3 (Sketch)
We have seen in 2 and 3 that H1 and H2 have concrete group-theoretic
interpretations. I t turns out that there are also group-theoretic interpreta-
lions of the functors H" Cor n > 3, discovered independen t1y by several
people. (Many of the re)evant references can be found in MacLane [1979].)
We will confine ourselves here to a sketch or the theory in the case or H3.
The essential ideas in this case go back to MacLane [1949], although he
did not give the precise c1assification theorem 5.4 below.
Let E and N be groups. Suppose we are given an action or E on N, denoted
(e, n) H en, as well as a homomorphism IX = N ...., E satisfying
(5.1 )
and
O:(JI)n' = nnJn -I (n, nJ E N)
(5.2)
lXen) = eet(n)e-1 (e E E, n E N)
We tben say that N (together with a and the action) is a crossed module
over E.
The canonical example of a crossed mod u)e is thar where E is the fuU
automorphism group Aut N and a(n) is the inner automorphism associated
to n. Thus 5.1 is true by definition, and 5.2 is easi)y verified. Crossed modules
also arise naturaIly in topology. and it is in this context that they were
first introduced (Whitehead [1949J). Namely, if X is a topoJogical space and
Y is a subspace, then tbe relative homotopy group 1t2(X, Y) is a crossed
module over 1tl  with IX being the boundary map ò: 1l"2(X, Y) --+ 7tl ¥.
For another example. suppose E is the total space of a fibration with fiber F 
then one can make 1r.1F a crossed module over 7r}E.
Note that any ordinary (abelian) E-module can be viewed as a crossed
module, with iL being the trivial map. At the other extreme, any norma)
subgroup or E is a crossed module with E acting by conjugation and Cl being
the inclusion.
Let N be a crossed module aver E. We set A = ker a. and G = coker a;
the latter makes sense because ìm a. is normal in E by 5.2. Nate that A is
centra) in N (by 5.1) and that tbe actìon .or E on N induces an action or G
OD A. Thus we have a 4-term exact sequence
: -."
,'..3:
..
(5.3)
,
; o: 'I[
O - A -l' N -. E --io G -.. 19
..
"
..
-,
"
.f,
\.
'L
'":.
-
I
..
"
.
.,
"
.
.'
..
, -
, ,
,
.
,
. '.
,
...,
!

>
,
103
5 Crossed Modules and H3 (Sket.ch)
where A is a G-module. lt turns out that exact sequences of this form are
c1assified (up to a suitable equivalence relation) by H3(G. A).
More precisely, suppose we start wit an arbitrary group G and an
arbitrary G-module A. Consìder ali posslble exact sequencs of the form
5.39 where N is a crossed module over. E such that the actlon of E on N
induces the given action or G on A. We Impose on these exact sequences tbe
smallest equivalence relation such that
O....... A -.. N -+ E ---J G -+ l
is equivalent to
o -. A -+ N' -+ EJ -+ G -+ l
whenever there is a commutati ve diagram
N
E
.
o
. 1
in which the vertical arrows are compatible with the acti0,ns of E nd E'
00 N and N'. [Warning: The vertical arrows need not be lsomorphlsrns.]
Wethen bave:
(5.4) Theorem. There is a l-l correspondence between equivalence c/asses as
above a,ul elemenls DJ H3(G, A).
We ornit the proof, w hich can be found in the references cite in acLane
[1979]. We will, however. explain how an exact sequence.5.3 gJves nse to an
element of H3(G, A). Choose a set-theoretic cross-s:ct1on s: G -+ E of 7t.
Its failure to be multiplicative is measured by a functton f : G x G -+ ker ]t
such that
(5.5) s(g)s(h) = f(g, h)s(gh).
As in 3, the associativity of the product in E forces a "cocycle condition"
OD f. which takes the form
(5.6) J(g, h)f(gh. k) = bl)f(h, k)f(g, hk),
where sfJl)f(h k) = s(g)f(h,k)s(g)-l. [5.6 is a non-abelian nalogue .of 3.9;
it is proved by computing the triple product s(g)s(h)s(k) In two dlfferent
ways.]
Since ker 1r = im IX we can lirt f to a function F: G x G -+ N. and we
can ask whether F stisfies tbe analogue of 5.6 (which now involves the
crossed-module action of E on N). The failure of F to do so is measured by a
function c: G x G x G -. A such that
(5.7) 8(g)F(h k)F(g, hk) = ì(c{g, h k»F(g. h)F(gh, k).

104
IV Low.Dimensional Cohomo]ogy and Gr()up Extensions
One then hows that c is a 3-cocyc1e w hose cohomoJogy class is independent
of3the cholces of s and F. and this cohomology class is the desired element of
H (G. A).
6 Extensions with Non-Abelian Kernel (Sketch)
The main references ror this section are MacLane [1963] and Eilenber _
MacLane [1947]. g
. Let N be a oup and recall from 5 that N is an Aut(N)-crossed moduIe
VIa the canomcal map a: N -+ Aut(N) and the canonicaJ action of Aut(N)
on N. The kernel of o: is the center C of N, and the cokemel of 11 is, by definition,
the group Out(N) of outer automorphisms of N. The resulting exact sequence
(6.1) O -+ C -+ N  Aut(N) -+ Out(N) -+ 1
plays a fundamentaJ role in the study of extensions
(6.2)
1-+NE!!.G-+l
with kernel N.
Th first observation to make is that such an extension gi ves rise not to
an actlon of G OD N, but onJy to an "outer action,' i.e., a homomorphism
1/1: G -+ Out(N). This is induced by the conjugation action of E OD N:
N
 E
1

G
l
N .. Aut(N)  Dllt (N).
We therefore fix a homomorphism t/1: G -+ Ou t( N) and try to understand
the t (G. N, Vi) of equivalence c1asses of extensions giving rise to rjJ. Note
that lt 18 not even obvious for a given tjJ whether or not G(G, N,W) is non-
empty. In particuJar, we do not have a semjdirect product N  G in
8(G N, 1/1) unless VI lifts to a homomorphism G -+ Aut(N).
Suppose now that an extension 6.2 18 given. As in 3 choose a set-theoretic
cross-section s: G -+ E of 1t. This determines as usual. a function f; G x G
-+ N measuring the failure of s to be a homomorphism. In add ition. it
detnnin.es a set-theoretic lifting tp: G -+ Aut(N) or tJt; namely, lf1(g) is
conJugatlon by s(g). Tbe functions f and tp are related by
(6.3)
cp(g)cp(h) = a.(f(g:. h))f{J(gh)
where a. is as in 6.1. Mo reo ver, j satisfies a 'cocycle condition" (cf. 5.6)
(6.4)
j(g. h)f(gh k) = fiI(fI}f(h. k)f(g. hk).
,
'.
.<
i
'o
,
,
"
 .'.
.;
00
;

<
i
.
.,
,
,
,
.p
'.0
:1'
,-
.
'..
;:
'o
.
I
",I.
,
;
...;.
.;

.,
.
.
-
. .
).
.
; ,
6 Extcnsions with Non-Abelian Kernel (Skek:h)
105
Conversely. given f and qJ satisfying 6.3 and 6.4, one can construct an
extension
(6.5)
l --+ N - E/." -+ G -+ 1,
where Ef is N x G with a product that can be written down explicitly in
,9'
tenns of f and cp. One sees in this way that extensions are c1assified by pairs
(f, CI') as above, subject to some equivalence relation.
This description or 4(C, N, .p) can be drastically simplified once we
o bserve that tbc lifting qJ of 1/1 can be taken to be tbe same in ali extensions
under consideration; for by cbanging the choice or s, we can change Qur
lifting rp to any other lifting. Thus we can fu cp and just consider the extensions
Ef,qn where f ranges over ali "'cocycles" relative to f/J. It follows from 6.3
that any two such !'s will differ by a function G x G -+ C, and the latter is
an honest 2-cocycle of G with coefficients in C, where C is a G-module
via JjJ. One shows in this way that any two elements of 4(G, N rjI) bave a
well-defined "difference" in H2(G, C). This leads to:
(6Ai) Theorem. T he set 8( G, N, 1jJ) admits a free, transitive action b y the
abelian group H2(G, C). Hence either 8{G, N, Vi) = 0 or else tlrere is a
bijection tf(G, N, 1jJ)  H2(G C). This bijection depends on the choice DJ a
particular element oJ 8(G, N, tJI).
To complete tbe classi ficat ion, we must say when 8(G, N, Jfi) '# 0.
Recall from 5 that tbe sequence 6.1 yields an element U E H3(Out(N), C).
Applying the cohomology map 1/1* = H3(t/J, C), we obtain an element
t/1*u E H3( G, C). I claim that "'.u is the . obstruction " to the existence of an
element of 4(G, N, IjJ). To see this we need the following explicit description
of ljJ*u, which is easi1y deduced from 5: Choose a set-theoretic lifting
fj): G - Aut(N) or t/1, and choose a function f: G x G --+ N such that 6.3
holds. Then the failure of f to satisfy 6.4 is measured by a function G x G x G
-+ C which is a 3-cocycle representing tjt*u..
In particular, if 8(G. N, rjJ) .p 0, then we have already seen that q> and f
can be chosen to satisfy 6.4, so tbe 3-cocyde is zero. Conversely, if "'.u = O
then we can change the choice or f so that 6.4 holds and we can then construct
an extension 6.5. This pro ves :
(6.7) Tbeorem. A homomorphism : G -+ Out(N) gives rise lO an ,. obstruc-
tion" in H3(G, C) which vanishes if and only if tB'(G, N,) #- 0.
This obstruction, incidentally, does not aIways vanish. Indeed, ror any
group G, any G-module C, and any element v E H3( G, C), one can construct
a group N with center C and a map : G -+ Out(N) whose obstruction is
equal to v.

106
IV Low-DimensionaI Cohomology and Groul' Extensions
Final1y, we mention one special case where tbc theory or extensions is
particu1arIy simple :
(6.8) Corollary. If N has a lrivial center then there is exactly ol1e extension
ofG hy N (up lo equivaience) correspoflding lo any homomorphism G ---f> Out(N).
This, of course, is an immediate consequence or 6.6 and 6.7, but it also
admits an easy direct proof whiçh uses no cohomology theory (cf. exercise 1
belo w).
EXERCISES
I. Give a direct proof of 6.8. [Hint: If N has trivial center then any extension of G by N
fits into a diagram
]
)N
, E
l
). G
l
.. l
1
... N
) Aut (N)
.. Out(N)
) 1
with exact rows. Such a diagram Is necessarily a pull-back diagram (cf. 3. exercise la).]
2_ Let 1 - N - E -+ G - 1 be an extension or finite groups such that I N I 8nd I Giare
relatively prime. We proved in 3.13 that such an extension must spHt if N if abeJian.
and we remarked that this result could ])e generalized to the non-abeJian case. One
might hope 10 deduce this generalization directly [rom 6.6 in view or the vanishing .
of H2( G. C). Why doesn't thjs work?
..
;.
I
'.
"
A
CHAPTER V
Products
1 The Tensor Product of Resolutions
If G and G' afe groups and M (resp. M') is a G-modllle (resp. G'-module)
then M @ M' is a G x G'-module in an obvious way: g')' m  m') =
gm C8> g'm'. Note that if M is projective over 'l.G and M. lS proJectlve oer
lLG' then M @ M' is projective over Z[ G x G']. In fact, It suffices to velfy
this in the case where M = ZG and M' = ZG', in which case the assertIon
follows from tbe obvious isomorphism 7L.G 0 lLG'  Z[G x G'].
Now let G: F -+ li.. (resp. €': F' - lL) be a projective resolution of 7L over
?LG (resp_ 7LG'), and consider the complex F @ F'. This is a complex f
projective Z[ G x G']-modules, and it is augmented over 7L y 1: @ e :
F @ F' -+ ?L @ 7L = 7i. Moreover, 8 @ f.' is a weak equivalence. Thls follows,
for instancet from I. 7.6; for the latter shows that f: F - Z and s': F' -- Z
afe homotop)i equivalences if we ignore the G-action, so the same is true of
£ @ t' (cf. exercìse 7ç of I.O). [Alternatively use the Ktinneth Formula.]
We have therefore established:
(1I) Proposition. lf e: F --+ 7L and 6': F' -+ 7L are proeclive .resolutins .0/
7L. ot'er ZG and ?LG', respectively then E; @ r:: F C8> F - 7L lS a proJectwe
resolution oJ ?L over Z[ G x G'].
This result sbould be thought of as the algebraic analogue or the obvious
fact that the cartesian product of a K(G. 1) and a K(G', 1) is a K(G x G',1).
(1.2) Corollary. 1/ B: F ---f> 7L and e': F' - 7L are projecive resolutions oJ .1'-
aver lLG. then so is E: @ B': F @ F' --io Z, where G acts dUlgonally on F @ F .
101

108
V Products
This follows from t.1 by restriction of scalars with respect to the . diagonal'
ernbedding G --+ G x G given by 9 t-+ (g g).
Note that if F = F' in 1.2 then we bave two obvious maps F @ F -+ F
or resolutions, namely, F @ e: F @ F -+ F @ Z = F and E (8) F:F @ F--+
7L (8) F = F. There is no obvious map in the other difectioo, but wc know
from I. 7.5 that there exist augmentation-preserving G-chain maps :
F -+ F (8) F, and that any two afe homotopic. Any such map A wiU be called
a diagonal approximation. In case F is the standard resolution, for example,
there is a well-known diagonaJ appro xi matio n, called the Alexander-
Whitney map. In terrns of the homogeneous description of F, this map is
given hy
"
(1.3) tJ.(go,.. ., glI) = L (go, . ..,01') (8) (gp,..., gn)'
p=o
Translating this into the bar notation, we find
JJ
(1.4) ,6,[g11.. .jgJ = L [gli. - .19p] @ 9. ... gp[gp+ 11... JgJ.
p=O
EXERCISE
Le. G be a finite cyclic group of order m with generator l and let F be the resoJution
16.3. In particular. Fn = lO for alI n  o. Let: F -+ F 0 F be the map whose (P. q)_
component Ap.r: F"+Q -+ F" @ Fq is given by
l @ l
Apq(l) = l @ t
I ti (8) ti
OI<j:S",,-1
p even
podd. q even
p odd. q odd.
Verify that  is a diagonaJ approximation.
2 Cross-products
Let G, G', F, and F'. be as in the previous sectioD. For any G-module M
and G'-module M' there is an obvious map
(2.1) (F @6 M) @ (F' @G' M') .... (F @ F') @G )(G' (M @ M'),
given by (x (8) m) (8) (x' (8) m')  (x @ x') (8) (m (8) m'). [One justifies tbis
rigorously by noting that
(F (8)G M) (8) (F' @G' M') = (F  M)G @ (F' @ M')6' .
which is a ccrtain quotient or F @ M C?9 F' (8) M'; making this quotient
explicit. one can even see that the map 2.1 is an isomorphism.] If Z E F @G M
..
,
i
, .
r
1-
,.
.,
.,
3 Cup and Cap Proouc(s
109
I.:
and z' E F' (8)6' M', then we will denote by z x z' the image of z @ z' under
this map. Clearly we have d(Z x z') = òz x z' + (-l)Pz x òz' if deg z = p;
hence the product of two cycles is a cycle, whose homology c1ass depends
only on the classes ofthe given cycJes. We therefore obtain, in view of 1.1, an
induced product
Hp(G, M) @ H¥(G' M') .... Hp+q(G x G', M (8) M').
called the homology cross-product and still denoted z @ z' H Z X z'. This
product is welI-defined, independent of the choice of thc resolutions F and
F'.
Similarly given cochains U E.J't'o-mG(F, M) and u' eJlfo-mQ.(F', M').. we
'define Il x u' eJItPomG)< G.(F @ F', M  M') to be the tensor product of the
maps u and u' as defined in exercise 7 of I.O; thus (u x u', x (8) x') =
(_l)degu' 'degx(u. x) (8) (u', x'). It is easy to verify that o(u x u') =
(m x u' + (-I)Pu x 8u' if deg u = p. [This i8 a speciaJ case of exercise 7a
of 1.0.] There is therefore an induced cohomology cross-product
HP(G M) (8) Hq(G', M') --+ Hp+q(G x G', M (8) M').
'.
EXERCISES
1. Show that the map 2.1 is an isomorphism. Deduce, under suitable hypotheses.
that there is a Ktinneth formula [or computing H*(G x G', M @ M') in terrns of
H .( G, 1\1) and H.( C', M').
*2. Give hypotheses under which the cochain cross-product
G(F, M) @.wbm(;.(F'. !vI') -+ KO>!IfG)( G.(F @ F'. M @ M')
is an isomQrphism, and deduce a cohomoJogy KUnneth formuJa.
3 Cup and Cap Products
The previous section dealt with . external H products, in voI ving three grou ps
G, G', and G x C'. In this section we wilJ discuss internai products invo1ving
the homology and/or cohomology of a single group G.
Given U E HP(G, M) and v E Hq(G, N). we define the cup producr ofu and v
(denoted u U v or uv) to be the element d*(u x v) E Hp+q(G, M (8) N),
where d: G .... G x G is the diagonal map. Here M @ N has the diagonal
G-action (as il must il dltl: H*(G x G, M @ N)H*(G. M @ N) is to make
sense).
Checkìng the definit,ions, we see that the cup product is induced by the
following cochain cup product: Let F and F' be projective resolutions of 7L
over lLG, and recall (1.2) that F  F' with diagonaJ G-action is also a pro-
jective resolution or Z over 7L.G. Given u e.Yt'omG(F, M) and v E.J'f'omG(F', N)

110
V ProduclS
define u u v to be u x v as defined in 2, bui now regarded as an element of
.JfIbneG(F @ F', M @ N). Alternatively, if we prefer to use a single resolution
F, then we choose a diagonal approximation d: F  F (8) F (cf. I) and
seI u u v = (u x v) o 6 E ..1l'G(F. M (8) N) [or li E £AnG(F, M) and v E
Jft'O»lG(F, 1\1). Far example, if F is the bar resolution and 11 is the Alexander-
Whitney map (1.4), then the product of U E C"(G, M) and v E Co!l(G N) is the
element u u r E Cp+ q( G, A-I @ N) given by
(u u V)(gh ..., 9p+q) = ( -l)pqu(gl' . . ., gp) @ 91 . . . 9pV(YP+I' ...,91'+.).
We now list some fonnal properties of the cup product.
'.
I.
(3.1) Dimension O: The cup product HO(G. M) (8) HO(G, N) --... HO(G, M @ N)
is the map MG @ NG --+ (M (8:1 N)G induced by the inclusions MG 4 M
and NG 4 N.
(3.2) Naturality with respect to coefficient homomorphisms: Given G-module
mapsf: M -+- M' and g: N -io N'and elements U E H*(G, M) and v e H*(G, N),
we have
(f @ g).(u u v) = f.u v g.v
in H*( G, M' @ NJ), where f.. = H*( G, f), etc.
3.1 and 3.2 are immediate from the definitions.
(3.3) Compatibility with {): Let O -+- ME -+ M -+- M" -+ O be a short exact
sequence of G-modules and let N be a G-module such that the sequence
O -io M' @ N  M  N ....,. M't  N -io O is exact. (For example, this holds
for any N if O - M' -io M --fo MII --+ O is split exact as a sequence of Z-mod-
ules.) Then we haveb(u uv) = tJu u vfor any U E HP(G, M") and v E HII(G, N).
In other words, the square
HP(G, M") lJ ). HJI+ I(G, M')
" ,
-u'l l-u,
HP+'l(G, M" @ N) " . HP+Il+ 1(G, M' @ N).
. '.
commutes.
PRODF. Consider the commutati ve diagram
I
,
I
O ) C*(G. M') · C*(G,M) ). C*(G. M") .0
1 1 1 ,
..c.".
..
..
D · C*(G, M' @ N) I C"(G M @ N) ). C*(G, M" @ N)  O,
..
.,
,..
..

 ".
,
3 Cup and Cap Products
111
.<
.".
where the vertical arrows are given by cup product on the right with a
fixed cocycle in cq(G N) representing v. These vertical maps commute with
the coboundary operators lJ in C*(G,-); far the formula b(a u b) =
ba u b + ( _l)dCiDa v cb reduces to (a u b) = lJa u b if b is a cocycle.
The result now follows from the naturality of connecting homomorphisrns
with respect to maps of short exact sequences (1.0.4). D
<
,
Similarly, one proves:
(3.3') If O -io N' -io N --... N" -+- O is a short exact sequence such that O-io
M (8) N' --... M @ N -+- M @ N" ....,. O is exact then (j(u uv) = (-l)pu v <5v
in HP + q+ 1(0, M @ N') for any U E HP(G, M), v E Hq(G, N").
3.3 and 3.3' alIow one to use dimension-shifting arguments in the study
of cup products. For we have seen (cf. exercise 3 of IlI.7) that we can use a
l-split injection M -io Ai for the dimension-shifting argument.
(3.4) Exislence of identity: The element l E BO(G, Z) = Z satisfies l u lo! =
U = U U ] for ali U E H.( G, M), where we make the obvious identifications
7L (8) M = M = M @ 7l. of coefficient modules.
This folIows from the definitions together with the following two observa-
tions: (a) l E HO(G, Z) is represented by the augmentation map é, regarded as
a O-cocycle in Jfb.mG(F, Z); and (b) the rnaps F @ G and l; (8) Fare maps or
resolutions F (8:1 F - F (cf.  1) and hence induce the 'identity map" on
cohomology. Alternatively, use tbe Alexander-Whitney formula.
(3.5) Associativity: Given Ui e H*(G, Mi) (i = 1, 2, 3), we have (U1u2)u3 =
U1(U2U3) in H*{G, M l (8:1 M2  M]).
Indeed, associativity holds on the cochain leve! as an identity in
JtbmG(F (8:1 F @ F, MI @ M 2 (8:1 M 3)' Alternatively, use the Alexander-
Whitney formula.
(3.6) Commutativity: For any U E H"{G, M), v E HII(G, N), we have uv =
(-l)"t.(vu), where t: N (8) M -+ M @ N is tbe canonical isomorphism.
PROOF. Let t: F (8) F....,. F @ F be the chain automorphism such that
'[(x @ y) = (_1)deKx.dtK'y (8) X, cf. exercise 5 of lO. We have a commutative
diagram
:Kb.mc;(F, M) (8) JfOmG(F, N) v ..JYG(F @ F. M  N)
1 l'" ,)
<HnG(F, N) @.JIFG(F, M) u ).J'fl(;(F @ F, N @ M)
where the vertica1 arrow 00 the left is given by u @ V 1--+ (_I)dc8u'deRtlv @ u.
[When you check the commutativity of trus square, the signs may seem to
.
.1
,

112
V Products
come out wrong al first; but dont forget that <ux) = O unless deg u =
deg x.] Since -r is an augmentation-preserving chain map it induces the
identity in cohomology. Hence the vertical arrow on the right induces t. in
cohomology, and this yields 3.6. D
It follows from 3.4-3.6 that H*(G l) is an anti-commutative graded
ring and that H*(G, M) is a graded modu]e over this ring. More generally.
if k is an arbitrary commutative ring (with trivial G-action for simplicity)
then we have a product
H*(G. k) @ H*(G k)  H*(G, k @ k) -+ H*(G, k)
which makes H*(C, k) a graded anti-commutative k-algebra. (The un-
labelled arrow above is induced by the multiplication map k @ k -+ k.)
Similarly, H*(G, M) is an H*(G k)-module if M is a kG-module.
(3.7) Naturality with respecc to group homonwrphisms: Given a: H -+ G
we have cx*(u u v) = a*u u a*v for any U E H*( G M) V E H*( G, N).
This is immediate from the definitions.
As a special case of 3.7,0:*: H*(G, k) -+ H*(H k) is a ring homomorphism.
(3.8) TrGllsfer formula; Suppose H c G is a subgroup of finite index_ For
any U E H*( G, M) and ve H*(H, N), we have
cor*(res(u) u v) = u u corZ v.
This says, in particular, that the transfer map H*(H, k) --+ H*(G, k) is a
homomorphism of H*(G k}modules, where H*(H, k) is regarded as an
H*(G, k)-module via the restriction homomorphism H*(G, k) -+ H*(H, k).
PROOF OF 3.8. Let F be a projective resolution or l over lLG; we wiU prove
the stated formula on the cochain leve!. Given U E #1t(F, M)G and
v E .7&hZ(F, N)H, we have in J't'om(F 0 F, M @ N)G:
co..g(res(u) u v) = I g. (u @ v)
(lE GIH
= I u0gv
gGIH
[because u is G-invariant]
.
= u@ I gv
(/EGIH
= U U corg.v_
D
There is a second internaI product, called the cap producr, which is useful
in the study or duality (cf. VI_7 and VIII. 10). If F is a projective resolution
of 7L. over 7L.G, then there is a chain map
y:.JtMnG(F, M) @ «F @ F) @oN)-+ F @G(M @ N)
.
,
"
.,
.
,
ì
3 Cup and Cap Products
113
r
.
given by u @ (x @ y @ n) 1--+ ( _1)de8I1ode8xx @ u(y) @ n. [Tbe reader can
check directly that 'V is weJl-defined and is a chain map. Alternatively, we
can appeal to tbe exercises or I.O According to exercise 7b of the latter
we have a chain map .Jfb.mG(F M) -+ (F 0 N) @G F, (F @ N) @o M)
given by U I----'P idF I8iN @ u. This corresponds by exercise 6b or I.O to a chain
map
.
-.
.,
.'
..7&mo(F M) @ «F @ N)  F) ..... (F 0 N) @Q M,
which is precisely y, modulo the identifications (F @ N) @G F =
(F @ F) @G N and (F @ N) @G M = F @G (M @ N). These identifica-
tions can be derived by manipulating triple tensor products as in the proof
of 111.2.2.]
For u E.1f'omG(F, M)" = .J'fomG(F M)_p and Z E (F @ F)q @G N y(u @ z)
is an element of Fq_" @G (M @ N) denoted U fì Z and called tbe cap product
of u and z. The same notation and terminology are used Cor the induced
product
-,
H"(G. M) 0 HIl(G, N) -+ Hq_p(G, M 0 N).
As with tbe cup product, one can use a diagonal approximation a: F _
F @ F to compute tbe cap product in terms of a mngle resolution F. Namely,
one composes }' with the map
id 0 (a @ id):G(F M) @ (F @G N) -+ .J'tOmG(F, M) @ «F @ F) 0G N).
The cap prodllct, which may seem strange at first can be motivated by
the fact that it is adjoint to the cup product, in a sense which we now explain.
Consider the .. evaluation map'
.Jf'hma(F, M) 0 (F@G N) -+ M 0G N
given by u @ (x 0 n) 1--+ u(x) @ n. We denote by < z) the image of U @ 7
under this map. [Except for the fact that we are carrying along the factor
N, this is tbe same as the evaluation map introduced in exercise 3 of I.O.]
The evaluation map is a chain map i.e., (c5u, z) + ( _l)de'Pl< oz) = O;
so there is an induced pairing
R"(G, M) @ H ,,(G, N) -+ M @G N
stili denoted ('.), which is independent or tbe choice of resolution. One
now checks, directly from the definitions, that the folIowing adjunction
formula holds for any U E H"(G M l), VE Hq(G M2), z E H,,+iG, M3):
(3.9) (u u v, z) = <u v n z).
(Note: 80th sides or this equation are in (M l @ M 2 (&I M 3)G.) In particular,
taking u = 1 eHO(GZ) we find:
(3.10) For any v E Hq(G, M) and z E H,J..G, N), we have v (""'1 z = (v, z) in
Ho(G, M @ N) = M @GN.

114
V Products
The cap product has properties analogous to 3.1-3.8. We leave it to the
interested reader to write these down.
EXERCISES
L Given m E HO(G, M) = MG and U E H4(G, N), show that m u u = !",(u). where
1m: H*(G. N) -- H*(G. M @ N) is i[)duced by the coefficie[)t bomomorphism
n I-!o m @ Il. [Hi[)t: Use 3.2 and 3.4.] Stale and prove a similar i[)terpretation or the
cap product HO(G, M) <8J Hq{G. N) - HqCG. M @ N).
2. Using the diagonal approximation given i[) the exerClse oC SI. compute ali cup
products in H*(G, -) if G is fl[)ite cyclic.
3. Let G be a finite group which acts freely on 82. - l as i[) 1.6.
(a) For a[)y G-module M. show that there is a[) iterated coboundary map d:
Hi(G, M)........ Hi+2I:(G, M) which is 3D isomorphism Cor i > O and an epimorphism
for i = o. [Hjnt: Tensor the sequence 1.6.1 with M, break it up into short exact
sequences, and use the dimension-shifting argume[)t.]
(b) Show that there is an element U E H2"(G,l) such thal the ""periodicity map"
d or (a) is given by d(v) = u u v for alI v E H.(G. M). [Hint: By 3.3. d(w uv) =
d(w) u v for a[)y w e H.(G. Z). v E H*(G, M); now se! w = L]
(c) Using (b). c.alculate the ring structure O[) H*(G.I) for G finite cyclic.
4. Let G be cyclìc of order n. For any m E Z!.., there is an endormorphism O!(m) of G,
give[) by a(m)g = gn. Calculate m).: H*(G. Z) -+ H*(G. Z). [Hint: In view or
whal you know about the ring structure O[) H*(G Z) from exercise 2 a[)d/or 3c.
il suffices to calcu1ate o:(m)* on H2(G,2'). This can be done non-compulalionally
by using the universa! coefficient isomorphism H2(G.Z)  Ext(H1G..l) or by
using the interpretaHon of H2 in terms or group extensions.]
5. Using exercise 4 and Theorem 11110.3. calculate the integrai cohomology or the
symmetrìc group on 3 letters.
.
I '
, .
4 Composition Products
We observed several tirnes in Chapter 111 that there afe a number of chain
complexes that can be used to oompute H .(G, -) and H.(G, -). Here is
one more example of this:
(4.1) Lemma Let e: F - 7L al1d i/: F' - 7L be resolutions 017L over ZG such
thal F is projective and F' is 7L.-free. For any G-module M, the map r,' <8> M:
F' @ M .... Z <8> M = M i"duces weak equivalences F Q% (F' <8> M) -+ F <8>G M
and .Jt'o,mG(F. F' @ M) -+ .Jft?omG(F. M). ilence
H .( G, M) R:: H .(F <8>G (F' <8> M»
..
..
4 Composition Products
IIS
and
H*(G. M)  H*(.Jft?-tm1G(F, F' (8) M».
PROOF. We already noted in III.2 that t' (8) M: F' (8) M...... M is a weak
equivalence. The lemma therefore follows from 1.8.5 and 1.8.6. D
.-
Note that the last isomorphism in 4.1 can be wriUen in the form
H*(G, M)  [F F' <8> M]*, where [F, F' @ M]JJ = [F. FI @ M]_" is the
group of homotopy classes of chain maps of degree - n from F' to F' <8> M
(cf. I.O). The significance ofthis is that it allows us to construct products by
composition of chain maps. Taking M = Z and F = F, for instance, the
isomorphism takes the form H.(G, Z)  [F, F]*. Since homotopy dasses or
chain maps can be composed, we obtain a composition product
.'
H*(G. Z) @ H*(G, Z) - H*(G,Z).
More generally, let r,: F - 7L, r,'; F' -+ Z, and r/'; FII -io 7L be resolutions
of 7L Over lG such that F and F' are projective and F" is 7L -free. [In practice
we will either take FI' projective or F" = Z.] For any G-modules M, N.,
there is a cochain product
(4.2) Jlt'omQ(F'. FU @ M) @ G(F, F' (8) N) -io JeomG(F, F" <8> M <8> N)
given by u <8> VI-+(u@ idN) o v for u e.Jft?on-tG(FI. F" @ M), v e.J'&»nG(F, F' <8> N):
F  F' @ N "@i F" @ M (8) N.
One easily verifies that 4.2 is a chain map (cE exercise 4 of I.O); in view of
4.1, there is an induced çohomology product
(4.3)
H*(G, M) (8) H.(G, N) - H.(G, M  N),
called the composition product. It is well-defined, independent of the choices
or resolution.
Similarly, there is a product
(4.4) :J'f.omG(F', F" @ M) (8) (F <8> (;CF' <8> N» -io F @G (F" @ M @ N)
given by
u (8) x @ x'  n 1-+ (_l)dCill' dgxX @ u(x') (8) n
for U E Jlt'omG(F', F" <8> M), x E F, x' E F', n E N. One verifies as in the defini-
tioo or the cap prod uct t hat 4.4 is chain map; in view of 4.1, there is an induced
prod uct
(4.5)
H*(G, M) <8> H .cG N)..... H .(G M <8> N).
The following result, although not unexpected, is useful:
(4.6) Theorem. The producls 4.3 and 4.5 coincide with thecupand cap producls.

116
V Products
PR<X>F. Consider the cochain composition product 4-2 with F = F' and
FI' = Z:
(.)
-*hmG(F, M) (8) :KmnG(F. F C8> N) - -*hmG(F, M (g) N).
This induces the product 4.3 via the weak equivalence a; = G(F, l; @ N):
G(F, F @ N) -+ IG(F, N} of 4.1. In order io compare 4.3 to the cup
product, we would like to find a weak inverse of IX, i.e., a map rl: .tfl:.mG(F, N) 
.J'tbmG(F, F @ N) which induces in cohomology the isomorphism inverse to
that induced by a. This would allow us to convert (* ) to a cocbain product
( . *) K.omG(F, M) @ :1f!'mnG(F, N) -+ .Jt"'omG( F, M @ N)
by composing (*) witb
id t8 a.': 3'emnG(F, M) @ ffC4nG(F, N) -+ :Jt'omG(F, M) @ .J'fomG(F, F @ N).
It turns out thar a diagonal approximation A: F -+ F @ F gives rise to
such an r1.'. Namely, we take cl to be the composite
id.. (8) - - Q t!.
Q(F, N) · .J'&mG(F @ F, F @ N) ) .1YomG{F, F @ N),
i.e., cc'(u) = (idy @ u) 0.6.. Computing a.O:':G(F, N) -+KomG{F, N), one
finds that it is the map
u......... (f: @ idN) o (idF @ u) () A = (t: @ u) o A.
But this is simply u 1--+ E U u; since the cocyc1e f represents l E HO(G, il),
it foIlows that a' is indeed a weak inverse of Il.
We now bave a cochain producI (**) which induces the composition
product in cohomology 7 and it is given explicitly by
u @ V t--+ (u @ idN) o a'(v) = (u @ idNJdF @ v)A
= (u @ v)A
=uuv
,
where tbe latter is the cochain cup product de:fined via A. Thus the composi-
tion product in cohomology equals the cup product.
The proof that 4.5 equals the cap product is similar, and even easier.
This time we consider tbe rnap 4.4, with F = F' and F" = 71., and we seek a
weak inverse p': F 0G N -+ F <8>G (F (8) N) of the weak equivalence fJ =
F @  @ N: F @G(F <8> N) - F @GN of4.1. Iclaim wecantakef3' = A  N.
For the composite (F @ f 0 N) .., (A @ N): F @a N  F @G N is a map
of the form 1: @ N, where '[: F --+ F is an augmentation-preserving chain
map, hence it is homotopic to the identity. We now use p' to convert 4.4 to a
product
omG(F, M) @ (F (8)G N) -+ F @G (M @ N),
and it is transparent that this map coincides with the chain level cap product
defined by /J.. O
...
,
I
!
I
t
f
.


I
f"
.
.
5 The pontryagin Produc(
117
We dose by remarking that the methods of this section can be used to
define composition prod ucts
(4.7)
Ext(M', M") (8) Ext(M, M') -+ Ext(M, M")
and
-
.:.
..
(4.8) Ext(M', M") @ Tor:(M, M') -+ Tor(M, M")
far any G-modules M, M', M". One uses suitable resolutions F. F', F" of
the three modules and ane defines maps JfG(F', F") <8> JIlbmc;(F, F') _
.7tbml;(F, F") and KomG(F', F") @ (F @e; F') -+ F @G F'/ similar to 4.2 and
4.4. The products 4.7 and 4.8 are obviously closely related to 4.3 and 4.5,
but it does not seem that either type is a special case ofthe other. For instance
4.3, which we now know is the same as the cup product, can be written in tbe
forro
Ext(.z. M) <8> Ext(.l, N) --+ Ext(.l, M  N).
This fits into the framework of 4.7 in the important speciai case where
N = Z, but not in generaI. In tbe other direction, we know that the Ext
groups in 4.7 can be expressed as cohomology groups if M and M' are Z-free
(cf. 111.2.2), so we expect 4.7 to be describable in terrns of cup product8 in
this case, but not in generaI.
Note, incidentally, that 4.7 and 4.8 can be defined for Ext and Tor over
arbitrary rings, not just group rings. One need only be carefui abaut which
mod ules are Iert modules and which are right mod ules.
EXERCISES
l. Carry out the details ofthe definitions or 4.7 and 4.8.
2. Describe 4.7 in terrns of cup products if M and M'aIe Z-free. Describe 4.8 in terrns of
cap products jf M' is Z -free and M is Z -torsion-free. [Hìnt; Use Theorem 4.6.]
5 The Pontryagin Product
If G is an abelian group then the multiplication map G x G --+ G is a grou p
homomorphism. U sing tbis map we will defioe an internaI homoIogy
product, called the Pontryagin product. For simplicity. we will take the
coefficient module to be a commutative ring k, with trivial G-action. See
exercise l below for a more generaI product.
The Pontryagi,z product on H.( G, k) for G abelian is defined to be the
composite
)(
H.(G, k) C8> H.(G, k) - H.(G x G, k @ k) !:; H ..(G, k),

.!
118
V ProdUClS
where Il: (G x C, k @ k) -io (O, k) is thc muItiplication map (g, g')  09',
A (g) A' 1--+ lÀ'). To compute this product in terrns of a projcctive resolution
F of Z aver ZG, wc need to choose an augmentation-preserving chain map
T: F @ F --+ F compatible with the multiplication map G x G _ G, Le.,
satisfying T(gX <8> g'y) = gg'1:(x @ y). In other words. writing xy = f(X @ Y)
far x. y € F, what we bave is a ZG-bilinear product on F such that (a) f(XY) =
E(X)t(y) and (b) ò(xy) = òx o y + ( _I)dellxx o cy. (Far brevity, we will calI a
ZG-bilinear product on F satisfying (a) and (b) an admissible product.)
Sucha product inducesa k-bilinear product OD F <8>G k, which in turn induces
the Pontryagin product OD H.( G, k). As in 3.4-3.6, one verifies that this
product makes H ..( G, k) an associative. anti-commutative, graded k-algebra
with identity.
[o ease F is tbe standard resolution, tbere is a canonical product on F
called the shujjfe product. This has its origins in the Eilenberg-Zilber theorem
or algebraic topology (cf. MacLane [1963], VIII.8, X.I2), and in the present
context it takes the following formo Let !l'Il be tbe group of permutations or
{l, - - _, n}, and let 9'" act IG-linearly on F" by
o-[g 1 1- - .1 gn] = ( -1 tgna[OI1- 1(1) I. .01 (}u- I(JI)].
If n = p + q then an element U E !l'" is said to be a (p, q)-shu.ffle if o{i) < r.r{j)
for I < i < j < p and for p + 1 < i < j < p + q. [Such a permuta tion can
be viewed as a way of shuffiing a deck of p cards with a deck of q cards.] The
shutHe product on F is now defineò. to be the ZG-bilinear product such that
-
.-
'.
,
.,.
.'.
.
,-
,
.
t

,
,.
..
..,
<'
,
.'
t-
,.
-
.)
, '.
.. .
..
lo..
"
.,
::,
I '. ,"
....
,
l' ..-.
'"0 i
'. ..-IC
, -
.'I!
1- ,..
[gli.. .I(}p] . [gp+ t 1- - .IBp+q] = L a(gll- - -Ig,+..,],
ti
.-
I:
,
.,-
where a ranges over a11 (p, q)-shumes. One can verify that, in addition to
being adrnissible, this product is associative anti-commutative, and has an
identity. Since F is l-torsion-free, anti-commutativity implies striet anti-
commutativity, ie., x2 = O if deg x is odd. It follows easi1y that F <8>0 k is
strictly anti-commutativc_ Consequently:
, -
".I"
"
 .n
"r:'
,-
>
I . '.
.
-,
"\:
(51) Proposition. Far any abelum group G and commutlJtive ring k the ring
H.(G. k) is strictly anti-commutative.
. ",". :..
I......
. 
. ,-I
, .-
., .
... !;, ".'
o',.
::
. '-,
(By contrast, the cohomology ring H.(G, k) is not strictlyanti-commuta-
tive in generaI. An attempt to prove striet anti-commutativity by tbe method
used above fails because the Alexander-Whitney cochain product is only
anti-commutative up to homotopy.)
We now give some other examples of resolutions with adrnissible products.
'(.:.
,., 1
. -
'.-
.. ... .
:'
.: i. "
.; / .:
....'..
"-:!::
". !
(5.2) Let G be infinite cyclic and let F be thc resolution of 1.4.5. Let I and x
denote tbe ZG-basis elements in dirnensions O and l, respectively. We set
1 .1 = l. l. x = x-l = x and x2 = O. and it is trivial to verify adrnissibilityo
. .
;
.,
.
-
,
5 The Pontryagit1 Product
119
(Note that F, as a ZG-algebra. is the exterior algebra !\.<x) with one generator
x or degree 1.)
Dur next example will use the notion of divided poi ynomial algebra
which is defined as follows: Consider the elements li) = pi/i! (i > O) in
the polynomial ring Q[y). We have y(iJy(J) = (i j)y<i+ J1, where (i, j) is the
binomial coefficient (i11 = (i + j)!ji!j!, so the Z-subrnodule of Q[y]
generated by the li) is a subring. This rìng is denoted r(y) or r z(y) and is
called the divided polynomial ring (over Z) in one variable y. More generally,
il R is an arbitrary ring, then the R-algebra R <8> r(y) is called the divided
polynomial algebra over R in one variable y and is denoted r(y) or r R(Y).
We wilI regard r(y) as a graded ring, with deg y = 2.
(5.3) Let G be a finite cycIic group and let F be thc resolution 1.6.3 with one
ZG-basis element ei in dimension i, Be2i = Ne2i-t (i > O), and òeU-l =
(t - l)eu-l' Define a ZG-bilinear product on F by:
(i) e2ielj = {i, })e2i+ lj
(ii) eZie2j+ 1 = (i, J)e2H2j+ 1 = e2j+ Jeli
(iii) e2H le2j+ 1 = O.
Thus F, as a ZG-algebra. is simply !\(eI) @ZG r(e2)' [Recall that if A and B
are graded algebras over a commutative ring R, then their graded tensor
product A <8> R B is a graded algebra, with
(a @ bXd @ b') = ( -l )deibodel"a' ad @ bb'.]
One can check that thìs prod uct on F is admissible; it is also associative and
anti-commutative and has an identity.
Remark. The motivation for the definition above or the product in F is as
rollows: Let x = e 1 and y = e2' Suppose we want our product to be associa-
tive and anti-commutative to have eo as identity, and to satisfy (iv) xe2i =
e2i+ l' Let ]i = CieZi, where C, is an element of 7L.G to be deteIDÙned. Then
a(yi') = ci<1(e2i) = CiNe2i-l' On thc other hand, admissibility and (iv)
imply that
Ò(yi) = iyi-1òy = ici-te2i-2Nx = iCi-lNe2i-l'
So we must have Ci N = iCi-lN. The most obvious way to satìsfy this is to
take Ci = i, so that e21 = y'jiL Formulas (i)-(iH) now follow at once.
(5.4) Suppose F and F' are resolutions with admissible product for G and
G'. Then it is easily verified that F @ F' is a resolution with adrnissible
product for G x G', where, as usua1,
(x C8I x')(y (g) y') = (_l)dea""'deuxy C8I x'y'.

120
V Products
Combining examples 5.2-5.4 one can write down a resolution with
adrnissible product for any finitely generated abelian group G.
Finally we use 5.4 to relate the cross-product lo the Pontryagin product.
If F and F' are as in 5.4, there is a chain isamorphism
(5.5) (F C8>G k) @II (F' C8>G' k) -+ (F C8> F') @G x G' k.
given by (x @ A) 0 (x' C8> A')  (x @ x') @ li'. For Z E F (8)G k and z' E
F' @a' k we denote by z x z' the image of z C8> z' under this map. As in 2t
there is an induced homology cross-product
(5.6) H*(G, k) C8>k H.(G'. k)  H.(G x G', k).
t

(5.7) Proposition. Il G and G' are abelian, then the cross-product 5.6 is a
k-algebra homornorphism. Moreover, Z x z' = i.z - iz' for any Z E H.(G k)
and z' EH.( G', k), where i: G -+ G x G' and i': G1 - G X G' aTe the inclusions.
PROOF. Tbe first assertion is an easy consequence of 5.4. Since z C8> z' =
(z 0 I) . (1 (8) z'), it now rollows that z x z' = (z x l) - (1 x Z/). I claim
that z x 1 = i. z; indeed, by naturality or the cross-product it suffices to
check this for G' = 1, in which case it is obvious. Similarly, 1 x z' = iz',
whence the proposition. D
We can now state the following Ktinneth formula:

, ,
,
!
,
;
!
;
;"
:
1
,
r
(5.8) Corollary..lf G and G' are abelian and k is a principal ideai domain, then
there is a split-exact sequence
0-+ E9 H,,(G, k) @"H,lG',k)  H,,(G x G'k)
p+q""..
-+ E9 Tor1(H p(G, k), BiG', k» -+ O
,p+q=n-l
t
I
!
where }l(: @ Z') = i.2' i z'.
PROOF. In view of the isomorphism 5.5 and the Kiinneth theorem for chain
complexes, we have a split-exact sequence as above with li equal to the
cross-product map 5.6. [This is true even if G and G' are non-abelian.] Now
apply 5.7. D
l . :'
!
EXERCISES
1. RecaJl that if R is a commutati ve rìng, then M 0R N has an R-module structure for
any two R-modules M and N defined by r - (m @ n) = rm G9 n = m @ rn. Thìs
applies in particular if R = :le with G abelian. The resultjng tensor product M @jfG N
should not be confused with the tensor product M (29G N = (M (8) N)G whìch we
introduced in III.O for arbitrary G. (By contrasto the present tel1sor product M 0.lG N.
as an abelian group. is equal to (M @ N)o where G acls anti-diagonally OD M (8) N:
9 . (m @ n) = g -1m @ gn.)
..
,.

.'
6 Appli(:ation; Calculation or the Homology or au Abelian Group
121
I
(a) Ir G is abelian, show that there is a PontryaginproductH...(G. M) @ H.(G. N)4>
H .(G. M @.l(ì N).
(b) Give an example where M (g),w N  M @G N. [Rint: Take G cyclic of arder 3
and M = N = 71.." with non.triviaJ G-action.]
2. State and prove a result analogous to 5.7. expressing the cohomology cross-prodUC1
in terrns of the cup producI.
,
l (a) Prove that homology commutes with dìrect limits. More predseJy. let (G(%)ad
be a direet system of groups. where D is a directed sei, and let G = ljm Gli!' For any
-
G-module M. wc bave a compatible Camily of maps H.(GlZ. M)........ H.{G, M) (ti E D),
hence a map lp: !!!!l H...( G,.. M) - H.( G. M). Prove that lp is an isomorphism. [Hint:
Use the standard resolution.]
(b) Prove 5.1 without using the shuffle product. [Hint: Use (a) to reduce to the case
where G is fÌnÌlely genera[ed.]
.
"
,
-.
4. Let n = p + q + r. Define the notion of (P. q. r}-shuffie and prove that the 3fold
shuffle product in the bar resolution ìs given by
[gli. . .Igp] . (gp+ 11. . .19,.+Q] . [gp+q+ 11.. 'Igp+'l+r] = L u[gt!. . .IOp+q+rJ,
6
where (J ranges over the (p, q, r)-shumes. Generalize.
6 Application: Calculation of the Homology of an
Abelìan Group
For any abelian group G we have H1(G,k) = H1(G) @ k = (j @ k. We can
therefore construct elements or H.(G, k) by taking products in the sense of
5 or elements or G @ k. (If k = 1., for example, the reader has already seen
this construction in exercise 1 of I1.3.) Our first goal in this section is to
show (at least if k is a principal ideai domain) that there are no relations
among these products other than the relations imposed by the fact that
H.( G. k) is a strictly anti-commutative k-algebra. We begin by reviewing
the notion of exterior algebra, so that we can state this result precisely.
Let k be an arbitrary commutative ring, let V be a k-module, and let
TP(V) = V @... @ V (p copies of V), where @ = @J:' (By convention,
TO(V) = k.) There is an obvious k-bilinear product on T.(V) given by
juxtaposition of tensors. making T.(V) a graded k-algebra. We define the
exterior algebra of Jt:. denoted A. *(V) or I\:(V). to be the quotient of T.(V)
by the two-sided ideai generated by the elements v @ v E T.1(V). Explicitly,
tben, /\P(V) is the quotient or TP(V) by the k-submodule generated by the
elements V1 @ . . . @ vp such that Vi = V, + 1 far some i. Tbe product in
1\ .(V) is sometimes denoted x A y.

122
V Products
We have /\ o(V) = k and A l(V) = v; moreover, /\ l(V) generates
/\ *(V) as a k-algebra. Since the generators v E V = f\/(V) satisfy v2 = O
by construction, it follows easily that /\ *(V) is strictly anr;-commutative.
Moreover, it is immediate from the definition that 1\ *(V) has the following
universal property:
.:.
'-
..;
.
"
".0 .
(6.1) If A* is a strict1y anti-commutative graded k-algebra, then any k-
module map V -+ Al extends uniquely to a k-algebra map A*(V) -+ A*:
V , A*
[ //
/\ *( V)
Thus A *(V) is the strictly anti-commutative k-algebra 'freely generated"
by V. Tbe following two properties of tbc exterior algebra functor wiU be
crucial in our study ofhomology:
:..;
"
.'1
, '
:."
"i:
.,
'I.
. ,
, :.
,
.
"
'.:
.'.
o
,.
:.
(62) A *( VI (B J-2)  1\ *() @ 1\*( JS). More precisely, let i l: 1\ *( VI) -
1\ .(VI tE> J-l) and ;2: A *(V2) -/\ *(Vt EB V2) be induced by the inclusions
VI '+ f; tE> V2 and J'2  JI1 $ V2, and let cp: /\ *( VJ) (8) 1\ *( V2) -+
A *( VI EB V2) be defined by cp(x  y) = i1x. i2 y. Then qJ is an isomorphism.
PROOF. There is a k-map VI EB V2 - /\ *(VI) C8> !\ *(V2) given by VI 1-+
V l  l, V2 H 1  V2 (Vi E Vi)' By 6.1 this extends to an algebra map
/\ *(V1 ffi V2) -+ A .(Vt) 0 A *(V2),
.'
"
"
which is the inverse of qJ.
D
(6.3) Let (Va)aeD be a direct system of k-modules. where D is a directed set.
Then A *( )  !!!!1 A .( ). More precisely, the canonical maps J.:--..
!im  induce a compatible family or maps 1\ *() -+ A *( ) and hence
a map qJ : !!!!l/\. *() --+ A *(lli!l ); this map f/' is an isomorphism.
PROOF. The inclusions J::  A *() induce a k-map li!!1. Va -+ fu!} A *().
This extends by 6.1 to an algebra map 1\ *(!!!!l-v..) --"'li!!l A *(), which is
the inverse of lp. D
"
'.
,
.
. .'
Using 6.2 and 6.3, one can give the following concrete description of
1\ *(V) if V is a free k-module: Choose a basis (Xi)i.. I with l simply ordered;
then I\P(V) has a k-basis consisting or the monomials Xi. ... XiI' with
il < ... < ip. ODe often writes 1\ *(V) = A .(X)iEl'
Retuming now to the study of H.(G, k) for G abelian. the isomorphism
G @ k -+ HI(G, k) extends by 6.1 to a k-algebra map tjJ: I\*(G @ k)-+
H*(G k), wrnch is an isomorphism in dimensions O and 1.1t is an obvious
but important fact that tJt is a natural map or functors of G.

6 ApplicalÌon: Ca]cu]ation of the Homology or an AbeJian Group
123
L
(6.4) Tbeorem. Assume thar k is a principal ideai domain.
(i) The map .p: 1\ *(G (81 k) -+ H.(G k) is injective JOT everyabelian group
G and is a split injection if G isfinitely generated.
(ii) Suppose that every prime p such that G has p-torsion is invertible in k;
then '" is an iso11wrphism.
(ili) Il k has characterisric zero (e.g., if k = Z) then '" is an isomorphism in
dimension 2.
Note that the hypothesis or (ii) holds, in particular, if k = Q, or if k = 7L p
and G is p.torsion-free, or if k = Z and G is torsion-free.
PROOF OF 6.4. Suppose first that G is cyclic. Then I\P(G 0 k) = O for p > l,
so (i) holds trivially. Under the hypothesis of (H) we also ha ve Hp(G, k) = O
for p > 1 by tbe computation in III.l, so tjJ is an isomorphism. The same
computation shows that H 2( G, k) = O if char k = O, so (iii) holds.
Now suppose G ìs finitely generated and hence a finite direct product or
cyclic groups. Arguing by induction on the number of cyclic factors, we may
assume that G = GI X G2 and that tbe theorem is known for GI and G2.
Consider the diagram
!\ *(01  k) @t /\. *(G2  k)
"<G,) '" O(G,I l
H.(G1. k) @Ir H.(G2, k)
= ) 1\ .(G @ k)
10(Q
p ) H *(G k),
where qJ is the isomorphism of 6.2 and JJ is the spIi t injection of 5.8. Looking
at tbe definitions or rp and Il and using the naturality of, we see that this
diagram commutes. Since tjI(GI) and tjJ(G2) afe split injections by hypothesis,
it now follows that r/1(G) is a split injection, whence (i). Now assume that the
hypothesis of (ii) holds. Then it also holds for GI and G2 so t/!(Gl) and
",(G2) are isomorphisrns. Since G1 @ k is a free k-module (with one basis
element for every infinite cyc1ic factor of Gi), it follows that H .(Gh k) is
k-free. Therefore the Tor terro in the Ktinneth formula vanishes, so Il is an
isomorphism and (ii) follows at once. Similar1y, Il is always an isomorphism
in dimension 2 because the Tor terro involves Ho(- k) = k. Assuming
now that char k = O, we know that Vi(G.) and rjJ(G2) afe isomorphisrns in
dimensions < 2, whence (Hi).
Finally any group G is the direct limit ofits finitely generated subgroups
Grt" We therefore bave adiagram
, !!!!lA *(G" C8> k) rp ) A*(G  k)
'I::
,  O(G.) l 10(G)
H.(Gfatk) =: . H.(G,k),
.,

124
V Products
where ({J is the isomorphisrn of 6.3 and the unlabelled isomorphisrn is that or
exercise 3a of 5. As above, this diagram commutes because '" is naturaI
and the theorem follows at once. O
If G has p-torsion and p is not invertible in k, then the situation is more
complicated. F or simplicity we will confine ourseives to the ease k = 7l..p
and even in this case we will be somewhat sketchy. See Cartan [1954/55] for
more details and far the calculation of H .(G, Z).
RecaIl from 5.3 that if G is a finite cyclic group with generator t then there
is a free resolution F of the form f\(x) @ZG r(y), where deg x = 1, deg y = 2,
OX = (t - 1).1, and òy = Nx. li p is a prime dividing IGI, then F @G?Lp =
f\{x) 0 r(y) with iJx = O and òy = O, where f\, 0, and rare now over
:l,,; hence H .(G, Zp)  f\(x) @ r(y). Thus we ha ve, in addition to tbe
exterior algebra part of H.(G, Zp) which we understand via 6.4, a 2-dimen-
sional generator y and its divided powers y(;). This suggests that the homology
or an abelian group should have, in addition to its ring structure, a . divided
power" structure. We now make this precise.
Let A = (A,,},,;o:o be a strictly anti-commutative graded ring. By a system
oJ divided powers on A we mean a Camily of runctions A2" .... A2ni for n > O
and i > O, denoted x 1--+ X(i}, with the following properties:
(a) x(O) = l, X(l) = X.
(b) x(i)x(Jl = (id)X(i+ h, where (i, j) = (i + j) Vi!jL
(c) (x + y)(i) = L xU1lk).
j+l=-i
(') {O
(d) (xy) I = xiy(i)
if deg x and deg y are odd and i > 2
if deg x and deg y are even and deg y > o.
(e) (X(i»W = EUX(ij) if i, j > O, where
é:ij = (i, i - 1)(2i, i - 1) ... (U - l)i, i - l).
If A has a differential il (which is always assumed to satisfy ò(xy) =
òx. y + (_1)de8x - òy), then we require also:
(f) ÒX(i) = X(i-l}ÒX for i > O.
Note that this implies that X(i) is a cycIe if x is a cycle. But there is not
necessarily an induced system of divided powers on H.A; see exercise 3
below.
In working with divided powers it is often convenient to introduce tbe
formai power series e'7; = Li2:0 X(i)ti, where t is an indeterminate. Formula
(c) then takes the form e'(x+)'} = r!e'Y, so x t--+ e'x is a homomorphism from
the additive group A;lJt (n > O) to the multiplicative group or formaI power
series 1 + ylt + )l2t2 + ... with YiEA2nf'
...
-
;
\
,
)
\
i
,
.
..
.
i.
,
6 Application: Ca]culation cf the Homology of an Abeljan Group
125
t
,
(6.S) EXAMPLES
I. If A is a strictly anti-çommutative graded Q-algebra, then A admits a
system of divided powers with x(l) = xi/il. Moreover, this is the unique
system or divided powers on A; for (a) and (b) imply that, in any algebra witb
divided powers, xxu- l) = jx{J) and bence xi = i !X(f).
2. Let a.: R -+ S be a homomorphism or commutative rings and let A
be a (strictly anti-commutative) graded R-algebra with a system or divided
powers. Then the S-algebra S @ R A obtained by extension of scalars admits
a unique system of divided powers such that (1 @ x)(i) = l @ x(f}. Indeed
we must have (s @ X)<i) = i @ X(I) by (d), so uniqueness follows from (c).
To prove existence, one can use the uni versaI mapping property of the
tensor product to define ez for Z E S @R A2" by et(@Jt) = LI (Si @ X(i)tI;
details are left to the reader.
3. The divided polynomial algebra r R(y) defined in S adrnits a system or
divided powers. Indeed, in view of example 2, it suffices to check tbis if
R =  in which case the assertion follows from the faet that r(y) c: Q[y]
is closed under the divided power operations [cf. identity (e)]. In exactly the
same way we can construct a divided polynomial algebra r R(Yt)ie I in
several variables; if 1 is simply ordered, then this has an R-basis consisting
of tbe "monomials  yld . . . y:") with il < . .. < i" and nj > O. If V is a free
R-module with basis (Y,),d, then we set r(V) = r R(Yi); this has V in dimen-
sion 2 and has a universal mapping property (in the category of R-algebras
with divided powers) analogous to 6.1. It follows, in particular, that r(V) is
well-defined up to canonical isomorphism, independent of the choice or
basis. -
4. The Pontryagin ring H.( G, k) of an abelian group G admits a natural
system or divided powers. To see this, consider the bar resolution F with the
shuffie product. If x = [gli.. '192"] (n > O), then we ha ve (cf. 5 exercise 4)
Xi = L u[g. 1-. .IB2wl.. 'Iod. ..lg2n],
D  y,.
where the . block" gli. . . I g2" is repeated i times and [/ A C Y'2ni is the set
of (2n, . . . ,2n)-shuflles. Now there is an obvious embedding 9'i 4 Il' 2nh
obtained by letting!/i permute i sets of 2n elements blockwise. Since II'i c:
!/ 2"i is a group or even perrnutations, it is clear that f/i fixes
[gli. . ,192" I. . .1 gli. . '192"]'
Moreover :fA is closed under right multiplication by elements of !/j;
the above formula for Xi can therefore be written
Xi = i I a-[gll- ..192111.. .Iul [.. .lg2,J.
a" Ytl.19'i
It follows at once (via identity (c» that F c O C8> F is closed under the
divided power operations and hence admits a system or divided powers.

126
V PtodUCIS
This yields (cf. example 2) a system of divided powers OD k @G F, and I
claim that there is an induced system or divided powers on H .(G, k). To
prove this, it suffices (cf. exercise 3 below) to show that Z{i) is a boundary if
Z E k @G F 2n is a boundary. Far tbis purpose we view k @G F as (k @ F)G
and note thal k C8> F is acyclic in positive dimensions (because 8: F -+ 7L. is a
homotopy equivalence if we forget the G-action). Now any boundary
Z E k @G F is tbe image of a boundary w E k 0 F, and W(i) is a cycle by
identity (f). But then W(i) is a boundary by acyclicity, and hence its image
Z{l) is also a boundary.
,
:
'.
:
:
. --
I
!
..
..
.
!
Returning now to the study of H *(G, Zp), consider the split-exact universal
coefficient sequence
0- HzG C8> Zp -+ H2(G, lp) - Tor(H1G, 7L,,) -+ O.
We have H2 G  f\ z(G) by 6.4(iii), anq it is easily seen that 1\2(0) @ 7l.p 
1\i.,.{Gp), where Gp = G C8> Z" = GjpG. Also Tor(HiG, Zp) = Tor{G. Zp) =
pG, where the lattee denotes {g E G; pg = O}. Tbe sequence above therefore
takes tbe form
-

I
 ;
.
.
','
t
t
, ,
0-+ /\2(G9) -+ H]{G, Zp) -+ pG - O.
Choose a splitting pG ---t' H2(G, 7L.p) of this sequence. (If p is od We may use
the canonical splitting given in exercise 4b below.) Ths extends to a Zp-
algebra homomorphism ({J: r(,pG) -+ H.(G, Zp) compatible with divided
powers, cf. example 3 above. Combining q> with the map rJi: f\(Gp)---t'
H.(G, Z,p) studied in 6.4, we obtain an algebra map p: f\(Gp) fi!J r(pG)--iI>
H .(G, Z,p) given by p(x @ y) = tjI(x)q>(y). We can now state:
"

;
<
,
.
,
;-

i
I
.
,.
<
. <
(6.6)Theorem. The map p: f\(Gp) @ r(pG) - H.(G, Zp) is an isomorphism.
II is natural ifp '* 2.
Ir G is cyc1ic this follows easily from our earlier computations. Tbe
generaI case can now be deducea by using the Ktinneth formula and direct
limits as in the proof of 6.4. See CartaQ [1954/55] for more details.
.'
EXERCISES
;
1. If G is an abelìan group (written addjtiveIy) and " E Z. compute thc endomorphism or
H...(G. Q) induced by the endomorphism g a-ng or G.
2. Let A and B be strictly anti-commutative graded k-algebras. Show that A <81.. B Is
the sum of A and B in the category of strictJy anti-commutative geaded k-algebras,
via the maps a 1--+ a @ l, b.......,. 1 18) b (a E A. b E B). Thus 6.2 can be interpreted as
saying that thc exterioT algebra functor prcserves sums.
..
6 Application: Calculation of the HomoJogy of an Abelian Group
127
3. Let A be a strìctly anti-comnmtative graded fing with a differential il and a system
or divided powers.
(a) Suppose that X(i) is a boundary whenever x is a boundary. Show that there ìs an
induced system of divided powers OD H... A.
(b) Give an ex.ample to show that the hypothesis of (a) need not hold.
4. (a) Show that the iojection t/t: /\ 2(G 12> k) - H leGo k) of 6.4 takes (g @ l) A (h @ 1)
to the dass of [glh] - [hjg].
(b) 1£2 is invertible in k, show that the map C2(G. k) -+ 1\ 2(G (&) k) given by [g Ih] f-+
(g @ l) A (h C8> 1){2 induces a map H z( G. k) -+ 1\ 2( G (8) k) wruch is a Ieft inverse
of.
5. Let G be abelian and let A be a G-moduJe with trivial G-action. In view of tbe iso-
morphism H2G  /\2G the universal coefficjent theorem (cf. exercise 3 of III.l)
gives us a split exact sequence
0- Ext(G. A) -+ H2(G. A) !a. ;t(om(/\ 2G. A) - o.
Deduce from exercise 4a thar B coincides with the map called () in exercise 8 of
IV.3. hence the latter is a (spIit) surjectiol1. Thus every altemating map comes from
a 2-.cocycle. (A DO n-cohomologicaJ proof or this surprising fact can be found in
Hughes [1951] proof or Theorem 2.) lncidentally, it also follows that Ext(G. A) 
tt,,(G. A). [More generally. it is known for any ring R that Ext(M. N) is isomorphic
to the set of equivalence c1asses or R-module extensions or M by N. whence the
name "" Ext." A proofofthiscan befound inalmostybookon homoJogicalalgebra.]

CHAPTER VI
Cohomology Theory Or Finite Groups
'.
,
;
,
..
..
I
,
I .
I ...
'.
, ,
,
..
I
...
.
..
1 Introduction
.;
Homology and cohomology are usually thought of as dua1 to one another.
We have seen in Chapter I1I, for example, that homology has a number of
formai properties and that cohomology has "dual properties.1f G is finite,
however, then homology and cohomo]ogy seem to bave similar properties
rather than dual ones. F or example since every su bgroup H of a finite group
G has finite index, we have restriction and corestriction maps for arbitrary
subgroups, in both homology and cohomology. For another example, the
distinction between induced modules and co-induced modules disappears,
so we have a single class J of G-modules (namely, tbe induced modules
7LG 0 A) with tbe roIlowing properties; (a) Every M E J is acyclic for both
homology and cohomology. (b) For every G-module M there is a module
]J E  such that M Is a quotient of M and M can be embedded in M.
Tate discQvered an ingenious way to exp)oit these similarities between H.
andH* for Gfinite. Namely he showed that tbere is a quotient no or HO and a
sub-functor ilo c Ho such that the functors...  H2, H 1 ilo, no. H1, H2, ...
lìt together to fonn a "cohomology theory" fl. involving functors Br for alI
iEZ:
-
,
,
,
1
,
..
. .
t.
,
, .
'" H2
H1 Ro HO HI H2 . ..
I . .
,
,
J
R-2 H-l fJo fJl fl2 . __
. .. fJ-3
Tbc purpose ofthis chapter is to develop this Tate cohomology theory and to
illustrate its usefu1ness by discussing (a) the Nakayama-Rim theory of 00-
.28
2 Relative Homological Algebra
129
homologica11y trivial modules (8) and (b) thc theory of groups with periodic
cohomology (9).
2 Relative Homological Algebra
We indicated in exercise 2 or I.7 that the fundamenta11emma 17.4 and its
corollaries could be genera1ized in various ways. In particular, we will need in
this chapter a "relative-dua1" version of those results, which we work out in
the present section.
Throughout tbe section, G wiU be a fixed group (not necessarily finite) and
H a fixed subgroup. For the purposes or the present chapter, the case of
interest will be that where G is finite and H = {l}. But the generaI case will be
needed later, in Chapter X.
An injection i: M' 4 M or G-modules will be called admissible if it is a
split injection when regarded as an in jection of H -modules, i.e., if. there.is an
H-map n: M -+ M' such that ni = idr./" An exact sequence M' -.4 M -.4 M"
wil1 be called admissible if the inclusion im j r:,. M" is admissible. An acyclic
chain complex C of G-modules will be called admissible if each exact sequece
C1+ 1 -- Ci -+ Ci-1 is admissible; in view of 1.0.3, tbis is equivalent t saymg
that C is contractible when regarded as a complex of H-modules. Fmally, a
G-module Q is relatively injective if it satisfies the following equivalent
conditions:
(i) Every mapping problem
M' JM
1
Q
M"
,
,
.
,
.
I
,
,
/01.
with admissible exact row can be solved.
(iO Every mapping problem
M'C i )M
l 
,
,
.
.
.
.
Il:
Q
with i an admissible injection can be solved.
(Hi) The contravariant functor HomG( -, Q) takes admissible injections of
G-modules to surjections or abelian groups.

130
VI Cohomology Theory. or Fjnite Groups
3 Complete Resolutions
131
In particular, every injective module is certainly relatively injective. But
there are many relative injectives which are not injective:
By a relative il1jective resolution of a G-module M we mean a non-negative
cochain complex Q of relative injectives, together with a weak equivalence
f'J: M -+ Q such that the augmented complex
O -+ M -+ QO _ Ql -+ .__
is. admissible. As an immediate corollary or 2.4, we ha ve:
(2.1) Proposition. For any H-module N the G-module Coind N is relatively
il'ljective.
PROOE By the universaI property of co-induction (III.3.6) we have
HomG( - Coind N) :=:::; HomH(Res( --), N).
The functor on the right clearly takes any admissibIe injection of G-modules
.0 a surjectjon or abelian groups. o
(2.5) Corollary. Any rwo relative injective resolutions oJ Mare canonically
homotopyequivalel'lt.
(2.2) Corollary. F or an y G-module M there is a cQnonical admissible injection
M c.;. M where M is relatively injective.lf(G: H) < co then this construction
has thefollowing properties: (a) li M isfree (resp_ projective) as a ZH-module
then 1J isfree (resp. projective) over ZG. (b) Il M isfinitelygenerated as a G-
module, then so is M.
It is clear that relative injective resolutions exist for any M; indeed, we can
take 1]: M  QO to be the canonical adrnissible injection or 2.2, then apply 2.2
again to get an adrnissible injection coker   Q l , etc. Moreover if (G: H) < 00
and M is projective as a ZH -module, then I c1aim that the modules Qn w hich
occur in this resolution will be projective over ZG. This is clear for QO by
2.2a. Since '1 is H-split, it follows that coker '1 is projective as a ZH-module,
hence 2.2 implies that Ql is projective over 7l.G. Continuing in this way, one
proves tbe claim. Similarly, if M is finitely generated, then so is each Qn.
Summarizing, we have shown:
PROOF. Take M = Coind Res M and use the canonical H-split injection
M c.. M ofIlI.3.7.1f (G: H) < 00, then we have CoindZ( -)  Ind2{ -), and
OOOOW. D
(2.6) Proposition. Suppose (G: H) < 00. II M is a lG-module which is pro-
jective (resp.finitely generated arul projective)asa ZH-module, chen M admitsa
relative injective resolution '1: M --+ Q such that each QJt is a projective (resp.
finitely generated projective) 7LG-module.
(2.3) Corollary4 Suppose (G: H) < 00. Then any 7l.G-projective module is
relatively injective.
PROOF. It is easy to see that a direct summand of a relative injective is relatively
injective, so it suffices to consider free modules. N ow if F is a free li. G-module
,
tbcn clearly F :;:;: ZG 0zH F' = IndZ F' where F' is a free 7LH-module of the
same rank. But IndZ F1  Coind F', which is relatively injective by 2.1. O
EXERCISE
I.
Show that the resolution Q obtained in the proof or 2.6 ìs a complex or free ZG-modules
if M is free as a 7l.H -module.
We now record the "relative dual. ofLemma I. 7.4, in the form in which we
will need it:
(2-4) Proposition. Let C ami C' be chain complexes 0/ G-modules and let r be an
integer. Suppose rhal C; is relatively injective for i < r and thar Ci+ 1 -+ Ci---+
Ci - l is exact and adrnissible for i < r.
(a) Any family (i: Ci - C;)i> r DJ maps commuting with boundary operators
extends to a chain map C ---+ C'.
(b) Let J.. g; C -+ C' be chain maps and leI (hi: Ci -+ Ci + di 2:,-1 be a family oJ
mapssuchthatò;+thi + hi-1Ò; =h - gifori > r. Then{hJ)i2:r_l extends
lo a homotQPY from f lo g.
Tbe proof is virtually identical to that or I.1.4 except that alI arrows are
reversed. D
3 Complete Resolutions
We now specialize the relative homological algebra of2 to the case where G is
finite and H = {l}. As a consequence of 2.6 we obtain the perhaps surprising
result that the Gmodule Z admits a "back wards' resolution O -+ 7L ---+
QO -+ Q l -+ . . '. where each Qi is finitely generated and projective. [This
becomes less surprising. however. when one realizes that such a backwards
resolution can be o btained by taking the dual of an ordinary projective
reso]ution of finite type; cf. 3.5 below.] Setting Fi = Q-i-l  tbis takes the
rorro
(3. l)
O-+7LF_I-+F_2-+""
...

132
VI Cobomology Theory or Finite Groups
Now. splice 3.1.onto an ordinary projective resolution e: (FI'I)"o -+ Z, to
obta1n an acycbc complex
. . .
F2
) . . .
.
, Fl
) Fo
) F_)
· F_'2,
\/
1L
A complex of this type is called a complete resolution for G. More precisely,
a complete resolutiol1 is an acyclic complex F = (Ft)fez or projective ZG-
modules, together with a map E: F o - 7L such that 8: F + -+- Z is a resolution in
tbe usual sense, where F + = (F;)j2:o. It follows from tbe definition that
ò: F o...... F -1 factors uniquely as iJ = 'lE, wOOre '1: li...... F  1 is a monomor-
phism. It also follows that the resulting complex 3.1 is a relative injective
resolution or Z. In fact, we know from 2.3 that each Fi is relatively injective,
and adrnissibility follows from exercise 3b or 1.8, which we repeat here:
(3.2) LeIlUlUl. Any acyclic chtJin complex C 01 free abelian groups is con-
tractible.
PROOF. The abelian group z" or n-cycles is free, being a subgroup of a free
abelian group" so the sequence O -+ z,,+ 1 - C,,+ 1 ...... Z" -+ O splits. D
We do not require in tOO definition of complete resolution that F be of
finite type, i.e., that each Fj be finitely generated, but we saw at the beginning
or this section that there do exist complete resolutions of finite type.
As with ordinary resolutions" we may view the map 8 in a complete resolu-
non as a chain map F --.-.Jo lL. (Note, however" that it is not a weak equivalence.)
Given complete resolutions e: F - Z and e': r -+ 71., a chain map 1': F - F'
is augmentation-preserving if e'1' = 8.
(3.3) Proposition.lf g : F -+ 71. and E': F' ...... Z are complete resolutiom, then there
exists a u,zique homotopy class of augmentation-preserving mapsfrom F to F'.
These maps are homotopy equivalences.
PROOF. By 1.7.4 we can find an augmentation-preserving chain map t"+:
F + -+ F'+. Since F is acyc1ic and admissible (by 3.2) and F' is dimension-wise
relatively injective 1 + can be extended to negative dimensions by 2.4.
Similarly, given two augmentation-preserving maps t't 7:': F --+ F't 1.7.4 gives
us a homotopy between T 4- and T'+ , which can then be extended to negative
dimensions by 2.4. It is clear from the uniqueness that any map of complete
resolutions is a homotopy equivalence. D
Finally, we want to show tOOt the negative part of a complete resolution or
finite type is the dual or an ordinary projective resolution of Z over ZG.
Recall that a G-module M has a dual M* = HornG(M, ZG), which we studied
...
,
, .
I
i

,
l
,
,
.}.
.
i
3 Complete Reso]utions
133
in 1.8.3 for finitely generated projective modules. [Note that M* is naturally a
right G-module if M is a left G-module, but we can convert it to a left module
as in III.O. Thus (gu)(m) = u(m)g-l for 9 E G, U E M*, m E 1\-1.] On the other
hand, ODe can also consider the dual Hom(M  Z) or M as a Z-module. This is a
G-moduJe via the diagonaJ action (cf. III.O), where G acts trivially on Z.
Thus (gu)(m) = u(g-lm) for g E G, U E Hom(M, Z), m E 1. It turns out that
Hom(M. Z) coincides with the dual M* when G is finite:
(34) Proposition. For any finite group G and any (left) G-module M, there is a
G-module isomor phism
t/1 : Hom(M, Z) --+ M* = HomG(M, ZG).
given by t/1(u)(m) = LE'G u(g-lm)g for u E Hom(M, Z)t m € M.
PROOF. One can simply verify this by straightforward definition-checking, but
in fact it follows from things we bave already done. Namely, 7LG is the induced
module lL.G (8IlL., so it is isomorphic to the co-induced module Hom(lG, Z) by
111.5.9. Tbe universal property 111.3.6 of co-induction therefore implies that
HomG(M, 'ZG)  Hom(M, 71.) (as abelian groups), and aD examination ofthe
proofs of 111.3.6 and 111.5.9 yields the specific isomorphism f/J. lt is easy to
verify that 'il is compatible with the G-action. D
As an application of this, we will give a concrete interpretation of the
'backwards projective resolution" 7L...... Q discussed al the beginning ofthis
section. By tOO dual of a chain complex F (over an arbitrary ring R) we mean
thc complex F = JlfqmR(F, R). Thus F" = F _Il = HomR(F", R) = (PII)*.
(3S) Proposition.lfe: P -+ Z is afmite type projective resolution of71. over 7LG
(G finite), thel'Il.: 7L* = 7L. ...... P is a backwards projective resolution nwreover,
up to isonwrphism every finite type backwards projective resolution is obtained
in this way. Consequently any finite type complete resolution is obtainedfrom
two ordinary finite type projective resolutions p' -+ 7L and P - Z by splicing
together P' at1d 'EP, where r,p is the suspension of P (I.O).
PROOF. The augmented chain complex associated to e: P - Z is contractible
as a complex or abelian groups (by 3.2, for instance). It therefore remains
contractible when the duality fUDctor ( )*  Hom( -, Z) is applied, so
eli<: 71.* = Z _ P is a backwards projective resolution. This proves the first
assertion. Similarly, given any finite type backwards projective resolution
Z ---+ Q, its dual Q -+ 7L is 3D ordinary projective resolutio and Q can be
identified with tbe dual of Q by 18.3d. The remaining assertion is c1ear. D
EXERClSE
If G acts rreely on S21:- l as in 1.6. show that there js a complete resolution F Cor G which
is periodic or period 2k. Write this out explicitly if G is finire cyclic.

134
VI Cohomology Theory of Finite Graups
4 Definition of lì.
The Tate cohomology of a finite group G with coefficients in a G-module M is
defined by
fJi(G, M) = Hi(J'f'omc;(F, M»
for alI i E lL, where F is a complete resolution for G. By 3.3, Hi is well-defined up
to canonical isomorphism.
Let F + = (Fi)iO and F - = (Fi);< o, and let C+ (resp. C-) be the complex
(F +, M) (resp. J&mG(F _, M)). Then we bave (cf proof ofI1.5.1)
Ri(G M) = {H(C+) i> O
, HI(C-) i < -1
and there is an exact sequence
0-+ R-I(G, M) --+ H- I(C-)  HO(C+) -+ fìO(G, M) -+ O,
where rt is induced by the coboundary operator lJ: C-l -+ Co. More preciseIy,
IX is determined by the diagram
C-I
.'!
CO
(*)
1
J
H-l(C-) l:[ ) HO(C+).
Now E: F + -+ Z is a projective resolu tion or.z, so Hi( C +) ::;:; Hi( G, M). And
ifwe assume (as we may)that F is offinite type, then 3.5 implies that F _ = I:.P
for some projective resolution p -+ 7L of finite type. The duality isomorpbism
I.8.3c now yields
c- = .Ytb.mG(F _ M) = .H'OHtG(I:P, M)  r,p @G M
so Hi(C-)  H-i(LP @c;M) = H_1-I(P@c;M) = H-i_I(G, M). Wethere-
fore ha ve
Ri(G, M) = {HG, M)
H -i-I(G, M)
j> O
i < 1
and there is an exact sequence
o --+ R l(G, M) --+ Ho(G, M)  HO(G, M) --+ fJo(G, M) -+ O.
f claim that o: is the norm map N: Ma -+ MG defined in III.l. To prove
this, it is convenient to assume that tbe projective resolutions F + and P above
both ha ve ZG in dimension O and both start with the canonical augmentatioD
lLG -+ 71.. [We can certain1y assume thist since the resolutions can be taken to
....
.
'i
!
!
,
...
"
'.
,.
<
.,
.
4 Definilion of O.
135

be arbitrary finite type projective resolutions.] Then '1 = f;*: 7L --+ 7L.G is
given by 'l(I) = L.. a g. Since HornG(ZG. M) = M, the diagram (Il') becomes
M
l
N
,
M
J
Ho(G, M) a HO(G, M),
and it is easy to check that the vertical maps are tbe canonical maps M .... M G
and  .:.. M. This proves the cIaim.
We bave now proved
R-l(G, M) = ker N c: Ho(G, M)
and
fJO( G, M) = coker N *"- HO( G, M
For example, fl-l(G, l) = O and flo(G, Z) = ?L/I G I. ?L.
We can summarize the results above by means or the following diagram:
A-3
......n
R2 .. .
H1
H1... .
R2
R-I
f
RO
R1
... H2
H1
Ho
One çan also define Tale homology groups, by R*(G, M) = H*(F @GM),
where F is a complete resolution. Arguments analogous to those above show
that
R.=
I
H.
J
ker N
coker N
H-i-I
i> O
i = O
i = -1
;<-1.
In other words, fli = R- i-l, In view ofthis, it may seem pointless to introduce
R. since it, like R*.just consists ofthe functors Hj and H1 for i> O together
with modifications or HO and Ho _ As we wiU see in 7, however, the n equality"
Di = R- i- I should really be viewed as a duality theorem, and, as such, it has
important consequences.

136
VI Cohornology Theory ofFinite Groups
5 Properties or lì.
Because Tate cohomology was defined in terrns of resolutions, it is easy to
prove that many of the formaI properties of H* hold also for fJ*. F or example,
one pro ves as in III.6.1(ii'):
(5.1) A short exact sequence O - M' --+- M --+ M" --+ O of G-modules gives rise
to a long eXBct sequence
. . .  fJi(G, MI) -+ Bi(G. M) -lo fJi(G. M") !. fìi+ l(G, M') -..... _ . . .
Similarly, the proof of Shapirds lemma (111.6.2) goes through in the
present context, since a complete resolution for G can also be regarded as a
comp]ete resolution Cor any H  G. Since co-induction is the same as induc-
tion (because G is :finite), Shapiro 's lemma takes the form;
r
I
I
(542)IfH c G and M isanH-module, thenfl*(H, M)  fJ.(G, ZG <&zHM).
Taking H = {l} and noting that R*({l}, -) = O, we obtain:
(5.3) R.(G, 7LG @ A) = O for any abelian group A. Consequently, each Ri is
both effaceable and co-effaceable.
Note that 5.1 and 5.3 allow one to use dimension-shifting. Namely, given a
G-module M we can find G-modules K and C (as in 111.1.1 and 111.7.2) such
that:
(5.4) fJi{G, M)  RH l(G K) and fJi(G, M)  RI-l(G, C) for alI i E Z.
This shows, at least heuristically, that the Tate cohomology theory is
completely determined by any one of tbe £unctors fli. Tbe interested reader
can make this statement precise, as in 111.7.3 and 111.7.5.
Given H c: G, a complete resolution F for G and a G-module M, we bave
cochainmapsJfO.m<r{F, M)  JIlhmH(F, M)andF, M) -+:Wh.mG(F, M),
where the 8eCond map is a transfer map, defined as in definition (C) of II19.
Consequently:
(5.5) For any H c G and any G-module M. one has a restriction map
R*(G, M) --+ R.(H, M) and a corestriction map fl*(H, M) _ R"(G, M).
These maps bave formaI properties analogous to those stated in 1119.5.
It £ollows. in particular, that the analogues of 111.10.2 and 111.10.3 hold
for fJ-_
Finally. we will show that the theory of cup products extends to R*:
(S.6) There is a cup product R"(G. M) @ /lfl(G, N) --+ Rp+4(G, M @ N), wìth
formaI properties analogous to 3.1-3.8 ofChapter V.
}':
....
:.
!

. : 
i
.1
"
: .

.
"
.-
,-
.
'.
 Propertit5 o( It..
137
(Note: To statethe analogueofV.3. 7, onehas to assumethat CI:: H  G is an
inc1usion, since that is the only case where we bave a map a*: R-( G) 
1l*(H).) ." . h .d .
In particular, R*(G, Z) is an antl-commutatlve graded nng Wlt 1. entIty
element 1 E ZII GI. Z = RO(G, 1.), and B*(G. M) is a module over Il (G, Z)
for any M.
The construction of the cup product requires some work. Recall that wc
based our treatment or cup products in ordinary cohomology OD tbe faet that
the tensor product of resolutions is a resolution. 00 the cochain level, then,
the cup product was simply the map
(5.7) :Jf{},mG(F, M) (8I.1YomG(F, N) -+ G(F @ F, M @ N)
given by tensor product of graded maps. We pointed oul that ne cold al80
define tbe cup product in terms of a single resolution F by choosmg a dgonal
approximation h.: F --+ F @ F and composing tbe product above wlth
G(.à, M @ N): G(F @ F', M @ N) ..... :J't'omG(F, M @ N).
Note that, from this point of view, the cup product
Jfe€mtG(F, M)11 @.Jthma(F, N)'  .JftbmG(F, M @ N)P+f
depends only on tbe (p, q)-component I1pq: F p+.q --+ F".  FQ of A.. .
Now suppose F is a complete resolution. Tbe first dlfficulty IO trYlng to
imitate tbe procedure above is that F (81 F does not appear to .be a complete
resolution. In particular, (F @ F)+ is not equa] to the rsolutton F.+ @ F +.
ConsequentJy, a definition of the forro (5.1) would not In any ObVIUS ,:ay
induce a cohomology product flp @ flq -+ fJp -+-41, so a diagonal approXlmatton
now appears to be crucial, rather than a mere convenien. Secondly" a
moment's thought shows that F @ F is not really the appropnate target for a
diagonal approximation. Indeed, for an  E 7L thee are nfinitely many (p, q)
such that p + q = n, and dimension-shiftmg conslderatlons suggest that the
corresponding cup products should ali be non-triviai. Thus A should bave a
nontrivial component Apq for ali (p, q), so the target of A should be the
graded module w hicb in dimension n is Op + q:; le  P @ F Il ther than
eD F fVI F This discussion motivates tbe followmg defimtlons.
Q/p-+-q=.. p \01 q' A , .
H C and C' are graded modules, their completed tensor product C (81 C IS
defined by
(C @ C')n = n Cl' (81 C.
,,+q=n
Given two other graded modu1es D, D' and maps u: C -. D of degree r and
v: C' -+ D' of degree s, there is a map u c&> v: C lE> C' -+ D @ D' of degree
r + s defined by
(u @ v)ft = n (-l)1"'up (81 v,,: n c" @ C --+ n Dp+r <81 D+$.
p+.=" ,,+q=ft JI'+IIJl=n

138
VI Cohomology Theory or Finite Groups
Now let e: F ..... 7L be a complete resolution, and let d be the differential in F.
Then F @ F has the "partial dìfferentials" d @ idF and idF @ d; these are of
square zero and anti-comrnute, since (d @ idF)(idF @ d) = d @ d whereas
(idp @ d)(d @ idF) = -d @ d(cf.exercise8 of1.0). So the " totaldifferentia) "
il = d @ idF + idF @ d is of square zero and makes F @ F a chain complex.
Moreover, we have the . augmentation" e @ E; F @ F --+ 7L @ 7L = lL. By a
complete diagorral approximation we mean aD augmentation-preserving chain
map d: F - F @ F. It is by no means obvious that complete diagonal
approximations exist, bu t we will prove below that they do. Accepting tbis for
the moment, we can deftne a cochain cup product
Jeo.-mG(F, M) @&m6(F, N) :I&l/jnG(F, M @ N)
by u u v = (u @ v) o .6..
One verifies tbe usual coboundary formula
l5(u u v) = bu u v + ( -l)Pu U OV
and it follows that there is an induced product flp(G M) @ flt(G, N)-+
fIp+Jl(G, M @ N). It is immediate that this product is natura) with respect to
coefficient homomorphisms (as in V.3.2). and one pro ves exactly as in V.3.3
and V.3.3' that it is compatible with connecting homomorphisrns lJ in long
exact cohomoIogy sequences. Moreover, the fact that 6 is augmentation-
preserving allows us to calculate the cup product flo Q9 fIo --+ {J0. and we
find as in V.3.1 that it is ìnduced by the obvious map MG @ NG -+ (M @ N)G.
[This makes sense because flo(G -) is a quotient of (_ )G.]
We can now use dimension-shifting to prove that the cup product is
independent of the choice or F and h.. More precisely:
(5.8) Lemma. There is al most one cup product on H*(G, -) satisfying the
analogues 01 V.3.I, V.3.3, and V.3.3'.
PROOF. Given a module M, let M be the induced rnodule ZG @ M and let
0-+ K ..... M --+ M --+ O be the canonical l-split exact sequence. For any
G-module N the sequence O -+ K Q9 N --+ M @ N -+ M @ N --+ O is exact.
and the module M @ N is induced (cf. III.5. exercise 2a). We therefore have
dimension-shifting isomorphisms
lJ: fIi(G. M)  fIH leGo K) and : {Ji(G M Q9 N)  fJi+ l(G, K @ N);
moreover. any cup product satisfying the analogue ofV.3.3 is compatible with
these isomorphisrns, in the sense tha t
Rp(G, M)
- v.l
HP + I(G, K)
l-v,
HP+Il(G, M @ N)  Rp+q+ 1(0, K Q9 N)
......
--
.
-
,

"
,
.,
"
.'
,
.
.'
.i
5 PrQperlies or lì.
139
..
commutes for any v E fl4(G N). Now suppose we have two cup products
fJJI @ fl4 --+ Rp+q as in the statement ofthe lemma. By hypothesis they agree
wben p = q = O, so the square above allows us to prove by descending
induction on p that they agree for p < O and q = O. Similarly, writing N as a
quotient of an induced module, we can extend this by descending induction
OD q to the case p  O, q < O. Next we embed M in an induced module and
prove inductively that the cup products agree for p arbitrary and q < O, and
finally we embed N in an induced module to prove that they agree for ali p
 D
..
'.
Stili assuming the existence or 6., we can now easily prove the remaining
properties of the cup product: The fact that l E ZII G I . 71.. = flo( G, Z) is n
identity is proved as in V.3.4 or by dimensìon-shifting [it is obviously true 1D
dimension O]; associativity is proved by dimension-shifting [it ìs obviously
true in dimension O]; commutativity can be deduced from 5.8 [u u v and
(-I)pqt*(v u u) are two tUP products with the required properties] or,
alternatively, there is a direct proof analogous to that ofV.3.6; finally, the fact
that the cup product behaves properly with respect to restriction and 00-
restriction follows directly from the definition as in V.3.1 and V.3.8, or by
dimension-shifting.
To complete tbe construction of tbe cup product. we must prove he
existence or a complete diagonal approximation 6. We will use the followmg
three lemmas:
(5.9) Lemma. F <2$) F is acyclic and dimension-wise relatively injective.
PROOF. Relative injectivity follows from the fact that an arbitrary direct pro-
duct of relative injectives is relatively injective. To prove acyclicity. let
h: F --+ F be a contracting homotopy for F, regarded as a complex or abelian
groups (cf. 3.2), and let H = h  idF. I c1aim that H is a contracting ho,?otopy
for F 0 F (regarded as a complex of abelian groups). Indeed, recalImg the
definition of the differential a in F <8> F, we have
aH + Ha = (d  idF + idF 0 d)o(h @ id,,)
+ (h  idF)o(d @ idF + idF @ d)
= dh  idF - h @ d + hd @ idF + h @ d (cf. exercise 8 ofI.O)
= (dh + hd) @ idj:
= idp <2$) idF
= idFF'
whence the c1aim.
D
(5.10) Lemma. Let (C, ò) and (C'. ò') be two acyclic chain complexes oJ 7LG-
modules. Assume thal each Ci is projective and each CI is relatively i,rjective. Il

140
VI Cohomology Tbeory of Finite Oroups
1'0: Co -+ C is a map such that O'tOÒl = O, then 1'0 exrends to a chain map
t': C ..... C'.
PROOF. Since C is projective and C' is acyclic, we can construct (t'i)lO as in
the proof of the fundamentallemma 1.7.4 so as to commute with boundary
operators; indeed, the condition 00 1'0 al = o is exactly what is needed to
start the inductive constructio cf. 1.7.3a. Similarly, since C is acyclic and
admissible (by 3.2) and C' is relatively injective, (1' i)i  D can be extended to
negative dimensions by 2.4. O
(5.1]) Lemma. Let C be an admissible acyclic complex oflLG-modules. For any
projective ZG-module P, the complex P @ C (with diagona/ G-action) is con-
tractible as a complex ofZG-modules.
PROOE It suffices to prove this for P = lLG. By 111.5.7, we have ZG (8) C
isomorphic to the induced complex ZG @ C', where C' is C regarded as a
complex of abelian groups. Since C' is contractible by hypothesis, it follows
that l6 @ C is contractible. D
In view of5.9 and 5.10, tbe construction of A: F -+ F @ F is reduced to the
construction of a map IX = (ap): F o -+ npliOz F fJ @ F _ p such that (i) oa.1 B = O
and (ii) (e @ f) = E, where Bo c F o is the module of boundaries. Let
a = dI' @ Fq: Fp @ FlJ - Fp-1 (8) Fll and lei o = (-l)PFp @ d(l= Fp @ F'l
-+ F II @ F 11- 1. Then oa: F o -+ n,... z F p- l @ F _ p has components ò. _ p al' +
ò;_ l, 1 - JJ IXp  1. So (i) is equivalent to (i') «3' al' + o" a" _ 1) I Bo = O, where we
have omitted the subscripts on ò' and o" to simplify the notation. We now
start constructing tX by taking IXD: F o - F o @ Foto be any map satisfying (ii);
this is possible because F o is projective. Assuming that p > O and that IX p-l
has been defined, we wish to define !X,p: F o - F p @ F _ p so that the diagram
Bo
Q:plBo ..", .'' 1
_, - PIBo
-- '
11r"'
 b
F,,0F_p .Fp-1 @E_p .Fp-z@F_p
commutes, where p = -o" a. p_ t. I claim tha o' fJ I Bo = O. In fact, if P > 1 then
we can assume inductively that (o' ap_ l + Ò" cxp_ 2) I Bo = O, so that on Bo we
have
ò'fJ = - a' éY'r:J.P-1
= ò"iJ'a.P-l
= -éY'ò"al'-2
- O.
- t
by definition or p
because a' and ò" anti-commute
by tbe inductive hypothesis
...
!
l
5 Properties or It"
141
and if p = 1 then on Bo we have
ò'fJ = -i)'i)"cxo by definition or p
= -(do @ do>ao by definition of a' and ill
= -(f] @ ")(1; @ f)ao because do = flt
= -(1,1 @ '1)8 by (ii)
= o because slBo = o.
This proves the claim. By 5.11 the complex (F. @ F _ p' ò') is contractible. We
can therefore choose a contracting homotopy h and set fXp = hp to complete
tbe inductive step. A similar argument constructs IXp for p < O by descending
induction, and the proof of 5.6 is complete. D
<
Fina1ly we remark that there are also cap producrs in the Tate theory. No
new difficulties arise here so we confine ourselves to a brief discussion. There
18 a map
JItomu(F, M) -+ .1fom«F <8> F) <& N, F 0G (M @ N)
given by u....... id  u @ idN. This corresponds to a map
}':G(F, M) @ «F  F) @c N) -+ F (8)G(M (8) N).
We now define the cap product
.Jfb-mG(F, M) @ (F @G N)  F @G (M @ N)
to be y composed with
id @ (1\ (8) id): .JIt'omG(F, M) @ (F @G N) -+ Jl&rnG(F, M) (8) «F 0 F) @G N),
where A: F -+ F <8> F is any complete diagonal approximation. This induces a
well-<lefined cap product
11* (8) R* -+ Il.
with the usual properties.
EXERCISES
1. Show that the cup product on R. is compatible wjth that defined in Chapter V on
H*. More precisel}', we have a natural map H* --+ fl- whìch is an isomorphism in
posjtjvedimensionsand an epimorphism in dimension O; show that this map preserves
products.
2. Let fl*(G)(p} be the p-primar}' component of fl*(G) = R*(G. Z). so that we have
8*(G) = EBpIIGIR*(G)!P) by the analogue of 111.]0.2.
(a) Show that each il*(G)(p, is an idea] in R*(G), hencc so is EBI,ITP H*(G){Q). Con se-
quently. R*(G}(p, is a quotient ring of n*(G) via the projection R"(G) -- n"'(G}(PI.
[Note: il*( G)(p) is nor a subring or 11*( G); indeed. although the inclusion 11*( G)(P) 
11'*(G) does preserve products. it does not preserve identity elements.]

142
VI Cohomology Theory of Finite Grcups
(b) Show that, ther:e is a rin isomorphism fl3{G)  Dp!lGlfl*(G){P)9 where each
factoT on the nght IS a nng Vla (a), and multiplication in the product ìs done com-
ponentwìse.
'..'
6 Composition Products
 n
.
l ,.
..:
,
,
As in ordinary cohomology (V.4) there is a second method or defining
products, based on composition of chain maps. We begin witb the analogue
or V.4.1 :
(6.1) Proposition Let G be a finite group, let F be an acyclic chail'l complex 01
projective llG-modules, and let e': FI -+ 7L he a complete resolution.
(a) For any G-module M, the map 8' @ M: F' @ M -+ M induces a weak
equivalenceJt'o.mc;(F!t F' @ M) -+ .1'&m(;(F, M). In particular, ifF is a complete
resolutwl1 then R*(G, M)  H*(KomG(F, F' (S M) = [F, F' @ M]*.
(b) lf F L't 01 finite type then r.' (S M induces a weak equivalence
-, ...
F @G (F @ M) --t F @G M = F 0G M.
In particular, if F is a complete resolution then
R.(G, M) :=:;:; H.(F @u (F' @ M».
PROOF. For(a)wemust prove [F, F' (9 M]"":::' [F, M]"forallneZ.Replacing
F by its n-fold suspension I:/fF, we reduce to the case n = O; thus it suffices to
show [F, F' @ M]  [F, M]. Let u: F -+ M be a chain map. This means that
u is a map F o  M such that ud J = O, where d is the boundary operator in F.
Since Fo is projective, we can lift u to a map 'l'o: Fo -+ F @ M such that
(rl @ M)t'o = u. Recall that the boundary opecator do: Fc -+ F'_ t adrnits a
factorization do = fI'e', and consider (d @ M)Tod1:
F dI F
l ) o
-.
}
 :.
, .
'-
,
}
;-
.
I- j
-
!
- - -
'. l

, -' ;
Fo@M
]
"0M M r- ,,'0M F' 101 M
n .. -1\CJ .
d@M T
:
Wc have
(d @ M}r:od1 = (l'J' @ M)(e' @ M)-rod1
= (l'I' @ M)udJ
= O.
,
so we may apply lemma 5.10 to conclude that 'lo extends to a chain map
T: F -+ F' @ M. [Note that Fj @ M is relatively injective; indeed, itsuffices to
observe that if L is a free ZG-module then L @ M is an induced module by
..

6 Composition Products
143
,-
111.5.7.] This proves the surjectivity of [F, F1 @ M] -+ [F M]. Now suppose
'f: F -+ F' (S M is achain map such that (8' @ M)-r: F -+ M is null-homotopic
Le., such that (l @ MJto = sdo for some s: F - l -+ M. Lift s to a map h_ 1 :
F _ 1  Fé. @ M satisfying (t' @ M)h_1 = s, and consider (do 0 M)h_1 do:
F 40 F
o  -1
..1 :/ l'
Fo@M
l
(' 0 M   M r- ':' 0 M J F'_ l @ M.
T
4"0011I
We bave
(do @ M)h_Jdo = ('1' (9 M)(£' @ M)h_1do
= ('l @ M)sdo
= ('1' @ M)(t' 0 M)to
= (do @ M)-co.
As in the proof or the fundamental lemma 17.4, this is precisely what is
needed to extend h_ l to {hik: - l satisfying hd + (d' (9 M)h = t. cf. I.7.3b. We
can now use Proposition 2.4b to extend (hai O!: -1 lo a null-homotopy h or 'f.
Thus [F, F' @ M] -+ [F, M] is injective, whence (a). To prove (b), consider
the dual F = :l&»nG(F, ZG). This is again projective and acyclic (cf. proof or
3.5), and we bave a duality isomorphism F G -  .1'&mG(F, -) by exercise
1 below. So (b) rollows from (a). O
Remark. It is perhaps surprising at first that we obtain the same conclusion
here as in L8.5 and 1.8.6, since our map t' @ M is not a weak equivalence. On
tbe other hand. we have a very strong hypothesis on F (acyclicity), and this
compensates for the failure of e' @ M to be a weak equivalence.
It is now a routine matter to imitate the definitions in V.4: Let e: F -+ l
and f.': F' -+ 7L be complete resolutions9 and let rt: F" -+ 7L be either a com-
pIete resolution or tbe identity map Z -+ Z. Then there are chain maps
o-mG(F', F'I 0 M) (9mG(F F' 0 N) -+ G(F, F" @ M (S N)
and
G(F', F" @ M) @ (F @G (F' @ N) --. F @Q (F" @ M 0 N),
defined as in V.4.2 and V.4.4. These induce composition products
{J.(G, M) (S fl*(G, N) -+ fl*(G, M @ N)
and
11*(G, M) @ R.(G, N) -+ fJ .(G, M @ N).

144
VI Cohomology Theory or Finite Groups
The proof or V.4.6 now goes through essentia1ly verbatim, and yields:
(6.2) Theonm. The composition products defined above coincide with the cup
and cap producrs.
EXERCISES
1. Let R be a ring.let C be a chain complex offinitely generated projective R-modules.
and let C be the dual complex JfbmR( C. R) or finitely generated projective right
R -modu[es. Prove the folJowing analogue or 18. 3b :
For any chain complex C of left R-modules. there is a chain isomorphism
rp: C @R C mll(C, C). [Hint: Define CfJp9.: Cp @R C...... HomR(C _P' C) by
< qJpq,(u @ x'), x) = ( - l )pq(u. x)x' for li E Cp = HOffiR(C _". R). x' e C. x E C_P'
It is an 1somorphism by 1.8.3b. Now set lp" = Op+I1="(,O"\!: np....Ij"'.. Cp 0 C--
f1p+q=.. HornR(C -p' C) = ..m.:.omR(c' C)".] Similarly prove analogues or 1.8.3c and
1.8.3d.
2. Let C and C be as in exercise l. For any Z E (C @R C)" and any chain compJex C.
there is a graded map f/J : Jtb"-'Il(c. C) -+ C @ C of degree n. given by I/J,,(u) =
(jdl"@ u)(z). Sbow that there is a cycle Z E (C R C)o such that for any C t/lz is the
inverse of the isomorphism (f): C @R C'  .Jtb#lR(C. C) of exercise 1. [Hint: Let
Z E (C @ R C)o correspond to idc under r:p: C @  C .; Jf$mR( C. C). Then z =
(Zp)PII!Z' where Zp E C_p @R C". == (C,,)"- @R Cp corresponds to (-1)" idcp under
the canonical isomorphism (Cp). @ CI'  HomR(Cp. Cp). Checking the det1nition
or rJI z. you should find that 1jJ.t is ind uced by ma ps l./t pq: HomR( C _ p' C) -+ C p @ R C
given by I/J pq( u) = ( -1 )(q... P"(idcp @ u)( z _ ,). Exercise 7 or I.g now shows thal
'" pq == tp l .]
*3. Is the finiteness hypothesis in 6.tb oecessary?
-4. Let F be a complete resolutìon.lel n be an arbitrary integer. and let Z be the module
Zn(F) of n-cycles. Show that there is a ring homomorphism HornG(Z, Z) - [F. F]
which is surjec'ive and has as 1t5 kernel thc group I of maps Z - Z which extend to
maps F" - Z. Deduce that I 1s a l-sided ideaI in HOffiG(Z, Z)and that there is a ring
isomorphism HomG( Z. Z)/ 1  lìÙ( G. Z) = l/I G 1.1l... [H int: U se the technìq ues of
the proof or 6.1.]
7 A Duality Theorem
It follows from the existence of finite type complete resolutions that the
groups R;( G, 1:) are finitely generated. On tbe other hand, they are annihilated
by I G I  so they are finite. The main purpose or this section is to show thai
Ri(G, Z) and R- i(G Z) are dual finite abelian groups; more precisely, we wilJ
show that the duality belween them is given by the cup product
fli(G Z) <8> fl-i{G, ) --? RO(G, Z) = Z/IG I . lL.
I .
.
7 A Duality Theorem
145
.,
We will give a proofwhich uses Tate homology theory and the cap product;
a different proof w hich uses onIy cup prod ucts and dirnension-shifting. ean be
found in Cartan-Eilenberg [1956], XII. 6. We begin by reviewing the rele-
van t duality theory for a belian groups.
For any abelian group A wc set AI = Hom(A, O/lL). Sinee Qlli. is injective,
thc (contravariant) rUDctor ( )' is exacr. In case nA = O for some n > O, wc
bave Hom(A, QJI/Z) = Hom(A, n-I Z/Z)  Hom{A, Z/lIlL}; hence we ca[)
identify A' with Hom(A, lL,J In particular, if A is cyclic of order n then so is A'.
Consequently, A'  A (non-canonically) for any finite abelian group A. As
usual, we must pass to tbe double dual to get a canonical isomorphism: There
is a natural map A -+ AlI given by a 1-+(1  f(a» which is an isomorphism
if A is finite.
A map p A Q9 B ,. Oj7L gives rise to a map p: A --+ B'; we will say that p is
a dualit y pairing if p is an isomorphism. or course p also gives rise to a map
p: B -+ A', which is equal to the composite B -+ BU £+ A'. It follows easily that,
if A and Bare finite, p is a d uality pairing if and only if p is an isomorphism.
Thus a duality pairing betweenfinite abelian groups provides an isomorphism
of each one with the dual of the other. Thc canonical example of a duality
pairing is, or course, tbe evaluation map A' (9 A -iO C/lL, where A is arbitrary.
Finally, if nA = o and nB = O, then wc can similarly speak of duality pairings
A (9 B -+ 7ln.
If G is a group and M is a G-module, then M' = Hom(M, O/lL) inherits a
G-action in the usual way: (gu)(m) = u(g-tm) for g € G, U E M', m E M. There
is an evaluation pairing p: Hi(G. M') @ Hi(G, M) -+ O/l., obtained by
composing the pairing <" .) of V.3 with tbe evaluation map M' @G M -+
O/Z.Similarly,ifGisfinite,thereisapairingp: fti(G, M') @ R;(G, M) -+ Q/l..
,.
'.
'.
(7.1) Proposition. The pairings p ami /J are duality pairings. Thus Hi(G, M') 
HI..G, M)' for any G and M, and R'(G, M')  fl1(G, Mr fG isfinite.
PROOF. Let F be a projective resolution of 7l over lLG. Then (F, M') =
JleomG(F, Hom(M OjZ)  Jf'omG(F @ M, Q/lL) = .1t'bm(F @G M, 0/71) =
(F Q9G M)'. Since ( r is exact we can pass to homology to obtain H*(G, M')
 H.(G, M)'. It is easy to check that this isomorphism corresponds to the
pairing p. The argument for p is thc same. D
Assume now that G is finite. As we remarked briefty at the end of, there is
an isomorphism Jìi(G, M)  ft-1-i(G, M). We wish to make this more
precise by showing that there is an isomorphism of this type given by cap
product with a L4 fundamen tal c1ass':
(7.2) Proposition. There is an element Z E R _ 1 (G l) such that the cap product
map n z: fìi(G. M) -+ il-1-i(G, M) is an isomorphismfor any G-module M.
PROOF. Let (F, à) be a complete resolution of finite type, with do = J']£ as in 3,
and let F be the dual complexK(F, ZG). Thus F1 = (F _j)*. Then F is stili

146
V1 Cohomology Theory or Finite Groups
proje<;tive and acyclìc (cf. proof Or 3.5), and thc boundary operator F l -+ F o
is the composite
(F -1)*  71 4 (Fo).,
Except for indexing. then. F is a complete resolution; more preci seI y , F is tbe
suspension LE or a complete resolution E. We now apply the duality isomor.
phismJfO?nt}(F, M)  F C8>G M of I.8.3b. This yields
Ri(G, M)  H -i(F C8>G M) = H -j _ l(E @G M) = fl-i-t(G, M).
<
t -
.
.
,
$
i
..
-
"
, 
<.
".

-,
.
I. ..
.,
t
.
.-.!'
.'

-
:
,
.
,
<,
'-
"
.

'.-
I claim that this isomorphism is given, up to sign, by cap prodUCI with a
universal element Z E fI - 1 (G, l). There is an easy abstract proof of this,
outIined in exercise 1 below. We wiII give here a direct but somewhat tedious
proof, based on tbe composition prod uct and the exercises of .
Let Wo be the O-cycle in F 0G F which corresponds to idF under the
canonical isomorphism F  F  :¥Ii»ìlG(F. F) of exercise 1 of 96. Using the
notation and resuIt or exercise 2 of 6, we find that the isomorphism
.:¥fl.wzQ(F, M)  F  M
used above is the map wo' Now F @G F = I:.E @o F. so Wo can also be
regarded as a (-l)-cycle w -1 in E @G F. Moreover, one discovers by
cheçking definitions that "'W-I: :Kbmc(F M)i -+ (E @G M)i-l is equal to
(-lttJ1W(}:-*,&'Ft(i(F, M)i - (F Q9G M)t. Thus the isomorphism Ri(G, M) _
ft - i - l (G, M) of the previous paragraph is induced up to sign, by I/J w _ J' On
the other hand, Vt W-I is precisely the composition product with w -1 as
...
defined in ; it therefore induces the cap product with the cJass Z E H-l (G, Z)
represented by w _ l" This proves the claim and the proposition. O
Remark. The element z above is necessarily a generator of the cydic group
R _ 1 (G, Z) = Zj I G' Z; for the cap product isomorphism n z: RO( G 1.) -+
R -l(G, Z) takes l E flo(G,.l) = .l/I GI7L to t n z = z.

(7..3) Corollary. For an}' G-module M, tlle composite
fìi(G, M') @ R-I-i(G, M).:+ Hl(G, M' C8> M)  fJ-I(G, Q/lL.)
<:
, 
is a duality pairing, where ex. is induced by lire canonical pairing 0:: ME C8> M
- QjZ.
L
..
,
7 A DuaJity Theorem
147
,
. IX: M' C8>cr M -+ QjZ is induced by 1:(. Since a<u v n z) = a(u u v, z) =
(a*(u uv), z), it follows that the composite
fli(G. M') rg; fl-l -i(G. M)::::' fI-l(G, M' @ M)  fJ-l(G, O/7L) <-'%1 Q/Z
is a duality pairing. To complete the proof, wc need only note that < - z)
maps fl- l (G, Q(lL) isomorphically onto I G ,- l71j71. Indeed, < -, z> is the
composite
fJ-1(G 0/1:.)  Il -l(G, 71)' -+ Q/li.,
where the fiest map is given by 7.1 and the second is evaluation at the generator
Z E R -l(G, 71). And it is obvious that if A is a finite cyclic group then evaIua.
tion at a generator or A gi ves an isomorphism A'  I A ,- llLjlL. D
We can now prove oue main result:
(7 4) TheoreDl. 1he cup producI fIi( G, Z) @ fl- i{ G, Z) _ fIo( G, Z) =
Zj l G I . 7L is a dualit y pairing.
PROOF. Since R*( G, Q) = O, tbe coefficient sequence O - 7L -+ C  Q/IL -+ O
yields an isomorphism o: Ri(G, O/li)  Rj+1(G, Z) for allj. Moreover, we
bave (cf. V.3.3) a commutative diagram
fJi-l(G, 0/1') @ R-i(G, Z)
,@+
Ri(G, Z) @ fl-i(G, Z)
u ) R-1(G, Qj7L)
'l-
u  flO(G, Z).
Since the top row is a duality pairing by 7.3 (with M = Z), so is the bottom
row. D
Remark.. The fact that fll(G, Z) and R-i(G, l) afe dual to ODe another can be
proved quite easily by means of the uni versaI coefficient theorem and the
interpretation of fJ* in terms of H* and H. ' [Recall that Ext(A. Z)  A' if A
is finite, cf. exercise 2 below.] What is not obvious, however is that the cup
product provides a duality pairing. This is the essential content of 7.4.
I EXERCISES
(N ote that the statement makes sense beca use
fì-I(G. QjZ) = ker{N  (4JjZ)G -+ (Q/Z)G} = IG r-l7i.jZ
and fJ.(G. -.) is annihilated by I GI.)
PROOF. By 7.1 and 7.2 we haveRi(G M/)fì.iC.G. M)'R-I-i(G. M)'. The
composite isomorphism is given explicitly by U H (v 1---+ o:<u, v n z», where
,
;
,
I. The purpose or this exercise is to give an alternate proof that the isomorphism
R i( G. M) .!:'t fj _ l _ i( G. M) establìshed in the first part of tbe proof or 7.2 is given, up
to sign. by C'dp product with a fixed element Z E H _ leGo Z). Cali that isomorphisrn tp
and let z = (f'{l).
(a) Show that (f) ìs natumJ and that it iscompatible wìth cormecting homomorphisrns
in long exact sequences.

148
VI Cohomology Theory or Finite Groups
(b) Show that ({J; n{J(G M) - R -l(G. M) is given by cap product with z. [By
deflnition or z, cp and n z agree on 1 E flO(c. Z). lf now M and LI E fIO(G, M) are
arbitrary, there is a coefficient hornomorphism 7L -+ M such that l  u under the
induced map RO(G. Z) -. f/o(G, M). so tp(u) = u tì Z by naturanty.]
() Now use dimension-shifting to show that ({J and n 2 agree in alJ dimensions. up to
sign.
2. For any abelian torsion group A, pro\'e that Ext(A, Z)  A'. [Hint: Look at the Jong
exact Ext*(A. - )-sequence associated to the short exact sequence O -+ 7l. -+ Q ......
Cj71 - O. Equivalently, compute Ext(A. Z) by means of the .jnjective resolutìon
O -+ 7L - D -+ O/L - O of Z.]
3. If M is a. G-module which is free as 8:11 abeJian group, show that
flt(G. M*) (8) n-I(G. M) .':;.1l0(G, M* @ M) -+ RO(G. Z)
is a duality pairing.
. -
4. Let k be an arbitrary commutative cing and Q an injective k-module. Let A' =
HOffi,.,(A, Q) for any k-module A. If M is a kGmodule, show that the pairing
fìi(G. M')!Z> R-l-r(G. M)  fì-l(G. M' 0 M) --+ n-1(G, Q) 4 Q
induces an isomorphism Ri(G, M')  /ì-l-i(G. M)'.
8 Cohomologically Trivial Modules
The theory to be presented in this section is due to Nakayama and Rim. Our
treatment is based on Serre [1968].
A G-module M (where G is finite) is said to be cohomologically trivial if
{P(R, M) = O ror ali i E Z and ali H c G. For example, any induced module
lLG @ A is cohomological1y trivial by 5.3, since ZG @ A is stili induced when
regarded as an H-module for any H c G. [This is an easy special case ofthe
double-coset formula III.5.6b.] In particular, if k is an arbitrary commutative
ring, it follows that any free kG-module is cohomologically triviaJ and hence
that any projective kG-module is cohomologically trivial. Due goals in this
section are (a) to prove that, conversely, under suitable hypotheses cohomo-
logically trivial kG-modules are projective; and (b) to find simple criteria Cor
checking that a module is cohomologica1ly triviaL These results bave applica-
tions to class field theory (cf. Serre [1968]) as well as to algebraic K -theory
and homotopy theory.
We begin with the case where G is a p-group for some prime p. In this case
everything will be deduced from the following simple result:
I.

(8.1) Proposition.lfG is a p-group and M is a G-module in whù:h everyelemenr
has order a power o/p, then MG -:F o.
I
,
,
!.
8 CohomologicaJly Trivial Modules
149
<
PROOF. We may assume that M is finitely generated as a G-module, hence also
as an abelian group. But then M is finite, of arder a power of p. Since every
G-orbit in M has arder pD for some a > O, it follows that I MG I = I M r  O
(mod p), so that MG '# o. O
.>
(8.2) Corollary.lfk is afield oJ characteristic p. G is a p-group, and M is a simple
kG-module, then M  k, with trivial G-action.
.'
(Recall that a simple module is a non-zero module which contains no
proper non-zero submodule.)
PROOF. SinceMG :# O by 8.1, we must bave MG = M, so G acts trivially on M.
Simplicity now implies that dimlc M = 1. O
Let l be the augmentation ideai of kG, i.e., tbe kernel of the k-algebra
homomorphism B: kG -+ k such that e(g) = 1 for ali 9 E G. We can a1so
describe I as the annihilator of the kG-module k.
(8.3) Corollary. lf G is a p-group and k is a field oJ characteristic p, then the
augmentation idealI oJ kG is nilpotent.
PROOF. Choose a composition series O = Jo C J1 C ... c Jn = kG for kG,
regarded as a left kG-module. Thus eac:Q.J(ifa left ideaI and JJJt-t is simple
for 1 S i :S; fI. [The existence of a.composition series follows from the finite-
dimensionaJity of kG aver k.] By 8.2, l annihilates J JJi- l' hence l Ji c J i-I'
ConsequentlYt In = ['Il,. ç::; Jo = O. D
If G is cyclic of order q = pQ, for example, we can easily see directly that I is
nilpotent; for I is generated by t - l where t is a generator of G (cf. 1.2,
exercise lb)t and (t - 1)'1 = (1 - 1 = O since char k = p.
(8.4) Corollary. Let G be a p-group and k a field DJ charac teris tic p. If M is a
kG-module such that Ho(G, M) = 0, then M = O.
PROOF. It is easy to see that Ho(G, M) = M/1M (cf. first paragraph or II.2).
Thus Ho(G, M) = o=:- M = 1M  M = ]nM for ali n. In view of 8.3, tbis im-
plies that M = O. ' O
We can now achieve, for kG-modules as above, the two goals (a) and (b)
stated at the beginning or this section:
(8.5) Theorem. Let G be a p-group and k afield oJ characteristic p. The jollowing
conditions on a kG-module Mare equivalent:
(i) M isfree.
(ii) M is projective.
(iii) M is cohomologically trivial.
(iv) HI(Gt M) = O.
(v) f/i(G, M) = O for some i € Z

150
VI Cohomology Theory or Finite Groups
PROOF. Clearly (i) =:o. (ii) =- (iii) => (iv)::::;.. (v). To complete tbe prooC we wiU
show (iv)  Ci) and (v)  (iv). Choose a k-basis Cor the vector space M G, and
tift the basis eJements to elements mj U E J) in M. Let F be a free kG-module
with basis (ej)jEJ and define f: F -+ M by f(ej) = mJ' By construction, then,
Ho(G, f): Ho(G, F) -+ Ho(G, M) is an isomorphism. By right exactness or
Ho(G, -), it follows that Ho(G, coker f) = O, and 8.4 now implies that
coker f = O. Consider now the long exact homology sequence associated to
0-+ ker f -+ F.-. M -+ O. If H1(O, M) = O then the fact that Ho(G, f) is an
isomorphism implies that H o( G, ker f) = O. Applying 8.4 again, we conclude
that ker f = O, SO f is an isomorphism. Thus M is free and we bave proven
(iv) =- (i).
Assume now that (v) hoIds. By dimension-shifting we can find a kG-module
N such thatll"(G, N)  fln+i+Z(G, M)forall n E 7L. [Notethat thedimension-
shifting argument does not take us out ofthe category of kG-modules, ause
lG (8:J A is a kG-module if A is a k-module.] In particular, H l(G, N) =
fl-2(G, N) = fJi(G M) = O, so N is free bytheprevious paragraph. But then
R*(G, N) = O, hence fJ*(G, M) = O, hence H 1 (G, M) = O. Thus (v) :$ (iv).
D
Note that the theorem implies, in particular, that Hi( G, Zp) ':f:: O for alI i > O
if G is a (non-triviaI) p-group.
Remarks
1. Corollary 8.2 can be interpreted as saying that l is the Jacobson radical
of kG. From this point or view, 8.3 is simply the well-known result that the
Jacobson radical of an Artin ring is nilpotent, and 8.4 is a special case of
"Nakayama's lemma." Tbe eq uivalence of (i), (ii), and (iv) in 8.5 could also be
stated in a more generai context, as a theorem about modules over a local
ring. [It follows from 8.3 that kG is a (non-commutative) Iocal ring, with l as
its unique maximal ideaI.]
2. The proof above yields the following more precise version of the
implication (iv)  (i): For any kG-moduIe M, dim. H o( G, M) is the minimal
number of generators of M and dimi; H 1 (G, M) is tbe minimai number of
defining relations among any minimal set of geilerators.
Next wc wiII consider more generaI G-modules M, stili assuming that G is a
p-group. Let MI' = M @ ?Lp = MjpM.
(8..6) Lemma. Let G be a p-group and M a G-module which is p-torsion-free. F or
any i E ?L. Ri(G, MI') = O if and only if fli(G, M) = fJi+ I(G M) = O.
PRODE From tbe short exact sequence O -. M .!,. M -+ M" -+ O we obtain an
exactsequence
Ri(G M) .f.. Rl(G, M) -+ Ri(G. M,,) -+ fJH I(G, M)  RH I(G, M).
.
..
....
'''.
.,
..:"
.':
;
.
! .
.
I
, .
I.
i
8 Cohomologica.l1y Trivia] Modules
151
If fll{G, M) = Il'+ l(G, M) = O, then clearly this implies that Bi(G, M,,) = O.
Conversely. if Ri( G. M p) = O, then we conclude that f/i( G. M) is p--d.ivisible
and fJi+ l(G. M) is p-torsion-free. But fl*(G, M) is annihilated by I G I" which
is a power of p, so fli(G, M) = Ri+l{G. M) = O. D
(8.7) Theorem.. Il G is a p-group, then the following two conditions are equivalent
10,. any G-module M:
(i) M is cohomologicall y tri vial.
(ii) fJi(GJ M) = O lor two consecutive inregers i.
lf M is p-torsion-free, then (i) and (ii) Qre also equivalent to:
(iii) fli( G, M,,) = O lor some i.
(iv) M p is cohomologically trivial.
(v) M p isfree over Zp[G].
PROOF. Suppose fust that M is p-torsion-free. Then (i) => (ii) trivially;
(ii)  (iii) by 8.6; (iii) ç;> (iv) <=:> (v) by 8.5; and (iv) => (i) by 8.6applied to G and
ali ofits subgroups. If M is DOW arbitrary, then we can prove the equivalence
or (i) and (ii) by using dimension-shifting to reduce to the previous case; for
we can choose an exact sequence O -+ M' -+ F -+ M - O with F free Qver 7L.G
and then apply the previous case to M'. D
An important consequence of 8.7 is that cohomological triviality can be
checked (if G is a p-group) by looking only at R*(G, M), even though the
definition involves fl*(H, M) for arbitrary subgroups H c G.
We turn now to the case where G is an arbitrary finite group. Far each
prime p dividing I Gl, choose a p-Sylow subgroup G(p).
(88) Proposition. A G-module M is cohomologically trivial if and only if its
restriction lo G(p) is cohomologically trivial for each p.
PROOF. Tbe U onIy if"' part is triviaI. To prove the 'if" part, note first that ifthe
restriction of M to G(p) is cohomologically triviaI then fl*(p, M) = O for
any p-group P c: G. For we know that gPg- l £; G(p) far some g E G, so there
is a conjugation isomorphism R*(p, M)  f/*(gPg-l, M) = O. If R c G
is now an arbitrary subgroup, then we know [rom transfer theory (cf. IIl.IO)
that fl*(H, M) = EBp fJ*(H, M)(p) c.. EBp fì*(H(P). M) = O, where p ranges
over the primes dividing 1 H I and H(P) is a p-Sylow subgrou p or H. Thus M is
cohomologically tri vial. D
Com bining 8.8 with our previous results on cohomoiogicai triviality over
pegroups, we obtain a solution to probIem (b) stated at the beginning of the
section. In particular. we have:
(8.9) Theorem. A G-module M is cohomologically trivial if and Q1'Ily if jQr each
prime p there are two consecutive inteoers i such that 1t'{G(p), M) = O.

152
VI Cohomology Theory of Finite Groups
The solution to problem (a) (with k = Z) is given by the foUowing theorem
of Rim:
(8.10) Theorem.lfM is a G-module which is Z-free and cohomologically trivial,
then M is ZG-projective.
PROOF. Choose a short exact sequence O --. K --. F --+ M --+ O with F lLG.
free. We will prove that this sequence sp1its, so tha t M is a direct summand or
F and hence is projective. I c1aim that the o bstruction to splitting the sequence
lies in H1(G, Hom(M, K»). More preciseJy, we ha ve a short exact sequence or
G-modules 0-+ Hom(M. K)...... Horn(M, F) -+ Hom(1'J, M) --+ o since M is
Z-free, and this yields an eXac sequence HomG(M, F) --. HornG(M, M)-4
H1(G. Hom(M, K». [Recall that
Hornc( -. -) = Horn( -, _)G = HO(G, Hom( _, _ ».]
Hence tbe extension splits if and only if (j(idM) = O. Il will thererore suffice to
prove:
(8.11) Lemma. Let M and K be G-modules which are lL-free.lf M is cohomo-
logical1y trivial then so is Hom(A:l K).
PROOF. By 8.8 we need ooly consider the case where G is a p-group, in which
case it suffices by 8.7 to prove that Hom(M K)p is cohomologicalJy trivial.
From the exact sequem;e O -+ K -E. K -+ Kp -. O we get (since M is Z-free)
an exact sequence O -+ Hom(M K) .4 Hom(M, K) -+ Hom(M, Kp) -+ o. Thus
Hom(M, Kp)  Hom(M, Kp) = Hom(Mp Kp)' ButMp is Zp[G]-rree by 8.7,
so Hom(M p' K p) is induced (cf. III. 5, exercise 2b) and hence cohomologically
trivial. D
Remark. The group H1(G, Hom(M. K» which arose in the proor or 8.10 is
isomorphic to ExtG(M K) by 111.2.2. The reader familiar with tbe theory of
extensions of Itlodules will therefore not be surprised that this group contains
the obstruction to splitting the extension O -+ K -+ F -+ M -+ O.
Final1y, we will deduce from 8.10 a characterization (also due to Rim) of
cohomologically trivial modules which are not necessarily Z -free. If R is a ring
and M is an R -module, the projective dimension or M, denoted proj dim M or
proj dirnR A1, is defined 10 be the infimum of the set of integers n such that M
admits a projective resolution O -+ p" -+ ..., - po - M - O of length n. (In
particular, proj dim M = CI) if M does not admit a projective resolution or
finite length.)
(8..12) Theorem. The following conditions on a ZC-"wdule M ore equivalenr:
(i) M is cohomologically trivial.
(ii) proj dim M < 1.
(iii) proj dim M < 00.
9 Groups with Peciodic Cohomology
153
. .
PROOF. (i)  (ii): Suppose M is cohomologicaJly trivial, and choose a short
exact sequence 0-+ P1 -+ Po - M -+ O with Po ZG-projective. Then p. is
cohomologically trivial and Z-free, so P l is projective by 8.10 and (H) holds.
(H) =- (iii) trivially. (iii)::::;o. (i): Supposeproj dimM < 00 and let O -+ p" -+ .,.
-+ p o -+ M -+ O be a projective resolution of finite length. Breaking up this
resoIution into short exact sequences O -+ Zi --. Pj -+ Zt-l  O. one sees by
descending induction on i that Zi is cohomologically trivial. In particular,
M = Z _ 1 is cohomologically trivial. D
Rirn's resuIts can be used to provide examples of projective modules over
ZG. For example, suppose J c ZG is a Ieft ideai offinite index m, where m is
relativeIy prime to IGI. Then ICI is invertible in M = ZGjJ, so M is ço..
homologicalIy trivial. It now follows as in the proof of (i) => (ii) above that
J is projective, (Se e also exercise 7 of VIII.2 for an alternative proor that
proj dirn M < I when I G I is iovertible in M.)
EXERCISES
1. Give an example wbere n*(G, M) = O but M is not cohomologicalJy trivial [Hint:
Ta.ke G to be cycIic.]
2. Prove the folJowing improvement or 8,11: Jf M is cohomologicaUy trivial and lL -free
then Hom(M. K) is cohomologically trivial for any G-module K. [Use 8.10. Alter-
natively, choose an exact sequence o - L ---... F ---... K -+ o wJth F free. and apply 8. Il lo
Hom(M, L) and Hom(M, F).]
3. (a) U M and P are ZG-modules sueh that M is lL-free and P is lLG-projective, show
that any exact sequence O -+ P ---... E - M -+ O splits. [Hint: Argue as in the proof or
8.10. or, more simply, use the faet that P is relatìvely injective.]
(b) Using (a) give a direct proof (without using the resu)ts or this section) that
(iii) => (ii) in 8.12.
.
4. Lei G be a group such that there exists a free IÌnite G-C W- complex X with H .( X) 
H .(5U- l). Prove that G admits a complete resolution which is periodic of period 2k.
[Hint: lf B is the module or (2k - I }-boundaries or C ..(X), then C 211. _ 1 (X)j B is
lL-free and has finite projective dimension.]
\.
9 Groups with Periodic Cohomology
A finite group G IS said to bave periodic cohomology if for some d#;O there is
an element U E R4( G Z) which is invertible in the ring R*( G  Z). Cup product
with u then gives a periodicity isomorphism
u u -: Rn(c, M) .::. fln+4(G, M)

154
VI Cohomology Theory of Finite Groups
for ali n E 7L. and all G-modules M. In particular, taking n = O and M = lL, we
see that Rd( G, Z)  ?LI I G I . 7L. and that u generates fl4( G  Z).
If we know that a group G has periodic cohomology then the task of
computing R*(G) is obvjously enonnously simplified. It is therefore of
interest to find criteria for deciding whether G has periodic cohomology. and
that is what we will do in this section.
 .
.
, 1
-
",
-
\: '.
'.
..
r -
;. 
- "

,
.L:
(9.1) Theorem. The following conditions are equivalent:
(i) G has periodic cohomology.
(ii) Thereexistintegersnandd,withd -# O,suchtharflll(G, M)  fJlI+d(G, M)
far ali G-modules M.
(iii) Eor some d =F 0" fJd(G lL)  ZII GI. lL.
(iv) For some d =F O, fld(G, Z) contains an element u oJ order I GI.
PROOF. (i) =- (ii) trivially. If (ii) holds then we have fJ"(G, M)  fJrt+4(G, M)
for ali n E Z by dimension-shifting; taking n = O and M = ?L. we obtain (iii).
(iii) => (iv) trivially. Finally, suppose (i v) holds. Then there is a map fld( G 1::) -+
Q/lL such that u 1--+ l/I GI mod lL (cf. proof of III.4.3). By the duality thcorem
7.4, this map is given by
fl4(G, lL) -U(J flo(G lL)  IGr ll/Z '* Q/l
for some v E fl-4(G, lL). Thus uv = l, whence (i). O
.,
..
.
! "
.
,
..
,
"
.-
,
';
.
,.
For example, if G acts freely on S2k-l as in 16, t-hen G obviously admits a
periodic complete resolution of period d = 2k, so 9.1(ii) holds and G has
periodic cohomology. [Alternatively, one can show directly, without using
9.1. that fJd(G Z) contains an invertible eJement; cf. V.3. exercise 3.]
One way to obtain examples of this is to start with a linear action of a finite
group G on an even-dimensional real vectOT space V, such that G acts freely on
V - {O} (i.e." such that l is not an eigenvalue of any non-trivia1 e1ement
g E G); such an action is called a fixed-point-free representation of G. By
choosing a G-invariant sphere S  V - {O} (e.g. the unit sphere relative to a
G-invariant inner product on V)" we obtain a free action of G on an odd-
dimensional sphere. In order to apply 1.6 of course, we rnust verify that S can
be chosen to ha ve a G-equi variant C W -structure. This follows from a general
triangulation theorem., but in the present case there is a much more elementary
argument, due to Illman [1978]: Let (ei)l <i<2k be a basis for JI and let C be
the convex hull ofthe finite set {+ gej: g E G, 1 < i < 2k}. Then C is a convex
celi containing O as an interior point, hence its boundary S is a topological
sphere in V - {O}) with a CW-structure given by thenatural (rectilineaT)faces
of C (see, for example, Hudson [1969] Chapter I). Since G acts linearJy on V
and maps C into itself it is clear that S is a G-complex.
-.
I
r
-..
i. '.
. .
, ,.
, ..
, .
.: ,;
. ,
,-
r
." .'
I
I.. .
, '
- .
;.
. ,
Remark. The question of the existence of an equivariant CW-structure is
aCUlal1y a ree! herring. For one can show, without assuming an equivariant
..
,;
"
1
9 Groups with Periodic Cohomology
155
c W -structure, that if G acts freely on Sn - l then 11*( G) is periodi c of period
2k. A proof wiU be outlined in exercise 4c of VII. 7.
We can now give some specific examples of groups with periodic 00-
homology:
(9.2) EXAMPLES
1. Any finite cyclic group G adrnits a 2-dimensionaJ fixed-point -Cree
representation as a group of rotations, hence fì*(G) is periodic of period 2.
2. Let [H] be the quaternion algebra IR EB lRi $ IRj $ IRk. If G is a finjte
subgroup ofthe multiplicative group IHI* then the multiplication action of G
on IHI yields a 4-dimensional fixed-point-free representation of G, since!HJ is a
division algebra. Thus fl*(G) is periodic or period 4. The most obvious
examples of finite subgroups or !HI* are the generalized quatemion groups
Q4,., (m > 2) which we studied in *IVA. In addition to these (and or course,
the cyclic groups), there are precisely three finite subgroups of IHJ*, up to
conjugacy (cf. Wolf [1974], 2.6): the binary tetrahedral group (of order 24),
the binary octahedral group (or order 48), and the binary icosahedral group
(of arder 120). The latter group G is particularly interesting because it is a
perfect group, Le., Gob = O. Using Poincaré duality, one can deduce from this
that the quotient manifold S'3/G has the same homology as S3. This was
discovered by Poincaré [1904]; it was the first example of a 3-manifold
homoJogicaJly equivaIent to S3 but not homeomorphic to S3.2 Another
interesting fact about the binary icosahedral group G is given in the following
surprising theorem ofZassenhaus (cf. Wolf[1974], 6.2): Up to isomorphism,
G is the unique perfect group which adrnits a fixed-point-free representation.
3. Let m n. and r be positive integers such that r" = l mod m. Assume that
the following two conditions hold: (a) m and n are relatively prime; (b) if k is
the multiplicative order of r mod m (so necessarily kl n), then every prime
divisor or n divides n/k. Let G be the semi-direct product lm><l lLft  where the
generator of ZII acts oh Zrn by multiplication by r. [ claim that G admits a 21£-
dirnertS10nai fued-point-free representation and hence fJ*(G) is periodic of
period 2k. To see this. note first that the subgroup A c lL" of index k acts
triviaUy OD lLmtso G contains C = 7l.", x A, which is cyclic by (a). From (b) we
see that A contains every subgroup of Zn of prime order and it follows easily
that C contains every subgroup of G of prime order. Choose a fixed-point-free
representation of C on a 2-dirnensional vector space W. and fonn the induced
2 Poincaré had asked jn an earHer paper whether such a 3-manifoJd could exist. In view of this
exampJe. Poincaré reformulated his queslion. ad(Hng the oondition that the manjfold be sìrnply-
conneçted. Tbjs question, Of ratheT tOO assertion that there can be no such manirold. has since
become known as the Poincare oonjecture.

156
VI Cohomology Theory of Finite Groups
module V = ZG @zc W = IRG (8)RC W. Since C <J G, we have Resg V =
EBl'le G/C gW by 1I1.5.6b, and each gW is cleady a fixedpoint-free representa-
tion of C. Consequently, V is a fixed-point-free representation of C. But then
V is also fixedpoìnt-free as a representation or G, for a non-trivial isotropy
group Gt; (v E V - {O}) would contain a subgroup of prime order and hence
would meet C nontrivially.
See Wolf [1974] for a complete classification or groups which admit a
fixed -point -free representation.
..
.,'
I
"
.'
':
I-
.}
:
'.
'.
Remarks
-
......
1. If G has periodic cohomology then so does any subgroup H. This follows
from tbe fact that the restriction map fl*(G, Z) ...... ft*(H, lE), being a ring
homomorphism must take invertible elements to invertible elements.
[Alternatìvely, use Shapiro's lemma.]
2. If G is abelian but not cyclic, then G does not have periodic cohomology.
Far G rnust contain a subgroup isomorphic to l" x ?Lp JOT some prime p, and
direçt calculation (using tbe Kiinneth formula, far example) shows that
f!"('Zp x 7Lp; ?Lp) has Zp-dimension n + l for n > O and hence is not periodico
l"

We retum now to the generaI question or finding criteria for a group to
have periodic cohomology. In case G is a p-group, the question is completely
sett1ed by the foIlowing result. Reca)) that an elementary abelian p-group of
rank r > O is a group isomorphic to Z = ?Lp x ... x 7Lp (r factors).
(9.3) Proposition. lf G is a p-group for some prime p, t hen Che following con-
ditiol1s are equivalent:
(i) G has periodic cohonwlogy.
(iì) Every abelian subgroup ofG is cyclic.
(iii) Every elementary abelian p-subgroup ofG has rank :s; l.
(iv) G has a unique subgroup oJ order p.
(v) G is a c yclic or generalized quaternion group.
.
\' :
. .
.-
"
.'
(The generalized quatemion groups, or course, can only occur if p = 2.)
PROOF_ We have (v) "* (i) => (H) =-- (iii) by the examples and remarks above.
(iii)=>(iv): Since G is a p-group, one knows that Gcontains a centraI subgroup
of order p (cf. Hall [1959]. Theorem 4.3.1). If there were any other subgroup of
arder P. the two would generate an elementary abelian p-group of rank 2,
contrary to (iii). Finally, the implication (ìv) => (v) is Theorem IV.4.3. D
,
.
To pass from the p-group case to the generaI case, we ha ve:
(9.4) Proposition. Afinite group G has periodic cohomology if and only if every
Sylow subgroup has periodic colwmology.
,
,
9 Groups with Periodic Cohomology
157
PROOF. The 6'only if" part folIows from Remark l above. To prove the '4if"
part, fix a prime p and let H c G be a p-Sylow subgroup. By hypothesis
fJ*(Ht .l) contains an invertible element U of degree d #:- O, and l clairn that
some power of u is G-invariant in the sense of III.1 O. In facl, let e > O be an
integer such that ae = 1 for a11 a E (?L/I H 1- Z)*, the latter being the group of
units in 7L11 H I . Z. F or any (J E G, the elements v = resZ n tlHtI - 1 U and W =
res'J,Hg-l gu afe invertible in fJ*(H f1 gHg-1  lL) and hence generate the
cyclic group fld(H n gHg- l. Z). We therefore have v = aw for some a E
(1L1 I H I . Z)*, since tbe map (1L1 I H I . Z)'" -+ (lLj I H' I. Z)* is surjeçtive for any
B' ç H. Thusve = w,anditfollows thatue E fìde(H, Z)isinvariant. Similarly
U-e is invariant. In view ofthe Tate cohomology version ofTheorem 111.10.3,
thìs proves that the ring fl*( G, Z)Cp) has an in vertible elernent u(P) or non-zero
degree d(P). Finally, replacing u(P) by a suitable power u(p )e(P), we may assume
that d(P) is the same for ali primes p Il G I ; since f!*( G  Z) is the direct product
ofthe rings fì*(G, lL)(p) (cf. exercise 2 of5), it follows that fJ*(G, Z) contaìns
an invertible element of positive degree. D
Combining 9.3 and 9.4, we have:
(9.5) Theorem The following conditiofls are equivalent for a finite group G:
(i) G has periodic cohomology.
(ii) Every abelian subgroup of G is c yclic.
(Hi) Ever y elementar y abelian. p.subgroup of G (p prime) has rank :s;; 1.
(iv) 1he Sylow subgroups ofG are cyclic or generalized quaternion groups.
U sing the criterion (iv) of this theorem, we can nQW gìve some examples of
groups with periodic cohomology, beyond thase of 9.2.
(9.6) EXAMPLES
l. If m and n afe relatively prime integers, then any semi-direct product
lm  7l.ft (arbitrary ac!ion) has cyclic Sylow su bgroups and hence has periodic
cohoology. (ConverselYt a theorem ofBurnside (cf. Wolf [1974] 5.4) implies
that any group with cyclic Sylow .subgroups is a semi-direct product of this
type.] This generalizes example 3 or 9.2.
2. Let p be a prime and let G = SL2([F p), tbe group of 2 )( 2 matrices of
determinant l aver the prime field IF p. Then G has periodic cohomology. To
see this note first that I G I = p(p2 - 1)_ [There are p2 - 1 possibilities for the
first column of a mRtrix in G; given the fust column, tbe number of possibilities
for the second is (p2 - p)/(P - l) = p.] Thus a p-Sylow subgroup of G is of
oeder p and hence is cyclic. N ow let l be a prime # p_ Then the polynomial
x' - 1 is separable over F Il' so a matrix of arder l is diagonaJizable over the
finite extension k obtained from IF" by adjoining the l-th root8 or unity. Two

158
VI Cohomology Theory or Finite Gnmps
commuting elements of G or arder l are then sìmultaneousl y diagonalizable
and hence generate a group isomorphic to
{( l,):lEk.l' = l},
which is of order l. Thus G does not contain a subgroup isomorphic to
Zl x Z, and hence G has periodic cohomology. [See exercise 8 below for a
different proof, which consists of explicitly exhibiting cyclic or quaternionic
Sylow subgroups.J
\ r.
, ..
.
i 
l
Remarks
1. Thc groups SL2(1F,,) for p = 2,3, 5 have occurred earlier in our examples.
Namely, SL2(1F2)  713><371.2, SLz{1F3) is isomorphic to the binary tetra-
hedral group, and SL1(lFs) is isomorphic to the binary icosahedral group.
2. A complete c1assification of the groups with periodic cohomology has
been obtained by Suzuki [1955], extending earlier work of Zassenhaus. (See
also Wolf[1974], 6.1 and 6.3.) Not ali ofthese groups can actfreely OD a spbere.
In fact, Milnor [1957a] showed that a group which acts freely OD a sphere has
at most one element of order 2. (Thus, for example, the dihedral group D 211
for n odd cannot act freely on a sphere, although ali of its Sylow subgroups are
cyclic.) Conversely, Madsen Thomas, and Wall [1976] have proven that ira
has periodic cohomology and at most one element of order 2 then G acts
feely on a sphere. This applies, for example, to the groups SL2(1F px.p odd)
smce
{
,.
"
. j
. i
, I '
.', I .':
r
1
,  
'.-
l
,.- -.
(-1 O)
O -1
is the only element of order 2. These groups do not, however, adrnit a free
orthogonal action on a sphere (i.e., a fixed-point-free representation) except
for p = 3 and p = 5.
J ,.
,
:
. .
,-
.- .,
''-
The condition tp.at allSylow subgroups ora group G becyc1ic or generalized
quaternion is obviously very restrictive. What is much more commODt
however, is for there to be at least one prime p such that the p-Sylow subgroups
are cyc1ic or generalized quaternion. It is therefore of interest to observe that
the results of tbis section can be "Iocalized'" at a singIe prime p. Thus we say
that G has p-periodic cohomology ifthe ringl/.(Gt Z)(p) contains an invertible
element ofnon-zero degree d. Cup product with such an element then gives
periodicity isomorphisms fl'*(G. M)(p)  fln+d(G, M)iP) for any G-module M.
The reader can easily state and prove analogues or 9.1,9.4, and 9.5 for this
situation. In particular, ODe can prove:
(9..7) Theorem. F or any finite group G and prime p dividing 'G I, the follow;ng
conditions are equivalent:
.,
.,
'.
9 Groups with Periodic Cohomology
159
(i) G has p-periodic cohomology.
(ii) The ring n.( G, ?Lp) contains an invertible element oJ non-zero degree.
(iii) Every elementary abelial1 p-subgroup oJ G has rank < 1.
(iv) The p.S ylow subgroups oJ Gare cyclic or generalized qutlternion.
We dose this section by stating without proof a beautiful generalization
of this theorem, which was conjectured by Atiyah and Swan and proved by
Quillen [1971]. Fix a prime p and let H( G) denote thecommutative lLp-algebra
E9n2: o H2J1(G, ?Lp). H the maximal rank or an elementary abelian p-subgroup
of G is 1, then 9.7 implies that H(G) is a finitely generated module over a
polynomial subalgebra 7L,,[u]. Consequent1y, H(G) has Krull dimension L
(See, for example. Serre (1965] for a treatment of dimension theory or
commutati ve rings.) Quillen's generalization is:
(9.8) Theorem4 Far any finite group G, the Krull dimension oJ H(G) is equal to
the rnaximal rank oJ an. elementary abelian p-subgroup oJ G.
RoughJy speaking then, the complexity of the ring H*( G. lLp) is determined
by the complexity or the p-su bgroups of G.
EXERCISES
l. If G is non-trivial and has periodic cohomology or period d. prove that d ìs even.
[Hint: Use anti-commutativity.]
2. Prove that G has periodic cohomology or period 2 if and on]y ìf G ìs cydic. [Hint:
What can YOti say about H 1G if fì.(G) has period 21]
3. Compute H"(G. Z) and H-lG. Z) for aH n if {J-(G) ìs periodic of period 4. Note, in
particular. that HiG) = fl- 3(G) = Rl(G) = Hom(G, Z) = O. so we obtain a ncw
proof of the resu]t of exercise 7a of II.5.
4. If G has periodic coi!omo]ogy. show that fti(G. Z) = O for i odd. [Hint: It suffices to
do this when G is a p-group.]
5. Give a direct proof that (iv) => (ì) ìn 9.3. [Hint: Use induced representatìons 3S in
example 3 of9.2.]
6. Let G = 1'..1/1 ><J Zn. where m and n afe relatively prime and 7l.pt acts on Zm via a
homomorphism lLn --+ Z: whose image has order k. Prove that the mìnìma] perìod
or fl*(G) is 2k. [Hint: If pl n then fl+(G. Z)(P)  fì.(ll.... Z)(P)' If P I m then n*(G)(PI o;;:;:
ft.(Zm)fp\.J
7. Let F Il be a fie]d wìth q elements, where q is a prime power. Show that SLn(lF q) does
not ha ve periodic cohomology if" > 3 or if q is nOI prime.

160
VI Cohomology Thwry or Finite Groups
8. Let k = IFq. where q is an odd prime power.
(a) Lei M be the subgroup or SL2(k) consisting of tbc monomial matrices
( 1  l) and ( _-l).
.it Ek*.
Show that M  QZ('l-11 if q =# 3.
(b) Let K  k be a q uadratic extension (so K = If q). We regard K a.s a 2-dimensionaJ
vector space over k and denote by GL(K) (resp. SL(K») the group of vector space
automorphisms or K (resp. the group or automorphisms of determinant 1). Thus
GL(K)  GL2(k) and SL{K)  SL,Jk). We have Gal(Kfk) c GL(K) and K* <+
GL(K), where the latter embedding corresponds to the multiplication action or K*
on K. Show that Gal(Kfk) and K* generate a subgroup of GL(K) of arder 2(q2 - l).
whose intersection with SL{ K} is isornorphic to Q2(q+ 1 j' [Hint: The composite
K. 4 GL(K)  k* is simply the narm map or Galois tbeocy. which in tbis case is
given by À 1--+ ÀIj'+ 1. Also. if cr E Gal(K/k) is the non-trivial element. then det O' = -1.]
(c) Using (a) and (b) and the fact Ihat I SL2(k) I = q(q2 - l), show that SL2(k) has a
quaternionic 2-Sylow su bgroup and a cyclic 1.Sylow su bgroup for any odd prime
11= char k.
, .
9. Suppose that G has p-periodic cohomology_ Let P c:: G be a subgroup of order p,
tet N(P) (resp_ C(P» be the norma1izer (resp. centralizer) of P in G, and let W =
N(P)jC(P). Prove tI'lat R.(G, M)(I'I ::::: n*(N(p). M)(p)  1ì"(C(P), .M):;;). [Hint:
For the first isomorphism. choose a p-Sylow subgroup H :::> P and show that every
N{P)-invariant in H*(H) is G-invariant. For the sewnd isomorphism, nOie that
P%IWI.]
10. For any finite group G. show that che augmentation ideal J c:: lG is a cyclic G-
moòule itT G is a cyclic group. [Rint: lf l is cyclic, then G admits a pefÌodìc resolution
of period 2.]

,
.
-
j
;
"
\
i,
,
1
,o.
.
.,
CHAPTER VII
Equivariant Homology and Spectral
Sequences
l Introduction
If a group G operates on a topologica] space X, then one can define equi-
variont honwlogyand cohomology groups, which CM be thought of heuristic-
allyas a 'mixture" of H(G) and H(X). This equivariant theory provides a
powerful tool for extracting homological information about G Crom the
action of G on X. It is in this waYt for example, that QuiUen proved his
theorem about the Krull dimension of H*(Gt 'Lp} for G finite (VI.9.8).
The main purpose of this chapter is to construct the equivariant homology
theory and give its basic propertiest including two spectral sequences. This
theory will play a crucial role in the remaining chapters of this book. For
simplicity, we wiU confine ourselves to the case where X is a G-complex in the
sense of I.4. See Grothendieck [1957] or Quillen [1971] for a more generaI
point or view.
We begin with a brief treatment of the theory of spectral seq uences.

2 The Spectral Sequence of a Filtered Complex
If C is a chain complex and c' is a subcom plex, then there is a long exact
seq uence wruch gives informatioo about H.( C) in terms or H.( C') and
H.(C/C'). Now suppose we are given, instead of a single subcomplex C',
a sequence of subcomplexes {F p C} p"z  with F p _ l C cF" C. It is reasonable
then, to try to get information about H.(C) in terms of the groups
H .(F p e/F p_ 1 C). In this section, we will describe a method for doing this.
161

162
VII Equivarian[ Homology and SpectraJ Sequences
2 The Spectra! Sequel] of a Filtered Complex
163
W:t,1at one obtains, roughly speak.ing, is a sequence of successive approxima-
tions E r > O? to H *( C), such that E J consists of tbe groups H .(F pC / F p _ l C).
We. w1I omlt most of te proofs in tis section. The reader can either supply
the nussmg proofs (whJch are routme) or consuIt any text which treats
spectraJ sequences, e.g., Spanier [1966] or MacLane [1963].
Let R bean arbitrary Ting. (In our applications we will usualI y take R = Z.)
By aD increasing filrration OD an R-module M we mean a family oi sub
modules F pM (p E Z) such that F p M c: F p'" 1 M. Tbe filtration is said to be
finite if F p M = O for p sufficìently small and F" M = M for p sufficiently
Jarge. Given a filtration on M. thc associated graded module Gr M is defined
by GrpM = FpM/Fp_1M. One thinks of M, tben, as being built up from the
"piecesnGrp M. The following elementary lemma shows thal, in some sense,
we do not lose too mllch information by passing from M to Gr M:
C. This is a sequence {E'},,? o or ..successive approximations'" toGr H(C).
Let Z; = FpC r-. è-1Fp_rC. (More precisely,
Z = F"Cp+q n a-1pp_,Cp+q_l.)
Let Z'; = FpC n Z. We have FpC = Z :::> Z 2 ...  Z';. Since the
6ltration {F pC} is dimension-wise finite, this sequence or inclusions stabil-
izes dimension-wise to a sequence or equalities. i.e., fOT fixed (p, q) we bave
Z = Z,,+1 = ... = Z for r sufficiently large. Let B; = FpC lì iJFp+-lC
= ÒZ+(l' and let B: = FpC n B. Then B ç B c... cB;, and again
thjs stabilizes dimension-wise to a sequence of equalities. We now have
Bo c B1 C . . . c Bro c ZQ;) c ... C Z. C ZO = F C
p- p- - 1'- p- - 1'- P p'
and we set
(2.1) Lemma. Lei f : M -+ M' be a jiltration-preserving map, where M and M'
are nwdules with .finite filtrations. II Gr f: Gr A1 -+ Gr M' is an isomorphism,
then f is an isomorphism.
E = Z(B + Z-=- \) = Z(B + (F p-l C n z;))
and
Another elementary (but important) observation is that if we have a
notion of urank" for R-modules (e.g., if R is a field or R = Z), such that
rk M = rk M' + rk MN for every short exact sequence O -+ M' -+ M .......
M't -+ 0, then the rank of a finitely-filtered module can be computed from the
associated graded module:
E; = Z';f(B;' + Z;-d = GrpH(C)
(cf. 2.3). F or fixed (p, q), we ha ve
Er - E+ 1 - ... - Et()
pq- pq - - pq
If the filtered module M is itself graded (and each F p M is a graded sub-
module), then we have for each n E 7L a filtration {F pM,J on M". and hence
there is an obvious way or associating a bigraded module lo M. The usua)
notational convention in this case is to set Grpq M = FpMp+,/Fp_IM"+,,.
An element of Gr pq M is said to ha ve filtration degree p, complementary
degree q, and total degree p + q. T o simplify the notation, we will sOlDetimes
suppress tbe second subscript and simply write GrpM = FpM/Fp_tM.
Now let C = (Cn)IIEZ be a filtered chain complex (with each F l'C a sub-
complex). For simplicity, we wiII a]ways assume that the filtration is di-
mension-wisejinite, i.e., that {F"C,,}peZ is a finite filtration ol CII for each n.
There is an induced filtration on the homology H( C), defined by F pH( C) =
Im{H(FpC)....... H(C)}. We can identify FpH(C) with (FpC rì Z)/(FpC (ì B),
where Z (resp. B) is the module of cycles (resp. boundaries) of C. The associated
bigraded module Gr H(C) ìs given by
.
for r sufficiently large, so the sequence {E"} ., converges ., to Gr H( C) as r -+ co.
(One often suppresses the '.Gr" here and simply says that the spectral se-
quence converges lo H(C) or that H(C) is the abutment of the spectral se-
quence.)
The modules E' are particularly easy lo describe for r = O and 1:
(2.4) E = FpC/Fp_IC = Grl'C
(2.5) E = (FpC (Ì a-1pp_1C)/(iJFpC + Fp-tC)  H(FpCfFp_1C).
(More precisely, E  Hp+qfFpCfFp_1C).) Thus £1 is the homology or EO,
relative to the differential induced on EO by ò. More generally, one can show
that ò induces a differential dr on Er of bidegree ( - r, r - l) (i.e., d: E -+
E;-,.,q+r-l), and that Et+ l  H(Er). An important consequence ofthis is:

(2.2)
rk M = L rk Gr p M.
peZ
(2.3) Grp H(C) = (FpC n Z)j«FpC n B) + (Fp-1C lì Z».
(2.6) Proposidon Ler t: C -+ C' be a filtration-preserving cooin map. where C
and Cf have dimension-wise finite fiitrations./f the induced map Er( t): Er( C) _
E(C') oJ spectral sequences is aJ1 isomorphismfor some r. then H("C): n(C) 
H(C') is an isomorphism.
PROOF. If Elr) is an isomorphism then so is E"tr) for s > r, since ES =
H(E5-t). Hence E«>(T) = Gr R(i) is an isomorphism, and the proposition
now follows rrom 2.1. D
We now descrìbe the spectral sequence associated to the filtered complex
Another important fact is that spectral sequences can be used to compute
Euler characteristics. Suppose. for example, that R = 7L or R is a field. For

-
'.
164
VII Equivariant Homology and Spectral Sequences
any graded R-module A = (Ali) (resp. bigraded module A = (APQ»t let
X(A) = L ( -: 1:r rk A" (resp. X(A) = Lp. q ( - 1)1' +,. rk AI14), provided alI the
ranks afe fimte and almost alI of them are zero. In case A has a differential
.
it is well-known that X(A) = X(H(A») (cf. Spanier [1966], 4.3.14). In par-
ticular, if C i8 a filtered complex as above and ìf x(EF) is defined for some r, then
it follows that
f
,
"
r
 '.
(2.7) X(E') = X(F:+ l) = ... = X(ECO) = X(H(C)),
the last equality being a conseq nence or 2.2.
Finally, we briefiy describe the notational conventions in case C is
indexed as a cocMin complex (CIC)" e z, with differential or degree + I. In tbis
case the filtration is denoted {FPC} and is assumed to be decreasing (i.e.,
FPC  FP+ l C). The terms or the resulting cohonwlogy spectral sequence are
denoted E. and one has E: = Gr" H(C) = FP HP +'l(C)/FP + l Hp+q( C). Tbe
differentiaI d, is of bidegree (r,  r + I), so that dr: Er -+ Ef+r.fl- r+ 1. If we
think or Eq as sitting OD tbe lattice point (p, q) of the pIane, then dr can be
visualized as an arrow pointing to the right and downward for r > 2. By
contrast, the differentials dr in a homology spectral sequence point to the
Iert and upward.
The canonical example or a filtered cochain complex is m(C, M)t
where C is a fiJtered chain complex. One sets
FP .Tthm(C, M} = C/F ,,-1 C, M).
.-
EXERCISE
...
Let O -+ c.' - C - eli - o be a short exact sequence ofchain compJexes. LeI {FpC} be
tbc fiJtratlon such that FoC = O, FJC = C, and F1;C = C. Desccibe the modu1es E
and deduce from the spectral sequence the Camiliar long exact homology sequence.
[Thus the spectral sequence of a filtered complex can be regarded as a generalization of
the long exact sequence associated to a chain complex and a subcomplex. 3.5 we implied
in the intcoductory paragraph or this section.]
.
(
3 Double Complexes
By a double complex we mean a bigraded module C = (Cpq)P,qEZ with a
"horizontal differential ò' or bidegree (-1;0) and a "vertical" differential
il' of bidegree (O, - I), such that a' ir = o" a' :
Cp-1,t1 c il Cpq
l l..
Cp-l,-l'" i! Cp.,,_I.
3 Double Complexes
165
Note that a double complex can be regarded, in two different ways as a
"chain complex in thc category of chain complexes." Thus for each q we have
a horizontal chain complex C.. q with differential iJ and we are given chain
maps it: C.,q --il C.. q-l such that ò"ò" = O. Similarly, for each p we bave a
vertical chain complex C P.. with differential ò", and we are given chain maps
ò': C P.. -+ C 1'_ 1,. with a' il = O.
A double complex C gives rise to an ordinary chain complex TC, called
the toral complex. as follows: (TC)n = EB,. + q'" li Cpq" with differential a given
by òlCpq = fJ' + (-I)Po", The tensor product oftwo chain complexes C' and
C" provides a familiar example or this construction. lndeed, we ha ve a
double complex C with C pq = C @ C;. and TC is simply the usual tensor
product C' @ C'I of chain complexes.
We now filter TC bysetting F p(TC)1I = EBi':1;P Ci,lI-i. This is adimension-
wise finite filtration provided C has only finitely many non-zero modules
Cpq in any given total degree p + q. (This holds automatically, for example,
il C is afirst quadrant double complex, Le., if Cpq = O when p < O or q < O.)
Thus we bave a spectral sequence {E'} converging to H.(TC). It is immediate
from tbe definitioDs that £ = Cpq and that dO = + o". Consequently, El
is the vertical homology or C, i.e., E = Hq( Cp. *). The differential di : E -+
E_ 1. ti is easily seen to be the map induced by the chain map il: C P.. -+
C p_ l..; for an element of E is represented by an element c E c such that
iJ" c = O, and for such a c one has oc = a' c. Thus E2 can be descnbed as the
horizontai homology or the vertical homology of C.
One could equally well filter TC by F JTC)n = E8J" C,,_ j.j' We obtain,
tben, a second spectral sequence converging to H(TC), this time with
E = Cqp, E = Hll(C* p) and da: E -+ E-l.q equal (up to sign) to the
map induced by a": c., p - C.. p-l'
[Warning: Even though the two spectral sequences have tbe same
abutment H*(TC) they do not in generai ha ve the same EOO-term; for we
have two different filtrations on H(TC) and hence two different Ecn-terrns
Gr H(TC).]
A similar discussion applies to double cochain complexes C = Cpq. One has
a total cochain complex Te with two decreasing filtrations. and hence two
spectral sequences IÙf cohomologicàl type converging to H*(TC) (provided
tha C bas only finitely many non-zero modules in any given total degree).
Details are left to the reader.
EXERCISE
Let C be a first-quadrant double complex such that one of the associated spectra[
sequences (the first one. say) has E = O for q ¥' o. Let D be thc chain complex El.o
with differential dI.
(a) Show thal there is an isomorphism cp: H.(TC)'; H",(D).

166
VII Equivarìant Homology and Spectral Sequences
(b) .Make ths resu]t more precise by sbowing that there is a weak equivalence 1': TC -io D.
[Hmt: Take T to be the cal1onica] surjeçtion which comes from the fact tbat D is the
verti] o-d,imensional t1oo]ogy or C. Show that 1".. = fP by directly examining the
defiltlons 111 the constructlon of the spectra] sequence. Altcrnatively, you can avoìd
gettmg your hands dirt)' by observing that 'f can be viewed as a map of double complexes
 -io D (where D is regarded as a double complex cODcentraled OD the line q = O); the
mduced map of spectral sequences is an isomorphism at Ihe E l-level so T induccs an
isomorphism H*(TC) -io H.(TD) = H.(D).] ·
I
,
!
, :
.,
-
.
.'
"
.'
4 Example: The Homology of a Union
Let X be a CW-complex which is the union of a family or non-empty sub-
complexes X:.:, where IX ranges over some total1y ordered index set J. In case
J = {l. 2} one has a short exact sequence of chain complexes
0-+ C(X 1 n X2) -+ C(X 1) E8 C(X 2) .... C(X) ...... O
(where C( ) denotes the cellular cbain compIex), from which one obtains
the familiar Mayer-Vietoris sequence. In the generai case, we will show that
the exact sequence above is replaced by an exact seQuence
... -+ EB C(X (ì Xfl n X) -+ EBC(X(I (Ì Xp) -+ EB C(Xa) -+ C(X) -+ O
CC<: (1< j' l'J.< fJ (I
and that thc Mayer - Vietoris sequence is replaced by a spectral seq uence.
The detaiIs are as follows.
Let K be the abstract simplicial complex whose vertex set is J and w hose
s1mplices are the non..empty finite subsets a of J such that the intersection
XfJ' = n aefJ' X" is non-empty; K is called tbe nerve of the covering {Xcc}'
For p > o let Cp be the chain complex EBO'eK(p)C(Xu)' where K(p) is the set
of p-simplices of K. If G has vertices ClO < . . . < ap' we denote by
ÒjCf (O < i <p) the (p - l)-simplex {ao,.... tij,..., (lp}' The inclusions
C(X.,.) '"-*' C(X..\a) !nduce a chain map Oj:Cp - Cp-1 for p > l, and we set
ò = Ir o (-l)'t1i. Similarly, theinclusionsC(X) 4 C(X)induceachainmap
8: Co -+ C(X). We have, then, an augmented chain complex
(4.1) . - - -io Cp  Cp-1 -io . - . -+ Co  C(X) -+ O
in the category of chain complexes, and hence a double complex C with C pq
equal to tbe group of q-chains of Cl" Le., CP4 = EBO'05KtPI C'l(XcJ).
I claim now that 4.1 is exact. To see this, we give an alternative description
or C pq' F or any cell e or X  let Ke be the su bcomplex of K consisting of tbe
simplices t1 such that e c X (1" Then C P4 has one basis element for every
pair (O", e) such that e is a q-cell of X and (J is a p-simplex or Ke. In other
words, C pq  EE>o['eX 9) C p(Ke). where X(Q') is the set of q-ceIls of X. Moreover,
an examina tion or the definitions of the maps ò and E in 4.1 shows that ò: Cpq -+
.
I
I
j
4 Exampte: The Homology of a Union
167
Cp-1.q is equal to EBeeX() {a: Cp(Ke)  CJl-l(Ke)} and similarly for &.
Thus the q-dimensionaJ part of 4.1,
. . . -+ Cpq -+ . . - -+ COli -+ C'I(X) - O
is isomorpbic to EBeeX(..n C(Ke) where è(Ke) is the augmented simplicial
chain complex of Ke. But Ke is acyclic. Indeed, its simplices are ali or the
finite subsets ofthe non-empty set J e = {o: E J: e c X o:}, so Ke is the 'simplex"
spanned by the (possibly infinite) set J I? Thus EBe è(KI?) is acyclic, which
proves the claim.
Consider, now, the two spectral sequences of the double complex (Cpq).
In view of the exactness of 4.1, the second spectral sequence has
{O ifq#=O
E = Hq(C.,p) = Cp(X) if q = O.
Taking the homology with respect to p, we obtain
E2 _ {O
pq - Hp(X)
if q =#= O
if q = O.
The spectral sequence therefoTe H collapses" to yield:
(4.2)
H .(TC)  H .(X).
The first spectral seq uence, on the other hand, has
(4.3)
E = Hq(Cp) = EB Hq(X6).
64;;K(F}
Moreover, in view of the isomorphism 4.2, we can regard H.(X) as the
abutment of the spectral sequence, i.e., EOO  Gr H.(X) relative to some
filtration on H .(X). This fust spectral sequence then is OUI desired general-
ization of the Mayer-Vietoris sequence. It approximates H * X in terms or
the homology ofthe Xo: and their intersections.
To describe tbe E2-term we need the notion of "coefficient system" OD a
simplicial complex K. By this we will mean a family .d = {A6} of abelian
groups. where (J ranges over the simplices or K, togetherwith a map fn"'C'= AcJ...... AI
whenever L is a face of (J (written 'r c a). sucb thatfrp ffN = J"P if p c 'r c; (l.
There is an o bvious way to construct a chain complex C(K, si), with CJ..K, d)
= EBO'E I: (M All, and one therefore has homology groups H.(K. çf). (See
Godement [1958], 1.3.3, ror a detailed treatment of the cohomological
version of this.)
Retuming now to the situation where K is the nerve of the covering {X oJ
we have for each q > O a coefficient system  = {Hq(Xu)} on K, where
ffJ'f: Hq(X41) -+ Hq(Xr) is induced by the inclusion XtJ 4 Xf- It is immediate
from the detinition (hat the E1-tcrm in 4.3 is equal to Cp(K, Jt;) hence
£;q = Hp{K. ). We summarize this discussion by writing
E = H p(K) => H p+q(X)

168
VII Equivarianr HomoJogy and Spectral Sequences
where the syrnbol "'''*'. indicates that tbe spectral sequence çonverges to
what appears on the right-hand side.
In case each X fT is acyclic, we ha ve £0 = Z and  = O for q :F O. The
sptral sequence therefore collapses at E2 to yield the following result,
whlch seems to be essentially due to Leray:
(4.4) Theor. Supose X is the unioll o[ subcomplexes Xli such that every
non-empty fnlersectlOn X«o n ... n Xp (p > O) is acyclic. Then H.(X) 
H ..JK), where K is the nerve oJ the cov€'o
EXERCISES
I. Make 4.4 more precìse by showing that, under the given hypotheses, there is a chain
complex T which admits weak. equivalences T - C(X) and T _ C(K). [Take
T = TC. where C is the double complex used in this section. and use part (b) ofthe
exerdse or 3.]
*2. Make 4.4 slilI more precise by showing that there is a CW-complex Y which maps to
both K and X by maps inducing homology isomorphisrns. [Hint: Let Y be tbe
union oftbe ceUs u )( e in K x X sucb thar e c X a (or, equivaIentty. such thal a is
in Ke); note that C(Y) ca.[) be identified with the complex T or exercise I.]
Remark. Tl1e map Y -+ X is in fact a homotopy equivalence. This can be seen. [or
example. by an inductive argument over tbe skeleta of X, using the fact that each
compIex Ke is contractible. Consequently. one obtains a canonical homotopy class
of maps X - K inducing a homoJogy isomorphism. In case each X oT is contractible
(and not just acyclìc), one can simiIarly show that the map Y -+ K is a bomotopy
equivalence, so that X and K are homotopy equivaJent. This sort of result was first
proved by Weil [1952]. 5; see also BoreI-Serre [1974], . and QuilJen [1978J
1.6- 1.8.
5 Homology of a Group with Coefficients in a Chain
Complex
Recall that H.(G. M) is defined to be H.(F (8)G M). where F is a projective
resolution of 7l. over lLG. It is usefu] to geileralize this by aJIowing a non-
negative chain complex C = (C,,)"O of coefficients. Thus wc set
H.(G, C) = H.(F (8)G C);
as usual. this is well-defined up to canonical isomorphism. If C consists of a
single module M concentrated. in dimension O, then H.(G. C) reduces to
H.(G. M).
i
. ..
,
,
,
,
,
-
.
.
I
,
5 Homology or a Group with Coefficients in a Chain Complex
169
Since F (8)G C is the total complex of the double complex of alian
groups (F" 0G Cq), we have by 3 two spectral sequençes convergm to
H.(G. C). Tbe first of these has E = HiFp @G C.) = ,,@G HiC) smce
F (8)0 - is an exact fUDetor. Taking now tbe homology wlth respect to p, we
obtain E;'" = H,,( G. H Il C). Thus the spectral sequence has thc form
(5.1) E = Hp(G. BqC) =- Hp+q(G, C).
An importanl consequence or this is that H .(G, C) is an invariant of the
4weak homotopy type" or C:
(5.2) Proposition. Il t': C - C' is a weak equivalence oJ G-chain complexes,
then T induces an isomorphism H.( G. C)  H.( G, Cl
PROOF. 1" induces a map or spectra] sequences which is an isomorphism at the
F.;2 -level, hence t ind uces an isomorphisrn OD the abutments by 2.6. O
A
[This result also follows, of course, rrom 1.8.6, but il still provid a Dice
ilIustration ol spectral sequence techniques.]
The second spectral sequence has E = Rfl(F. @a Cp) = HiG, C,,).
Thus we bave:
(5.3)
E = Hfl(G. Cp) => Hp+q(G, C).
Tbe group E can therefore be described as the p-th homoloy grop of he
complex obtained from C by applying the functor H q(, ) dlmenson-Ise.
80th spectral sequences, then. can be thought or as glvmg approxlmatIons
to H .( G, C) in terrns or ordilUlry homology groups H *( G, M)..
To get a better feeling fOT H.(G, C), le!'s look al some speclal cases. Sp-
pose, first, that G acts trivially on C. Then F (8)G C  F G @ C, so tbere IS a
Kunneth formula
0-+ EB H"G @ HqC -+ Hn(G C) -+ EB Tor(HpG, HqC) - O
p+q p+q=n-l
-
expressingH.(G C) in terms of H.G and H.C. Similarly. ifkisa field and C
is a complex of k-vector spaces (with trivial G-action) then
(5.4)
H.(GC)  H.(G, k) c&:H.C.
At the other extreme, suppose that each C p is a free ZG-module, or, more
general1y, an H.-acyclic G-module (e.g., projective or induced). The El_
term in 5.3 is then concentrated OD the line q = O, and E.o = (Cp}G- The
spectral sequence therefore collapses at E2 to yield:
(5.5)
H.(G, C)  H.(CG).

170
,
-.
."
'.
'.
,
f
'!
<
'.
,
.' .,
-
,
:
.
,
VII Equivariant Homology and Spectra! Sequences
Tbe abutment ofthe spectral sequence (5.1) can therefore be identified with
H*(CG); this proves:
(5.6) Proposition. Let C be a non-negative chain complex oJ G-modules such
that each CII is H . -acyclic. Then there is a spectral sequence DJ the form
E = Hp{G, HqC) => Hp+q(C(J)'
Remark. This spectra) sequence is a special case or the . universal coefficient
spectral sequence,,'" cf. Godement [1958],1.5.5.1.
,.
t

.
We can also define cohomotogy groups H*(G, C), where C = (C")ItO is a
non-negative cochain complex; nameJy, we set
H*(G" C) = H*("'NG{F, C»,
where F is a projective resolution or 7L over ZG. Now JfhvI1{;(F. C) is in fact
the total complex associatcd to a double complex with cpq = HomG(Fq, CP).
Indeed. wehavem:G(F, C)" = .}f';»nG(F. C)-IT = Op+q-n HornG(Fq. C_p) =
ffiv+q=" Hom(ì(Fq" CP). And if we take (/ to be the coboundary operator in
féi,.ffl(j(Fq, C) [where q is fixed and the module Fq is regarded as a chain
complex concentrated in dimension O] and " to be the coboundary operator
in .Jt:b;1Jft;(F, CP). then the resulting total coboundary operator is the same as
that in &-'mG(F, C). One can now easily deduce cohomology analogues or
5.1-5.6.
Similarly" if G is finite, one can define Tate cohomology groups H*(G, C)
by using a complete resolutìon F. We assume, here, that CII = O ror n» 0"
so that the double complex HornG(F p' Cq} has only finitely many non-zero
groups in each total degree.
FinalIy, we remark that alJ or the formaI properties of homology and
cohomoJogy g]ven in Chapters III and V extend without diffi.culty to homol-
ogy and cohomoIogy with coefficients in a (co-) chain complex. In particular.
one has long exact sequences, Shapiro's lemma, restriction and corestriction
maps. cup products, etc. Mo reo ver, the cohomology spectral sequences
analogous to 5.1 and 5.3 are compatible with thc C\Jp products. More pre-
cisely, suppose C and D are G-coçhain complexes, and consider tbe cochain
cup product
,
,
<
,.
"
.
'..
"
:.
,
,
..-....
 
"
'.
'-

.
L . ;'
'.
,\ '.",
'.
"
, .
". . ...
'".'
,
KMnQ(F, C) @K-MnG{F. D) --+ ft'Q,lH.G(F C @ D)
defined via some diagonal approximation F -+ F @ F. One checks easily
that this cup pcoduct is compatible with tbe filtrations defining the two
spectral sequences i.e. that it induces a prod uct
FP Jeom(;(F, C) x FPJl-'mn(;(F D) -I> FP + PJymH.G(F, C @ D).
I t follows that there is an induced cup product
E(C) @ Ef'q'(D) -I> E!+P'.q+i'(C @ D),
. 
,',
, ,
6 Example: The HochschiJd-Serre Spectral Sequence
171
where {E,( )} denotes tbc cohomology version of either 5.1 or 5.3. We will
see applications ofthis product structure in Chapter x.
6 Example: The Hochschild-Serre Spectral Sequence
,
J
i
r
Let 1 --+ H -+ G -+ Q --+ 1 be a group extension. U F is a projective resolution
of Z over ?LG and M is a G-module, then F (8IG M = (F @ l'J)G can. be
computed in two steps (d. II.2, exercise 3) by st dividing out by the acÌlon
of H on F @ M and then dividing out by the actlon of Q:
F @G M = {(F (8) M}H)Q = (F @n M)Q'
Thus
(6.1) H.(G, M) = H.(CQ)"
where C = F @H M. Note also that we have a Q-module isomorphisrn
(6.2) H .(H, M)  H.(C),
where the Q-action OD H.(H, M) is that of 111.8.2. Finally, I c1aim tbat the
Q-modules C p = (F p @ M)H are H * -acyclic. In fact, it suffices to show tat
(ZG 0 M)H is H. -acyclic; and for this one need only O bserve (by usmg
111.5.7. for ex ampie) that (ZG @ M)H is an induced Q-module l'Q  A.
We are DOW in a position to apply 5.6 to the Q-complex C. In vlew of 6.1
and 6.2 we obtain:
(6.3) TheorelD (Hocbschild-Serre [1953]). For any group extension I --+ H -+
G -+ Q --+ l and any G-module M. there is a spectral sequence of the form
E = H ,,(Q, Hq(H, M»  H l'+q(G, M).
(There is, or course, an analogous spectral sequence in cohomology" de-
rived from the observation that J'ftmlG{F" M) = (J'f£1mH(F. M»Q.)
AD important consequence of 6.3 is the following 5-term exact sequence
oflow-dimensional homoJogy groups, generalizing the 5-term exact sequence
of exercise 6 of n.5 :
(64) CorollarY4 Under the hypotheses of6.3 rhere is an exact sequence
H2(G, M) -+ H2(Q" MH) -I> H1(H. M)Q -+ Hi(G. M) --+ H1(Q, MB) -+ O.
PROOF. Since E = Gr H{G" M), we have a short exact sequence
0--. E0.1 --. H1(G. M) -+ Eo -+ O.
Now there are no non-zero dilferentials involving E'i.o (r > 2), so
EI:() - E2
1.0 - 1,0.

172
VII Equìvariant Homology and Spectral Sequences
And the only possible non-zero differential involving £0.1 or £2.0 is
d2. E2 E2
. 2.0 --. 0.1'
so there is an exact sequence
0--+ Ef,o --+ Eto 4 E.1. -+ E0.1 --+ O.
The short-exact sequence above now yields an exact sequence
0- E2',o --+ E.o -to E.1 - H l(G M) -+ Ei.o --+ O.
Since E = Hp(Q.. Hq(H, M») and E2.o is a quotient of H2(G.. M), tbis yields
the desired 5-term exact sequence. D
7 Equivariant Homology
We now specialize the tbeory of 5 to tbe case where the chain complex C
is the cellular chain complex C(X) of a a..complex X (cf. I.4). The resulting
homology groups H.( G.. C(X)) are denoted H(X) and ca1led the equivariant
homology groups of(G, X). MOre generally if M is a G-module.. tben there is a
diagonal G-action on C(X, M) = C(X) @ M, and we set
(X, M) = H.(G, C(X, M».
SimilarIy, the equivariant cohomology groups are defined by
H(X, M) = H.(G, C*(X, M).
In case X is finite dimensional and G is finite, we can also define equivariant
Tate cohomology groups by
ilt(Xt M) = il*(Gt C-(X, M».
Finally, if Y c X is a G-invariant subcomplex, then there are relative equi-
variant homology and cobomology groups, defined by means of the relative
complex C(X) Y) = C(X)/C(Y).
For simplicity, we will confine ourselves in tbis section to a discussion or
the (absolute) homology groups (X, M). There are, of course, relative
versions, as well as cohomology analogues, of alI tbe results.
Note first that H(pt., M) = H.(G, M). Since any G-complex X admits
a (unique) G-map to a point, we deduce tbat there is a canonicaI map
(7.1)
H(X, M) -+ H.(G, M).
Next we record the spectral seq uence 5.1 in the present context:
(7.2) E = HJ.G, H,tX. M» => Ir;+(iX, M).
{Note that the E2-term here involves the diagoTllJI acr;or1 or G on H .(X, M),
induced by tOO action or G on X and M.)
....
,
-
!'
"
1-
,
.

'-
.!
. .-
. ,
7 Equivariant HomoJogy
113
"
..
;
As a consequence of 7.2, we have (cf. 5.2):
(793) Proposition. 1/ f: X -+ Y is a cellular map of G-complexes such that
f.: H.X -+ H. Y is an isomorphism) then f induces an isomorphism H(X, M)
.=. H(Y) M) for any G-module M. In particular. if X is acyclic thn the
canonical nuzp 7.1 is an isonwrphisrn H(X, M).; H*(G, M).
,
..
..
.,
To analyze tbe second spectral sequence (5.3), we decompose CP(X) @ M
as in III.5.5b. F or each p-celi Cl of X we have a G rT -module la which is ad-
ditively isomorphic to lL.,on which G a operates by the  orientation character"
la: G" -+ {+ l}. Let
Ma=ZfS@M;
thus M a is a G,,-module whicb is additively isomorphic to M, with the Gft-
action "twisted' by '¥.fS' Let X p be tbe set of p-cells or X and let :Ep be a set of
representatives for X ,JG. We then have an additive decomposition
(7.4) Cp(X, M) = Cp(X) @ M = ED Ma.
fSe Xp
.-
.
from whicb we obtain a G-module decomposition
(7.5)
C,,(X, M):=:::: EB IndtM.,..
a e J:p
Shapiro's lemma now yields
(7.6) Hq(G.. CJ.X, M)  EB Hq(Ga, Mw),
ft e J:p
so tbat 5.3 takes the form:
(7.7)
E = EB Hq(Gft. Ma) => H+q(X, M).
ft e J:,.
Suppose, for exampIe, that tbe G-action is free.. so that eacb G ft = {l}, and
assume for simplicity that M = Z. The spectral sequence then collapses at
E2 to yield (cf. 5.5):
(7.8) H:(X)  H .(C(X)G) = H.(X/G).
Combining 7.8 and 7.2.. we conclude (cl: 5.6):
(7.9) Theorem. If X is afree G-complex, rhe'1 rhere is a spectral sequence oJ
the form
E = H ,,(G, Hq X) =-- Hp+q(X/G).
Remark. Tbis spectral sequence is called the Cartlln-Leray spectral sequence
associated lO lhe regular covering map X -+ X/G. Theorem 7.9 remains
true if we introduce an arbitrary G-module M or coefficients. but the resulting
homology groups H.(X/G, M) have to be interpreted as homology groups
with local ooefficients if G acts nontrivia11y on M.

174
VII Equivariant Homo1ogy and Spectra] Sequences
Consider now the situation where X is acyclic (but the action of G is not
necessarily free), so that
(X M)  H *(G M)
by 7.3. The spectraJ sequence 7.7 can then be rewritten:
(7.10)
E = E8 Hq(GfS' Mft}  H'Hq(G, M).
a e 1:"
This is an important computational tool in tbe homology theory of groups.
To use it, of course, one must find an interesting acyclic space X on which G
acts. If G is an amalgamated free product, for example we will see in  that a
suitable choice of X leads to tbe Mayer-Vietoris sequence whicb we derived
earlier from a different point of view (11.7.7 and tbe exercise or I1I.6).
EXERCISES
1. Conslder the "left-hand edge" E.* ofthe spectra] sequence 1.7. Srnce d'is zero on
this edge for r > l, the spectral sequence gives us maps
EB H*(GJ.. M) = E.*...". E* =GroH(X. M) 4 H(X. M).
I:o
[We havc written H .(Gv. Al) here insread of H *(Gv, MI') because a zero--cell has a
unique orientation. so that MI' is canonicaUy isomorphic to M as a Gv-moduJe.] The
composite
E8 H..(Gv' M) -+ H(X, M)
l't: l(J
ìs one or the two .. edge homomorphisms" associated to the spectral sequence. Show
that its restriction to H*(G.,. M) is the map
H..(G", M) = H"(v. M) - lf2(X. M)
induced by the indusion (GIJ' v) c, (G, X). [Hinc Thìs can be donc by straightforward
(but tedious) definition-checking. Jt is easier. however. to use a naturaJjty argument
to reduce to theca.se where X consists or the single vertex v. in which case the definition-
checking is trivìaI.]
2. Le. X be a G-complex such that for each ceH (T of X. the isotropy group G(f fixes (1
pointwise. In this case it is easy to see that the orbit space X /G inherits a C W-structure.
If. in addition, each Ga is finite, show that
H(X. Q)  H*(X/G. Q).
Deduce. in particular. that
H*(G. Q)  H.(XjG. Q)
if X is contraclible. [Remark: The hypothesjs that Gq fixes (J pointwise is not very
restrictive in pracrice. In the case af a simpliciaJ action, for examp1e. it can always be
achieved by passage to the barycentric subdivision.]
.
-
..
-
-
,

.,
(
! ..

-
i .. :<:
I .
r ,
;
 ,.
..
.
-
:
..
:} -
.,-
<
.
,
,
..
:
,

..
o.
\
-

J -
-
.,

,
"
l

<-
I.
<
. ,
-.
."
..
.,
g Computatìon or di
175
3. Show that H(X)  H ..(X X G E). where E is a free. contractible G-complex (.e.. the
universal cover of a K( G, l)) and X x G E denotcs X x E modulo t he dlagonal
G-action. [Hint: There is a G-map X x E - X which is a homotopy equiva]ence.
and G acts freely 00 X x E.]
Remark. The isomorphism of exercise 3 is often taken as a definitìon. F rom thls poim
of vlew one obtains a speclral sequence of the form 1.2 by noting that there IS a tiber
bundle X x G E _ K(G l) with fiber X.
4. (a) Let X be a G-space (noI necessariJy a G-complex). Use singular chains to detìne
eQuivariant singular homo]ogy H(X) and construct a spectra] sequence of the form
7.2.
(b) Suppose that G acts freely on X and that Ihe projection X --. X/C is a covering
map. (11 is then a regular G-cover in the sense of tbe appendix to Chapter L) Prove
tha! H(X)  H .(X/G). [Hint: 5.5.] Deduce that there is a Cartan-leray spcctral
sequence (as in 7.9) in singular homology.
.(c) Suppose X = S2t.- 1 in (b). Prove that G has periodic cohomology or period 2k.
[Method l: X/O is a finite dimensional manifold and hence has vanishing homology
in high dimensions, with arbitrary Iocal coeffidents. The differential in the spectral
sequence therefore yic1ds H.(G. M)  Hi+ 2l(G, M) Cor i »0. Mcthod 2: We have ,a
sphere bundle 82"- 1 X G E - K(G. 1) (wlth E as in exercisc J) whose [Oial space lS
homotopy equiva]ent to SU-l/G. Up to homotopy. then, the inclusion or the fiber
in the totaI space is a map of degree I G I or orientable (2k - l )-manifolds.. The Gysin
sequence now shows that the Euler c1ass orthe sphere bundle is an element of order
IGI in H2k.(G, Z). so the result fo]]ows from VI.9.Hiv). See Jackowskj [1918] for
further results obtainable from the study or sphere bundles over K(G. 1).]
5. Let X be a G-complex.lf N is a normal subgroup of G which acts freely on X. show
thal H(X)  H/N(X/N) with any G/N-modu1e of coefficients. [Hint: AppJy 7.1 to
(G, X) and (G/N. X/N).]
6. LeI Y be a connected CW-complex with?ti Y = O for 1 < j < n as in Il.5.2. Deduce
from the Carlan-Leray spectral sequence of Ihe universa] cover or Y that there is a
5-term exact sequence
HIt+ 1 Y --. H.. + i?t - (1t.. Y),. -. H" Y - Hn1r. -+ O,
where :Jt = 1t 1 Y.
8 Computation of dI
Let X be a G-complex and M a G-module. In this section we will compute
the differential dI in the spectral sequence 1.7.
li a is a p--cell or X and T is a (p - l)-cell. we denote by Ò.,.,,: Ma -+ M the
«(1, t)-component or the boundary operator ò: C,,(X M) -7- Cp-l(X M)
.

176
VII Equivariant Homo]ogy and Spectral Sequences
(cf. 1.4). Let  = {'r: iJr:a #= O}. Tbis is a finite set or (p  I)-cells and is
G(Y-invariant. Writing GOT = Go n GT9 we conclude that (Ga: G'() < 00 for
t' E g;;,. There is therefore a transfer map
tfT'[: H.(G(J' Ma) -+ H.(Gt1P Ma).
Next, note that ÒOT: M" -+ MT is a GUt-map, since ò is a G-map; so òar:
and the inclusion G aT c.. G t induce a map
U(1'{: H.(Gur:, Mt1) -+ H .(GT. M'f)'
Finally. let t'o E I:p_ 1 be the representative of the G-orbit of 't', and choose
g(t) E G such that g(t)T = t'o. Then the action or g(t) OD C,,-t(X9 M) gives
aD isomorphism M"( -- Mto compatible with the conjllgatioD isomorphism
G'( -+ GTO given by 9 H g(r)gg(t')-l; thus there is an induced isomorphism
VT: Hrtc(Gp MT) -+ H*(G,o' MTO)-
[As the notation suggests, ODe can show that VT depends on)y on T, and not on
thc choice of g(T); but we will not need to know this).
We can now define a map
<p; EB HiGI1, Ma) -+ EB H.(Gn Mr:)
crel:p téI.. _ 1
hy
fPIH.(G"" M(1) = L VtUI11:terf9
tejO'
f7
where , is a set ofrepresentatives for 'ujG(f'
(8.1) Proposilion. Up lo sign, the map q> is the differential dI: E.. -+ E;-l..
in 7.7.
PROOF. What wc must prove is that the diagram
EB H.(GtI. Ma)
(f lE 1:.1'
fP ) EB H.(G, M'[)
te£l'-l
1 l
H .(G, Cp(X. M» H..(G.)) H..(G. Cp_1(X, M»
commutes, where the vertical isomorphisrns come from 7.6_ We will in fact
prove commu tativi ty or the corresponding diagram of chain complexes.
More precisely. if F is a projective resolution or Z-over lLG. we wiU compute
tp OD the chain level as a map (stili denoted ({)
E8 F (8h.. M -+ EB F @G.. MT9
e I::p te1:6- I
-
1
.'
8 Coroputation or dt
177
and we wilt prove that the diagram
ffi F.. MIJ
ti E' I.,.
9' ) EB F @G M T
u,:Ip_1
.,
-1 l.
F @G CiX, M) Ffl;c J. F  Cp-l(X, M)
commutes. Here 8 and V1 are the chain isomorphisms underlying 7.6; ex
plicitly, a glance at the proof of Shapiro9s lemma (111.6.2) shows that (J and tJI
are simply the maps induced by the inclusions M '+ Cp(X) @ M and
Mr  Cp-1(X) @ M.
We begin by computing the transfer map tct on tbe chain level by means of
Definition (C) of III.9. F or x E F and 111 E Ma' we find
t,,"x @ m) = L 9 - l X @ g - 1 m E F Q!)Gor: M" .
fleGt/IGt/
Next, the map "er. is given on tbe chain level by F @ Dar:: F @GIn M(1 -+
F @G Mr:. Thus
r:
ucrrcglx @ m) = I g-lx @ ÒOt(g-lm)
/lE G OIG"T
L g-lx @ g-lòq.sl.m),
f/E 6alGISr
where the second equality comes from the fact that a is a G-map. We apply
now tbe map Vr, which can be computed via the diagonal action'" of g(T)
(cf. gIII.8, paragraph preceding 111.8.1), and we sum over t E ,; this gives
lp{x @ m) = L L g(t)g-lx @ g(y)g-lòO,fJJm).
1:1E (JEGqlGcrt
We can now establish tbe desired commutative diagram of chain com-
plexes. Using the tensor product relations gu @ gv = u @ v in - @G -, we
deduce from the formula for cp above that
VJtp(x @ m) = I L x @ Ott.,-r(m).
n 5'; geG(Y/Gerr
But it is clear from the detìnition of .'F that tbe right-hand side of this equa-
lioo is simply
r X  otl!..m) = X @ om.
re/Fo
Thus .jJ(j) = (F @ ò)8. as required.
D
Tbe description of E1 and d1 simplifies greatly if X has the property that
GIJ fixes t1 pointwise Cor every celi u, as in exercise 2 of 7. Indeed, one ean
tben orient each celI of X in such a way that tbe G-action preserves orien ta-
tions, and such a choice of orientatioDS dett;1iuines for each u an isomorphism

178
VII Equivarian1 Homologyand Spectral Sequences
Mrr ;:::::: M of Go--moduJes. We can therefore write Hq{Ga, M) instead of
Hq{G(TJ Ma)' More significantly, we have G.,c = Go for '( E, since such a T
is necessarily in tbc closure of (J and hence is fixed by Go. Thus the transfer
map t(lT is the identity map, and aIso:F = :F(r
Roughly speaking, then, we have a .. coefficient system n
 = {Hq(G(I' A-I)}
00 the orbit complex XjG, and (EJ, dI) is tbe chain complex af x/a with
coefficients in this system. The E2-term or the spectraJ sequence can therefore
be thought of as Hp(XIG, J¥"q). We will not attempt lo make these remarks
precise.
9 Example: Amalgamations
Consider an amalgamation G = H * A K. where A  H and A 4 K. Let X
be tbe tree associated to G (cf. appendix to Chapter II), and recall that G
acts on X with no "inversions," Le., no element of G interchanges the vertices
of a l-simplex or x. [Thus we are in the situation described at the end of 8.]
There is a single l-simplex e which maps isomorphical1y onto tbe q uotient
graph X /G., and the isotropy groups of e and its vertices vand w are given by
Gv = H, Gw = K, and GI.' = A.
We therefore have for any G-module M, a spectraI sequence converging
lo H.(G, M) (cf. 7.10), with
H.(H, M) E9 H.(K. M)
E.. = H.(A, M)
O
if p=O
if p = ]
if p > 1.
(We have taken, here, 1:0 = {v, w} and I = {e}_) The spectral sequence
therefore collapses at E2 to yield a Mayer- V ietoris sequence
(9.1) ... -+ Hn(A. M)  Hf/(H, M) ffi H,lK, M) !. HII(G., M)
-+ H Il - l (A, M) -+ .. . .
The map p here is the edge homomorphism discussed in exercise 1 or
7, and hence it is the map induced by the inc1usions H 4 G and K  G-
The map Cl is the differential di given by 8.1. The latter is particularly easy
to apply here because the rnaps tUT and VI' of 8 are identity maps. Since
o : M -+ M is idM and () v: M --+ M is - idM. we conclude that IX is simply
e.w e.
the difference of the maps H .(A) --I> H .(H) and H *(A) --+ H .(K) induced by
the incJusions A  H and A  K.
-
I
t
,
;
,
I:
I
, ,
>
L.
r
l,
I
,
,-
I
,
,

. -
\
'"
9 Example: Amalgama1tOns
179
Next we wish to briefly describe a generalization of the notion of amalga-
mation for which one can also construct a Mayer-Vietoris sequence. Bya
graph oJ groups we mean a çonnected graph Y together with (a) groups Gv
and Gl" where v (resp. e) ranges aver tbe vertÌces (resp_ edges) of  and (b)
monomorphisms 00: Ge 4. G(.I and (Jt: Ge C;. Gw for every edge e, where v and
w are the vertices or e. [Note: We aIlow the possibility that v = w, but 00
and ()l may stilJ differ.] One can then define the fundamental group G of the
graph of groups by using a suitable nation of' palh t.,: one regards an element
of G v as a path from v to v. and one imposes the relations
8o(a)e = e61(a)
for every a E G, i.e_. one identifies the composite paths
BoCa) O,.
t'
, .o and o
w v

) wO 81(a)
';
from v to w. (See Serre [1977a] for more detaiJs.)
For example, the amalgam H * A. K is thc fundamental group of the graph
A
-
Q
H
K
--{)
where we ha ve labeUed the vertices and edge with the associated groups. (In
this case 80 and (Jt are the inclusions A 4 H, A  K.)
Another importa'nt group theoret1C construction which fits into this
framework is the HNN extension H *.4. Rere we are given a group H, a
subgroup A, and a monomorphism (J: A -io H, and H ..4 is obtained by ad-
joining an element t to H subject to the relations t -lat = O(a) (a E A). This
group can be realized as the fundamental group of the graph
AOH
with one vertex and one edge, where 80 and 61 are the inc1usion i: A  H and
tbe given map 8: A -+ H.
Just as in the case of amalgamation, tbere is a tree X associated to a graph
of groups, and tbe fundamental group G acts on X witbou t inversion. The
originai graph Y is the orbit graph X IG. and the groups G v and G e are the
isotropy subgroups of G at suitable liftings to X of the vertices and edges of ¥_
ConsequentlYt one obtains from 7_10 a Mayer- Vietoris sequence
(9.2) ... - EB H,.(Ge, M) -+ EB HII{Gv, M) - Hn(G, M)
ee'Y vero
-+ EBHft-1(Ge, M) -+ -.',
ee Ya
generalizing 9.1 (Here li is the set of i-cells or 1':) This exact sequence was first
written down explicitly by Chiswell (1916a].

180
VII Equivariant Homology and Spectral Sequences
[This exact sequence can, of course, be derived without thc use or spectraJ
sequences. Namely one argues as in the exercise of III.6. using the exact
sequence of G-modules
(9.3)
0--+ EB Z[G/G.] -io E9 Z[G/Gv] -. Z --+ O
eeYl .-eYo
given by thc augmented chain complex or the tree X-J
Consider, for example, the H N N extensioD G = H * A desçribed above.
The sequence 9_2 then has the form
Il p )
(9.4) ... -io Hn(A M) -io H,,(H, M) ...... HrzCG, M --+ HII_I(A. M) --+....
As in 9.1, f3 is induced by tbe inclusion H -1> G. To evaluate Cl = di, we need
to know the following fact about the tree X: the unique edge of Y lifts to an
edge e of X whose vertices have the form v tv. Taking LO = {v} and:tl = {e}
in the notation of8. we find that vT (t = tv) is the conjugation isomorphism
c(t-l).H.(tHt-l M) -io H.(H, M). Since the composite
(A, M) -io (tHt-l, M) (\1-1) (H, M)
is equal to (8. t-l) it follows that IX = (8 t-l). - i$'
lO Equivariant Tate Cohomology
Recall from 7 that tl:;(X, M) is defined il G is finite and the G-complex X
is finite dimensional. This theory was introduced by SWan [196Oa], who in
fact constructed a topo logica l version of the theory (Le., X is not required to
be a CU'-complex).
The usefuiness of equivariant Tate cohomology comes from tbe fact that
it provides a machine for systematically ignoring free actions:
(1041) PropositioDa Let Y be a G.invariant subcomplex oJ X such that tbe
isotropy group Gr1 is trivial for every cell II oJ X that is not in Y. Then the
inclusion Y c.. X induces an isomorphisrn fl"t;(X, M) ..::. fl(1: M).
PROOF. Consider the Tate cohomology version of the spectral sequence 7.7
for (O, X) and (G, Y). By hypothesis, fJ*(GI1, M) = O if (J is celi of X - Y.
Thus the inclusion Y  X ind uces an isomorphism of spectral sequences
and hence an isomorphism of the abutments R(X M) and flt;(Y, M). D
As a simpie ilIustration of 10.1, we prove tbe rollowing result (cf. VI.8,
exercise 4):
(10.2) Proposition. Ler G be a finite group vhich admits a finite dimensional
free G.complex X such that H.(X)  H.(S211-1). Then G has periodic eD-
homofogy oJ period 2k.
-
t
f
,
I
I
.:
[.
I
r
t
 .
.
IO Equivariant Tate Cohomology
181
PROOF. By 10.1 (with Y = 0), fl(X M) = O.On the other hand we ha ve a
spectral sequence (cf. 7_2)
(10.3) Eq = flp(G, Hq(X M» =-1lt;+q(X. M).
Since the spectral sequence is concentrated on the horiwntallines q = O and
q = 2k - t it follows that the differentiai d 2k is an isomocphisrn
(10.4)
fìp(G, Hn-J(X, M»  f1P+2"(G M).
.,
In view of the universa] coefficient isomorphism
H211-I(X. M)  Hom(H2"_lX M),
which is natural and hence an isomorphism of G-modules, the proof will be
complete if we show that G acts trivially on the infinite cyclic group H 21- 1 X.
This can be deduced from a generaI Lefscbetz fixed-point theorem (cf.
IX.5, exercise 1), but in the present situation there is a much easier proof.
It suffires to prove that every cyclic subgroup of G acts trivially OD H 2J: _ IX,
so we are immediately red uced lo the case where G is cycIic (and non-trivial).
In tbis case we apply 10.4 with M = 7L and p = O to conclude that
flo(G, Hlk-l(X Z))  1ì2k(G, Z)  l'liGI, 7l.
But this implies that H2k-l(X, lL)G ::F O, which can only happen if G acts
trivially OD HZk-tX. D
.-
?
Remark. The advantage of this spectral sequence proof of 10.2 is that it
applies verbatim to the topological situation, where X is not assumed to be a
CW-coniplex cf. Swan [1960a].
'.
As another applicati OD of equivariant T ate cohomology theory (a180 due
to Swan), we wilI prove some classical results of fixed-point theory. (See
Bredon [1972] or Bore] et al [1960] for the standard proofs of these results.
based on 'Smith theory_")
(10.5) Theorem. Let X he a jinite-dimensional G-complex, where G has prime
order p. Assume that the jìxed-point set XG is a subcomplex.
(a) II H*(X, ..lI') isfinitely generated, then so is H*(XG, ,l,,).
(b) II X is acyclic mod p (i.e., H.(X lL,J  H.(pt., lp», then so is XV.
(C) lf X is a cohonwlogy sphere mod p (i.e. H.(X lp)  H*(S'\ lLp)for some
n > O), tben so is XG  provided XG '# 0.
(The hypothesis that XO is a su bcomplex holds, for ex ampIe, if X satisfies
the hypothesis of exercise 2 of 7.)
PaOOF. We have
fJ1;(X ?Lp)  1tt;(xG, lLp)

182
VII Equiv8riant Homology and Spc:ctral Sequences
by 10.1. On thc other hand. since G acts trivially on XG, we have a Ktinneth
isomorphisrn
R(xG, Zp)  R*(G 7L.p) @zp H*(XG, Zp),
cf. 5.4. Sincedimzp iìrr(G ?Lp) = l for alI n E li, it follows that
(10.6) dimzp RG(X, Zp) = L dimzp Hi(XG, Zp)
iO
for ali n. Assuming now that H*(X, lLp) is finitely generated. the spectral
sequence 10.3 (with M = 'Zp) implies that fì(X, Zp) is finitely generated, so
(a) follows from ] 0.6. Similarly, if X is acyclic mod p then the Ieft side of 10.6
is equal to l, hence so is thc right side; this pro ves (b). The prooi of (c) is
identical to that of (b), except that one uses relative cohomoIogy groups of
tbe pair (X, v), where v is some vertex or xG. o
EXERCISES
1. Where did the proof or 10.5 use the assumption that I G I = p? [There are two places.]
2. Extend 10.5 to the case where G is an arbitrary p-group and XEI is a subcomplex of
X for every H c G. [Hint: V se inductiol1 on I G I, noting tl1at XG = (X"'ti/lò if
N <;! G.]
3. USÌng the method of proof or 10.2, show that if X is a tìnite-dimensional rree G-
complex (G finite) with H*(X)  H*(Sn). then every non-triviaJ element of G acts
nOI1-'rivial1y on H 2k(X), and hence I G I ::;: 2.
-
l
I
,
;
,
!
.'
"
.

CHAPTER VIII
Finiteness Conditions
1 Introduction
Reçall that the definition of H.(G, M) and H*(G, M) allows us to choose an
arbitrary projective resolution P = (PI)j'() of 7L over lLG. Similarly, if we
wish to take tbe topological point of view, tben we can compute H .(Gt M)
and H*( G, M) in terms of an arbitrary K( G, 1 )-complex ¥. Since we have this
freedom of choice, it is reasonable to try to choose P (or Y) to be as " small "
as possible, and this Ieads to various finiteness conditions on G.
For example, if we interpret '.small" in terrns of the length of P (or the
dimension of Y), then we are led to the notion or cohomological dimension.
Or ifwe interpret small to mean that each Pi should be finitely generated (or
that Y should have onIy finitely many cells), then we are led to the so-called
 F P" and . F L" conditions.
Our goal in tbis chapter is to introduce these and related finiteness
conditions and to give some examples. Our treatment will by no means be
complete; for the most part we will give complete proofs only for those
results which will be needed in Chapters IX and X. See Bieri [1976] and
Serre [1971,1979] for a much more thorough treatment ofthe subject and a
guide to the literature. See also Wall [1979] for a list of open questions
conceming finiteness conditions.
Finally, a word about notation: In the theory of discrete subgroups of Lie
groups, which is the source of many of our examples, it is customary to
denote the Lie group by G and the discrete subgroup by r. In order to be
consistent with this we will use tbe letter "rto from now on (instead of"O")
to denote a typical abstract group.
li1

184
VIII Fìniteness Conditions
2 Cohomological Dimension
We begin with a basic lemma which characterizes projective dimension in
terrns of the vanishing of cohomology functors. Recall first (cf. V18) that
if R is a ring, M is an R-moduIe, and n is a non-negative integer, tben
proj dimR M < n if and onIy if M admits a projective resolution
O -t' PJI --+ . .. -io Po -+ M - O
of length n. Recall aiso (cf. III.2) that the Ext functoTS are defined by
Ext(M, -) = Hi(.1fh.mR(P -.-»,
where P is a projective resolution or M. In particular.
t .
Extzr(.Z, -) = Hf(r, -).
(2.1) Lemma. The following condirions are equivalent:
(i) proj dimR M < n.
(ii) Extk(M. -) = O for i > n.
(iii) Ext+ l(M,. -) = O.
(iv) lf 0-+ K -io P,,_ J -+ . . . -+ Po -+ M -+ O 1s any exact sequence oJ R-modules
with each Pi projective, then K is projective.
PROOF. It is obvious that (iv)  (i) => (ii) =- (Hi), so we need only prove
(iii) => (iv). Given a partial resolution as in (iv), complete it arbitrarily to a
projective resolution
. . . -+ p It + t -+ P" -+ p  - 1 -+ . . . -+ p o -+ M -- O.
\J\J
L K
For any R-module N, an (n + l}-cocycle in:KomR(P N) is a map Pn+l -+ N
whose composition with PII+2 -+ P,,+ 1 is zero. Such a cocycle, therefore, can
be regarded as a map rp: L -+ N. Tbe cocycle is a coboundary if and only if ({J
extends to a map P It --+ N. Thus (iii) implies that every map on L extends to
P". In particular, tbe identity map on L extends to P", so Pli  L G) K
and hence K is projective. O
The implication (i) ::;.- (iv) of 2.1 is very useful. It shows that if tbere exist
projective resolutions of length n,. then we don't ha ve to be clever to find one-
any partial resolution or length n-l can be completed to a projective
resolution of length n. We will give another proof ofthis important fact later
(remark 3 following Lemma 4.4).
We now specialize to the case R = zr, M = Z. The cohomological
dimension of r 7 denoted ed r, is defined to be tbe smallest integer n such tbat
-
,
,.
; .
2 Cohomological Dimension
185
tbe conditions of tbc lemma hold, provided tbere exist such integers n;
otherwise we set ed r = 00. Thus
ed r = proj dimzrlL
= inf{n: 71.. adrnits a projective resolution oflength n}
= inf{n: H(r, -) = O for i> n
= sup{n: H"(r, M) #:- O for some r-module M}.
.
There is an obvious topological analogue of ed r: The geometric dimension
of r, denoted georn dim r, is defined to be tbe minimal dimension of a
K(r, l )-complex. Since thc cellular chain complex of tbe universal cover or a
K{r, 1) complex Yyields a free resolution of71. over Zr(oflength equal to tbe
dimension ofY), we clearly have:
(2.2) Proposition. ed r < geom dim r.
We will return to this in 7, where we will prove that equality usually
holds.
EXAMPLES
1. ed r = O if and onIy if r is the trivial group (cf. exerçise l beIow).
2_ If r is free and non-trivial then ed r = l (cf. 1.4.3). Conversely, a deep
tbeorem of Stallings [1968] and Swan [1969] says that every group or
cohomological dimension 1 is free. In view of Chapter IV, tbis result can be
restated as fo Il ows : If r is a group which admits no non-split extension with
abelian kernel then r adrnits no no n-spii t extension at alI.
3. Let r be the fundamental group of a connected dosed surface Y other
than S2 or P2. Tben Y is a 2-dimensional K(r, l) (cf. II.4 example 2) so
ed r < 2_ And H2(r, Z2)  H2(y. Z2) ;I:. O, so ed r = 2.
4. More generally,. suppose r is a one-relator group whose relator is nat
a pro.per power. Then ed r < 2 by Lyndon's theorem which we quoted in
1I.4, example 3.
5.1£ r = l'I then the n-dimensional torus Y = S1 X ... X SJ is a K(r, l)
with HIJ(Y, Z)  Z =F O, hence ed r = n.
6. Let r be the grou p or 3 x 3 strictly upper triangular matrices with
integraI entries:
'.
] .. ..
r= O 1 .
O O 1

186
VIII Finiteness COt1ditions
We will show that ed r = 3. Let r' be the central subgroup
lO.
O 1 O
O O l
of r, and note that the quotient r" = r jr' is free abelian of rank 2. Consider
the Hochschild -Serre spectral se q uence
Er = HP(r", Hq(r', M»  HP+q(r. M),
where M is an arbitrary r -mod ule. Example 5 shows that this spectral
sequence is concentrated in the rectangle O < P < 2, O < q < 1. Conse-
quently, Hi(r, M) = O for i > 3, so ed r  3. Moreover. since the differential
dI' maps E to E+r.q-r+ l, there can be no non-zero differentials involving
the upper right- hand corner E;' l. This beìng tbe onIy non-tri vial terro or total
degree 3, it rollows that
H3(r, M) = E; l = E' l = H2(r", H1(r', M».
In particular. taking M = 7l. and reealling that r' is centraI in r, we see that
H3(r, Z)  li (cf. exercise l of I1I.8); hence ed r = 3.
7. More generally, let r be an arbitrary finitely generated, torsian-free,
nilpotent group. One can show that r admits a centrai series
r=rOrl =:>"':::::Jr,.={l}
with free abelian quotients ri/ri + l' The sum of the ranks of these quotients
is independent or the choice of centraI series and is called the rank (or Hirsch
number) of r. Arguing as in example 6t one finds
ed r = rank r.
Details are omitted. See Bieri [1916]. 7.3, and Gruenberg [1970], 8.8, for
more details and for rurther results of this type. See also example 2 of 9
below for an indication or a different proof of this result.
The rest of tbis section will be devoted to some elementary properties of
cohomological dimension.
(2.3) PropositiOll. lf ed r < 00 then
ed r = sup{n: W(r. F) #; Ofor somefree .zr-module F}.
PROOF. Let n = ed r. In view or the long exact cohomology sequence
(III.6.1(ii'») the functor HII(r, -) is righI exact. Since H"(r, M) =# O for
some M. it follows that HJI(r. F) ::j; O for any free module F which maps
onto M. D
I
..
i
,
!.
;
I '.
,
f '
"
.
"
,.
"
,
..>
"
.
,
,
.
,
.
,-
2 Cohomological Dimensiofl
187
(2.4) Proposition.
(a) Ifr' c r then
.,
ed r' < ed r;
equality holds if ed r < 00 and (r: r') < 00.
(b) li 1 -+ r' -+ r -+ r" -+ 1 is a short exacr sequence 01 groups, llten
ed r < cd r' + ed r".
(e) lfr = r1 *..4 r2 (where A '+ rat then
ed r s; max{ed r., ed r2, 1 + ed A}.
PROOF. The inequality in (a) follows immediately from Shapiro's lemma or,
alternatively, from the fact that a projective resolu tion or Z over zr can
also be regarded as a projeçtive resolution ofZ aver zr'. To prove the second
part of (a), suppose ed r = n < :::o. By 2.3 there is a free Zr-module F with
HII(r, F) i= O. If F' is a free zr' -mod ule of the same rank. then F  IndF F',
so Shapiro's lemma yields H"(r', F')  HIf(r, F) #< O. Thus cd r > n.
whence (a), [Exercise Where did we use the hypothesis that (r: r') < oo?]
(b) is an immediate consequence of the Hochschild-Serre spectral sequence,
as in Example 6 above. Finally, (c) follows from the cohomology version of
the Mayer Vietoris sequence (VII.9.1 or exercise of 11I.6). D
(25) Coronary. 1f ed r < 00 then r is torsion-free.
PROOF. If r is not torsion-free then r contains a nontrivial finite cyclie sub-
group r'. Such a r' has cd r' = 00, since H21c(r', Z) #:- O ror alJ k (cf.
III.l, Example 2) so 2.4a implies that ed r = 00. D
If r c r and (r: r') < 00, then 2.4a shows that ed r' = ed r unless the
following occurs:
(.)
ed r = co and ed r' < 00.
Easy examples show that (.) can in fact occur; e.g. ta.ke r of order 2 and
r' = {l}. More generally, if r is any group with torsion, then ed r = 00
by 2.5, but r may very well have torsion-free subgroups r' of finite index
with ed r' < 00. We will prove in the next section a theorem of Serre which
says that ali examples of (>ti) are of this type, Le., that (*) cannot occur if r is
torsion-free.
Our last result shows that, as far as cohomologieal dimension is çoneemed.
wc never need to use projective resolutioDS which are not free:
(2.6) Proposition. For any group r there is a free resolution DJ 7L over zr oJ
length equal lO ed r.
Tbe proof requires tbe following U Eilenberg trick":

188
VIII Finiteness Conditions
(2.7) Lenuna.lf P is a projective module over an arbitrary ring R, then there is a
free module F such that P EB F  F_
(Warning: F will be ofinfinite rank, in generaI, even if P is finitely generated.)
PROOF. Since P is projective there is a module Q such that P E9 Q is free. Let
F be the countable direct sum
(P a7 Q) a7 (P a7 Q) <1> . . . .
Then F is free" being a direct sum of free modules. But F can also be described
as the sum of a countable number of copies of P and a countable number of
copies of Q. Adding ODe more copy or P does not change this so P $ F  F.
o
PROOF OF 2.6. Let 11 = ed r. We may assume O < n < 00. Choose a partial
free resolution
iJ
Fn-l -+ ... -+ Fa -+ Z -+ O
of length n - I and let P = ker{o: Fn-l - Fn-2}. (Here we set F -1 = 1.
if n = 1.) Then P is projective by 2.1, so 2.7 gives us a free module F such that
p EX> F is free. Thus if we replace F 11_ l by F n- I Ef) F and set a I F = 0, we obtain
a partial free resolution of length n-l with ker iJ free, whence tbe pro-
position. D
EXERCISES
1. Prove that the trivial group is the on1y group or cohomoJogical dimension zero.
[Hint: ed r = O.ç;;. Z is a projective zr module .e> the canonical surjection
e: zr --. 7L sp]jts_]
2_ Give an example lO show that tbe hypothesis ed r < 00 is necessary in 2.3.
3. (a) Show that geom dim r. *A r2 $ max{geom dim rl. geom dim r2, 1 +
geom dim A}.
"Cb) If l - r' - r ...., r" ...., 1 is exact. show that geom dim r < geom dim r +
geom dim r".
4. Prove the following generaljution of 2.4c: If r is ao arbitrary group and X is an
acyclic rcomplex. then
ed r :s;; sup{cd re + dim oJ.
(J"
where (J ranges over a set of representatives for the ceUs or X mod r. (Taking X to
be the tree associated to an amalgamo we ff;Cover 2.4c.) [Hint: U se the cohomology
version of the spectral sequence VII.7.10. Or see Serre [1971] for an alternative
proor. ]
t
..
,

I
, . .
\
I
I
2 Cohomo]ogica] Dimension
189
.,
5. The purpose or this exercise is to give a cohomologica.1 criterion for finite-
dimensionality (up to homotopy) or an arbitrary chain complex. In case the chain
complex is a projective resolution, this result «b) beJow) reduces lO the equivalence
(i)..;:;.(it)<:>(tv) or Lemma 2.1. Let C be a chain compIex over an arbitrary ring R.
(a) For any integer 11, prove that the following two conditions are equivaJent:
(i) H" + 1 (.KomR(c. M)) = O for alI R.modules M.
(ii) H n + . C = O and the mod ule B" or n-boundaries is a direct summand or Cn.
[Hint: Examine cocydes and coboundaries in .J'&mR(C, M) as in the proof of2.1.]
(b) For any integer 11. prove that the folIowing three conditions are equivalent:
(i) C is homotopy equivalent to a complex C such that C = O for i > n.
(ii) Hi(.J't'omR(C. M» = O for aH i > n and aH R-modules M.
(iii) Ir C is the quotient of C defined by
.'
C. if 1<1'1
I
c= CJBII if i = n
l
.. O if i> n.
then the quotient map C ---+ C is a homotopy equivalence.
Ifthese condìtions hold and C LS a complex of projective R-modules. prove that the
complex C' in (iii) i5 also a complex or projectives.
[Hint: If (ii) holds, deduce from (a) that C  C EE> C'. where CH is CQntractible.]
. .
*6. Suppose ed r = n < 00, and let r' erbe a subgroup of finite index.
(a) For any r-module lvf, show that the transfer map tr: H"(r'. M) -io Hn{r. M) is
surjective. [Hint: Compute tr on the chain leveI, using a projective resolutjon of
Z over zr oflength n; no(e that the chain map is smjective. AlternativeJy. see Serre
[1971], 1.3, Lemme 2.]
(b) Suppose that r is normaJ in r. Ir M is a r -module and d is an integer such that
H'(r'. M) = O for i > d, show that Hi(r. M) = O for i > d and that tr induces an
isomorphism
.
Hd(r'. M)rr  Hd(r, M).
In particular. this holds with d =  for an y M. [See Brown - Kaho [1977]. Prop. 1.2_]
1. (a) Show that induced r-modules.z:r (g) A have projective dimensìon < 1.
(b) If proj dimR M S; n. show that proj dimJ;: M' < Il for any direct summand M'
of M. (Hint: 2.1.]
(c) Deduce from (a) and (b) the following result (which is also an immediate conse-
quence of VI.8.12): If r is finite and M is a r -module in which I r I is invertible. then
proj dim.zrM < 1. [Hint: M is a dicect summand oran induced module.]

190
VIII Finiteness Condi[10ns
3 Serre' s Theorem
(3.1) Theorem (Serre [1971]).lfr is a torsion{ree group and r is Q subgroup
01 finite index, then ed r' = ed r.
PROOF. In view or 2.4a, we need only show that if cd r' < 00 then ed r < 00.
It is possible, as we will indicate below, to give a purely algebraic proof of
this result; but we wil] give instead a topological proof, when will yield
important information that will be needed in Chapters IX and X. This
topological proof requires the following result of Eilenberg and Ganea
[1957], which we will prove later as part of Theorem 7.1: If r is a group
sllch that ed r < oo then there exists a finite dimensional K(r, 1 )-complex.
Returning now to the proof of 3.1, we are given that ed r' < 00, so there
is a finite-dimensional K(r', l). Its universal cover X' is then a finite di-
mensional, con tract ible, free r'-complex. To prove ed r < cc, we will
construct rrom X' a finite dimensional. contractible, free r -complex X. The
construction, which is a straightforward analogue of the co-induction
constrllction for modules, goes as follows:
The underIying set of X is defined by X = Hom.....(r, X'), where f' acts
on r by Ieri translation and Hornr,( , ) denotes maps in the category or
left r' -sets. Since the right action of r on itself commutes with the Ieft action
of r' OD r there is an induced (Ieft) actioD of r on X, given by (ro!)(V) =
f(y'fo) for f E X, l', ')lo E r.
If we ehoose a set of coset representatives }'h''', l'n far r'\r, then we
obtain a bijection
rz
cp: X  f1 X',
i= 1
given by eval uation at V l' . . . , Y 11. Since the prod uct OD the righi has a natura)
CW-structure (with the cells being tbe products of the ceIls of tbe factors), we
can use qJ to give X a totiology and a CW-structure. [Note: The product is to
be given the CW-topology, ie., the "weak topology" witp. respect to tbe cells;
this agrees with tbe usual product topology if X is countable or locally
compact, cf. Lundell-Weingram [1969], 1I.5. In any case, the two topologies
agree on alI compact subsets.]
This structure is independent of the choice of coset representatives; for if
we replaee 'Vt, . . . , VJI by }Il Y l. . . . , V 'VII (y E r'), then the new <p is obtained
from the old one by composition with the CW-isomorphism
R n n
fl ì'l: n X'  n x'.
i= l i= l i",,; l
The structure is also indcpendent of the ordering of tbc cosets.
It now follows that the r -action OD X preserves thc C W-structure.
I
I.
..
,.
.
r
l
.
'.
. .
.
-'
I
I
4 ResoJutions of Finite Type
191
Indeed, for any y E r we have a commutative diagram
X y  X
\ ;.
"
n X'
j=-= 1
where (f) is defined via coset representatives (Yi)1 S;iS;JI and ql is defined via
(Vi}')t iS;l'I' Thus X is a well-defined r-cmplex, which is clearly contractible
and finite dimensional.
To complete the proof that ed r < 00, we will show that r acts freely OD
X. There is a canonical map X .... X', given by evaluation at 1 E r. This map
is r'-equivariant and takes cells to cells. Since r' acts freely on X', it follows
that r' acts freely OD X. For any cell (1 of X, then, we have r (I" r"'\ r' = {l},
hence r ti is finite. But r is torsion-free, so these finite isotropy groups r ti are
triyial. D
Remark. Tbe interested reader can transJate the proof above into a purely
algebraie proof. One works direetly with projective resolutions, and one does
the 'eo-induction" construction using tensor products or chain complexes
rather than cartesian products of CW-complexes. More details can be found
in Swan [1969], Theorem 9.2.
EXERCISE
Note that the proof of 3.1 is valid even if r has torsion, except for the last sentence of the
proof. Consequently. if r ìs a group which contains a subgroup r' of finite index such
that ed r' < X). then we can construct a contractible. finite dimensional r.complex X,
with finite isotropy groups r.... Show that this complex has the followìng additional
property: For every finite subgroup H c:. r. the fixed-point set XH is contractible.
[Hint: Show that X, as an H-complex, is isomorphic to the product of(r: r') copies of
the complex X', with H acting by permuting the factors according to the (free) right
action or H OD r\L One can see this eìther by using the double coset fonnula for co-
ìnduction or simply by direct ìnspection, using coset representatives or the fOnTI
{}'.h} l s;i:od.IleH, where d = (r: r')/1 H I. Hence XH is homeomorphic to the product or
d copies or x'.]
4 Resolutions or Finite Type
Our next goa1 is to study a different sort or finiteness condition where we
require that there be a projective resolution P with each P, finitely generated.
In this section wc collect some genera] facts about such resolutions over an

192
VIII Finiteness Conditiorul
arbitrary ring R. Then in tbe next two sections we specialize to resolutions of
7L over 7Lr.
We begin hy reviewing the theory of finitely presented modules;
(41) Proposition. The !ollowi'1g conditions on an R-module Mare equivalent:
) There is an exact sequence Rm -io Rtf -+ M - O for some integers mt n.
(Il) There is an exact sequence P 1  P o - M ---f- O for some finitely generated
projectives Po, P l'
(in) M is finitely generted, ami for every surjection f: P "* M with P finitely
genero.led and pro}ective, ker f isfinitely generated.
The proofis based 00 Schanuers lemma":
(4.2) Lemma. Let 0-+ K -io P -+ M -+ O and O -io K1 -+ P' --+ M' -io O be
exact sequences with P and P' projective. Then p $ K'  p' Ef) K.
PROOF. Let Q be the pullback of the given maps
p
1
P M,
]t
ie., Q is the suhrnodule of P x P consisting or those pairs (x, x') such that
n(x) = n'(x')_ One then verifies easily that there is a commutative diagram
O O
1 l
K1 K'
1 l
O )K I Q .p' . O
l 1
O .K ,p )M o
l l
O O
with exact rows and columns. Since P and pl are projective, tbe two exact
sequences involving Q must split.. yielding P ffi K'  Q :::: p' ffi K. D
PROOF OF 4.1. Clearly (iii)  (i) => (ii). To prove (ii) ""> (iii) note fust that M
is certainly finitely generated if (ii) holds. since Po is finitely generated. Now
I
I
I
I .
I "
.
<
4 Resolutions or Finite Type
193
apply Schanuel to get P EB ker{P o -+ M} :::::::; P o Ef) ker e. The lefthand side
being finitely generated by hypothesis, it follows that P o Ef) ker l; is finitely
generatedt hence so is ker . D
M is said to be finitely presenred if the conditions of 4.1 hold. An exact
sequence as in (i) is said to give a finite presentation of M with n generators
and m relations. Note that the implication (i) => (iii), applied when P is free,
reduces to the following well-known fact: If M adrnits some finite presenta-
tion. then every finite set xl".' t Xl of generators of M has the property that
the relations among them (i.e_, the k-tuples (rl, - . - T r,,) such that I rixi = O)
f<?rm a finitely generated submodule of Rk.
It is natural to generalize finite presentation as follows: A resolution or
partia! resolution (Pi) is said to be offinite type if each Pl is fìnitely generated.
A module M is said to be of type PP" (n > O) if there is a partial projective
resolution P" -io . . . -io P o -io M -+ O or finite type. Thus the F P o condition is
simply finite generation, F P 1 is finite presentation, and the conditions
F p 2 , F P 3, . . . are successive strengthenings or finite presentation.
Generalizing 4.1 we bave:
(4.3) Proposition. F or any module M arul integer n > O the following con-
ditions are equivalent:
(i) There is a partial resolution Fn -+ . . . - Fo -+ M --+ O with each Fj free
of finite rank.
(ii) M is oftype FPn.
(iii) M is finitely generated, and for every partial projective resolution
P"  .. . -+ Po --+ M --+ O of finite type with k < n, ker{P" -+ PIì.-l} is
finitely generated.
F or the proof we wi11 need tbe following generalization of Schanuel's
lemma:
(4.4) Lemma Let
o -+ p -+ p 1 --+ . .. --+ Po -+ M ---f- O
;q pr-
and
O -+ Pn -+ p- 1 -+ . . . -+ Pa -+ M --+ O
be exact sequences with Pi and p projective Ior i  n-L Then
p o Ef) P'l Ef) P 2 E9 P3 EB . --  P'o 63 P 1 <1> P2 Ef) P 3 Ef) . .. .
Consequently, ifPl and  arefinitelygenerateJfor i < n - l, lhen PII isfinitely
generated if and only if P',. is finitely generated.

194
VIII Finiteness Conditions
PROOF. We argue by induction OD n_ Let K (resp_ K') be the kernel of Pn- 2 -+
p lJ - 3 (resp. p - 2 -+ p- 3)- By the induction hypothesis we have
K ffi Q  K' EB Q'
where
Q = P-2 Ef> Pn-3 ffi ...
and
Q' = Pn-2 Ef> P-3 Ef>.. . .
On the other hand, we have exact sequences
O -+ PII --lo P,,-l Ef> Q -.. K 6) Q --+ O
li
O -+ P: ---l' P-l Ef> Q' --+ K' ffi Q' -+ o.
Since P lJ- l ffi Q and p - l Ef) Q' are projective 4_2 implies that
Pn ffi P-l 6) Q' ;:.-:; p ffi PR-l 6) Q,
which is the desired isomorphism.
D
PROOF OF 4.3. (i) => (ii) trivially. (ii) => (iii): If M 1s of type F Pn, then for any
k < n there is a partial projective resolution P" -+ . -. -+ Po ..... M --+ O of
finite type with ker{P" - Pt- d finitely generated. It follows from 4.4
that any other partial projective resolution P - . . . --+ Po --+ M -+ O of
finite type (and the same length k) has ker{P --+ _ d finitely generated,
so (iii) holds. (iii)  (i): If (Hi) holds then we can construct the desired
F n -+ - - . - F O -+ M --+ O step by step_ D
Remarks
1. ODe can remember the isomorphism of 4.4 by formally "transposing"
terrns to obtain an U Euler characteristic" equality
p o - P 1 + p 2 - .. . = 1>'0 - P + P2 - ....
2. If we regard 4.4 as a oomparison theorem for resolutions, it is natura]
to ask if il can be deduced from the results of I. 7. This can in fact be done;
here is an outline: By 1.7.4 we can choose an augmentation-preserving chain
map P' -+ P; Iet C be its mapping cane. Then C is acyclic and hence breaks up
into short exact sequences. Using the fact that C is projective except in the
top two dimensions, ODe can prove inductively that these short exact
sequences split (cf. exercise 3a of I. 8). Thus C 1  C o EB Z 1 t C 2  Z 1 Ef> Z 2 ,
C3  Zz Ef> Z3, etc., whence .
Co ffi C2 ffi . .. :::::: Cl EB C) EB . . . .
Recalling that Cl = Pi EB Pi- h we obtain tbe desired isomorphisOL
I
I
:-
r
"..
-j
.
4 Reso]utions or Finite Type
195
3. It follows from the isomorphism of 4.4 that P" is projective if and only if
P: is projective. Thus 4.4 yields an alternative proof of the implication
(i) => (iv) of 2.1.
We wil1 be primarily interested in the caSe where the conditions of 4.3
hold for alI integers n > O. This situation is characterized as follows:
(45) Proposition Thefollowing conditions on a module Mare equivalent:
(i) M admits afree resolution offinite type.
(ji) M admits a projective resolution of finite type.
(iii) M is oftype FPn!or alI integers n  O.
PRlXJF. (i) '* (ii)  (iii) trivially. If (iii) holds then we can use 4.3(iii) to
construct a free resolution of finite type step by step, so (iii)  (i)_ D
We say that M is of type FP 00 if these conditions hold.
We dose this section by mentioning some useful formaI properties which
tbe homology and cohomology functors Tor:(M -) and Extt(M,-)
satisfy when M is of type F PII_ (The reader should keep in mind, during this
discussio the main caSe of interest: R = le M = ?L. In this case the T or
and Ext functors are simply H .(r, -) and H*(r -).)
Note first that the homology functors Tor:(M, -) always commute
with direct limits, in the following sense: Let {NaiEl be adirect system or R-
modules, where l is a directed set. and let N = limiEI Nj. Thus N is the univer-
---+
sal target or a compatible family of maps Ni --+ N (i E I). These maps induce a
compatible family of abelian group homomorpbisms Tor:(M, Ni)--+
Tor:(M, N), from which we obtain a map
tp  Tor.(M Ni) -+ Tor.(M, N);
iE'
tbe assertion, then, is that cp is an isomorphism. This follows directly from the
definition of T or:(M, -) as H tlcCP @ R ), where P is a projective resolution
of M, together with the fol1owing two facts:
(a) P @R - commutes with direct limits (cf. Spanier [1966] 5_1.9);
(b) H.( -) commutes with direct limits (cf. Spanier [1966] 4.1.7).
Functors or the form Jf'OI1tR(P, -), however, do not in generaI commute
with direct limits, so we cannot expect Ext:(M -) to commute with direct
Iimits. But we can prove that this does hold under suitable F P n hypotheses.
For simplicity, we will confine ourselves to the case n = 00.
(4.6) Proposition. Il M is ofrype FP 00 then Ext;(M. -) commutes with direct
limits.
This follows immediately from:

196
VIII Finiteness ConditioDS
(4. 7) Le. lf  is afinitely generated projective R-module rhen HomR(P _)
commutes wlrh dlrect limits.
OO. The duality isomorphism I.8.3b shows that the functor HOffiR(P, _)
18 eqUlvalent to the functor P* @ R -, and we know that the latter commutes
with direct limits. D
(See Exercise 4 below for a generalization or 4.7 and an indication of an
alternative proof.)
. Si.miIarly, one can show that Tor:(M, -) commutes witb direct products
if M 18 ofpe F P w (whereas Extl(M  -) commutes with products for any M).
Surpnmgly these formaI properties of Ext(M, -) and Tor:(M,-)
charactenze the F P  property. ln fact, one can prove:
(4.8) Tbeorem. The following conditions are equivalent:
(i) M is ofrype FP 00'
(ii) Ext:(M, -) commutes with direci limits.
(iii) Tor:(M, -) commutes with direct products.
We omit the proor, since we will not be making serious use of this result.
The equivalence of (i) and (ìii) was first proved by Bieri and Eckmann [1974].
See Brown [1975a] for the equivalence of (i) and (ii), as well as for a general-
ization or 4.8. See also Strebel [1976] for further results or this type.
EXERCISES
1. (r  a group. Show that 1'- is finitely presented as a zr -module if and only if r
IS a fimtely generated group. [Hint: Use Exerdse Id of I.2.]
2. Let M' and M" be moduJes and let M = M' $ M". Show that M is of type F p.. jf and
only if M' and M" are oftype FP.._ [Hint: Suppose p' and P" are finite type partial
resolutions or M' and M" of length k. and let P = P' ffi P". Note that P has finitely
generated kemel li and only if P and P' do.]
3. Let C and C' be non-negative chain complexes of projective modules. If C and C'
are homotopy equivalent, show that
Co EI:) Ci ffi C2 €e .. .  Co EB C t ffi c; E9 ....
[Hint: Imitate the alternative proof or 4.4 outlined in Remark 2 following 4.4.]
4. If M is a finitely presented R-moduJe. show that HomR(M. -) commutes with direct
limits. [Hint: This can be deduced from 4.7. AlternativeJy. give a direct proofbased
Oh thc facl that a map from M to any other rnodule can be specified by gjving a finite
set of eJements of the target module which s8tisfy a certain finite set of relations.]
.,
I
l
,
..
I. .-
..
..
-

.'
.,
I
.,
....:
'.
.,-
. .
.-
5 Gr.oups or Type FP..
197
5 Groups of Type FPn
We now specialize the theory or  to the case R = zr, M = ?L. We will say
that r is of type F Pn (O < rI < 00) if 7L is or type F P n as a .lr -module. Thus
every group is of type F Po, and it is easy to see that r ìs or type F P l if and
only if it is finitely generated (cI. Exercise l of ). The FP 2 condition is less
well-understood, however. One knows that finitely presented groups rare of
type FP2. for if Y is a finite 2-complex with 7r1 Y = r then the cellular chain
complex of tbe universal cover of Y is a parti al free resolution of 7L over zr
of Iength 2 and or finite type. [See al50 exercise 4c of IV.2 for an algebraic
proof.] But it is not known whether the converse is true. See Exercise 3 below
for a reformulation of the problem.3
We will not pause now to discuss examples since there will be plenty or
examples or groups of type FP t() in , 9, and 11. The reader who wants to
see examples for n < 00 of groups of type F PII but not of type F PlI + l should
consult Bieri [1976], 2.6, and Stuhler [1980]; see also Exercise 2 below for
the case n = l and Stallings [1963] for the case n = 2.
The FP" conditions behave nicely with respect to subgroups af finite
index:
(s. l) Proposition Let r' erbe a subgroup o/finite index. Then r is oftype
FP" (O < n < oc-) ifand only lfr' is oftype FPn.
PROOF. Any (parti a]) projective resolution of 7L. over zr or finite type can also
be regarded as a (partia1) projectì ve resolution of 7L over zr. and as such it is
stili oI finite type since (f: r) < 00. This proves the only if" parto Con-
versely. suppose r' is or type FPn. We will show that the .lr-module Z
satisfies condition 4.3(iii). Let P be a partia] projective resolution or 7L over
?Lr of finite type and of length k < TZ. Regarding P as a partial resolution or
?L over zr', we can apply 4.3(iii) to conclude that ker{PIt -+ PA-l} is finitely
generated over zr'. But then ker{Pj: - P"-I} is certainly finitely generated
over ?Lr, so 4.3(iii) holds for Z over zr. D
Taking fI = 1, for example, we recover tbe well-known (but not completely
o bvious) fact that r is finitely generated if and ooly if r' is finitely generated.
Finally, we record for future reference an important consequence of 4.6:
(5.2) Proposition Let r be a group aJ type FP ro and let n be an integer such
that Hn(r .lT) = O. Then H"(r, F) = Ofor allfree lT-modules F.
PROOF. If F is or finite rank then the result follows from the addjtivity
or H"(r, -). In the genera] case, choose a basis (ei)iel far F and note
that F = lim Fj, where J ranges aver the finite subsets of I and FJ is the
-
3 For solvab!e r. substal1tia1 progress on th'is question has been made by D'ieri and Strebel. For
references, ree the appendix to the forthcomins second edition of Bieri [1976].

198
VIII Finìt.eDess Conditions
submodule of F generated by thc ei for i E J. Then F.I is free or finite rank;
using 4.6 we conclude thal HII(r, F) = lim HI'I(r, F J) = O. D
--
EXERCISES
l. Let r be a group oftype FP" and let M be a r-module which is finitely generated as
an abelian group. Show that HiL M) aIld Hi(r. :'\1) are finitely generated abe1ian
groups for i S; n.
2. Let r be an amalgamation r l . A r 1 where r I and r 2 are free of finìte rank and A is
free of infinite rank. Show tbat r is finitely generated but not of type F P 2. In partìcular.
r is not finitely presented. [Hint: Use the Mayer- Vietoris seqnence to show that H 2 r
is noI finitely generated.]
3. (a) U r is of type F P 2 and N is a perfect normal subgroup (i.e. a normal subgroup
snch that N = [N. N]). prove thal rjN ìs of type FP2' [Hint: Apply the runctor
( ).... to a partial resolu tion or Z over zr or length 2 and finite type; tbe resulting
complex wiH stili be acydic in dimension one because its one-dimensional homology
is H1N = O.]
(b) Lei r = F/R where F is a finitely generate<! free group. Prove that tbe foUowjng
condìtìons are equivalent:
(i) r is oftype FP2-
(ii) The relatioD module Rahìs a finitely generated I-module.
- ....
(Hi) r :;::; r IN, where r is finitely presented and N is a perfect nonnal subgroup.
[Hint: For (i) => (ii) use Il.5.4.]
Remark. In view of the equivalence of (i) and (iìì), we can reformulate as foUows the
question as to whether e"ery group of (ype F P 2 is finiteJy presented: Is a perfect
norma! subgroup of a finiteJy pcesented group necessariJy finitely generated as a
normal subgroup?
4. FOl any group r. note that the cohomoJogy groups H.(r, l:r) have a canonical
structure or righr r-moduJe. Indeed. Ihe coefficient module l:r. whlch is thoughl of
as a left module for the purpose of defining H*(r, Zr). also admits a right raction
(by right translation) which commutes with the left action; this rìght action induces a
right action of r on the cohomo]ogy groups H*(r.l'T). In this exercise you will
prove two "universal coefficient" formulas involving these right r-modules.
(a) If r is a group of type FP DO' prove that
H*(r, F)  H.(f. .lO @.u F
for any flat zr -module F. [Hint: U se L8.3b to rewrite K-mnr< p. F) as a tensor product.
where P is a projective resolution of finite type.]
(b) If r is of type F P <t:> and Q is an in jective zr -module. prove that
H*(r. Q)  Hornr(H"(r, .!l'T), Q).
[Here we should take Q IO be a righl r.module so tha.t the Hom makes sense.]
<J
".
.
6 Groups or Type FP and FL
199
5. Let r be a free produCi r 1 . r 2. where r l and r 2 are infinite and of. type F P"". F or
any ìnteger i show that there is a map of (right) r-modules
Hi{r, lT) -- Ind L Hi(r 10 zr l) (f) Ind H'(r 2.. zr 2)
whìch is an isomorphism if i > 1 and an epimorphism jf i = 1 with kerne1 a free
Zr-modue of rank 1. (Hint: Use the Mayer-Vietoris sequence, and appJy exercise
4a to compute H*(rj. ZT),j = 1. 2.]
6 Groups of Type FP and FL
We now combine the two types of finiteness conditions whicb we bave
considered in this chapter. A resolution is said to be finite if it is both of finite
type and of finite lengtb. A group r is said to be of rype FP if 7L admits a
finite projective resolution Dver zr.
(6.1) Proposition. r is oJ t ype F P lf Qrtd onl y if (i) ed r < 00 and (ii) r is oJ type
FP cc'
PROOF . The' on Iy ir' part is obvious. Conversel y, if ed r < 00 and r is of type
Fp tben we can construct a finite resolution as follows: Take a partial
00 ,
resolution P rr _ l ...... . . . -+ p o -+ 7L -- O of finite type, where n = ed r, and
let P n = ker { P rr _ 1 -+ p rr- 2}' Then p" is projective (2.1 (iv) and finitely
generated (4.3(iii». so we bave a finite projective resolution
O -+ P 1'1 -+ . . . -+ p o -+ 7L -+ 0- D
Note that it would have been enough in tbe proof above to assume that r
was or type FPII instead or FP 07;1' where n = ed r. It follows, fOT instance,
that a finitely presented group r with ed r = 2 is or type F P. Note also
that the parti al resolution (Pi)iS",-l above could have bee taken free.
Hence if r is of type F P tben there is a finite projective resoluuon
(6.2) O -+ P ...... F 1'1_ l -+ . . . -+ F o --+ 7L -+ O
with each F i rree. (See also exercise 2 below.) Bul there is no reason to
expect to be able to take P free. Thus, for the first time in ths boo thre
really seerns to be a difference between what can be done wlth proJectlve
resolutions and what can be done using only free resolutions.
We are therefore led to introduce a still stronger finiteness condition:
r is or type FL if 1. adrnits a finite free resolution over zr. [Warning: It is
a eommon mistake to assume that "FL  stands for finite length/' and
thereby to confuse the F L property with the much weak.er property of ha ving
finite cohomologica1 dimension. In fact, the . L" in .. F L" stands for "libre,"
not for "length.' One also fÌnds ., F F" C1ìnite .fÌ'ee resolution ') in tbe
literature instead of" FL.1

200
VIII Finiteness Conditions
It is obvious how to use topoJogy to obtain examples of groups of type
FL;
(6.3) Proposition. Il there exists a K(r. 1) which is a finite complex, then
r is oftype FL.
(W e will see in 7 that the converse of this is also trne, at least if we assume
that r ìs finitely presented.)
Looking a t the examples in 2. we immediatel y deduce the following
examples of groups of type FL: free groups of finite rank; surface groups;
finitely generated one-relator groups whose relator is nor a power; and free
abelian groups of finite rank. With a ]ittle more work (cC exercise 8 below),
ODe can also show that torsìon-free. finitely generated, nilpotent groups are
or type F L. We will reprove this result and give many additional examples of
groups of type F L in 9.
The FP property also admits a topological interpretation. far which we
need the following notion: A space Y is finitely dominated if there is a finite
complex K such that Y is a retract or K in the homotopy category (i.e., we
require maps i: Y .... K and r K -+ Y with ri "'-' idy).
(64) Proposition. lf there exists a finitely dominated K(r, 1), rhen r is DJ
type FP.
(Again the converse is also true, and wiJI be proved in 7, provided r is
finìtely presented.)
We will not be making any use of this result, so we confine ourselves to a
brief sketch of the proof:
Let Y be a K(r7 1 )-complex dorninated by a finite complex K. One can
oose K so that the maps Y!:+ K induce 1! cisomorphìsrns. Letting Yand
K be the universal covers, one deduces that the cellular chain complex
C(Y) is a retract of C(K) in the homotopy category of chain complexes over
zr. Since C(Y) is a free resolution of 7L and C(K) is a finite free complex, it
follows that H*(r, -) commutes with direct limits and that Hi(r, -) = O
for i > dim K. In view of 4.8, this implies that r is of type F P. D
Remark. Propositions 6.3 and 6.4 and their converses are special cases of the
following result due to Wall [1965, 1966]: Let Ybe a connectOO CW-complex
whose fundamental group n is finitely presented and let C be the chaìn
complex of the universal cover of Y, regarded as a complex of n-modules.
Then (a) Y is homotopy equivalent to a finite CW-complex if and anly if C
is homotopy equivalent to a finite free chain complex; and (b) Y is finitely
dominated if and only if C is homotopy equivalent to a finite projective chain
complex.
Ha ving carefully explained. the difference between the F P condition and
the F L condition. we are now forced to admit that there are no known
0.0
,
,
..
,;

J
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.
.:
,'.
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.'
..
,
i
.
"
,

.'
,';"
,.
I
-
.
-'
:
.
6 Groups or Type FP and FL
201
examples of groups of type FP which are not of type FL. [From the topo-
logical point of view we bave the followmg situation: Altough there are
plenty of known examples of finitely dominated spaces whlch do not have
the homotopy type of a finite complex. none of these known examples are
K(r, 1)'s.] ,.
In order to better appreciate tbe problem, let s see what the obstructlOD
is to proving that a group of type F P is of type F L. Suppose r is or ty F P,
and choose a finite projective resolution as in 6.2 which is free except 1D tbc
top dimension. Tbe projective P which occurs in the top dimension might
ha ve tbe property that P ffi F is rree for some free module F of finite rank.
In this case P is said to be stably free, and we can modify 6.2 exactly as in
tbe proof of 2.6 to obtain a finite free resolution. Conversely, if there exists
a finite free resolution, then we can compare il to 6.2 via 4.4 to deduce that
p is stably free. [Note: Tbe two resolutions might not have the same length,
but 4.4 is still applicabJe; for we can extend the sborter resolution by zeroes.]
Thus we have:
(6.5) Proposition. Let r be a group oJ type F P and let O -+ P -+ F n- 1 -+ . . .
.... F o .... 7L -+ O be a finite projective resolution 0/ 7L over zr with each F i
free. Then r js oJ type FL if ami only if p is stably free.
Thus the question as to whether there exist groups or type FP which are
noi of type FL has 100 to a more fundamenta} question= Do there exist
finitely generated projectives w hich afe not stably free? Over a generaI ring
the answer is certainly '6yes," and there are even known examples over
integrai group rings zr, tbe simplest example being with r = Z13 (cf.
Milnor [1971], 3). The surprising fact, however. is that there are no known
examples with r torsionfree. and a group of type F P is necessarily torsion-
free by 2.5.
We remark, finally, that there do exist resolu tions as in 6.2 in which P is
not free. Tbe fust such example was given by Dunwoody [1972], with r
equal to the trefoil group. In alI or tbe known examples, however, the group r
is known to be of type F L (and hence P is stably free).
In spite of this lack of examples, we will see later (cf. IX.6.4 and Reark 1
following its proof) that there are concrete results whose proofs reqUIre that
we consider the FP property, and oot just the FL property. One reason for
this is that we can prove tbe following result about groups of type FP, the
analogue of which for groups or type F L is not known:
(6.6) Proposition. Let r be a torsion-free group a,ul r' a subgroup oJ finire
index. Then r is oftype FP if and only ifr' is oJtype FP.
PROOF. This follows from 3.1 and 5,1.
D

202
VIII Finiteness Conditionfi
.In articula, if. r is a torsion-free group which contains a subgroup of
finIte mdex WhlCh lS of type F L. then we know from 6.6 that r is of type F P,
even though we don't know that r is of type F L.
. W e Iose this section by discussing some special features of the top-
dlmenslOnal cohomology of a group or type F P. First wc note the £ollowing
improvement of 2.3:
(6.7) Proposition. Ifr is ofrype FP then ed r = max{n: HtI(r, ZT) #:- O}.
PROOf. This follows from 2.3 and 5.2.
D
Recall that the cohomology groups Hi(r ZT) admit a canonical right
f-module structure (cf. exercise 4 of 5). For any (left) r-module M we can
therefore form the tensor product Hi(r, Zr) Q9zr M, and there is a canonical
map
qJ: H*(r, Zr) Q9zr M -- H*(r, M),
defined as follows on the cochain level: Let P be a projectìve resolution of
7L over zr; given a cochain u E.1'tò9nr<P ZO and an element m E M, we
send u @ m to the cochain x!--+ u(x)m (x E P) in .*'omr(P, M).
We can now prove the following "uni versaI coefficient theorem' for the
top-dimensional cohomology of a group of type F P:
(6.8) Proposition Ifr is oftype FP and n = ed r, then
cp: HIt{r, JEr) Q9.lr M -io Hn(r, M)
is un isomorphism for all r -modules M.
PROOF. We wi1l give two proofs, both of which are instructive.
Proof 1: Regard qJ as a natura! transformation between funetors of M. I
Since both functors are right exact, it suffices to prove that cp is an iso-
morphism w ben M is free. [The generai case is tben o btained by considering
an exaet sequence F' -io F --+ M -ti O with F and P' free.] Since both functors
afe additive and comm ute with direct limits, it suffices to consider the case
where M is free of rank l, ie., we may assume M = zr. But in this case
cp: HII{r, Zr) Q9zr zr -io H"(r, Zr) is simply the well-known isomorphism
u @ r  ur (with inverse u ...... u @ l).
Proof 2: Let P be a finite projective resolution of 7L over zr of length n
and let P be the dual complex .7t'mnr<P,:lI) of right r-modules. Then
H*(r, .zr)  H*(P). ]n particular, we have an exact sequence
pn- l --to pn -io HI'I(r, Zr) -- O
for the top-dimensional cohomology module. Tensoring with M and
applying the dualìty isomorphism I.8.3b, we obtain the diagram
'.
'0
r
.,
:.
- ,
.-
"
.!
.,.
>.
,.
.r
"
r
"
",
.
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"-
I:'
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6 Groups of Type FP and FL
203

.,.
1"'-1 Q9zr M
l
a H"(r, Zr) @zr M
-1
 Hn(r, M)
.0
.P@zrM
1
Homr(P l'I-lM)
I Hornr(P n' M)
" O.
w hich commutes and has exact rows. It follows that cp IS an isomorphism.
O

The isomorphism of 6.8 can be written in the following suggestive way:
Let D be the r-module H"(r, lr), so that HI'I(f,.lI) @zr M = D @-;.r M
= (D @ M)r = Ho(r, D @ M), where D Q9 M = D Q9z M with the diagonal
r -action. [Note: Since D is a rìght mod ule and M is a Ieft module, the diagonal
action is defined by 'Y. (d @ m) = dy-l Q9 ,m ror Y E r, d E D, m E M.]
Consequently, 6.8 can be viewed as an isomorphism
(6.9)
H"(r, M)  H ocr, D Q9 M).
We will retum to this point or view in 10.
EXERCISES
1. If r is or type F L and ed r = n, show that Z adrnits a finite free resolution over z:r
of length n.
,
.
2. Let r be of type F P and lel m be an a[bitrary integer > O. Prove lhal Z admits a
finite projeetive resolutàon P over £T such that Pi. is free ror i -1= m. Moreover, P
can be taken to have Jength equal to max{m, ed r}. [Hint: Given a resoiution. you
can modify it by taking the direct suro with a compiex of tbe form O -+ Q  Q - O.]
,.
3. If r is of type FP a[]d n = ed r, show that H"(r.1T) Is a finitely generated r-
module.
'0
.
.
*4. Let r be ortype FP, let n = ed r, and suppose Hn(f. .lT) is finitely generated as QrI
abeliml group. If r' c r is a subgroup of infinite index. show that ed f' < n. [Hint:
For any f' -module M'. Hn(f', M')  Hn( r, Coindf. M'). Now apply 6.9 and exercise
4b of 1I1.5.] Give an example to show that the hypothesis on H"(r. JT) cannot be
dropped.
5. In this exercìse il wiU be con\leniem IO thìnk of coeffident modules for homology a5
being righl r-modules and coefficient modules for cohomoJogy.as left modules.
Recall from *V.3 lhat there is an eva.ution pairing
Hi(r. M) (8) Hi(r, N) -+ M @.lr N.
In particular. takjng N = zr, we obtaìn a map
(110)
HI-r. M) @ Hi(r,lT) - M Q9zrT  M.

204
VIII Finitencss Conditions
(a) Show that this map is a homornorphism ofright r-modules. where r acts on the
domain via its action on Hl(r, T). Deduce a map
,p: Hi(r, M) - Hornr<H1(r.ZT). M).
. :
'.
, ':
.'
,
;
,
,
'.
..
'. "i
,
-
< ,
"
,.
.-
-
"
[Hint: Use the naturaJity or the evaluation pairing with respect to module homo-
morpbisms. Alternatively, simply check defioitions on the chain level.]
(b) ur is oftype FP and n = ed r, show that I/t is an isomorphism in dimensioD n.
Thus
(6.10)
H"(r. M) ::::; HomrtD. M) = H()(r. Hom(D. M)).
where D = H"(r. :t'T) as in 6.9. [Hil1t: Consider the natural map M @:lrP 
.1tomr<p. .M), wbere P = .w'om...{p,lr).]
(c) Under the hypotheses of Cb). let Z E HnCr. D) correspond under 6.10 to ido E
Homr( D. D). We calI z the fundamental cfass or r. By definit ion. it is characterized by
the equation
<z.u) = u
..;
.
Cor every U E H"(r. 1T) = D. where ( , ) denotes the evaluation pairing given
by (.) above. Show for any module M thal
r/J - I : Hornr<D. M)  H "(C M)
is gJVen by 1/1- 1(1) = Jz for f E Homr<D. M). where j = Hn(r. I): H,,(r, D)-+
H fI(r M). [Hint: This is true by the definilion or z jf M = D and f = id  tbe generaI
case follows from the nalurality or V1- l, c[ exercise 3a of I.7.] Deduce that the
isomorphism
>
I;
HO(r. Hom(D. M»  H.lr. M)
of 6.10 is given by cap product with z; more predsely. it is tbe composite
Ho{r Horn(D. M» RI!(r, Hom(D, M) @ D) -+ Hir. M),
where tbe second map is induced by the obvious coefficient homomorphism
Hom(D. M) @ D -+ M. [Hint: Use the description of t he cap product HO @ H n - H.. '
given in exercise 1 or V.3.]
(d) Show that tbe isomorphism
l'p - i: HII{r, M)  D @zrM
or 6.8 is given by evaluatiol1 OD tbe Cundarnental cJass z. [Hint: This is true by the
definition or z if M = zr; the generaI case Collows by generaI nonsense, as in tbe
first proof or 6.8.] Deduce that the isomorphism
H"(r, M)  Ho(r. D @ M)
of6.9 is given by cap product with z. [Hint: Use V.3.LO.]
l ".":
I
i
I
l
\
[ "
i
I
6. (a) Let r be of type F P and let D and n be as above. Show that Hn(f. D) #; O and
deduce that ed r = hd r. where hd r. the homological dimension of r is defined by
hd r = sup{n: H"(r M) #- O for some r-module M.}
.(b) Give an example oCa group r (necessarily noi oftype F P)such that hd r < ed r.
I
l"
l
7 TopologicaJ Interpretatìon
205
j
*7. If r is or type F P and ed r = n. show that there are spectral sequences
E = Tor(H,,-q(r. lT), M) ;::;.. Hn-{l'+q.{r -"v!)
and
E£' = Extf(Hn-Il(r. Zr). M);::;.. H"-(P"'4(r. M).
By considering the corner p = q = O. recover tbc isomorphisrns H ir. M) 
Hornr<D, M) and H"(C M) :::;; D @zr M. [Hint: Use the universal coefficient
spectral sequences cf. Godement [1958].1.5.4.1 and 1.5.5.1.]
*8. Prove that the FP and FL properties behave weH with respect to extensions,
amalgamations. etc. More precisely:
(a) Suppose 1 -fo r -+ r -+ r" - l is exact. If r' and r" are of type FP (resp. FL).
then so is r.
(b) Let X be an acyclic r-complex such that X has onIy finitely many ceUs mod r.
If each isotropy group r (r i5 of type FP (resp. FL). then so is r. In particular, an
amalgamation r l "A r 2 is oftype FP (or F L) ifr 10 r 2. and A are oftype F P (or F L).
(Hint: See Serre [1971] or Bieri [1976].]
7 Topologicallnterpretation
We ha ve seen (cf. 2.2) that the existence of a finite dimensional K(r, l)
implies that r has finite cohomological dimension. Similarly, 6.3 and 6.4
show that the existence of a finite (resp. finitely dominated) K(r, l) implies
that r is or type F L (resp. F P). The purpose of this section is to consider
the converse implications. We also want to give a topological interpretation
or the r -modules H*(r, zr) which arose in 6. We will require a tiny bit of
homotopy theory, namely, the Hurewicz theorem (which we quoted in
n.S).
The following theorem is due to Eilenberg-Ganea [1957] and Wall [1965,
1966]:
(7.1) Theorem. Let r be an arbitrary group and let '1 = max{cd r, 3}. Then
thel'e exists alt n-dimensional K(r, l)-complex Y. If r is finitely presented
and DJ type FL (resp. FP) then Y can be taken to be finite (resp. finitely
dominated).
(We allow here the possibility that ed r = 00. in which case the theorem
simply asserts the existence of a K(r,-l)-complex.)
As an immediate consequence of 7.1 we have:
(12) Corollary (Eilenberg-Ganea). 11 ed r > 3 thel1 ed r = geom dim r.

206
VIII Piniteness Contions
Of course we also have ed r = geom dim r if ed r = O (since r is then
trivial) or if ed r = l (by the Stallings-Swan theorem which we quoted in
Example 2 of 2). In view of Theorem 1.1, then, we always have ed r =
geom dim r except possibly if ed r = 2 and geom dim r = 3. Il is not
known whether this possibility can actual1y occur.
PROOF OF 7.1. We wiU construct the skeleta yrr. of the desired Y ind uctively. T o
start the induction, let y2 be the 2-complex associated to some presentation of
r (cf. II.5 Exercise 2); thus 1t1 y2  r. If r is finitely presented y2 çan
be taken to be finite. Note that its universal cover X2 has H i = O for O < i < 2.
Now assume inductively that yJr;-1 has been constructed and that its uni-
versaI cover Xk-I has Hi = O for O < i < k - 1. If r is tinitely presented
and or type F P  assume further thal yk - 1 is finite. Choose a set of generators
(z:z) for the r -module H k _ l Xk - I. By the H urewicz theorem we can find for
h {' SI: - 1 Xk - 1 h. h . h
eac Cl a map h.: -+ W IC represen ts ZQ  In t e sense t ha t
Hk-I(h): HIr.-1S"-1 --+ H"_IXIc-1 sends a generator of Hi_1SIr.-1 to ZQ'
We now set
y = yl-1 U U 
a'

where the k-cell e: is attached to ylr.-I via the composite
Sii. - 1  Xk - l --+ yk - 1.
Letting XJ: be the universal caver or yk, we must verify that HiXJ: = o
for o < i < k. Note first that we can view X"-I as tbe (k - l)-skeleton of
Xk; indeed, Xk is obtained from Xk -I by attaching k-cells via tbe maps k
and their transforrns under the action of r on Xt- I. It is clear, tben, that
HiX" = HiX"-l = O Cor O < i < k - 1 and that we bave an exact sequence
H (X" Xk- l) !. H X"- 1 __ H Xi O
t, l-l "-l -+.
It will therefore suffice to show that a is surjective.
Recall that Hi(X", Xi-I) (which is simply C1(X"). tbe k-th cellularchain
group) is a free ZT-module with one basis element ror each k-cell of Y",
i.e. for each index -. Explicitly, there is a basis (Va) obtained as follows: if
Xa: (E", SI; - l) _ (X.  X"- 1) is a characteristic map for the celi auached via
fa, then VI); E H1(XI:, Xl:- J) is defined to be tbe image under
H,,(Xr/.): Hk(E", st-I) -i> H,,'<Xk. Xk-l)
of a generator or H k(EJ:. S"- I) = lL. In view of the diagram
H,,(E", SA:- l)  HA:_1S"-1
H,(,,> 1 1 H, - "f.)
H1(XIr.,X"-I) c Ht._lX"-l.
il follows that òVrz = Zrz (assuming that the generators or Ht(EIr., 8"-1) and
H,._ l sk.- 1 bave been chosen compatibly), so that ò is indeed surjective.
,
. 
,
.'
.
'. ..
-
,
'.
.,

.
r. -
,
1.- ,
,
.'
;
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.'
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'-
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.,
1
7 Topologì(:alInterpretation
207
(Note for future reference that ìf Hk_tX"-l happens to be a free z:r-
module with basis (zJ, then ò: H ,,( X"  x"- 1) -+ H ,,_ 1 X"- 1 is an isomorphism.
It then follows from tbe long exact homology sequence ofthe pair (Xlr.. XIr.-I)
that H, X. = O for ali i > O. so that yk is a K{r. 1).)
To complete the inductive step, we must show that Y" can be taken to be
finite if r is finitely presented and of type F P, i.e., we must show in this case
that HA lXII-I is a finitely generated r-module. To see this, we need only
note that the cellular chain complex
CIc- 1 - . . . --+ Co - 7L. -+ O
of Xlr.-I is a partial froe resolution ofjinite type, since yl-1 was assumed to
be finite. Theefore H t-I = ker { C" - 1 --+ C" - 2} is finitely generated by 4.3
since r is of type F P.
If n = oo we now continue this inductive process indefinitely, and
y = U" yk is tbe desired K(r, I). If n < 00, consider X"-l. (This makes
sense because n-l > 2.) Its cellular chain çomplex
ClI-l -+ . . . -+ Co --+ lL -+ O
is a partial free resolution of lengtb Il - 1. Hence 2.1 implies that HII_1Xn- I
is a projective zr -module. By the Eilenberg trick (2.7) tbere is a free module
F such that H n- 1 XII-1 ffi F is free. We now replace yn-I by yn- t =
YII-1 V 8n-1 v S,.-l V ..., where there is one copy of 8"-1 for each basis
element of F. The effect ofthis OD C(X"-I) is simply to add F to CII-I, with
òlF = O. The universal cover Xn-1 now has Hn-l free. We may thererore
attach n-cells e: lo y"-l corresponding to basis elements Z:z of HI'I_IXII-I;
as remarked above, tbe resulting Y" = ¥"-1 V U e= will tben be an ndimen-
sional K(r, l).
Suppose now that r is finitely presented and of type F L. Then YII- t is
finite and the projective Hh-1X,.-1 is finitely generated. We know from 6.5
that HII_1Xn-1 is stably free, so that there is a free module F oCfinite rank
such that H,,_lXlI-1 G1 F is free of finite rank. We now proceed as in the
previous paragraph, and the resulting ¥" will be a finite K(r, l).
Finally, suppose that r is finitely presented but only ortype FP instead of
FL. On the one hand, the generaI inductive step above gives us a finite
complex
Y" = Y"  l V e" u .. . u 
whose uni versai cover has H i = O for O < i < n. Hence Xi YII :.::::: 7ti X" = O
for l < i < n. On the other hand, we know that there is a K(r, l) of the
form
y" Yn-l S"- 1 ..ti
= V V --.U u...
,
so that n, ¥" = O for alI i > l. I clajm that yn dominates YII. Indeed. the
required maps
ylt  YII  y"

208
VIII Piniteness Conditions
with ri  id are easily constructed as follows: i and r afe both defined to be
the identjty on the common subcompJex yn- l, and they are extended arbi-
trarily to the cells that were attached. (These extensions exist trivially for
each sn- l that was wedged onto ¥"- 1 in forming Y" and they exist for
each li' because 7[n-1 yn = O and 7[,,-1 YI'I = O.) Finally, tbe homotopy
ri  id is defined to be the constant homotopy OD Y"- 1 and is extended to
alI of Y" by means of the vanishing or 1tn _ l Y" and 7[" yn. o
For future reference we mention tbe following refinement of 7.1:
(7.3) Addendum. The complex Y in 7.1 can be taken to be a simplicial
complex_
PROOF. Assume ind uctively that Y" - l ìs simplìcial, where the notation is
that of the proof above_ By the sìmplicial approximation theorem, we may
then take each attaching map h.: S"- 1 -+ yt-l to be simplicial relative to
some triangulation of S"- l. Tbe resulting space y* is then triangulable by
Whitehead [1949], 9, Lemma 2. O
Finally, we give a topological interpretation of the right r-modules
H*(r, .l'T}, assuming that r is finitely presented and of type FL. We will
need the following o bservation :
(7.4) Lel1lll1s. Let r be (J group and M a left r-nwdule. Let Homc(M,l)
ç;; Hom(M, Z) consist DJ all abelian group homomorphisrns f: M ---t Z such
that..for every m E M,j(ym) = Ofor ali butfinitely many Y E r. Then there is a
natural isomorphism
HomM, Zr)  Homt'(M, Z).
Moreover, this is an isomorphism oJ right r-modules, where r acts on
HomM, Zr) via its right action on lr and r acts on HomC<M, Z) via its left
action on M (Le., (fI-')(m) = f(ym)for fE HOII\:(M, Z), 7 E r, m E M).
PROOF. A Z -rnodule map F: M -+ zr has the form
F{m) = I fim)y,
yeT
where h: M -+ 1L. and for each m E M, f;,(m) = O for almost all J' E r. One
checks that such an F is a r -modu]e homomorphism if and only if 1'I(m)
= fl(J,-lm) for all ì E f. We therefore have a map Homr<:M, ZT)-+
Homc(M. Z) given by F H il, and this map is an isomorphism with inverse
f.--..{m t--+ L j(y-Im)y}.
7Er
The reader can easily verify that this isomorphism is natural and compatible
with the right r -actions. D
,
!
'.
;:
!. 
-
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I
,
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I
L
'-
.
r
I.
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I -;
l'
"
-1
7 Topologicallnterpretation
209
Suppose now that r is finitely presented aod oftypeFL. By 7.1 there is a
contractible, free r -complex X with X tr finite. Then C(X) is a finite free
resolution of 7L. over zr, and H.(r ZI) is tbe cohomology of
.)t'bmr< C(X), ZI)o
In view or tbe lemma, we have
r<C(X), lT)  bm4:(C{X) Z) c JFmn{C(X), Z),
this isomorphism being compatible with the right r -action and the co-
boundary operators (tbe Iatter because of the naturality assertion in 7.4).
Recall that C(X) has a Z-basis with one element for each celi (J of X.
These basis elements are freely permuted by r and CalI into finitely many
orbits. It follows easily that f&mI"C(X), Z) consists or those cochains
f e.Jtbm(C(X), Z) such that f(a) = o for ali but finitely many cells fJ. The
cohomology of this complex is caUed the cohomology of X with compact
supports, and is denoted H:(X; l). We have now established:
(7 5) Proposition. Il X is a contractible,free r -complex with compact quotient
Xjr, then there is an isomorphism
H*(r, Zr)  H:(X; Z)
ofright f-modules, where the right action ofr on H:(X; Z) is induced by the
left action ofr on X.
In view of 6_7, this yields:
(7.6) Corollary.lf X is as in 7.5, then
ed r = max{n; H(X; Z) :# O}.
EXERCISES
L. If r is a countable group, show that there exists a countabJe K(r. I)-complex.
2. Give a tOpological interpretation or the F P,. condìtìons (3 < n :::;; 00) assuming r is
finitely presented.
3. Suppose ed r = 2. Show that there exists a 2-dimensional. acyclic, free r -complex.
[Hint: Write r = F/R. where F is free and R<J F. Let yl be a l-complex with
Xl = F,let Xl be the covering space corresponding to R, and argue as in tbe proof
of 7_1.] Thus for any r we can characterize ed r topologica1ly as the minimal
dimension of a free, acyclic r -complex.
*4. Prove thc following generalization or 7.5: Let X be a contractible r -complex with
finite isotropy groups r .,.and with only finitely many cells mod l. Then H.(r. ZI) 
H:'(X; Z). [Hint: Firsl show, by a spectral sequence argument COl' instance, that
H*(r, Zr) can be computed Crom r<C(X), .zr).]

210
VIII Finiteness Conditions
8 Further Topological Results
The purpose of this section is to see what finiteness properties of r can be
deduced if wc are given a K(r, 1) which is a numifold. We will see many
examples of this sìtuation in tbe next sectioD. The proofs to be given in this
section requirc more background in topology than we ha ve assumed e]se-
where in this book. The reader without this background, however. can still
read and understand thc statements or aH the results.
We begin by noting the analogues of 2.2 and 6.3 for K(r. 1 )-manifolds:
(8.1) Proposition. Suppose Y is a d-dimensional K(r, l)-manifold (possibly with
bourulary).
(a) cd r < d. with equality if and only if Y is ctosed (i.e., compact and without
boundary).
(b) 1f Y is compact then r is oftype FL.
PROOF. (a) Ir Y is smooth. as it wilI be in alI of our examples. then there are
severa 1 ways to show that Y has thc homotopy type ora CW-complex Y' or
dimension < d (so that ed r < d by 2.2): one can use Wbitehead's triangula-
tion theorem (cf. Munkres [1966] Ch. II), or Morse theory (cf. Milnor
[1963]), or tbe theory of nerves or coverings (cf. Weil [1952]). An alternative,
which works even if Y is Dot smooth, is to deduce from Poincaré d uality
with local coefficients that Hi(r, M) = Hi{y; M) = O for i > d and all
r-modules M. This pro ves the first part of (a). If Y is closed. then we have
H4(r. lL1) :::::: H4(y; lL2) = 71.2 * O, so ed r = d. If Y is not closed. then ODe
can deduce from Poincaré d uality with loca1 coefficien t8 that Hd(r, M) =
Hd( Y; M) = O for ali r -modu1es M, so that ed r < d; we ornit tbe details
since we wiU noi make serious use of this strict inequality. [Alternatively, if
Y is assumed to be triangulated, then there is a geometric proof that
ed r < d: one shows that Y adrnits a deformation retraction onto a su b-
compie" of dimension < d. We will describe such a deformation retraction
explicitly in an interesting example in. Example 3.]
(b) Suppose Y is compact. In the smooth case. the complex Y' in tbe proof
of (a) caD be taken to be finite, so r is of type FL by 6.3. In the generaI case
it is stilI true but considerably harder to prove. that Y has the homotopy
type or a finite complex (cf. Kirby -Siebenmann [1969] or West [1977]) so
(b) is proved. [It is worth Doting here that if we are content to prove that r
is oftype FP, then there is a quite elementary proof: we need only embed Y
in Euc1idean space and note that it is a retract of a compact polyhedral
neighborhood (cf. Dold [1972], proof or VA.ll); thus r is of type FP by
 D
Next we wish to reinterpret H*(r, ZO (cf. 7.5) in case there is a compact
K(r, 1 )-manifold 1': Let X be the universal cover or }: Since X is simply-
connected, it is certainly orientable, and we denote by n its "orientation
,
I. ,-
!
I .:

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'.
.,
1
.'
8 Furtber Topologìca1 Resulls
211
"
module." Thus Q is an infinite cyclic group whose two generators correspond
to tbe two orientations of X. The action of r on X induces an action of r on
n. with an element Y E r acting as + 1 according as tbe action or )1 on X is
orientation-preserving or orientation-reversing. Note that r acts trivially
on Q if and only if Y is orientablc. Finally, we make the convention that the
reduced homology of a space Z, denoted il *(Z) is the homology of the
augmented chain complex or Z. In particular, H-1(0) = Z. We can now
state:
(K2) Proposition. Let Y be a compact d-dimensional K(r, 1 )-manifold (possibly
with bourrdary). Let X be its universal caver and let il be the corresponding
ortentatiorl module. T hen there are r -module isomorphisms
H.i{r, lT)  H-i-l(ÒX) Q9 Q
ior ali i. l n particular7 if Y is il closed manifold then
H'(r, l'T) '" {
i #= d
i = d.
(8.3) Corollary  Under the hypotheses 018.2, there exists alleast one integer k
such that Ht(òX) -# O. Moreover, letting
l = l + min(k: il",(òX) =F O},
we have
ed r = d-i.
Tbe corollary is immediate from 8.2 and 6.7. Note that l > O if oY #: 0.
Thus 8.3 makes more precise the inequality ed r < d of 8.1a in this case.
PROOf OF 8.2. Since Y i8 compact and has tbc homotopy type of a finite
complex, we have
Hi(r, ZI)  H;(X)
by 1.5. On the other hand, there is a Poincaré-Lefschetz duality isomorphism
Ht:(X)  HIl- i(X, òX).
This, however, is not canonical; it depends on a choice or oTientation of X.
In particular, it commutes or anti-commutes with the action of an elernent
')' E r according as ì' preserves or reverses tbe orientation of X. Consequently,
we have a r-module isomorphism
H(X)  H4-ì(X. òX) @ n.
Finally. since li .(X) = o.
H,,-lX. òX)  i14-i-1(ÒX).
Thc proposition follows at once.
D

212
VIII Finiteness Conditions
Remark. The reader who is uncornfortable with Poincaré- Lefscbetz duality
for non-compact manifolds with boundary might prefer tbe following
alternative proof. which uses the duality theorem only for the compact
manifold Y (but with Ioeal coefficients) and which does not use the fact that
Y has tbe homotopy type of a finite complex:
Regarding zr and Q as local coefficient systerns on Y, we have
Hi(r. lr)  Hi(y; Zr)  H4-1..¥:' oY; zr @ Q).
Now zr @ il, with diagonal r-action, is an induced module (cf. 111.5.7),
and homology with coefficients in an induced module is easily seen to be
ordinary homology of tbe uni versai cover. One deduces
Hd-Y" 8Y; zr @ O)  Hd-I{X, òX;Q)  Hd_i(X.X) @ Cl,
and the result now follows as above.
EXERCISE
Let r be a group such that there exists a closed Ker. l ).mani fold. If r' c:: r is a subgroup
ofinfinite inde deduce from S_la that ed r' < ed r. [Hint: If p: f --+ Y is a covering
map of infinite degree. then f cannot be compact] See Exercise 6 or lO below for a
generalization of this result.
9 Further Examples
We saw a few examples in 2 of groups r with ed r < 00, and we noted in
 that some ofthose examples were oftype FL. We now want to introduce
some additional families of examples, tbe most interesting of which are the
4 arithmetic" groups. The study of such examples leads. aS we will see in the
next chapter, to some remarkable connections between group cohomology
theoryand number theory. Unfortunately, the assertions which we will make
about the examples in this seçtion are considerably less elementary than the
corresponding assertions concerning our previous examples and we will not
be able to give the proofs. The reader is therefore advised to casual1y read
through this section, taking on faith a number of deep results which we will
have to state without proof.
1. Let r be a classical knot group (i.e., the £undamenta1 group of the
complement or a non-trivial knot K 4 S3). Then r is or type FL and
ed r = 2. To see this" let Y = S3 - T, where T is an open tubular neighbor-
hood of K. Then Y is a compact 3-manirold wbose boundary is a torus, and
a deep theorem of Papakyriakopoulos [1957] says that Y is a K(r, 1). It
now follows from 8.1 that r is of type F L. To calculate ed r, we use another
result from knot theory namely 1hat 7t1(8Y) injects into Xl Y as a subgroup
. .
.'
"
. :.
"
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.',
-,
.,
l
,
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,
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,
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I
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:1
9 Further Exampies
213
"
of infinite index. If X is tbe universal cover of Y, it follows that ax is the
disjoint union of countably many copies of tbe universaJ cover [R2 of 8Y.
Thus the integer l in 8.3 is t. and we bave cd r = 3 - 1 = 2, as c1aimed.
Furthermore, we can use 8.2 to compute H*(r, Zr). In particular, we see
that Hì(r, ZO = O for i#=-2 and that H2(f', Zr) is a free abelian group of
countable rank.
2. Let r be the n x n strìct upper triangular group over lL. We saw in
Example 5 of IL4 how to construct a K(r, l)-manifold (without boundary).
Namely, take Y = r\ G where G is the n x n strict upper triangular group
over . It is easy to see that Y is compact (see the exercise below). so 8.1
implies tbat r is of type FL and that
ed r = dim Y = n(n - 1)/2.
(In particular, if n = 3 we recover the result of Example 6 of 2.) Moreover.
8.2 sbows that
o
. =F n(n - 1)
l 2
Hi(r. 1T) =
. n(n - 1)
l = 2 '
where r acts trivial1y on ?L. More generally, one can prove analogous results
for an arbitrary finitely generated. torsion-rree, nilpotent group r. For
Malcev [1949] proved that r can be embedded as a discrete subgroup with
compact quotient in a nilpotent Lie group G which is homeomorphic 10
Euc1idean space or dimension d = rank r. [One can view G as a '"tensor
product' IR @ r. cf. Bourbaki, LIE II, pp. 82-83, and LIE IIl, pp. 283-284.]
z
3. Consider now the group SL,,(Z), n > 2. This group has torsion, hence
cd(SL,,(Z) = 00. We know, however, that it has torsion-free subgroups r
offinite index, cf. Exercise 3 of I1.4. The intersection ofr with the strict upper
triangular group U has finite index in U and hence has ed = n(n - 1)/2 by
Example 2 and Prop. 2.4a. Thus
ed r n(n - 1)
> 2 .
(9.1 )
We will now outline a proof. based on tbc reduction theory or quadratic
forms, that r is of type FL and that equality holds in 9.1. A different proof
of these results will be discussed in Example 5 below.
Let X be the space which we called X o in Example 7 of IL4; thus X is
the space of positive definite quadratic forms on IR'\ modulo multiplication
by positive scalars. Recall that X is a contractible manifold (without
boundary) or dimension n(n + 1)/2 - l, that SLilR) (hence also r) acts on
X, and that x/r is a K(r 1). Recall also that if n = 2 then we can identify

214
VIII Finiteness Conditions
.
X with tbe upper half pIane (or, equivalently with tbe open unit disk), with
SL,.( IR) acting by linear rractional transformations.
In view or 8.la we bave
d r n(n + I) - 1
ç < 2 .
Moreover, one can show tbat X/f' is non-compact. so that 8.la implies that
striet inequality holds above. But we can do substantia1ly better than this by
giving a direct geometric construction instead of relying on the generalities
given in 8.1. Namely. wc will show tbat x/r admits a defonnation retraction
onto a subspace which is a CW-complex of dimension n(n - 1)/2 This will
show that ed r < n(n - 1)/2 and hence will prove our claim that equality
holds in 9.L
We begin by describing how this is done for n = 2. using the open unit
disk as our model for X. There is a tiling of X by ideal hyperbolic triangles,
which is compatible with the action or SL2(Z), and whicb is well-known in
the theory or modular forms (see, for instance, l.ehner [1964]). It is obtained
by starting with a single ideai triangle (Le., a hyperbolic triangle with vertices
on the unit circle) and generating further triangles by successive reflections
across the sides. This is illustrated in Fig. 9.2a. where alternate triangles oftbe
tiling are sbaded. Tbe vertices of tbe triangles or tbe tiling are called cusps,
and we denote by X* thc space obtained from X by adjoining tbe cusps; it
can be viewed as a simplicial complex with a simplicial SL2(Z)-action. [Note:
We give X. the usual simplicial topology, rather than tbe topology it inherits
as a subset of the piane. In particular, tbe set ax* of cusps is a discrete set in
/.. -
,
---
 . -
. -
-- .
-- --- - - - -- ..
I :¥-.:T_ .' - =,<=',>..-:' --= - - -:-'-'--:-:.---=::'::=>-. \
'''::.-'''_..r-':.' 7__' =.:.-- :..--_..-_-_ ':-==";...
.- -- -;;;;:a;:_....::_- _ ..11__ _". ..........1'
_4-::.... __._ - .:-_. :---
.-- - ---!
Figure 9.2a
,
l
,
,
,
,
>.
,-
.
"
,

I
.
9 Further Examples
215
.
.

i'f.
........:..=.:..;:  .
, %f  ,-s.   _ ,
I .; -', . -::''''--ijr., . -. .
-:: _+-'IL:r_ . :.:. -..... -=- I
')f,   :i'; :: -;;.  ::i;F 
) =H .). - ,
... Il!:. -:;u;o,_ . -==-=-=-- ---== _
:- -.".. - .....=- - - - £=-:"-' -:==---''': --:..
:" _.....
Figure 9.2b
-

-
-
-1
-
o
Figure 9.2c
(Fìgures 9.2a and 9.2c are reproduced from F. K]ein and R. Fricke. Vorlesungen fibe..
dìe Theorie dee ellìptischen Modulfunctionen. Band I. B. G. Teubner. Leipzig. 1890. by
permissìon ofthe pubJisher.)
the simplicial topology.1t can be shown however, that the simplicial topology
agrees with the usual topology on the open subspace X of X*.]
Now let T be the simplicial complement or i}X* in the barycentrìc sub-
division K or X*, i.e. T is the largest subcomplex of K disjoint from òX*.
.
(Explicitly, T consists of ali simplices or K none or whose vertices are in
òX..) The subcomplex T is shown in Fig. 9.2b. See also Fig. 9.2c where the

216
VIII "Finiteness Condjtions
entire barycentric subdivision K is shown in tbe upper half pIane model for
X. The reader is invited to trace out the subcomplex T in this picture. [Note:
Tbc vertices of tbe large triangle in Fig. 9.2a correspond to the points -l,
O, 00 in Fig. 9.2c; the barycenter of this triangle corresponds to the point
( -] + iJ3)/2 in Fig. 9.2c.]
I claim that this simplicial complement T is, in a canonical way a de-
formation retract ofthe geometric complement X = X* - ax*. Indeed one
deforrns X to T by pushing away from ax* along straight lines (in the
byperbolic or simplicial sense). More precisely, any x E X lies in a c10sed
2-simplex a or K with one vertex v in ax* and the opposi te face T in T. Since
x #= v, there is a well-defined ray from v to x, and the deformation moves x
along this ray away from v unti) it hits T.
Note that T and the deformation afe described purely in terms of the
simplicia1 structure OD X., so they are compatible with the action of SL2(71)
and its subgroup r. It follows that Tlf is a deformation retract of x/r, and
it has dimension l = 2(2 - 1 )/2, as required.
Before proceeding to the generaI case, we make some remarks about the
construction above. Let K be a simplicial complex. A subcomplex L is said
to be full if every simplex of K having alI of its vertices in L is itself in L. F or
ex ampIe. the subcomplex òX. above is not full in X., but it is full in the
barycentric subdivision of X.. (More gen era lIy , if K is an arbitrary complex
and L an arbitrary subcomplex, then the barycentric subdivision of L is a
CulI subcomplex af the barycentric subdivision of K.) Let T be tbe simplicial
complement of L in K; this is defined, as above, to be the subcomplex con-
sisting of ali simplices of K which have no vertices in L. If L is full in K, tben
every simplex of K which is not in L has a non-empty face in T such that the
opposite face (possibly empty) is in L. Il follows easily that the geometric
complement K - L admits a defonnation retraction onto the simplicial
complement T; one simpty pushes along straight lines £rom the L-£ace of any
simplex to tbe T -race.
What we did above then, can now be explained as follows: We first passed
to thc barycentric subdivision so that oX. would become full, and we then
had a derormation retraction of the geometric complement ofaX" onto the
simplicial complemento
Thc generalization to S L,,(lL) for arbitrary 11 is based on a theory due to
Voronoi [1907]. Voronoi constructs an enlargement X* or x, obtained by
adjoining certain positive semi-definite quadratic forms, namely, those whose
nullspace admits a basis consisting of vectors in Q". He gives X'" an explicit
decomposition into convex cells which are permuted by the action or
SL.,(lL). The subspace ÒX. = X. - X is a subcomplex. The cells of X are
not necessarily simplices but X. adrnits a barycentric subdivision K which
is simpliciai and which inherits a simpliciaJ action of SL,llL). It follows, as
abovc, that X = X. - Ò X. adrnits a deformation retraction (compatible
with thc SLn(Z}-action) onto tbc simpliciai complement T of ÒX. in K. One
sees by looking at V oronoi's construction that oX. contains tbe entire
,

.
9 FUrlher Examples
217
(n - 2}-skeleton or X.., and it follows easily that T has codimension >
n-L Thus
d. Tjr - d" T n(n + I) 1 ( l) _ n(n - 1)
1m -Im< --n--
- 2 2 
'.
as required. Finally  V oronoi pro ves in addition that X* has only finitely
many cells mod the SLn(Z}-action, so T /r is in fact a finite complex and r
is of type F L.
..
Remark. In case n = 2, the subcomplex T is a tree, being a l-dimensional
deformation retract ofthecontractible space X. This treewas first introduced
by Serre [1917a], 1.4.2, who used it to give an easy proof of the classical
theorem expressing SL2(Z) as an amalgamation 7L4 *Z2 lLfj. The existence or
an analogue of T for n > 2 (ie., a r-invariant, contractible subcomplex or
X of dimension n(n - 1)/2 with T;r compact) was proved by Soulé [1978]
for n = 3 and by Ash [1977] in generaI. Ash, in fact, proved an analogous
result Cor a more generai class of groups, using a generalization of V oronois
theory. The cohomological dimension of such gcoups had previously been
computed by Borel and Serre [1914], using different methods which we will
describe in Example 5 below.
The groups in Examples 2 and 3 are examples of . arithmetic" groups.
Before looking at generai arithrnetic groups, lets see what can be said about
arbitrary discrete subgroups of Lie groups.
4. Let G be a Lie group with only finitely rnany connected components,
let K be a maximaJ compact subgroup, and let d = dim G - dim K. If r is
any torsion-free discrete subgroup of G, I c1aim that
ed r < d
with equality if and only if r is co-compact in G, Le., if and only if G/r is
compact. Moreover, in the co-compact case r is of type FL and on has
H(f, .lf) = O for i #- d and H"(r, Zr) = 7L (possibly wjth non-trivial
r-action). To prove these assertions we need only recall (cf. II.4, Example 8)
that f\ GIK is a K(r, l )-manifold (without boundary) of dimension d; more-
over, r\G/K is compact if and only if f is co-compac!. The assertiODS now
follow at once from 8.1 and 8.2.
5. We now specialize Example 4 to tbe . arithmetic' case. We will confine
ourselves to a brief outline of this theory  referring to the survey paper of
Serre [1919] for more details and further references. Let G be a linear alge-
braic group defined over Q. In concrete terms this simply means that G is
a subgroup or some generai linear group GL", defined by polynomial
equations (with rationaJ coefficients) in the n2 matrix entries. For example,
G could be the group SL" (defined by the single equation det(aij) = 1) orthe

218
VIII Finiteness Conditions
strict upper triangular group (defined by thc eq uations au = l aij = O for
i > j). Thc rea) matrices satisfying the given polynomial equations then forro
a Lie group G(IR) (e.g., SLlI(iR», which is known lo have only finitely many
components; and the integrai matriees satisfying the equations forro a
discrete subgroup G(Z) ç; G([R) (e.g., SL,,(71..) c:: SL,llR). Thc group G(l) is
said to be an arithmetic group. M ore gen era lIy , il r c: G{Q) is a subgroup
commensurable with G(Z), Le., such that r fì G(Z) is of finite index in both
r and G(Z), then r is said to be an arithmetic subgroup of G(Q). In particular,
any subgroup of G(Z) of finite index is arithmetic_
Suppose now that r is a torsion-free arithmetic group. As in Example 4,
wc have a contractible manifold X = G(IR)jK on which r acts freely. so
that Xjr is a K(r, l)-manifold_ It turns Qut, however, that r is usually not
co-compact in G( JR) and henee t hat X jr is non-compact. In fact, there is a
'Q-rank" l > O attached to the algebraic group G, and r is co-compact in
G( IR) ir- and only if l = O.
[Here is the precise definition of I, for the benefit of the reader familiar
with algebraic groups: l is the rank of a maximal C-spIi t torus in G/RG,
where R G is the radical of G. If G is the striet upper triangular group, for
example, then RG = G, so l = O and r is co-compact, as wc saw in Example 2_
If G = SLn (n > 2), OD the other hand. then RG = {J} and a maximal spii t
torus in G/RG = G is given by the diagonal matrices; thus l = n - l > O
and r is not co-compact as we saw in Example 3.]
In spite of this failure or r to be co-compact, however, Borel and Serre
[1974] show that r is or type F L and compute H*(r, Zr). Their method is to
replace X by a contractible manifold X with boundary (with X as its interi or
if G is semi-simple) OD which G(O) operates_ They show that the aetion of r
is rree and proper and that the quotient X/f is compact. Thus x/r is a com-
pact K(r, l)-manifold, so that r is of type FL by 8.lb. Moreover, the coo-
struction of X is explicit enough that Borel and Serre afe able to identify tbe
homotopy type of the boundary: namely, ÒX has the homotopy type of a
countabIe bouquet of(l - I)-spheres where l is the Q-rank mentioned above.
[If l = O, this simply means that òX = 0.] Letting d = dim X, we conclude
from 8.2 that Hi(r. Zr) = O ror i -:# d -l and that Htl-r(r, ZT) is a free
abelian group of countable rank if l > O and of rank 1 if l = O. In particular,
ed r = d-l.
If G = SLn. far example, then d = n(n + 1)(2 - 1 and l = n - 1, so
ed r = d-l = n(12 - 1)/2, as we saw in Example 3.
6. Let S be a finite set of prime numbers and let lLs c Q be the localization
of 7L o btained by inverting the elements of S. If G is as in example 5, then a
subgroup. or G(O) is called S-arithmeric if il. is commensurable with G(Zs).
BoreI and Serre [1976] have shown that the results of example 5 extend to
torsion-free S-arithmetìc groups, provided G is reductive.
l
,.
,',
;
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,
;'
.
I
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,
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i
,
]0 Duality Gmups
219
7. Serre [1971] noted the following remarkable consequence of example 6:
If r is any finitely generated torsion-free subgroup of G LII(Q), then ed r < 00.
[Sketch of proof: Note that r c GL,{Zs) for some S, and apply example 6 to
the S-arithmetic group G Li7Ls) to conclude that any torsion-free subgroup
of the latter has finite cohornological dimension.]
,-
.
8. The result of example 7 is no longer true if C is replaced by a field K of
transcendence degree > 1 aver Q. F or one can find finitely generated torsion-
free subgroups of GL,lK) which contain unipotent subgroups that are free
abelian of infinite rank. [A subgroup U or GLn(K) is unipotent if every
e1ement of V has alI '2 of its eigenvalues equal to I. According to a famous
theorem of Kolchin, this holds iff U is conjugate to a subgroup of the striet
upper triangular group.] But Alperin and Shalen [1981] have shown that
this is essentially the only type of counter-example. More precisely, they
show that if r is a finitely generated torsion-free subgroup of GLn(K), where
K is an arbitrary field of characteristic O, then ed r < 00 iff there is a finite
upper bound on the Hirsch ranks or the finitely generated unipotent sub-
groups of r.
Remark. The reader may ha ve noticed that ali of the examples of grops of
type F L whieh we ga ve in this section ha ve the Collowing remarkabIe property:
There is an integer n (necessari1y equal to ed n such that Hi(r, Zr) = o for
i '# n and H"(r, ZT) is free abelian. The significance ofthis, as we will see in
the next section, is that it implies that these groups satisry a duality relation.
somewhat anaIogous to Poincaré duality for manifolds.
EXERCISE
Let G and r be as in Example 2. Let C = {(aij) E G; laul < 1 for alI i.i}. Show that
G = r. C and deduce tbat r i5 co+compact in G. [Hint: Consider the elementary matrices
in r. and interpret Iert multiplication by such a matrix as an eJementary row operation.]
lO Duality Groups
For any group r oftype FP, we have seen (6.9) that there is an isomorphism
H"(r. M)  Ho(r D @ M)
for any r-module M, where n = ed r. D is the right r-module Hn(r. :lI),
and D @ }.tI = D @z M witb the diagonal r -aetion. In this section we will
sludy those groups r for which the isomorphism above extends to iso-
morphisrns
Hi(r, M) ::::::: Hn-Ar, D @ M)

220
VIII Finileness Conditions
for ali i. These groups were first considered by Bieri and Eckmann [1973],
w ho proved the following characterization or tbem:
(10.1) Theorem. The following conditions are equivalent for a group r oJ type
FP:
(i) There exist an integer n and a r -module D such thal
Hi(r M)  Hn-t.r D @ M)
for ali r -nwdules M and ali integer s i.
(ii) There is an integer n such that Hi(r, zr @ A) = O for ali i =F n and ali
abelum groups A.
(iii) There is an integer n sucli that Hi(r, :lT) = O for alt i #= n antI Ir(r, ZI)
is torsion-free (as an ahelian group).
(iv) There are natura l isomorphisms
Hi(r, -)  Hn-i(r, D @ -),
where Il = ed r and D = HII(r, .zr), which are compatible with the
connecting homomorphisms in the long exact homology afld cohomology
sequences associated to a shorr exact sequence oJ modules.
PROOF. (i) => (ii): Apply (i) with M an induced module zr @ A. It is easy to
see that D @ M is also induced (cf. 111.5.7). Since induced modules are
H .-acyclic, it follows from (i) that H1(r, M)  Hrz-i(r, D @ M) = O for
i:#n.
(H) =- (Hi); Applying (ii) with A = lL, we find H'(r,lT) = O for i ¥' n.
Now apply (ii) with A = ZA: (k > O), and use the cohomalogy exact sequence
associated to tbe short exact sequence
O -+ zr -4 zr  zr C8> ?Li -+ O.
This yields
o = HJt- l(r, zr C8> :li;) -+ H"(r, lI)  H"(r, ZI),
showing that H"(r, zr) has no k-torsion.
(iii) =- (iv): Note fust that tbc integer n in (iii) is necessarily equal to
ed r by 6.7. We now give three different proofs that (iii)  (iv).
Firsl proof: Wc will use the generaI nonsense or III.7t which tbe reader
may want to review before proceeding further. Consider the cohomological
functor H*(r, -). Since ed r = n < 00, we can re-index and regard
H*(r, -) as a homological functor; indeed, an exact sequence O -+ M' -+
M -+ M" -+ O or r -mod ules yields an exact sequence
... -+ H,,-l(r, M") -+ H"(r, M') -+ W(r, M) -+ HII(r, M") -+ O,
-
so we obtain a homological functor T by setting 1; = W-1(r, -). Tbc
hypothesis.(iii) implies (via 5.2) that H'(r F) = O for ali i '# n and ali free
,
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l.
i
W Duality Groups
221
"-
"'
.zr-modules F. Consequently 7i is effaceable for i > O. Now consider thc
functor H*(r D @ -), where D = RII(r, Zr). This is also a homological
functor. For if O --+ M' -+ M -+ M'I - O is exact, then so is O --+ D (8) M' -
D @ M -+ D @ M" -+ O (since D is torsion-free by hypothesis), hence we
obtain a long exact homology sequence. Moreover9 Hj(r, D @ -) is efface-
able for i> O because D <29 M is induced if M is induced. Since To ;:::;
H ocr, D (8) -) by 6.9, Theorem II I. 7.3 yields an isomorphism T 
H .(r. D @ -) of homologial functors whence (iv).
Second proof: This is similar to the first proof, except that we wilI use cap
products instead of generai nonsense to get the map T -+ H *(r. D @ -).
By exercise 5d of 26. there is a "fundamental dass" Z E H,,(r. D) such that
cap product with z gives an isomorphism HJf(r, -)  Ho(r, D C8> -).
Since the cap product map
(Ì z: 1; = HII-i(r, -)  Hi(C D C8>-)
is compatible (up to sign) with eonnecting homomorphisms in long exact
sequences, it follows by dimension-shifting that this map is an isomorphism
Cor alI i.
Third proof: Let P = (PJO:!>iSR be a finite projective resolution of Z over
.zr of length n, and consider the duat complex P = Jfbm..{P t .zr). Since
Hi(r, Zr) = O for i #- n, P provides a projective resolution of D = Hn(r. Zr);
."
.
".
'"
.
-.
............ pn-l  ]>Il -.+ D-O'
,
more precisely, the l1-th suspension LIIP is a projective resolution or D. In
view of the duality isomorphism
Jt'omrlP, M)  P @r M
of I.8.3 it foHows that
(oli) Hi(r,M)  H-i(P(8)rM) = Hn-t{'L"P@rM) = Tor_i(D M)
for any r -module M. Since D is torsian free we can now apply 111.2.2 to
obtain
(**)
TorI_.j(D, M)  HII_i(r, D @ M).
Tbe reader can easily verify that (*) and (**) are natura} and compatible with
connecting homomorphisrns, whence (iv).
(iv) "'* (i) Trivia1. D
If r satisfies the conditions of the theorem then r is said to be a duality
group and tbe r-rnodule D = Hn(r, Zr) is called tbe dualizing. module of r.
li, in addition, D is infinite cyclic (as an abelian group), then r is said to be a
Poincaré duality grQup, and in this case r is said to be orientable if r acts

222
VIII Finiteness Conditions
trivially on D and non-orientable otherwise. Note that if r is an orientable
Poincaré duality group then (iv) takes the ramiliar rorm
Hi(r, M)  H"-i(r) M),
as in Poincaré duality for c1osed, orientable manifolds.
Remarks
I. I t is sometimes useful to note that if r is a d uality group, then there are
duality isomorphisrns Hì(r, M)  H"_i(r, D @ M) given by cap product
with a fixed elernenl Z E H,,(r, D). This was made explicit in the seeond proof
above, and it also follows from the third proof, exactly as in the proof of
VI. 7. 2.
2. Bieri and Eckmann originally defined a dualìty group to be a group r,
not necessaril}' oJ type F P, such that there are cap product isomorphisrns
nz: Hi(r, M)'; Hn-i(r, D Q9 M)
for ali i and M, where n is a fixed in teger, D a fixed r -mod ule, and z a fixed
element of H ,,(r, D). It is clear from the previous remark that a duality group
in our sense is also one in theirs. Conversely, if r is a duality group in their
sense, tben naturality of the cap product implies that there are natura l
isomorphisrns
Hi(r. -)  HfI-i(r D @ -).
Since D <8> - and H .(r, -) commute with direct limits, it follows that
H*(r, -) commutes with direct limits. It also follows that ed r < n < 00.
Thus r is of type F P by 4.8, and hence r is a duality group in our sense. It is
also worth noting that tbe integer n which occurs in the Bieri - Eckmann
definition is necessarily equal to ed r (this is clear from the proof of 10.1)
and that the module D is necessarily isomorphic to Hn(r. .lI). Indeed, the
given duality isomorphism with i = n and M = zr gives
Hn(C, .lT)  Ho(r, D @ Zr)  D @zrzr  D,
and this is com patible with the r -action by naturality of the duality iso-
morphism.
EXAMPLES
L If r is a group such that there exists a closed K(r, l}manifold  then,
as one would expect, r is a Poincaré duality grou p and is orientable if and
only if Y is orientable. This follows immediately from 8.1 b, 8.2, and the
criterion 10.1(iii) for duality. [Altematively, use Poincaré duality in Y. with
loeal coefficients, to verify directly that r satisfies the Bieri - Eckmann
definition of duality group which we quoted ahove, with D equa) to the
orientation module n associated to Y] For.example, the.free abelian grQUP
7L1I is an orientable Poincaré duality group. [Take Y to be tbe n-dimensional
toru8.] More generalIy. any finitely generated. torsion-rree, nilpoteDt group
-,
, -
,
;
.,
.
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,
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lO Duality Groups
223
is an orientable Poincaré duality group by example 2 of 9. [Alternatively
one can give a purely algebraic proof. cf. Bieri [1976]. Theorem 9.10.] StiU
more generally. if G is an arbitrary Lie group with only finitely many con-
nected components and r is a discrete, co-compact torsion-free subgroup,
then r is a Poincaré duality group and js orientable if G is connected (cf.
example 4 of 9)-
Remark. It is not known whether every Poincaré duality group adrnits a
c10sed K(r. 1 )-manifold. This is even unknown in the 2-dimensional case.
But Eckmann and Miiller [1980] have shown that every 2-dimensional
Poinearé duality group r which has the homology of a surface other than
Sl or pl is in fact isomorphic to the rundamenta] group of that surfaee. It
follows that if there is a 2-dimensional counter-example (Le., a 2-dimensional
Poincaré duality group r for which there is no dosed K(r, 1 )-manifold. then
there is even a counter-example r such that H.(r)  H.(S2). Such a r
would be very interesting indeed. In particular, the method or 1. Cohen [1972]
shows that r would not be of type F L.
2. If r is a free group OD k generators, l < k < 00, then r is a l-dimensional
duality group. Indeed, we know that ed r = l and that r is of type F L, so we
need only check that HO(r, zr <8> A) = O for any A; but this is easily seen to
hold for any infinite group r cf. III.5, exercise 4a. If k = 1, then r is an
orientable Poincaré duality group by example 1 (or by directly computing
H*(r, zr) from the resolution 1.4.5). Jf k > 1, however, then r is not a
Poincaré duality group; for we have
lL";:::, H1(r, l)  Ho(r, D @ Z) = Ho(r. D)
so D cannot be infinite cyclic. (In fact, it is nol hard to show that D is a free
abelian group of countable rank, er. exerdse 2 below.)
3. The results stated in example I or 9 show that knot groups are 2-
dimensional duality groups (but not Poincaré duality groups).
4. Tbe results of Borel and Serre quoted in example 5 of  show that
torsion-rree arithmetic groups are duality groups (but not Poincaré duality
groups except ìn the rank O case.)
5. If r is a finitely presented group of cohomological dimension 2 which
cannat be decomposed as a free product, then r is a duality group. Indeed. r is
of type F P by the commen ts following 6_1. so it suffices to show t hat
H!(r. lLr @ A) = O for i < 2 and aH A. This is clear for i = O. For i = l, it
is not hard to see (cf. Swan [1969], *3) that H1(r, Zr) is a free abelian group
of rank e - L. wherc e 1s the number of' ends" of r, and that H 1(r, lT (8) A)
 Ht(r. ,ilT) 0 A. BUi e = I (and hence H1(r..lr @ A) = O) under our
hypotheses. because or the following theorem of Stallings [1968]: A finitely
generated, torsion-free group with more than one end is either infinite cydic
or a rree product. (See also Scott-WaU [1979] far the theory'or ends and a
proof or Stallings's theorem_)

224
VIII Finiteness Conditjot)s
6. Further examples or duality grou ps can be construçted £rom known
examples by forming extensions and amalgamations. The precise results are
somewhat complicated, and we rerer to Sieri [1976] far details. [Warning:
These gro up theoretic constructions do not alwa ys preserve tbe property or
being a duality group. For example. a free product of duality groups is not a
duality group unless the groups are free. cf. exercise 3 below.]
Remark. The exampJes above suggest that if r is a d uality group then the
dua1ìzing module D is either infinite cyclic (in which case r is a Poincaré
duality group) or free abelian ofinfinite rank. It js not known whether this is
truet the problem being the freeness of D. It is known however, as a con-
sequence of results of Farrell [1975], that if D is not infinite cyclic then D has
infinite rank (i.e.. dimQ(D 0 Q) = 00). A proof can be found in Bieri [1976]
9.5 along with other interesting properties of D.
We dose this section by proving one useful property or duality groups:
(10.2) Proposition. Ler r be a torslon-free group and r' a subgroup o/finite
index. Then r is a duality group if and only ifr' is a duality group. Moreover, if
rami r' afe dualiry groups then they have the same dualizing module. (More
precisely, if D is the dualizing module for r, then the dualizing module for r' is
isomorphic to che module Res. D obtained by restricting che action ofr to rr.)
PROOF. By 6.6, r is or type F P if and only if r' is or type F P. Since H*(r, Zr) ::::::
H*(r', .lT') by Shapiro's lemma, the first part of the proposition follows from
the criterion 10.1(iii) for duaJity. T o prove the second part, we must check that
the isomorphism H"(r. Zr)  Bf/(r', .zr) is an isomorphism of righi r'
modules. By straightforward definition-checking (cf. III.8, exercise 2) one
sees that this isomorphism is the composite
Hr:I(r lT) n:s HII(r', 1:T) _ HII(r', lT'),
where the second map is induced by the coefficient homomorphism
zr  zr (2) . zr ) Hom .(lr Zr') c\'al at I t .zr'.
zr W['5.9) .-'
One checks that this homomorphisrn is given expJicitly by
{V }le r
Y"""'"
O otherwise,
and this is clear ly çompatible witb the right r' -actìon. Since res: HlI(r  ZT) -
Hr:I(r' Zr) is obviously compatible with the right r' -action (in fact. with the
right r -action), the proof is complete. D
EXERCISES
1. If r is an n-dimensioflal duality group with dualizing module. D. prove that D is a
.zr-moduJe or type FP (ie.. it admits a finite projective resolution) and ofprojective
dimension n. [Hint: Look at the third proof of (iii) => (iv) in lO. L]
I
...

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I. l Virtual NotRons
225
2. Let r be afree group offinite rank > 1. Prove that thedualizing module H1(r. Zf)
is a free abeJ.i;an group of infinite rank. [Hinl: Use exercise 5 of 5 and argue by
inductioD OD tbe rank or r. Alternatively, use the theory of ends.]
3. Show that a duaJity group of cohomoJogical dimension > 1 cannot be decomposed
as a free product. [Hint: Exercise: 5 or5.)
4. If r is an n-djmensiona duality group whose Qualizing module D is Z-free. prove
that
Hj(r. M)  H,,-i(r. Hom(D. Ai))
for any r -module M. [Hint: The starting point is exercise 5b or 96. Now imitate any
of the three proofs [hat (iii) => (IV) in 10.1. If you imitate the first or second proof, you
wiU need to use exercise 4b or 5.]
5. Show that if we are willing to use chajn complex coefficients (c[ VII.5) then every
group oftype FP behaves Hke a duaJity group. Namely. there is a ..dualizing com-
plex" !'J sucb that
Hi(r,)  H"_r.  @ E'J)
and
H.(r. CC)  H"-;(r.Kom{.}
for every complex tl! of r-modules. [Hint: LeI PJ be the dual (suitably re-indexed)
or a finite projective resolution of Z over zr. Look al the definitions or H!(r. W)
and Hr. ). and appJy exercise I orVL6.]
*6. (a) Prove thefollowing result ofStrebel [1977]: Ifr is a Pojncaré duality group and
r c: r is a subgroup or infinite indext then ed r < ed r. [Hint: Use exercise 4 or
'96.] Thìs result js motivated by the special case (which may in fact be tbe genera]
case) given in the exercise of 8.
Cb) Give an example which shows that the result of (a) does not hold for arbitrary
duaJity groups.
Il Virtual Notions
..
Up to now this chapter has beq1 concerned primarily with torsion-free
groups. Indeed, we observed in 2 that groups with torsion necessarily have
ed = 00, and hence they çertainly cannot satisfy the F L or F P conditions or
be duality groups. On the other hand, we have seen examples or groups
with torsion (such as SL,JZ» which have torsion-free subgroups or finite
indx that satisfy these finiteness conditions. The purpose of this section
is to introduce a convenient language for describing this situation.
In generaI we will say that a group virtually has a given property if some
.
subgroup of finite index has that property. Far instance, r is virrually torsiol1-
free ifr has a torsion-free subgroup offinite index.ln this case it follows from

226
VIII Finiteness Conditions
Serre's theorem (3.1) that ali such subgroups bave thc same cohomologica1
dimensìon (which ma y be finite or infinite); far jf r' and r" are two torsion-
free subgroups of finite index. then r n r" has finite index in both r' and r"
.
so Serre'8 theorem gives ed r' = cd(r' lì r") = ed r". The common co-
homological dimension of the torsion-free subgroups of finite index is called
the virtual cohomological dimension of r and is denoted vcd r.
Similarly. r is said lO be of t)'pe VFP (resp. VFL) if some subgroup of
finite index is of type F P (resp. F L). If r is of type V F P then il follows from 6.6
that every torsion-free subgroup of finite index is of type F P. Tbe correspond-
ing statement for groups oftype VFL is not known so one is led to introduce
the foJlowing apparently strongercondition: r is said to be oftype WFL ifr
is virtually torsion-free and every torsion-free subgroup of finite index is of
type F L.
To summarize. we have introduced four "virtual" finiteness conditions in
this section, which are related as follows:
WFL=*, VFL VFP(vcd < oc).
It is not known whether the. first two implications are reversible. Note also
that VFP  FP  by 5.l.
We saw in 7 that ed r < 00 if and only if there exists a finite dimensional,
contraetible free r-CW-eomplex. To state the 'virtual" analogue of this we
introduce the following weakening ofthe notion of'free action": We say that
a r -complex is proper if the isotropy grou p r fJ is finite for every celi fY. (This is
consistent with the notion of ,. pToper action ,. introduced in 1I.4, Example 6.
but wc will not need to know this.) Note that a proper r -complex is r -free for
every torsion-free subgroup r' c r. Conversely, if X is a r -complex which is
f' -free for some su bgroup r' of finite index (Le., if X is v irtually free), then X is
proper. Thus, if r is virtually torsion-free, a r -compJex is proper if and only
if it is virtually free.
We can now state the topological interpretation of finite virtual cohomo-
logical dimension:
(11.1) Theorem9 Let r be a virtually torsion-free group. Then vcd r < 00 if
and only ifthere exists afinite dimensional, con trae tibie. proper r -complex X.lf
there exists such an X whiclr has oMly finiteJy many cells mod r, then r is of
type W FL.
PROOF. The second assertion and the 'if" part of the fust assertion aIe
obvious from the remarks above. The.' only if' part follows from the proof or
Serre's theorem (3.1). Indeed, the enti re proof ofthat theorem, except far the
last sentence, is valid for any group r with vcd r < 00, and it yields the
desired X. D
There remai n several open questioos in connection with this theorem.lt is
not known, for example, whether X can be taken to bave dirnension equal to
I
I
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I
,
r'
. o
o'.

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o.
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.,
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-1
I] Virtual Notions
221
vcd r. Clearly we have vcd r < dim X for any X as in 11.1; but the proof
above alwa ys yields an X with dim X > vcd r except in the trivial case where
r is finite and X ìs a point. It is also not known whether the converse ofthe
second assertion of 11.1 is truc. In particular, the proof above will always yield
an X with infinitely many cells mod C unless r is finite. Thus we are very far
from understanding the relation between our algebraically defined finiteness
conditions and the analogous topologically defined conditions.
F or future reference, we record now some facts which can be deduced from
the proof of 1l.1:
(11.2) Addendum. Jj vcd r < co then the complex X in 11. I can be taken lo be
simplicilll (with simplicial r-action) and to have the following property: For
every finite subgroup H c: r. the fixed-point set XH is non-empty and con-
trae t ible.
PROOf. The assertion about tbe fixed-point sets follows easily from the proof
of Ll.l, cf. 3, exercise. The fact that X can be taken to be simplicial requires
some preliminaries concerning ordered simplieial complexes.
An ordered simplicial complex is a simplicial complex together with a
partial ordering on its vertices, sue h that the verticesof any simplexare linearly
ordered: Vo < v 1 < . .. < VII. The canonical example of an ordered simplicial
_complex is the barycentric subdivision K' of an arbitrary simplicial compie x
K. To see that K' is ordered, recatI that the vertices of K' afe the barycenters of
the simplices of K, hence there is an ordering 00 the vertices of K' cor-
responding to the obvious ordering (by theface relation) on the simplices of K ;
moreover the simplices of K' are precisely tbe finite sels ofbarycenters which
are linear1y ordered (cf. Spanier [1966]. 3.3, or Schubert [1968], 111.2.6), so
K' is indeed an ordered simplicial complex.
Next we recall the definition of the simplicial product K x L of two
ordered simplicial complexes K and L. The set of vertices vert(K x L) is
defined by vert(K x L) = vert(K) x vert(L) with the product ordering:
(v, w) < (v'. w')$> V < v' and w < w'. A simplex of K x L is defined to be a
linearly ordered set or vertices (vo, wo) < . . . < (v,p wn) such that {vo. . . . . v,J
is a simplex of K (possibly of dimension < n) and {wo. .... Wl!} is a simplex of
L. It is a c1assical fact (cf. Eilenberg-Steenrod [1952], II.8.9. or Milnor
[1957b]) that this abstract product complex K x L provides a triangulation
of the geometric product of K at1d L, i.e., that I K x L I  I K I x I L I. where
I I denotes geometric realization. [Here as in the proof or 3.1, some care
needs lo be taken in topologizing I K I x I L I.]
Look now at the proofof 3.1. The K(r', I)-complex Y' can betaken to be
simplicial by 7.3. Its universa] cover X' is then simplicial (with simplìcìaJ r-
action), cf. Spanier [1966], Theorem 3.8.3, or Schubert [1968]. III.6.9. Passing
to the barycentric subdivision if necessary, we eao assume that X' is an
ordered simplìcial complex and that the r -action preserves tbe orderiog. The
co-induction construction gi ven in the proof of3.! can therefore be done in the

228
VIII Finiteness ConditioDS
category or ordered simplicial complexes using the product described above,
and the resulting X will then be simplicial. D
EXAMPLFS
1. vcd f = o if and only if r is finite.
2. If r = r l * A r 2 where r t and r 2 are finite then r is of type W F L and
vcd r < l. More generally tbe same results afe true whenever r is the
fundamental group of a finite graph of finite groups. cf. VII.9. This is an easy
consequence of Serre's tbeory of groups acting on trees. which provides a
contcactible J -complex OD whicb r acts properly and with finite quotient. F or
details see Serre [1977a], Il.2.6, Prop. 11. [Note: Ifwe drop the requirement
that tbe graph of finite groups be finite and require instead that there be a
bound on the orders of the finite groups, then it is still true that vcd r < l, but
r will not necessarily be of type W F L.] Conversely  if r is a group such that
vcd r < 1, then the Stallings-Swan theorem (2 Example 2) shows that r has
a free subgroup of finite index, and it is then known that r is the fundamental
group of a graph of finite groups. This is due to Karrass- Pietrowski-Solitar
[1973] in case r is finitely generated (in which case the graph can be taken to
be finite) and to D. E. Cohen [1972] and Scott [1974] in the generai case.
Proofs can also be found in Dunwoody [1979] and Scott-Wall [1979].
3. If r is a finitely generated one-relator group, then r is or t ype W F L and
vcd r < 2_ Indeed, r is virtually torsion-free by Fischer-Karrass-Solitar
[1972]. And Lyndon [1950] established an exact sequence
o -+ Z[r le] -+ F  zr -+ 7l.. -+ O,
where F is a free zr -mod ule of finite rank and C is a finite cyc1ic su bgroup of f.
Our assertions follow at once, since Z[r/C] is c1early a free Zr'-module of
finite rank for every torsion-free subgroup rJ c r of finite index. [Note that
we bave in this case a contractiblet proper, 2-dimensional r -complex with
finite quotient. This follows from the topological version o£Lyndon s theorem,
which we stated in Exercise 2e or II.5-]
4. Let G be a Lie group and let X be the associated homogeneous space
G I K. where K is a maxirnal compact subgroup. Let r be a discrete su bgroup
of G. and assume that r is virtualIy torsion6free. [This is automatic if, for
instance, r is a subgroup or some G Ln( Z) cc. II.4. Exercise 3.] Then it follows
from Example 4 or 9 that
vcd r < dim X,
with equality il and only if r is co-compact m G. In the latter case, r is or type
W FL. More interestingly, r is of type W F L in the arithmetic case discusse<! in
I
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Il Virtual Notions
229
Example 5 of 9. even though r is usually not co-compact. and tbe Borel-
Serre results which we stated there give a precise formula for vcd r. For
instance
n(n - l)
vcd S Ln(lL) = 2 .
Finally, a group r is said IO be a virtual duality group if some subgroup of
finite index is a duality group. The arithmetic groups provide a large and
in teresting class or virtual d uality groups.
.-
,
.
(11.3) Proposition. A group r is a virtual duality group if al'ld only ifthe following
two condirions are satisfied:
(a) r is oftype VFP.
(b) There is al'l integer 1'1 such that Ht(r, 7l..r) = Ofor i =F rI and H"(f, Zr) is
torsion-free.
l n t his case every torsion{ree subgroup of finite index is a dualit y group with
dualizing module HII(r, l'T).
This is an easy consequence of Shapiro's lemma, as in the proor of 10.2.
EXERCISES
1. Suppose r is virlual1y torsion-free and let f' be an arbitrary subgroup. Show tbat
vcd r  veci r. wjth equaJity if (r: r) < 00.
*2. Study the behavior of the virtual finiteness properties with respect to extensions,
ama]gamations. etc. [Hìnt: Choose a torsion-free subgroup or finite index and
appJy 2.4b. Exercise 4 or 2. and Exercise 8 or .] Warning: These group.tbeoretic
constructions do nol preserve the property or being virtually torsion-free. cf.
Schneebeli [1978]. So the results you state wiH have to include a bypothesis that
the group under consideration is virtual1y torsion-free.

CHAPTER IX
Euler Characteristics
1 Ranks of Projective Modules: Introduction
Let r be a finite group and P a finitely generated projective ,ZT-module.
There afe several ways that one might try lo associate a 'rank n to P. One
candidate for this, whiçh we wilI denote by e(P) or i:rlP), is defined via
extensin or sca)ars with respect to tbe augmentation map zr -+ l. Namely,
we conSlder P r = 7L @ zr p. which is a finitely geoerated projecti ve 7L -module;
since projective 7L -rnodules afe rree, wc can set
(1.1) E(P) = rkZ(Pd,
the .right hand side being the rank in the naive sense (i.e., the cardinality or a
baSIS). Note that LI makes sense even if r is infinite.
. Ourseconddefinition ofa "rank" or P, which we will denote p{P)or PrlP),
18 based OD thc fact thal P itself is finitely generated and projective as a
Z-module; hence we can set
(1.2)
p(P) = rkz(P)/1 r I.
Here. of course" we do need r to be finite.
Both E and p give the 'right' answer n if P is the free module .zr". Other
than that, however, it is not obvious that they have anything to do with one
another. In fact it is not even obvious that the rationa1 number p(P) is an
integer. But it turns out that E; = p for al] finite groups r and ali finitely
generated projectives P. This fact, which is due to Swan, plays a crucia! role
in our theory of Euler characteristics. We will therefore devoe the first few
sections or the chapter to a proof or this equality.
Following Bass [1976, 1919J our proof"will go as follows: In 2 we will
introduce a third notion of rank, due to Hattori and Stallings and defined for
230
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2 The Hattori-- Stallings Rank
231
finitely generated projectives over an arbitrary ring. In 3 wewiU use standard
commutative algebra to give a concrete interpretation of the Hattorì-
Stallings rank in case tbe ring is commutative. This interpretation, together
with forma] properties or the Hattori-Stallings rank over group rings will
lead in  to a proof that E = p.
EXERCISE
The purpose of this exercise ìs to show that some restriction on a ring A is needed if
there is to be a ""reasonable" Z-valued rank for finitely generated projective A-modules.
Suppose there is a :l-valued function r on finitely generated projective A-modules,
satisfying:
(i) r(P EB Q) = r(P) + r{Q).
(ii) r(P) > O if P #= o.
(iii) r(A) = L
Prove then that A is indecomposabfe. i.e.. that A cannot be decompose<! as the direct
sum of two non-zero leFt ideals. [Hint: A decomposition A = l ffi J yieJds an equation
l = r(l) + r(J) in Z.J
2 The Hattori-Stallings Rank
If F is a finitely generated free lL-module, then the rank of F is equal to the
trace of the identity endomorphism of F; indeed. this is just a restatement or
the obvious fact that the trace or the n x n identity matrix is n. W ith this as
motivation, Hattori [1965] and Stallings [1965b] defined the 6'rankn or an
arbitrary finitely generated projective module aver an arbitrary ring to be
tbe trace or the identity map. In arder to make sense out of this, we need to
begin by developing a tbeory of traces for endomorphisms of projectives
which afe not necessarity rree over rings which are not necessarily com-
mutative.
Let A be an arbitrary ring F a finitely generated free A-module and
Cl: F -io F an endomorphism. One would like to define tr(cc) as the sum of
tbe diagonal entries ofthe matrix of Cl relative to a basis for F. Unfortunately.
this SUffi i8 not in generaI independent of the choice of basis for F. unless A is
commutati ve. (One can already see this for I x 1 matrices.)
There is, however, a well-definoo trace in the quotient T(A) = Aj[A, A],
where [A, A] is the additive subgroup of A generated by alI commutators
ab - ha, a, b E A. [Warning: [A A] is not an ideai, io generai, so T(A) i8
simply an abelian group" nOI a ring.] Namely we define tr(a) E T(A) for cc an

.
232
IX Euler Characteristics
n x n .matrix to be Li= 1 au, where exil is the image of aii in T(A). It is trivial
to verify that .
(2.1) tr{cxfJ) = tr(pcx)
whenever o: is an m x n matrix and p is an n x m matrix (so that afJ and f30:
are bot squae); and from this it follows that tr(po:f3- l} = tr(p-1 firx) = tr(a)
for any mvertIble p so the trace of an endomorphism is indeed a well-defined
element of T(A), independent of the choice of basis.
We would like now to extend the trace to endomorphisrns of finitely
generated projective modules. I claim that there is a unique way to do this
so that 2.1 continues to hold for maps P? P'. For let IX: P -+ P be an endo
morphism of a finitely generated projective, and express P as a dìrect
summand of a finitely generated free mod ule F via maps P  F wi th nl = id
Then o: gives rise to an endomorphism lo:1t of F. [In conrete terrns, this i
simply o: on the summand P and O on the complementary summand
ker m.] If we want 2.1 lo hold. then we are forced to define
(2.2) tr() = tr(l!Xn);
for2.1 would imply tr(lo:n) = tr(cun) = tr(aoid).
Il remains to show that the right side of 2.2 is independent or the choice of
p !::i F and that 2.1 continues to hold. Suppose we have maps
F  P  P 1(' F'
1t P' '7 
where F and F' are free, 1U = idp, and n',' = idr. Tben we can apply 2.1 to
matrices representing the composites
12'1I:
£1'" I F'
l'P'Jr
to concludethat tr(,o:1t'z' fin) = tr(l'{Jruo:n'). Since n'l' and nl are identity maps
trns yields  I
tr( la{Jx) = tre " {Jo:n').
Taking P = p and p = id, we conclude that tbe right side of 2.2 is indeed
independent of tbe choice of P !:i F. And taking p' and {3 arbitrary, we
conclude that 2.1 holds far arbitrary P  P'. This proves tbe claim. The
resulting trace tr(o:) will sometimes be deribted trA(O:),
Next we discuss the behavior of traces witb respect to extension and
restriction oJ scalars.
(2.3) Proposition. ù[ cp: A --+ B be a ring homomorphism and let T(cp): T(A)
-+ T(B) be the induced map. lf IX: P --+ P is an endomorphism oJ a jinitely
generared projective A-module, tllerI B @ A a: B @ A P --+ B @ A. P is an endo-
morphism 01 afinitely generated projective B-module, and we have
tr.J.B @A ex) = T(cp)(trA(cx».
.,
t .
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'. '.
,
,
.
.'
-
-.
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2 Tbc Hattori-Sta1Hngs Rank
233
i
PROOF. It suffices to consider tbe case where P is free. Then B @A P is also
,
free, and a matrix representing B @A a is obtained byapplying cp to the
entries of a matrix representing  D
(2.4) Proposition. Let cp: A -+ B be a ring Iwmomorphism such that B isfinitely
generated and projective as a left A-module. Then there is a rnap
trBIA: T(B).... T(A)
with the following property: Le! 0:: P -+ P be ari endomorphism 01 afinitely
generated projective B-module; then P is also finitely generated and projective ,
as an A-module, and
tr"ia) = trBjA(tr,Jet)).
The proof will use:
(2.5) Lemma Let A be an arbitrary ring and let P be a finitely generar ed
projective A-module which is a direct sum P1 EB ... EB PII' Let a: P -+ P be an
endomorphism with components aij: Pi --+ Pj' Then tr(a) = Li tr(ail)'
PROOF. This reduces easi1y to the case where eacb Pi is free, in which case tbe
result follows from an examination of tbe matrix of a. D
PRO:)f OF 2.4. For any b E B, tbe right multiplication map fJh: B - B is aD
endomorphism of the left A-module B, so it has a trace trAÙ1h) E T(A).
CODsider tbe additive map B -+ T(A) given by b 1--4 trA<P,,). Since fJh1b, =
Ilb2J1bl' tbis map annihilates commutators b1 b2 - bz bl. There is therefore
an induced map T(B) -+ T(A), which is tbe desired trB/A' To prove the for-
mula trA (a) = trBfA(tr,Jo:», note first that it holds if P = B (in whicb case a
is necessarily of the form Jlb)' In view of Lemma 2_5 it continues to hold if P
is a direct sum of copies of B, Le.. if P is a free B-module. In the generai case,
choose P  F with:!tl = id,., where F is free over B. Then tr(lo:n) = tr(Gtnl) =
tr(a) over both A and B, so tbe formula for a: P -+ P follows from the formula
for ICX7t: F -+ F. D
We can now define the Hatiori-Stallings rank of P, denoted R(P) or
RA(P), by
R{P) = tr(id,.).
Note that R(P) is not a number, but rather an element of the somewhat
mysterious abelian group T(A). In the next twO sections, however. we will
give concrete interpretations or R(P) for some interesting cIasses of rings.
Before proceeding to this tbe reader is strongly urged to look at tbe exercises
below; the results of exercises 5, 6. and 7 will be needed in .

234
IX EuJer Characteristics
EXERCISES
L Let Pbea finite]y generated projective (Jcfl:) A-module and Jet p. be its dual. Then P*
is a right A-module and we have P$ @__ P  HomA(P, P) by 18.3b. Prove that therc
is a commutative diagram
p. @A P  HomA(P, P)
I
T(A)
where ev. the ..evaluation map," is giveo by e\'(u @ x) = !.l(x). In some treaLmcnts of
the theory o traces.. this diagram is uscd to dcfine tr. cC Bass []976, [979]. CHint:
Consider p  A.]
2. (a) Let C be a finite projective chain complex over A and let 'l': C _ C re a chain map
with components "Cj: C; - Ci' Define the '.Lefschetz number" L(t) E T(A) by L(r)
= L ( -1 i tr(r,). Prove that L( 1:0) = L.('r I) if lO ,..,. T (. [Hint: Write down the
definìtion of homotopy and use 2.1.]
(b) Deduce that one can define a trace map rr: HomA(M, M) -+ T(A) whenever M
is an A-module or type F P by setting tr{ct) = L( y). where T is any lifting of.:x to a finite
projective resolution or M.
3. Suppose A is an gebra over a commutatì\'e ring k (e.g., A ìs always an algebra
over i.s center). Show that HomA(P. P) and T(A) are k-modules and that
tr: HomA(P, P) -- T(A) is k-linear. More generaJly. show that the same is true if P
is replaced bya module M of type FP as in exercise 2b.
4. Show by checking definitions that R(P) is equal to the trace of an ìdempotent matrìx
defining P. More precisely, rei F be a tinitely generated free module and e: F _ F an
(idempotent) projection operatot of F onto a direct summand isomorphic to P;
then R(P) = tr(e).
5. Let r be a group and <l': 1T - Z the augmentation map. Show tbat the numericaJ
'rank" tr< ) defined in 1 can be obtaìned rrom the Hattori..Stallings rank Rzrl. )
by the formula
er(P) = T(tp)(Rzr(P».
[Apply 2.3 with a = idp and note that T(qJ) takes values in Z because T(Z) = l.]
6. Let r be a finite group.
(a) Look atthe definìtion of trzr.-'z: T(Zr) - T(l.) = Z in the proof of2,4 and deduce
that tr ZI"!.I{I) = I r I and that tr zrlz(Y) = O for l =1= ì E r. Here )' is the image or r in
T{Zr). Consequently, there is a weU-defined homomorphism T: T(Zr) _ Z such
that t(I) = 1 and T(y) = oror J =1= rE r. and one has trZf"':'Z = In. T.
(b) Deduce rrom 2.4 app)ied to the inclusion Z. 4. zr (with (X = _idp) that rkz(P)
= Iq. t(Rn{P)) for any finiteJy gnerated projectivc Zr-module P. Hence the
'rank" p defined in 1 is gìven by
PrlP) = T{R.r..{P».
I
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3 Ranks over Commutati\le Rings
235
7. Exercise 6b shows that p..(P) E Z. On the other hand, it is ob\lious from thc defìnition
or fJ that fJr<P) :> o if P =1= O. Show. therefore. that zr ìs indecomposable when r is
finite. [Use the exerdse of L]
8. Show that the definìhon of T in exercise 6a makes sense even if r is infinite, ie.. there
is a well-defined homomorphism 'l': T(lT) -+ 7l. such that T(I) = 1 and T(Y) = O
ror l#-}' E r. We can therefore define a Z-valued '.rank" p over zr by p(P) =
t(R(P». In view of exercise 6b. this agrees with the p or 1 if r is finite. Moreover, it
turns oul for arbitrary r that P(P) > O if P :#: O. [This non-trivial resuJt is due to
Kaplansky proofs can be found in Montgomery [1969] and in Passman [1977].
Theorem 1.8 and Exercise 9 ofChapter l.J ConsequentJy. you can show as in exercìse
1 that zr is indecomposable for any group r.
3 Ranks over Commutative Rings
If A is an integraI domain with fie1d offractions K. then there is an easy way
to define a 7L -valued rank for finitely generated projective A-mQdules. Namely.
we set
rkA(P) = diml;:{K @A P).
Moreover, the Hattori-StaUings rank RA(P) is simply given by
(3.1)
RA(P) = rkA(P). l,
where 1 E A = T(A) is the identity elemento [T o prove this, we may regard
both sides of the equation as elements or K. The lert hand side is then equal
to R"JK @A P) by the compatibility or traces with extension or scalars (2.3),
and tbis in turn is equa) to dirnK(K @A P). I = rkA(P). 1 since K @A P is a
free K -module.J
Our goal in tbis section is to prove a generalization of 3.1 to indecom-
posable commutative rings which are not necessarily domains. [This restric-
tion to indecomposable rings should Dot surprise the reader in view of the
exercise of L There are however, interesting results along these lines for
decomposable rings; see the exercises below.]
Since A is no longer assumed -to be a domai we do not bave a field of
fractions to work with. We do, however, have a localization A.. for every
prime ideai p, and we can use this instead. [See Atiyah-MacDonald [1969],
Chapter 3. for the definitions and elementary facts concerning localization.]
The ring Ap is a iocal ring (it has a unique maximal idea)), and we will prove:
(3.2) Proposition. Any finitely generated projective over a Ioeal rjng is [ree.
We can therefore detine the rank oJ P al p to be the rank in the naive sense
or the free AII-module P» = A.. @ A P. Let Spec A be the set of prime ideals

236
IX Euler Characteristics
of A. Then what we have, so far, is a function !JlIA(P): Spec A --+ Z whose
value at :p is the rank or P at p.
Spec A has a welJ-known topology (the . Zariski topology"), whose
definjtion we will recall below, and we will prove:
(3.3) Proposition. Thefunction {JtlA(P) is locally conSlanr.
Finally, we will establish the (not surprising) geometric interpretation of
indecomposabìli ty:
(3.4) Proposi.ion. If A is indecomposable then Spec A is connected.
The function !ili A(P) is therefore constant if A is indecomposable, and
we set rk A (P) equal to its constant value.
Accepting alI this fOT the moment, we CM prove the desired generalization
of 3.1 :
(3.5) Tbeorem. Ler A he a commutative indecomposable ring, let P be afinitely
generaled projective A -module, and let rkA(P) be the rank Dj P at any prime ideai
p. Then the Hattori-Stallings rank RA,(P) is given by RA(P) = rk.A(P}, L
PROOF. The compatibility or traces with extension of scalars implies, as in
the proof or 3.1, that RA,(P) and rk...(P). 1 have the same image in A" for
every :p; hence they are equaL D
It remains to prove 3.2, 3.3, and 3.4. For 3.2 we wilI need 'Nakayama's
lemma M:
(3.6) Lemma. Let A be a loeai ring with maximal ideai m and residue field
k = A/m. and let M be a finitely generated A-module. Il k @A, M = O then
M= O.
(Note the similarity between this and VI.8.4. 80th results are special cases
of a generai "Nakayama's lemma.")
PROOF. Let mt, ..., mr be a minimal set of generators of M. and suppose
r > 1. Since k @ A M = M/mM, our hypothesis says that M = mM. In
particular. we can writem, = L aimiwithaiEm, so (1 - al)ml = l'i a.mi'
But 1 - al is invertible since it is not in the unique maximal ideaI of A; thus
M is generated by m2. . . _ . rnr. This contradicts the minimality of mi, . . . . mr,
so r must be O. D
PROOf OF 3.2. This is similar to the proof ofVI.8.5. LeI A be Ioeal with residue
field k, let P be a finitely generated projective A-module. and let
r = dirnJ;(k 0A P).
.'
-,
"
,-
..
I
i
!
.:
.
.-
..
:
.-
-
."
..:
"
.
.
i
l
..
..
..
I i
, ..
I
!
i
.
;'
i
3 Ranks over Commuta.tive Rinp
237
Then we can find a map f: A" -+ P such that
k @f:k! = k @ A" -J. k @ P
is an isomorphism. Then k @ cokerf= coker(k @f) = O, so cokerf= O by
3.6. We now have a short exact sequence
O -J. kerf...... A".4 p --+ O,
which splits because P is projective. The sequence therefore remains exact
after being tensored with k, so k @ ker f = O. Now ker I is finitely generated,
being a dìrect summand of A", so we can apply 3.6 agaìn to conclude that
kerf= O. Thusfis an isomorphism. D
The next step is to topologize Spec A. For any subset S c. A. we define a
subset V(S) c Spec A by
VeS) = {p: p 2 5}.

(3.7) Lemma.
(i) lf l is the ideaI generated by S, then VeS) = V(l).
(ii) VeS) = Spec A if and only if every element DJ S is nilpotenl.
(iii) lf l is an ideai, then V(l) = 0 if and only if l = A.
(iv) 111 and J are ideals, then V(lJ) = V(l) u V(J).
(v) II {l(l} is afamily ofideals and I is their sum. then V(I) = n:. V(la).
PROOF. The proofs are routine verifications which are left for the reader,
except for the . only if" part of (ii), which is proved as follows: Suppose
a € S and a is not nilpotent. Then the localization A[a - 1] is not the zero ring
and hence contains a maximal ideaI m. The contraction of m to A is then a
prime ideai not containing a, so V(S) >F Spec A. D
,.
"
.'.
It follows from (ii)-(v) that we can topologize Spec A by declaring that
the closed sets afe the sets ofthe rorm V(l), where / is an idea] of A. In view
of (i), every VeS) is then closed.
PROOF oF3.3. Suppose PII is free ofrank r over Ap. We must show that Pq is
free o£rank r over ACf for alI q in some neighborhood of:p. It is easy to see that
we can find r elements of P whose images in p l' form a basis for P P' In other
words. we can find a map f: F --+ P, where F is a free A-module of rank r,
such that Il': F:p -J. P p is an isomorphism. Therefore (coker f)p = 0, so every
element of coker fis annihilated by an element of A-p. But coker f is finitely
generated, so there is a single element S E A - P such that s . coker f = O. Thus
if we localize A by inverting s. we obtain a surjectionf[s-l]:F[s-lJ-+
p[S-I] which becomes an isomorphisrn when 10caIized at V. (This makes
sense since A" can be viewed as a localization of A [s - 1 ].)
P[s - 1] being projecti ve over A [s - 1], this surjection splits, and ker f [8 - 1]
is therefore finitely generated. Since it becomes zero when localized al p,

238
IX Euler Charactristics
it is annihilated by some element of A [s - 1] - p[.s  I] and hence by some
t E A-p. Letting u = st. we conclude that F[u-1]  P[u-1]. But then
F q .; p q for ali q such that u f. q, since A" ìs a localization or A[u - l] for such q.
Thus P is free of rank r for ali q in the neighborhood Spec A - V(u) of p.
q D
PROOF OF 3.4. Note first that a ring A is decomposable if and only if it contains
an idempotent element e # O» 1; for direct sum decompositions or tbe A-
module A correspond to l x 1 idempotent matrices. which are the same as
idempotent elements. What we must prove. then. 18 tha t if Spec A is discon-
nected then A contains an idempotent e #:- O. 1.
Suppose Spec A is disconnected, say Spec A = V(l) U V( J) where J and
J afe proper ideals. Since an element e is idempotent if and only if e(l - e)
= O, what we afe looking ror is a decomposition of 1 as e + j. w here ef = O
and e. f", I. Now we know (cf. 3_7(v) that V(I + J) = V(l) n V(J) = 0.
so l + J = A by 3.7(iii). Thus el + Il = I for some el E Ifl E J. We also
know (3.7(1V») that V(IJ) = V(I) u V(l) = Spec A. so every element or IJ
is nilpotent by 3_7(ii). In particular. elil is nilpotent. I claim now that we
can get our desired decomposition e + f = l by raising both sides of the
equation e1 + Il = 1 to a high enollgb power. Indeed, Cor any integer n > O.
we can group the terrns in the binomial expansion of (e l + fl)2n- 1 to obtain
I = (e1 + Il)21'1- I = e + f, where e E Aei c: l and f E Af  J (hence
e, f -# I). Since e1 Il is nilpotent, we will then have ef = O for large enough n.
D
EXERCISES
*1, GeneraJize 3.4 as follows: For any commutative ring A. the idempotents of A are in
1-1 correspondence with the su bsets or Spec A which are both open and c1osed.
*2. GeneraHze 3.5 as fol1ows: Let A be an arbitrary commutative ring and let P be a
finitely generated projective A -module. F or any imeger i > O let U, c Spec A be
ffU ..( P) - 1( i). Let e i E A be the idempotent corresponding to Vi (d. exercise 1). Then
ei = O for ali but finitely many i and R..(P) = Li2:0 iej-
+3. In view of exercìse 4 of 2, the resul. of exercìse 2 can be stated in the following
concrete terms: Let E be an idempotent matrix over a commutative ring A; then
there are idempotent elements e, EA such that tr(E) = Liò!:{J i;. It is natur] to
ask whether the ei can be computed explicitly in terrns of the entnes of the matnx E.
The purpose of this exercise is to gi\le an affirmative a[lswer due to Goldman
[1961] (see also Ahnk",ist [1973]), which gives a formula for ej m terrns of the co-
efficients or the characteristic polynomial of E.
(a) LeI E be an idempotent n x n matrix and Jet qJ(I) = det(L + tE) = 1 + tr(E)t
+ ... + det(E)r". Let P be tbe corresponding projective module Im{E), and le t
(ej)i2: o be as in exercise 2. Show thal lp( t - I) = Li  o ei (i. [int: At a prime ideaI
where P has rank i. both sides of this equation are equal to t.]
t
. .
4 Ranks over Group Rings; Swan's Theorem
239
(b) Writing lp(l) = Li atc. deduce tbat
_(ì + j)
ei =  (-1)1 ì aH)"
J
If n = 2. for exampJe. then e2 = det(E). el = tr(E) - 2 det(E), and eo = l - tr(E)
+ det(E).
:<
4 Ranks over Group Rings; Swan's Theorem
,
Far simplicity we will confine ourselves to integrai group rings zr although
it will be obvious that some of what we do in this section extends to more
generaI group rings kr.
In order to understand tbe Hattori-Stallings rank R(P) (also denoted
Rr(P» over Zf, we must first understand the group T(.lT). This group is
the quotient or the additive group of zr by the subgroup generated by the
commutators h, iJ = yi - y'y b, t E n. Now the commutators of tbis
form are precisely the elements 'YI'ly-l - li (y, 'VI E r); for we have [y, y'] =
y(')ly)y-l - (y'y) and 1Yl,,-1 - j11 = [y, l'1),-1]. It follows easily that T(ZIl
can be identified with the free abelian group on the conjugacy classes of
elements of r. Thus any element of T(Zr) has tbe rorm
r t(y) . [y],
}' IO 'if
y
-'
where ((j is a set oC representatives for the conjugacy classes in r, [y] denotes
the conjugacy class of y, and t: r -+ Z is a function whicb is constant on each
conjugacy class and is zero for almost all conjugacy classes. In concrete
terms. then. the trace of a matrix over Zf is obtained by summing the
diagonal elements and grouping together the terrns corresponding to
conjugate group elements.
Now let P be a finitely generated projective zr -module. Its Hattori-
Stallings rank Rr< P) E T(lr) has an expansion
Rr<P) = L Rr(P)(y) " [y].
YIOV
Thus Rr<P) can be viewed as a famiIy of integers RrlP)(y) (y E  associated
to P- From tbis point of view, tbe restriction formula 2.4 takes the following
form:
(4.1) Proposition. Let r be a group and r a subgroup offinite index. For any
y E r' there is an integer n{y) > O such that for every ji,ritely generated pro-
jective Zr-module P,
Rr'(PXy) = n(y). Rr(P}b}'
Mo,.eover nO) = (r: r').

240
IX Euter Characteristics
(See exercise 2 below for a precise formula for n()I).)
PROOF. We need to compute tr = tr1!r!zr': T(Zr) -+ T(.lT'). Fix Y E r, and
recall that tr([ ì']) is the trace over zr of the right multiplication map
Jl¥: lT -+ zr (cf. proof of 2.4). Let S be a set of coset representatives far
r'\r. Then S is a basis for zr as a lert 1:T' -module, and ll'l can be described
as follows in teTrns ofthis basis: Given S E S, we have Jty(s) = S7 = il, where
t E S is tbe representative of the coset ['s'Vand y' = syt-1 E r'. Thus we get a
non-zero contribution to tbe trace of P., if and only if t = s, Le., if and only if
SyS-l E r', and this contribution is then y' = sys-l. We therefore have
tr([}I]) = I [s-ys-l]r,
5ES
where
_ {the r' -conjugacy c1ass of sys - l if sys - 1 E r'
[S"IJS l] . =
I r O otherwise.
For any y E r', let n(y) be tbe number of cosets r's such that [sys-l]r
= [Y]r. Note that n(y) > l and that n(l) = (r: r'). Let f(} (resp. ') be a set
of representatives for the r-conjugacy classes (resp. r-conjugcy classes).
Then the formula of the previous paragraph yields the followmg formula
for tr: T(ZT) - T(Zr'):
tr( L r(Y)[)I]) = L n(y)r(Y)[ì]r"
}' e 'il' )' e '#'
Tbe proposition is now immediate from 2.4.
o
For example, if we take r finite and r' = {l}, 4.1 simply says rkiP) =
Ifl. R.-(PXl); thus the "rank" Pr(P} introduced in l satisfies
(4.2) pr<P) = Rr<P)(I).
[Note: We bave just rederived the result or exercise 6 Di 2.] On tbe other
hand, Qur other "rank" EP) can also be expressed in terms of Rr<P).
Indeed we have already done this in exercise 5 of 2 and tbe result of that
, . .
exercise, when translated into tbe notation of the present secuon, IS
(4.3)
ErlP) = L Rr<PK'l).
ì'eW'
We can now prove our main result. which can be viewed as sayng that the
three "ranks" G, p, and R that we have been talking about for finite r aIe alI
essentially tbe same:
(4.4) Theorem (Swan [1960b]). Il r is finite and p is il finitely generated
projective zr-module. then R..{P)(y) = o for y 1= 1. Thus there is al1 integer
r such thal Rr<P) = T' [1], and one has Er(P) = pr<P) = r.
I
I .
"i.
,
..
i
.
:
.
t;.
,
4 Ranks over Group Rings: Swan's Theorem
241
PROOF. Tbe second assertion follows from tbe first in view or 4.2 and 4.3. To
prove the first assertion, we may replace r by the cyclic subgroup generated
by 1'; for the restriction formula 4.1 shows that this does not change the
question as to whether RrlPKr) = o. In particular, we may assume that .zr is
commutatÌve. Also, as we pointed out in exercise 7 of 2, it follows from formai
properties of Prl ) that lr is indecomposable. Theorem 3.5 is therefore
applicable and shows that Rr<P) is an integra) multiple of [l], as required.
D

F or emphasis. we go back to the definitions of p and 8 and state explicitIy
what it means for them to be equaL
:.
(4.5) Corollary. Let r be a finite group and p a finitely generated projective
Zr-module. Then
rkz(P) = I rl. rkZ{Pd.
-.
,
Remark. Using some elementary representation theory (cf exercise 3 below),
one can restate Swan's theorem as follows: If r is finite and P is a finitely
generated projective Zr-module. tben GJ @z P is a free or-module. lt is
in this form that Swan stated and proved the theorem. Tbe rormulation and
proof in terms of Hattori-Stallings ran.ks afe due to Bass [1976, 1979].
Bass went OD to conjecture that 4.4 remains valid for arbitrary groups r.
[To make sense out of the second assertion of 4.4 foc arbitcary r, one should
take 4.2 as a definition when r is infinite, as we essential1y did in exercise 8
of2.] He proved the conjecture for a lacge class oftorsion{ree groups f. For
infinite groups with torsion, however, the conjecture is stilI completely open.
;;
.,.
ExERCISES
,.
l. Where did the proof of 4.4 use the assumption that r was finite?
2. Sbow that the integer l'I(y) ìn 4.1 is equa] to (Crb): Cro(Y»). where C.-{}') (resp. Cr'('P.»
is the centralizer of y in r (resp. r'),
3. Let r be finite. Jet k be a field of characteristìc zero. and let Y be a kr-module which is
finìte dimensional over k. Then V 18 finitely generated and projective (c£. I.8, exercise
5). hence ìt bas a Hattori-StaUings rank LR(y) - (y]. where R; r - k is a centrai
function. i.e. a function which is constant on conjugacy classes. On the other hand,
there is a classical way to use traces to associate a centra] functìon 1..: r --+ k to JI,
caIJed the charocter or JI. Namely. x(y) is the trace over k of the actìon of Y OD V. Thc
purpose of this cxercise is to relate 1.. and R and IO draw some consequences from
this.
(a) Show that the Hattori-Stallings rank of V ìs equal to
1 _ 1 X(1- 1 )
Irl)'X(y ). [y] = E I Cr(Y) I , b],

242
IX Euler Characteristics
i.e., that Rb') = l.(Y - 1)11 Cr<;) I. [Metbod 1; There is a C'dnonical way o writing V
as a direct summand of an induced module; namel)'. we have kr (8Jk V V. where
1r(j' @ v) = rvand
I" 1
I(V) = - I..r r @ r- v.
IrI)'<!;r
Choose a basis (ei) for V over k. so that (1 (g) e,) is a basis for kr @ V over kr. You
can now write down the matrix of m in terms ofthe matrices oiJ{ì') ofthe elements ofr
acting on V. Method 2: ICs cnough to prove this when k is algebraically dosed and V
is irreducible. 111 lhis case. the elernent
n
e = - L x<r'- tlì',
In 7ET
wherc tI = climI:. V. is the idempotent or kf which projects k r omo I[S summand of
type V (cf. Serre [t977b). 2.6). Since V occurs n timcs in kr. thc image of e in !(r)
is the H attor i - StalHngs ral1k or the suro of n copies of V. Method 3: By the restnctlOn
{onnu]a (ìn the precise form provided by exercise 2). we may assume r is cyclic, so
that the map JJ..,: V-V given by the actin or ì' is kr-linear. for each y E r. T?en
trk,...(P) = r. trlc:r(id) = y. Ru.(V) by exerclse 3 of 2. In parttcular. the coeffiClent
tr .u...(p.,)( l) of [ l] is equa1 lo R"r( V)( '! - t). On t hc other hand. the analogue of 4. l for
traces' gives X(f) == tr,,(,uy) = 111. tru-(J.tì)(l).]
(b) Using thc fact (known rrom representation theory) that the mod ule V is determined
up to isornorphism by its character. show that two fiiteJy gnerated kr-modules are
isomorphic if and on]y ifthey ha ve the same Hatton-Stalbngs rank.
«(;) Deduce the reformuJation of 4.4 given in the remark above.
4. Tale 4.2 as a definition. and prove that Pr'(P) = (f: r')Pr(P) for any group r.
subgroup r' of finite index, and finitely generated projective Zr-module P.
5. With r. f', and P as in exercise 4, prove that t:r'(P) = (r: f'}tr<P). [Hjnt Suppose ,
first that f' is normal il1 f. Let r = rjr' and let P = Pr, = zr (8)u p (cf. II.2.
exercise 3). l' is a finitely generated projective zr -m od ule, and the desìred resu]t
is obtained by applying 4.5 to il. In the generai case. choose a subgroup r"  f'
of finite index such that r" is normal in f; this ìs possible because r has onJy fimtely
many conjugates in r. so theìr interseçtion ìs stiIJ or finite index and is norma!.
Now compare er,. lo both Ero and tr.]
5 Consequences of Swan'8 Theorem
In this sectìon fin ali y , the reader will begin to see some Euler characteristics.
We will tben be ready in  to discuss Euler characteristics or groups.
For any finitely generated abelian group A we define the rank of A
(sometimes called the tQrsion-free rank of A) by
rkz{A) = dimQ(O @z A).
.
5 Consequences or Swan's Theorem
243
If A is free this obviousIy agrees with the naive rank (cardinality of a basisJ
which we ha ve been using throughout this chapter for free modules; in
generaI, rkz(A) as defined above is equal to the rank in this naive sense of
the free part' of A, i.e., of the quotient or A by its tarsion subgroup. In
particular, rkz(A) = O if and only if A is finite.
Let C be a non-negative chain complex of abelian groups. We say that C
is finite dimensional if Ci = O for sufficiently large i. Ir, in addition, each C
is finitely generated, then C is cal1edfinite. Suppose that C is tinitedimensional
and that H. C is finitely generated; then we define the Euler characteristic
x( C) by
x(C) = I (-1Y rkz{HiC).
i 2': o
In case C is the cellular chain complex C(X) of a finite dimensional CW
complex X, we write X(X) instead of x(C(X». Thus
l(X) = L (-l)j rkz(HjX).
iO
If X is actually finite, then x( X) is equa) to the classical Euler characteristic
Li (-l)lni, where '1i is the number of i-cells of X. This fo1lows rrom:
(!tl) Proposition.lfC is ajinite chain complex, then
x(C) = I (-I)i rkZ(Ci).
i
See. for instance, Spanier [1966]. 4.3.14, or Dold [1972], V.5.2, ror the
(easy) proof.
lt is alsa convenient sometimes to compute Euler characteristics using
mod p" homology. This is justified by:
(5.2) Proposition. Lei C be a finite dimensional free chain complex over lL
witlz H. C fi,ritely generated. Let p be (l prime number. Then
.
x(C)= I (-1Y dirnz.,(H,{C @ Zp»'
i
PROOF. Let ri = dimzp«HjC)p) and Si = dimzp(iHjC». (Reçall that Ap =
A 0 ?Lp = AjpA and that pA = {a E A: pa = O} = Tor(A, Zp) for any
abelian group A.] Using the universal coefficient theorem (or the long exact
homoIogy sequence associated to the short exact sequence O -+ C  C -
C @ lL" -t O). one finds that dimzp(Hj(C 0 lLp)) = ri + Si-l' On the other
hand since HiC is a direct sum or cyclic groups, one sees easiIy that ri =
rkz(HiC) + Si' Thus dimzp(Hj(C @ Zp)) = rkz(HjC) + Si + Si-h and the
proposition follows at once. D
We now prove the main result ofthis section:

244
IX Euier Characteristics
(5.3) Tbeorem. Lei G be a finite group and let C be a finite dimensional chain
complex oj projective ZGmodules. lf H.(C) is finitely generated then so is
H.(CG) and
x(C) = I G 1- X(CG).
(In other words C behaves like a finite free complex as far as Euler
charactecistics are concerned.)
PROOF_ In case C is a finite projective complex, this is immediate from 4.5,
since l can then be computed on tbe chain level (5.1). The genera] case is
reduced to tbe finite case by means of Lemma 5.4 below. D
Reca1J that a cing R is noetherian if every submodule or a finitely generated
module is finitely generated. For example 7l.G is noetherian ìf G is finite,
since a finitely generated ZG-module is finitely generated as a lL-module.
(5.4) Lemma Lel R be a noerherian ring and C a finite dimensional chain
complex DJ projective R-modules. Il H. C is finìtely generated over R, rhen C
is homotopy equivalel'lt to a finite projective complex.
PROOF. First we construct, step-by-step, a free complex F of finite type which
admits a weak equivalence 1:: F  C. The inductive step is carried out as
follows: Suppose we have constructed the n-skeleton or F togethec with a
chain map
Fn
l'
) Fa
1.
) .. .. .
I c", + l iJ t c,.
ICa
} iII 'Il .
such that H i 1" is an isomorphism for i < " and an epimorphism for i = n.
Then ker(Hn 1") is finitely generated since F Il is finitely generated and R is
noetherian. Let x l, . . . , X, E F Il be cyc1es representing generators or ker( H 111:),
let Yb . . ., y,. E CII + l be chains such that OYi = 't'Xii and let z 11 . . . , Zs E eli + 1
be cycles representing generators or tbe finitely generated module H 11+ 1 C.
Tben we can take F l'iI + l to be free with basis e l, ... , e" Il, . . . ,!s and set
òei = Xi' òli. = O, t"ei = Yi, and 1:fi = Zi- The resulting chain map
I
l.
,
I
l .
I
!
;
I.
FII+1 . Eli . . . ..
1- l. I
,
) CII+ 1 ) ClI .. .. . -
induces an Hi-isomorphism for i < n + l and an HII+ l-epimorphism. This
completes the inductive construction or F and 'f. Note that the weak equiva-
lence 1: is actually a homotopy equivalence by 1.8.4.
.t
S Consequcnces ofSwan's Thcorem
245
Now let n be an integer such that C  = O for i > n, and consider the
quotient F of F defined by
F. i<n
I
F-= F ,,/811 I = n
I
, O i> 11,
..
,
Bn being the module or n-boundaries. Then we still bave a weak equivalence
F ..... C, and the lemma wilI clearly follow if we can show that F JBn is pro-
jective. For any R-module M, we have
H"+ 1(R(F, M»)  HII+ l(KomR(C, M» = O
since F  C and C" + l = O. Examining cocycIes and coboundaries in
 R(F, M) as in tbe proof of VIII.2.1, we conclude easily that Brr is a direct
summand oi F n' and hence that F JBI1 is projective. D
T O illustrate the significanee of Theorem 5.3, we give two special cases.
(5.5) Corollary. Let G be a finite group and let X be a finite dimensional free
G-CW -complex with H * X finitely generated. Then H .(X/G) is finitely gener-
ated and X(X) = IGI. X(X/G).
PROOE Apply 5.3 to the cellular chain complex or x.
D
This cesuh is o bvious. or course, if X is finite since we can then compute
Euler characteristics by counting cells.
The second special case involves group cohomology. If r is a group 8uch
that H i r is finitely generated for alI i and finite for sufficiently large i, then
we set
x{r) = L ( -l)i rkz(H i D.
i
,
, .
[The notation i here is intended to suggest that tbis is not always the right"
Euler characteristic. We will explain this in 7.]
(3.6) CoroJIary Let r be a group with ed r < 00 and let r' be a norma l sub-
group offinite index.lj H. r' isjìnìtely generated then so is H .r, and i(f') =
(r: r') . iO;.
PROOF. Let P be a projective resolution or finite length of Z over zr, Jet
G = r/r'. and Jet C = Pro = ZO 0..ll" P (cf. II.2, exercise 3). T:hen C is a
finite dimensional complex of projective ZG-modules with H. C = H. r'
and H.(Co) = H.r; now apply 5.3. D
Finally, we will need a mod p analogue or 5.3:

246
IX Euler Characteristics
(5.7) Theorem. Let G be a finite p-group lor some prime p, let k he a field oJ
characteristic P. and let C he a finile dimensional complex oJ projective
kG-modules. lf H.C is oJ finite rank over k, then so is H*(CQ). and X(C)
= I G I. X(CG), where x( ) now denotes L (_l)i dimA;(Hi( ».
PROOF. This is identica l to that or 5.3, with one major simplification; namely.
the mod p analogue or 4.5 is now true for trivial reasons, since ali projective
kG-modnles are free (VI.8.5). D
One also has, obviously, mod p analogues of the corollaries 5.5 and 5.6
of 5.3.
EXERCISES
I. Let C and G be as in 5.3. For each g E G,let L(q) be the Lefschetz number of g acting
on C. i.e.,
L(g) = L (-I)l trQ (g acting 00 Boe <29 Q).
ì
Prove that L(g) = O for g #- l, and siate a topological coroJJary analogous to 5.5.
[Hine We mayassume C is finite, so that L(g) can be compuled on the chain ]evel.
Exercise 3 or t;4 now describes L(g) in terrns of Hattori-Stallings ranks, and the
latter are computed by Swan's theorem (4.4).)
2. Let r and r be as in 5.6. Let t}li be the character associated to the representation of
the finite group r/r' actingon Hi(r', O). Show that Li (-lYr/Ji isan integraI multiple
or the character of the reguJar representation of r jr'. [The regular cepresentation,
by definition, is the free ocr /r']-rnod ule of rank l  its character takes the value o on
ali non-identity elements of rjf'.]
6 Euler Characteristics of Grou ps :
The Torsion-Free Case
Euler characteristics of groups have been defioed by a number or people
under a number of different finiteness conditions. For our purposes tbe
following condition is the most convenient one to impose: A group r is said
to be ofjìnite homological type if (i) vcd r < O'J and (ii) for every r-module
M which is finitely generated as an abelian group, Hi(r M) is finitely gen-
erated for alI i. [Warning: This definition differs slightIy from that given in
Brown [1974].] Note that if r is torsion-£ree, then (i) impIies that ed r < CI:;
(cf. VIIl.3).
Remark The main examples 'or groups of finite homological type are the
groups oftype VFP, and the reader may prefer to replace "finite homologica]
,
..0
!
i 
I
i
. ,
"
6 Euler Characteristics or Groups: The Torsion-Free Case
247
type" by 4'type VFP'" in what follows; this results in occasionaI simplifica-
tions of the proofs. On the other hand, we will see that one cannot restrict
a ttention to groups Qr type V F L, even though alI known examples of groups
of type VF P are in fact of type V F L.
,
;.
(61) Lemma. lfr is a group and f' is a subgroup offinite index, then r is DJ
finite homological type if and only ifrl is oj finite honwlogical type.
PROOF. It is obvious that vcd r < 00 if and only if vcd r' < 00, so we need
only check that (ii) holds for f il and only if it holds for r. The '''only if90
part folIows from Shapiros lemma since coindf. M is finitely generated over
7L if M is. Con versely, suppose r' satisfies (ii). We know that r' has a subgroup
fl' of finite index which is norma' in r (cC , hint to exercise 5), and r" stilI
satisfies (ii) by what we have just proved. Replacing r' by r"', then we may
assume r' is nonnal in r. For any r-modu1e M which is finitely generated
over lL, tbe Hocbschild-Serre spectral sequence
E;q = Hirjr', H9.(r', M»)Hp+Jr, M)
now shows that H tr. M) is finitely generated for ali ;, as required. D
.-
Suppose now that r is or finite homological type and torsion-free. We then
define the Euler characteristic X(r) by
x(r) = L (-l)irlez(HiI)
i
[Thus x(r) = i(r) in this case, where X is as in 5-] In case r is or type F P.,
one has an analogue or the combinatorial formula for the EuJer character-
istic of a finite CW -complex. Namely, if P is a finite projective resolution of
lL over 71, then
(6.2)
x(r) = L ( -I )it(pa,
i
where E is the "rank" defined in 1. [fo prove this, we need only note that
H.r ean be computed from tbe finite chain complex Pr; the assertion now
follos from 5.1.] We also bave, trivially, tbe topological interpretation
X(r) = Z(K(r, 1)).
Here are a few examples where X(r) can easily be computed.
EXAMPLES
l. If r is a free group of Tank n > O, then there is a K(r  1) with one vertex
and n l--cells. hence x(r) = 1 - n.
2. If r is a free abelian group of rank n  l, then the n-torus S1 x ... X SI
is a K(r I) with Euler characteristic x(Sl) ... X(Sl) = O. hence x(f) = o.

248
IX Euler Characteristics
3. Let r be tbc commutator subgroup of SL2(Z). Tbe latter is well-known
to be aD amalgamation 714- *Z1 7L6. (See Serre [1977a J. 1.4.2, for an indication
of an easy proof ofthis; see also example 3 of VIlI.9 above.) In particular, it
fonows that SL2('Z)ull is of order 4.6/2 = 12. so that r is or index 12 in
SL2(lL). One can also check, using the normal form for words in an amalga-
mated free product. that r is free of rank 2 (cf. Serre [1977a]. 1.1.3, proof or
Prop. 4). Hence X(r) = 1 - 2 = - l by example 1.
We will see more examples later.
Tbe following property of x( ) is fundamental =
(6.3) Theorem. 1/ r is torsion-free und oJ finite homological type and r is a
subgroup oJ finite index, then
X(r') = (r: r) . X(r).
PROOF. It suffices to prove this when r' 1s normal in r, in which case it follows
from Corollary 5.6. D
Tbe proof of 6.3 simplifies slightly if r is of type F P; in this case we can
use 6.2 and exercise 5 or 94 instead of 5.6. And the proof simplifies drastically
if r is or type FL, since exercise 5 of  is trivial for free modules.
Theorem 6.3 has an interesting application to the theory of group exten-
.
810ns:
(6.4) CoroUary. Let 1  r -. E -+ G -+ 1 be a group extension such that r is
torsion1ree and offinite homological type and G is o/prime order p. l/p..r x(I),
then the extension splits.
(U r is free on n generators for instance. then such an extension must
spIit whenever p.r (n - 1).)
PROOF. The group E necessarily has torsion; for if it were torsion-free, then
6.3 would be applicable and would yield p I X(r). contrary to our hypothesis.
Let G be a non-trivial finite subgroup of E. Since r is torsion-free, (; n r
= {l}; thus e; maps injectively to G. But I G I is prime so li maps isomorphi-
cally and provides a splitting. D
Remarks
1. Even if we are interested in 6.4 only for groups r or type F L, the proof
stiJI req uires a theory of Euler characteristics for groups of type F P. F or we
need to apply 6.3 to E in the first partofthe proof(a ssumingE is torsionfree);
and alI we know a bout E. given that r is of type F L, is that it is of type F P
if it is torsion-free.
2. We will give a substantial improvement of 6.4 later (cf. 9.4).
.,
'.
'-
'.
, ..
i -
.
, ,
I .
I ,
t .

,
.,
-.
,
I
,
i' -
..
1
.
I
..
I
i
I
I. ,
7 Extenston to Groups with T orsion
249
7 Extension to Groups with Torsion
From the topological point of view, it may seem very reasonable to restrict
attention to torsion-free groups. Indeed. topologists usually consider Euler
characteristics only for finite dimensional complexes and we know that r
must be torsion-free ifthere is a finite dimensional K(r, 1). From the algebraic
point of view however, this seerns less reasonable; for it forces us to exclude
certain groups r (like SLiZ» just because they have torsion, even though r
may have subgroups offinite index for which X is defÌned.
There is a simple solution to this problem. based on an idea due to WalI
[1961]. Namely if r is an arbitrary group or finite homological type, then
we choose a torsion-free subgroup r' of finite index and set
x(r')
X(r) = (r: r') E Q.
This is motivated by 6.3, and. in fact, 6.3 is precisely what is needed to justify
this definition, Le., to show that the right-hand side is independent of the
choice of r'. F or suppose r" is another such subgroup and let r o = r' (ì r";
then
x(r') (62) X(r 0)/([': r o)
(r: r') (r: r')
x(r o)
(r: ro)
.
and similarly X(r")/(r: r'1) = X{r o)/(r: r o), w hich pro ves our assertion.
If r is finite, for instance, then we can take rl = {l} and we find
1
x(r) = 1fT
(7.1 )
Or if r = SL2(Z), then we can take r' to be the commutator subgroup
(cf. example 3 of 6). and we find
(7.2)
I
x(SL2(Z») = - 12-
These examples show that the rational number X(f) is noi generally an
integer. In particular, X(r) need not be equal to the naive Euler characteristic
i(r) defined in 5. which is always an integer. [Note that i(r) is indeed
defined ifr is offinite homological type. For we know Hj r is finitely generated
for ali i, and a transfer argument (111.10.1) shows that rkz(Hjr) = O for
i > vcd r.] This faiIure of X(I) to be integraI raises a number of interesting
questions, which we will take up in 9.
In our study of Euler characteristics we will attempt to get information
about X{r) .by studying actions oi r OD CW-complexes. For this purpose,
tbe equivariant Euler characteristic Xr(X) is a fundamenta) tool. It is definoo
as follows: Suppose X is a r -complex such that (i) every isotropy group r 

250
IX Eu]er Chat1tcteristics
is of finite homological type and (ii) X has only finitely many cells mod r;
then we set
x.{X) = L ( -1 )dimuX(r 17)'
(leI
where ti is a set of representatives for the cells of X mod r. Note that x(Il =
Xrtpt.), so tbe equivariant Euler characteristic can be viewed as a generaliza-
tion of the ordinary Euler characteristic.
We now dose this section by listing some useful properties of x( ) and
Xr< ).
(7.3) Proposition. (a) lfr is torsion-free and o/finite homological type then
X(r) = i(r) = L (_l)i dimzp(Hj(r, 7l.,»
i
for any prìme p.
(b) Ifr is offinite homological type and f' is Cl subgroup offinite index, then
X(r') = (f  f') . X(r).
(b') More generally, if Xr<X) is defmed and r' c r is a subgroup oJ finite
imlex, then Xr'(X) is defined and
Xr'(X) = (f: r). Xr(X).
(c) Suppose that XrtX) is defined and that each rq is torsion{ree. Tllen the
equivariant homology H(X) isfinitely generated and
X.-(X) = X.-(X)
where Xr(X) = Li (-li rkz(Hf(X».
(d) Let 1 - r' - r ......... r" - 1 be a short exact sequence oJ groups with r'
and r" oJ finite honwlogical type. II r is virtually torsion-free, then r is oJ I
finite homological type and
x(r) = X(r'}. x(r").
(e) Let r = r 1 . A r l' where r l' r 2' and A are oJ finite homological t ype.
Il r is virtuali y torsion{ree then r is of finite honwlogical type aniI
X(r) = X(rt) + X(r2) - X(A).
(e') More generally, suppose X is a cOl1tractible r-complex such that XrtX)
is defined.lfr is vtrtually torsion-free, then r is oJ finite homologicalrype and
x(r) = Xr< X).
PROOF. Tbe first equality of (a) is true by definition, and the second foJlows
from 5.2. (b) is immediate from the definition or x. To prove (b'), fu a celI a
or X and consider the se. ra of cells which are equivalent to (J mod r. Since
ru is in 1-1 correspondence with r/rw. it decomposes into finitely many
. 
<
'.
"
,
.'
:'
;-
,.
;.
, .
-.
:.
. ,
1
,
1 Extension to Groups with Tonioo
251
<
r' -orbits, represented by the cells ya where )' ranges over a set S or representa-
tives for rT\r jr w. Note that rCI = r' n r)'o = r' (Ì '}' r (J Y - 1  which is con-
jugate in r to y - 1 r'y n r 0'. The contribution of r (J to the sum defining
lr(X) is therefore
I (-l)dimy4'x(r' fì ,r O' 1-1) = ( _l)dimO' L X(y-I r', n r a)
}'ES )'ES
"
'-
,"
= (_l)dima I (rq: .,-lr'y n rO'). x(r..,.)
reS
"
= ( - l )dim (J(r: f') . X(r a).
(The last equality here is obtained by decomposing r'\r jnto orbits under
the right action of r 0'.) (b') nQW follows at once.
To prove (c1 consider thc equivariant homology spectral sequence
(VII.??)
'"
E = EB Hq(rcJ' Za)  H+q(X),
ae4J'
where 4" is a set or representatives for the p-cells of X mod r. Since r ti is
torsion-free and or finite homological type, we know that H.(rO') is finitely
generated. Hence H(X) is finitely generated, and we can compute Xr(X)
from the E l-term or the spectral seq uence (cf. VII.2.7). Assurning for the
moment that r CI acts triviaUy on 71..(1 for alI u. we find
i'{X) = L L (-1)"+11 rkz(Hir(f»)
p." O'EI
= I L (-l)PX(rl1)
" trelp
= Xr<X).
In the generaI case, we need only note that alI Euler cbaracteristics can be
computed from homology with 7L 2-coefficients (cf. (a) above). Since r O' acts
trivially on (1:2).", the result follows as above from the equivariant homology
spectral sequence with lrcoefficients.
(d) Let r o ç; r be a torsion-free subgroup or finite index whose image
re; in r" is torsion-free, and let fa = r o n r'. I claim that (r: r o) =
(r': Fo)' (f'J: r); for (r: r o) = (r: r'r o) . (rr o: r o), and the well-known
isomorphism laws of group theory imply that (r; r'ro) = (r": r) and
(r'r o: r o) = (f': ro). We may therefore replace the given exact sequence by
1  ro -t- r o -+ r - l. ie., we may assume that r', f, and r" are torsion-
free. Then ed r < co by VIII.2.4b, and the Hochschild-Serre spectral
sequence
E = H,,(r", H4(r'. M» => Hp+q(r, M)
shows that r is offinite homological type. Now take M = 7..2. Since H (rE, Z2)
is finite, there is a subgroup ro c: r" of finite index which acts triviaily on it.
Replacing r" by r and r by the inverse image of rc;, wc may assume that r'"

252
IX Euler Characteristics
acts trivially on H .(r'. 1:'2). Then E = Hp(r", Z2) @Zl Hq(r', Z2)' Comput-
ing Euler characteristics from this spectraJ sequence, we conclude that
X(r) = X{r'). X(r").
Fina1Jy, (e) follows from (e') appUed to the tree associated to r1 .Ar2
(cf. appendix to Chapter II), so it remains only to prove (e'). In view of (b'ì
it suffices to prove (e') when r is torsion-free. Since X is contractible, we bave
H.(r, M)  H(X, M) for any f-module M (VII.7.3), and similarly for
cohomology_ A spectral sequence argument (cf. proof of (c») then shows
that Hi(r, M) = O for i > dim X + max {ed r 11} and that H .(r, M) is finitely
generated if M is. Hence F is of finite homological type. Moreover,
X(f) = X(I) because r is torsion-free
= lr(X) because H.r  H(X)
= Xr(X) by (c). O
As an example of(e), consider SL2(l)  Z4 .zz 1L6' Then (e) gives
X(SL2(Z» = i +! - ì = --b,
in agreement with 7.2.
EXERCISES
1. The purpose of this exercise is to outJine an a]ternative method of definjng a
«rational Eu]er characteristic" under suitable finiteness hypotheses, This is due to
Bass (I 97l:i], Chiswell [l976b] and Stallìngs [unpubJished]. For any group f
and any finitely generate<! projective or-module p. we can define a ..rational rank""
P{P) [or p..{P)] by p(P) = RQ,.(P) (l) e O. [This js motivated by 4.2.] Note that
Pr'(P) = (r: r')pr{ P) ir f' s; r is a subgroup of finite index (cE 4.1).
(a) r is said to be or type FP over a if iO admits a finite projective resolution over
or. In this case we choose such a resolution P and set e(r) = I (_1)ip(Pj). Show
that e(r) is well-defined and tbat e{f') = (r: r)e(r) if r' s; f is a subgroup or
finite index.
(b) Show that any group of type VFP is of type FP over Q. [Hint: If P is a Qr-
modufe which is projcctive over or. where (r: r') < OC., then P is projective. This
can be proved by an averaging argument.]
(c) If r is of Iype JI F L. show that e(r) = x(r).
[Remark. It tS no( known whether e(r) = x(r) for arbitrary groups r of type
VFP.]
2. (a) If e(1) *" O, show that C(I). the center of r. ìs finite. [Consider the "complete
Euler characteristic" E(r) E T(or). defined by E(r) = L (-I)iRQ...(Pj). where (Pi)
is a finite projective resolution of Q over or. Note that E(T) is simply trQ...(idQ}. as
defined in excKise 2 or 2. and tbat E(r) (1) = e( O. If)' E C(r). tben we find (using
exercise 3 of 2) that E(r) = tr(idQ) = tr(y. idQ) = y' tr(ido) = ¥. E(D. Hence
e(r) = E(r)(},-l).]
(b) Deduce tbe following theorem or Gotlljeb [1965]: H Y is a finite K(r, 1)-
complex with X(Y) *- O..ben C(r) = {L}.
"-':",: ->. .:;".:.... \ .. :.: ..;. .
.':.::.......lic:. .
. _.___N_:'"" ._..___ -",
I
, ,
,

"
,.
. .
...
 Euler Characteristìcs and Number Theory
253
Remark. This proof ofGottlieb's theorem is due to StaJlings [1965b]. as reformulated
by Bass [1976].
.3. Suppose r is torsion-free and of finite homological type and let M be a r -module
which is finitely generated as an abelian group. Show that
I ( - l)i rkz(H;( r. M» = X(n . rkl'( M).
j
"
*4. If r js of finite hornological type and M is a r -module which is finite]y generated as
an abelian group. show that Hi(r. M) is finitely generated for ali i. [Hint: Reduce
to the case where M is Z-free. In this case let M- = Hom(M, Z) and note that
Xr<p. M)::::: K6m(P @r M*, Z).]
*5. Let r c SLz(Z) be a torsion-free subgroup of finite index k. Prove that r is a free
group on 1 + k/12 generators. Example: The princìpal congruence subgroup oflevel
3 has index 24 = I SLl1.3) I. hence it is free on 3 generators.
8 Euler Characteristics and Number Theory
The study of groups of integraI matrices is intimately related to algebraic
number theory. For examplc, if one tries to classify elements of order p in
G LJ7L) up to con jugacy, one quickly finds that ideaI classes of the p-th
cyclotomic fÌeld come into play. Recall now that many integraI matrix
groups are oftype VFL (cf. VIII.11). We therefore have rational numbers
x(Il associated to such groups, and it is natural to ask whether these numbers
have any number-theoretic significance. Thc remarkable answer is !hai X(f),
ror many arithmetic groups r, can be expressed in terms or values of zeta-
functions. In this section we wiU give a brief survey of results of this type. See
Serre [1971, 1979] for more dctaìls and a guide to the literature.
The starting point for the computation of x(f) is the Gauss-Bonnet
theorem of differential geometry, which says the following: Suppose Y is a
closed Riemannian manifold. Then there Is a measure p. on Y, constructed
canonically from the curvature associated to the Riemannian metric, such
that
x( Y) = 1l( Y).
[Note: J1- is not in generai a positive measure. If Y is a surface, for example,
then p. = K . dA/27C, where K is the Gaussian curvature function and dA
is the ca.nonical 'area" measure on ¥.J WC calI J1 the Gauss-Bormet measure
OD Y.
If Y is a K(r, l) and X is ìts universa] cover, then we can rewrite this
formula as
(8.1)
x(ll = p(X /r).

254
IX Eu1er Characteristics
Moreover, we ean interpret the right-hand side as [he measure of a funda-
mental domai n for the action or r on X, relative to the Gauss-Bonnet
measure OD X.
We now speciaIize to the arithmetic case. Suppose G is a semi-simple
algebraic group defined over Q and r is a lOrsìon-free arithmetie subgroup
of G(C) c G(IF!). Let K be a maximal compact subgroup of G(IR) and let
X = G(IR)/K. Since K is çompact, it is easy to see that X admits a Riemannian
metric invariant under the action of G([f;!); hence tbere is a G(II1:}-invariant
Gauss-Bonnet measure IL OD X, and 8.1 holds if r is co-compact in G(R).
U sing the compactness or K again, we can lin p, to a left-invariant measure
(stili called 11) on G(IIi) 8.1 then takes tbe form
(8.2)
x(r) = #J(r\G(IR».
The ad vantage of this forrnulation is that it remains valid even if r has tor-
sion. F or both sides of 8.2 get multiplied by (f: f') if we replace r by a torsion-
free subgroup r of finite index.
We derived 8_2 under the assumption that r was co-compact in G(IIi), and,
as we saw in VII19, this situation is far from typical. A remarkable theorem
of Harder [1971], however, says that 8.2 remains valid even if r is not co-
compact. Harder went on to explicitly calculate the measure Jl. for an inter-
esting family of groups G, the so-called "Cheval1ey groups.' [There is one
such group for every simple Lie algebra over C. Thus we have the special
linear groups SL", tbe symplectic groups SP2n, the special orthogonal groups
SO"' and five exceptional groups G2 . F., E6, Ei, and EB.]
For such G there is a canonical Haar measure (i.e., positive invariant
measure) f.la on G(). called the arithmetic measure, with the property that
Jl.a(G(IR)jG(Z» is expressible in terms of values of the zeta-function. qn the
other hand the Gauss-Bonnet measure #J, being invariant, must ha ve the
rorm J1 = CJ1lJ for some c E IR. What Harder did was to calculate the scalar c,
so that 8.2 would yield an explicit formula ror x< G(l» in tenns or values of
tbe zeta-function. We will give some examples or this f01muIa below, £ollow-
ing a brief review of the Riemann zeta-function.
Recall that CCs) is defined for Re{s) > l by
ro l J
(s) = I  = n -:;'
11= I n p prime l - P
Thus C(s) is an analytic function in the halr-plane Re(s) > I, and one extends
it to a meromorphic function in the entire complex piane. Moreover. the
function ç(s) = x-s/2r(sj2)(s) satislies the funetional equation ç(s) =
,(1 - s). We will be interested in the values of C(s) at negative integers. At
even negative integers this is trivial to compute:
C( - 2k) = O if k  1.
For ç(s) is hoIomorphic at s = 2k + l, hence also at s = 1 - (2k + 1) =
-2k. But r(sf2) has a pole at s = -2k, so '(s) must bave a zero there. At
:
;
.
,
..
-

j
1 ,
, "
,
/
;
!
1
..
8 Euler Char3cteristics and NumbeT Theory
255
".
odd negative integers, on the other hand, {(s) is non-zero and can be described
in terrns of the Bernoulli numbers whose delinition we now recall.
Tbe function zj(e2 - l) has a power series expansion of the form
Z _ l _ Z ItO B2/( 21
e% - l - 2 + k = l (2k)! z ,
where BlIe E O. The numbers B21e which arise in this way are called the
Bernoulli rlUmbers. Here are the first few of them. with their nwnerators and
denominators façtored into primes:
l
B2 = 2. 3
1
84 =-
2.'].5
8 _ 5
10-2.].11
I
B6 =
2.3.7
1
B8 =-
2.3.5
691
8u = - 2.3.5.7.13
43867
BIs =
2.3.7.19
103.2294797
B24 = -
2-3-5-7-13
-,
7
814 =
2.3
820 = _ 283.617
2.3.5.11
3617
B16 = - 2.3.5. 17
11.131.593
B22 = 2 . 3 . 23
."
]t is well-known that the Bernoulli numbers arise when one computes
(2k) for an integer k > 1:
Y(2k) = _ (_1)1e22t-1 821e 2k
1:0 (2k)! 1t.
,
.,
.,
J
.
Using the functional equation to rewrite this as a rormula for ((1- 2k), we
find that the factorial and the powers of - 1. 2 and 1t disappear; the result is
(8.3)
((1 - 2k) = _ k.
We can now state a few examples or Harder's formula for x(G(Z». First,
taking G = SL2, we have
(8.4)
X(SL2(Z) = ,( - 1).
Since C( -1) = - B2/2 = --b, this agrees witb 7.2. The group SL2 is part
of two infinite families of groups, namely, tbe speciaJ linear groups SL" and
tbe symplectic groups SP2"' [One has SPl = SL2.] Tbe generalizations of
8.4 to these ramilies are
(8.5)
"
x.{SLJ.7L» = n '(l - k)
k=2
i


256
IX Euler Cbaracteristics
and
(8.6)
"
X(SP2n(Z» = n (I - 2k).
1:=1
Of course 8.5 involves the factor ( - 2) as soon as ti > 3 so it simply says
X(SL"(Z) = o far n > 3. But 8.6 involves only non-zero values of (s) and
hence is more interesting.
For our last example we take G to be tbe exceptional Chevalley group
E7- Harders formula in this case turns out to involve, at -1. -5, -7, -9,
-11, -13 and -17, hence it involves B" for k = 2. 6, 8 lO, 12, 14, and 18.
Tbe precise formula turns out to be
69143867
Z(E7(l») = - 221 .39.52.73.11 . 13.19'
Finally, we mention that Harders formulas (such as 8.5 and 8.6) are
valid with 7L replaced by the ring of integers jn a number field F and , replaced
by the Dedekind (-function CF associated to F. And Serre has shown that one
can generalize even further, namely, to rings (!.!s or "S-integers"'. Here S
is a finite set of primes of F and e'S c F is the subring consjsting of the elements
of F which are integrai at an primes not in S. One then uses the '-function
CF.S obtained from (F by omitting from its infinite product expansion the
factors corresponding to tbe primes in S. If F = Q and S consists of a single
prime p. for example, then 'F.S<S) = (s)(l - p-S) and the generalization of
8.4 gives
(8.7)
p-I
X(SL2(l[1jp]» = {,,',s-( -1) = 12 .
9 Integrality Properties of X(r)
Let r be a group of finite homological type (e.g., a group of type V FP).
We know that x(r) = X(r) E 7L if r is torsion-free, so the non-integrality of
X(r) in generai is somehow due to the presence of torsion in r. The rest of
this chapter will be devoted to proving various precise versions ofthis vague
statement.
The most o bvious integrality statemen t that one can make in general1s
that (r: r'). X(r) E 7L if r is a torsion-.free subgroup of finite index; for
(r: r'). X(r) = X(r') = :Hr') E ?L. Consequently,
(9.1) d.X(r)EZ,
where d = gcd{(r: r): r' is torsion-free and of finite index}_ This simple
observation, in conjunction with Harder's deep results (8) already has non-
trivial applications. Far example Serre [1971] used it to prove integrality
I
,.
.'
T

:.

>
 .
',-
','
",
:.
','
'-
-
I...
, '
. -
. ,
,i.
,-
>


r
I.
,
I '
,

9 Integrality Properties or xen
257
results about values or the zeta-function of a number field. Another applica-
tion (also due to Serre) concerns the torsion in tbe exceptionaJ Chevalley
groups. It is based on:
.'
(9.2) Lell1ma. The prime divisors oJ dare precisety the primes p such that r
has p-torsion.
PROOF. Let H be a finite subgroup of r. If r' £ r is a torsion-free subgroup,
then H acts freely on r /r'  so (f = r'), if finite is divisible by I H I. Hence
IHI divides d- In particular, it follows that d is divisible by any prime p
such that r has p-torsion. Con versely. suppose p is a prime such that r has no
p-torsion. Let r o c: r be a tocsion-free norma! subgroup or finite index
and let r' be an intermediate subgroup (r o c: r c f) such that r' /r o is a
p.-Sylow subgroup of r/ro. Then (r': ro) is a power of p and (r: r') is
relatively prime to p. By the first part of the proof, any finite subgroup of r'
must be a p-group, so r' is torsion-free. But then di (r = r), so p ,r d. D
'.
Looking at formula 8.7 for X(E7(Z», for example, we conclude from 9.1
and 9.2 that E7(l) must bave p-torsion for p = 2,3, 5, 7, 11, 13 19.
The proof or 9.2 shows that d is divisible by the order of every finite
subgroup or r, hence d is divisible by the least common multiple m of the
orders of the finite subgroups. Moreover, it is cIear from 9.2 that d and m
have the same prime divisors. But in generaI d -1= m. If r = SL3(Z), for
-instance it can be shown that d = 24 . 3 w hereas m = 23 . 3.
- .
It is reasonable to ask, then, whether 9.1 can be improved to the statement
m . X(r) E 71... This question was raised by Serre, and it turns out to have an
affirmative answer:
(93) Tbeorem (Brown [1974]). Let r be a group oJ finite homological type
and let m be the least common multiple ofthe orders ofthefinite subgroups ofr.
Then m . X(r) E ?L. Consequently, if a prime power pO divides the denominator
oJ X(r), then r has a subgroup of order p".
The second assertion follows easily from the first and the Sylow theorems.
The proof of the fust assertion requires some topological ideas involving
finite group actions; it will be given in the next section.
Using 9.3 one can improve Serre's integrality results on the values or the
zeta function (cf. Brown [1974], 9). One can also obtain good estimates of
the amount of torsion in some of the exceptional Chevalley groups. For
instance, 9.3 and 8.7 show that E7(Z) has subgroups of order 221, 39, etc.
See Serre [1979] for more information on this. A third application is the
fOllowing improvement of 6.4:
(94) CoroUary. Ler. l -+ r -+ E --t> G -+ l he a gr0l4p exeension such that
r is lorsion-{ree and of finite homological type and G is a p-group for some
prime p. If P ...r Xer), then the extension splits.

258
IX Su1er Charactcristics
PROOF. We have X(E) = x(r)/I G I. and this fraction is in lowest terms since
p % X(r). Theorem 9.3 therefore implies that r has a finite su bgroup ti
with I G I = I G I. Since r is torsion-free, Ci maps isomorphical1y to G and
provides a splitting. D
lO Proof or Theorem 9.3; Finite Group Actions
We begin by interpreting Theorem 93 topologically. Since vcd r < oo we
know from VII!. l 1.1 that there is a finite dimensìonal contractible r-complex
X with finite isotropy groups. Moreover, I claìm that X eao be taken to be an
admissible r -complex, in the sense that the following condition bolds: F or
each cell U of X, the isotropy group rl1 fixes a pointwise. T o see this, we need
only recall that X can be taken to be simplicial (VIII. 1 1.2); passing to the
barycentric subdivision if necessary, we can assume that X is an ordered
simplicial eomplex and that r is order-preserving. Then c1early no elernent
of r can permute tbe vertices or a simplex non-trivially so the admissibility
condition holds.
Let r' c: r be a torsion-free norma 1 subgroup or finite index. Then r
acts freely on X and the quotient Y = X/f' is a K(r', l). Thus
i(r') X(Y)
x(r) = (r: r) = (r: r')'
where X(Y) = L (-IY rkz(Hi Y). What we are trying to prove. then, is that
m
.v(Y)e7L
(r: r') lo ,
where m = Icm { I H I : H is a finite subgroup of r}. In other wOfds we want
to prove that
k I X(Y),
where k is the integer (r: r')!m.
Let G be the finite quotient rjf'. Note that the action of r on X induces
an action or G OD Y. Moreover, I claim that tbe isotropy group Gt of any celi
"t of Y is simply n(f a), where (J is any celJ or X Iyìng over T and 1t: r -+ G
is the quotient map. For suppose g7: = 't and choose Y E r such that x(y) = g.
Then rO' and a both lie over T so y'yu = a for some y' E r'. Since n(y'y) = g,
tbis shows that Gt c n(ru). The opposi te indusion is trivially true, whence
the claim. (Incidental1y, 1t actually maps r q isomorphicaUy onto Gr; for
r ti n r' is finite and torsion-free, hence trivial.)
Two facts follow from this. First, the action or G on Y is adrnissible..
Second, every isotropy group G t has order dividing  hence the cardinality
I G III Gt I of the orbit or T is divisible by k = I G l/m
,
..
.
..
"
..
, .
"
T
I
,
,
,
,
.'
IO Prooe of Theorem 9.3; Finite Group Aclions
259
Theorem 9.3 now follows from:
(10.1) TheoreDl. Let G be a finite gTOUp and let Y be a finite dimensional
admissible G-complex such that H* Y is finitel}' generoled. r k is an integer
which dìvides Ole cardinalit)! of every G-orbit, then k I x( Y).
Note that it does not matter whether we interpret G-orbit to mean G-
orbit or cells or G-orbit or points; for the adrnissibility condition implìes
that GT = Gr Cor any interior point y of T.
Note also that the theorem is trivial1y true if Y is finite, since l.(Y) can
then be computed by counting ceIls. Tbe point of 10.1 is that it is true under
fairly mild homological finiteness hypotheses on Y. In fact. one can even prove
a version of 10.1 for spaces Y which are not CW..complexes (cf. Brown [1979],
Theorem 7.4). bu t the proof then requìres more machinery than is needed
for 10.1.
PROOF OF 10.1. For any subgroup H c G, let YH = {y E Y: Gy = H}. The
admissibility condition implies that YH is a union or (open) cells namely,
tbase cells T such that Gt = H. Note that g. 18 = 1;H9-1 for any 9 E G. In
particular YH ìs stable under tbe action of the normalizer N(H) or H in G.
Moreover, N(H)jH acts freely OD YH. We can now easily explain the idea ofthe
proof. Y is the disjoint union (set-theoretically) or the subspaces YH, sO
one might expect
(*)
x(y) = I x(Y,,}.
H£G
,
.
Now Y,gH9-1 = g. YH  YH so we can group together the terrns in (*)
corresponding to conjugate subgroups H, and we obtain
(**)
x(y) = I (G: N(H)}x(YH),
HE""
where q; is a set or representatives for the conjugacy classes of subgroups of G
which oceur as isotropy groups in Y. [W e ha ve used the elementary fact tbat
(G: N(H» is the number or conjugates of H in G.] Since N(H)/H acts freely
on YH. 5.5 should now impJy that X(YH) Is divisible by (N(H): H), so that
(G: N(H))x(Yn) is divisible by (G: H). But (G: H) is the cardinality of an
orbit, so every term on the fight-hand side of (**) is divisible hy k; hence
k I x( Y).
We now must clarify and justiey the steps in this "proof: In the fust
pIace, YH is not in generai a subcomplex or Y, since it need not be closed. (A
boundary point of a celi can have a bigger isotropy group than the interior
points.) But each fixed-point set yH is a subcomplex. and hence YH is a
difference of subcomplexes:
v _ YH Y>H
AH - - ,

260
IX Euler Cha.ra.cleristics
wherc y>H = UH'CjH yn'. Lets assume temporarily that each yR has finitely
generated homology. Then the same is true of y>H by a Mayer-Vietoris
argument, since the family {yH': H'  H} is closed under intersection.
Hence H*(yH, y>H) is finitely generated, and I daim that (...) becomes tfue
ifwe replace X(YH) by X(yH, y>H).
To prove this let H1..... HII be the subgroups of G ordered so that
IHtl > I Hi+ 11. We filter Y by subcomplexes
0= Yoc Y1 c...c = Y,
defined inductively by
Yi = li-l U yHi.
Note that 1';-1 n yHt = y>Hi. We therefore have an excision isomorphism
H*(, -l)  HiyHi, y>H.).
[Note that, because we are dealing with subcomplexes, there is no problem
justifying the excisioD. Indeed, we already have an excision isomoTpbism OD
the level of cellular chain complexes.] Hence
l'!
X(Y) = L x(, -1)
i-l
l't
= L X(yHi, y>HI),
1"'1
as claimed.
The remainder of eur initial heuristic proof is now valid, with YH re-
placd everyw here by (yH. y:> H). One needs, of course, to apply a relative
verSl0n or 5.5 to N(H)j H acting OD (yH, y:;.. H) bu t tbis presents no probJem-
simply appl y 5.3 to the celI ular chain complex C( yH. Y > H) which is a
finite dimensional complex of free Z[ N (H)/ H]-modules.
If we now drop the assumption that H...( yH) is finitely generated for
each H. then we can argue as follows. Suppose first that G is a p-group for
some prime p. Since H. Y is finitely generated, we can compute x( Y) from
H.(Y; Zp). We now repeat the argument above using H.( ; 'Z,,) and using
5.7 instead of 5.3. This time, however, we have a theorem (VII.I0.5 and
exercise 2 or VII.l O) which sa ys that H.( Y H  "Zp) is finitely generated, so we
do not need to assume this.
Finally, we can easi]y reduce the generaI case to the p-group case: To
prove 10.1, it suffices to show that if pii is a prime power w hich divides k
then p"lx(Y). Let G(p) be a p-Sylow subgroupofG. By hypothesis p"I(G: G..)
= tGI/IGrl for any ceJl T or Y. Looking at thc p-part or jGj and IGTI, we
conclude that pQ divides I G(p) 1/1 PI. where P is any p-subgroup of G'f' In
particular, p" divides I G(P)1I1 G(P)TI, which is the cardinality of a typical
G(p )-orbit. It nQW follows from tbe p.-group case that plt I x( Y). D
I
"
"
.'
o'.
.1
Il Tbc: Fractional Part or 1(1)
261
EXERCISE
Use the method ofproof or 10.1 (O prove the following.. Lefscnetz fixed-point theorern ":
Let G be a finite group and Yan adrnissible finite dimensiona] G-complex. Assume that
each fixed-point set yg (g e G) has finiteJy generated homology. Prove that the Lefschetz
number L(g} is given by
L(g) = X(Y").
[Hinl: You may assume G is cyclic. generated by g. Then g is non-trivial in N(H)jH
= G/H for every H ¥ G, so L(gl YII) = O by exercise l of 5. Now use the Lefschetz
number analogue of ( ..) and note that every term is zero except the one corresponding
to H == G.]
Remark. One can use this theorem IO prove tbe foUowing theorem about Euler charac-
teristics or groups (cf. Brown [1982]): Let r be a group such that the centralizer
C( '}') is or finite homological type for every y E r of finite order (including")! = l). Let 
be a seI of representatives for the conjugacy cJasses of elements of finite arder. Then 'I
is finite, and
X(T) = L x(C(y).
.."
Consequently, we have the foUowing formula, which expresses the deviation between
x(r) and i(l) in terms of the torsion in r:
x(r) = X(r) +  X(C(y»,
)'010'11"
where C{j' = re - {l}.
Il The Fractional Part of X(r)
U nder suitable hypotheses OD r it is possible to deduce from the proof of 9.3
precise formulas expressing Xer) - X(r) in terms of the torsion in r. Since
i(r) E 7L such a formula çan be regarded as a formula for the .. Cractional
part" of the rational n umber X(r). One formula of this t ype was j ust written
down, following the exercise of IO. Another formula which can be found in
Brown [1974], has the form
. (r) - - r) '" c(H)
l.. - X( + 7t I HI '
where H ranges over the non-trivial finite subgroups of r (up to conjugacy)
and c(H) is an integcr whose definition is too complicated to reat here.
Rather than state and prove this formula, we will use similar niethods to
prove some formulas which are less precise but easier to prove and apply.
If 8 Is a partially ordered set, then there is an obvious way to construct
from S an ordered simplicial complex I S l. Namely. thc vertices of 1.8 l are

262
IX Euler Characteristies
the elements Or s and the simplices of I S I afe the linearly ordered finite
subses So < .... < Sn o s. [_ e ha ve already seen one example or this con-
structlon: If K 18 any slmpbcml compIe:x and S is thc se. or simplices of K,
ordered by the race relation. then I S I is simply the barycentric subdivision
of K. c[ VIIl.ll_] If r acts on S (and preserves the ordering), then there is
an induced action or r on ISI, and we set
xr<S) = xr<ISI)
ir the equivariant Euler characteristic 00 the right-hand side is defined. More
generally, we will use tbe functor S '-'ISI to assign arbitrary topological
concepts to partially ordered sets. For instance we wiJI write C (S) and
H .(5) instead of C.( I SI) and H...( I SI), and we will say that S is conractible
or acyclic if I S I is.
(11.1) Theorem. Let r be a group oJ finite homological type. Assume that r
has only finitely many finite subgroups (up to conjugacy) and that the normalizer
N(H) of eachfinite subgroup H hasfinite homological type. Let S be the set oJ
non-trivial finite subgroups of r, regarded as a partially ordered set under
indusion, wirh r-action by conjugation. Then Xr(S) is defined and
x(r) = Xr(S) mod?L.
(In concrete terms, this gives the fractional part or x(r) in terms ol the
Euler characteristics of subgroups or r or the form N(Ho) n . . - n N(H,,),
where Ho c ... C HIt is a chain ofnon-trivial finite subgroups.)
PROOF. T o show Xr<S) is defined. we must show that I S J has only finitely many
simplices Cf mod r and that eacb r a has finite homological type. Let !F be a
(finite) set or representatives for tbe conjugacy classes offinite subgroups ofr.
Then any simplex H o c ... c H Il or I S I is equi valen t mod r to one such
that H n E ff. But there are clearly only finitely many such simplices, since
Hn is finite. To prove the assertion about tbe isotropy groups, note that
N(H) for any HES peTmutes the simplices Ho c ... C HIt with H" = H.
Since there are only finitely many such simplices, il follows that each such
simplex a has (N (H): N(H)a) < 00, hence N(H)" is of finite homological
type. But N(H)" = r (l' so tbe first part of the theorem is proved.
Now let X, r', G, and Y be as in the proof of Theorem 9.3. By VIII. I 1.2
we may assume that XH is contractible for each H E S. Let X o be the part of
X where the action or r is not free, i.e.
Xo = U xH.
HS
and let Yo = X o Ir' c Y. The analysis of the isotropy groups G'( that we
gave in the proof of 9_3 shows that Yo is the part of Y where the G-action is
not free. [Yo is in fact equal to the subcomplex called -1 in the peoof of
10.1.]
J
.
Il Tbc Fractional Part or x(T)
263
i
I claim now that H.( Yo) is finitely generated. and hence so is H.( Y. Yo).
Accepting this ror the moment, wc can apply 5.3 to C(Y, Yo) since G acts
rreely in Y - }Q. It follows that x( Y. Yo) is divisible by 1 G I. Thus
'.
x(y)
X(r) = IGI
X( yo) X( y. yo)
= IGr + IGI
X(Yo)
= ICI mod Z.
T o complete tbe proof, then, we will prove that H.( Yo) is finitely generated
and that x(Yo)/1 GI = Xr(S). Note first that H .(Yo)  H'(X o) since r' acts
rreely on Xo (cf. VII.7.8). Next, we apply the following observation to Xo:
.
.-
(112) Lemma. Let Z be an admissible proper r-complex such that each r(l'
is non-tr'ivial and eacll ZH (H E S) is acyclic. where S is the set DJ non-rrivial
finite subgroups ofr- Then. Z is homological'y equivalene to S, in thefollowing
sense: There is a chain complex C of r -modules which admits r -maps
C(Z) +- C -+ C(S) inducing homology isomorphisms.
.
(Note: By "acyclic" we mean 'having the homology of a point." In
particular. acyclic => non-empty.)
The proof is a straightforward analogue of that of VII.4.4. Details will
be given in tbe next SectiOD. [The intuitive idea is as follows: We have
Z = UHeS ZH, and tbe order relation on tbe index sei S completely deter;..
mines the intersection pattern of the acyc1ic complexes ZH. Namely,
ZHl n ZH] =1= 0 if and only if H1 and H2 have a least upper bound H in S,
in which case ZHl (ì ZHl = ZH. It is not surprising, then that we can use
I S I instead of the nerve of the covering in VII.4.4.]
Retuming to the proof of 11.1, we conclude (cf. VII.5.2) that H.(Yo) 
H'(X(J)  H'(S). In view of 7.3c and b', it follows that H.(Yo) is finitely
generated and that
x(yo) = Xr'(S) = Xr<S).
IGI 101
D
With very little effort, we can generalize 11.1 to the following result:
(113) Proposition. LeI r ami S be o.s in Il.1. Let Z be an admissible r -complex
sudi that Xr<Z) is defined. 1f ZH is acyclic lor each H E S. then
Xr{Z) - Xr(S) mod 7i.
(This reduces to 11.1 if Z is a point.)

264
IX Euter Chacacteristics
PROOF. Let X and rr be as above and apply the proor of Il.1 with X replaced
by X x Z. Note that the latter is finite dimensional. admissible, and proper,
and that (X x Z)H = XH X ZH is acyclic (or each H E S. Moreover,
H.((X x Z)/r')  H'(X x Z)  H'(Z)
since X is contractible. We therefore obtain. as in the proof or 11.1,
Xr.{Z)
l..{Z) = Cf: r)
X«X x Z)jr')
(r: r')
 X«(X x Z)o/r') od 7L
(r: f') m
= Xr<S).
D
Combining 11.1 and 11.3. we see that
(11.4)
X(f) - Xr<Z) mod Z
under the hypotheses of 11.3.
Finally. we will need the rollowing  IoeaI" version of 11.4, for which the
hypotheses on r can be weakened.
For any prime p let Z(p) = {a/b: a, b E Z p.r b}.
(11.5) Proposition. Let f be a group oJ finite honwlogical type and let Z be
an admissible r -complex such that Xr<Z) is defined. lf p is a prime such tlzat ZH
is acyclic for every non-t rivia l finite p-subgroup of r, then
x(r) = X.-(Z) mod Z(p).
PIlOOF. We contin ue to use t he notation or the proof of 11.1. Assume first [hat
r /r' is a p-group. Tben every finite subgroup of r is a p-group, so our hypo-
tbesis says that ZH is acyclicfor every H E S. We now repeat the proofs or 11.1
and Il.3 but taking alI homology groups to ha ve Zp-coefficients. We no
longer have hypotheses to guarantee that X.-(S) is defined, but instead we can
use VII.I0.5 as in the proof of 10.1 to cònclude that H.(Ya; 7l..p) is finiteIy
generated. Arguing as in the proof or 11.1 (and using 5.7 instead or 5.3), wc
find that H.(S; lLp) is finitely generated and that
i r'( S)
x(r) == (r: r') mod Z.
,
.-
.
12 Acyclic Covers; Proof or Lemma Il.2
265
where tbe equivariant Euler characteristic on the right is now understood to
be defined in tenns or Hr(S; 7L.p). Similarly.
Xr.(S)
Xr(Z) = (r: r) mod 7l.,
so
X(f) = Xr(Z) mod?L.
In thc generai case. where r/r is not necessarily a p-group choose a
subgroup r p ;2 r such that r p!r' is a p-Sylow subgroup or r /r'. B what
we ha ve just proved,
x{r p) = lr p(Z) mod ?L.
so that
(r: r p). X(r) = (r: r p). Xr<Z) mod ?L.
Since (f: r p) is relatively prime to p, this proves tbe proposition.
Knowing X(f) mod l(p) is equivalent to knowìng the J' p6fractional pat"
of X(r). i.e., that part of the partial fractions decomposition of X(f) wlth
denominator a power of p. Thus 11.5 says that x([} and Xr<Z) have the same
p-fractional parto We will give some substance to this result in  13 by giving
an interesting example of a complex Z satisfying the hypotheses or Il.5.
D
12 Acyclic Covers; Proof or Lemma 112
The proof of Lemma 11.2 will be based on the following variant ofVII.4.4:
(121) Proposition. Let X be a CW-complex and (X.:Jses a family of sub-
complexes indexed by a partially ordered set S. Assume that Xs :) Xt whenever
.5 < ,. Far each celi e of X let Se = {S E S: e c X,f}' 11 each subcomplex Xs is
acyclic and each partially ordered set Se is acyclic, then H .(X)  H.(S).
More precisely, there is a chain complex C, constructed cano'1ically from the
given data, which admits canonical weak equivalences C(X) - C -+ C(S).
PROOF. We will construct a double complex analogous to that used in VI1.4.
Let l S I(p) be the set or p-simplices of I S I. For eacb simplex o- = (so < ... < sp)
in ISI(p) let Xf1 = XI" Then Xq c Xt if 1" is a race or G. In partieular. we
ha ve inc1usion maps
Oi: C(X.,.) -+ C(X bi(1) (i = 0, . - . . p),
where Vi (J = (so < . . . < Si < ... < S,} Let
C p = E8 C(X (,.).
6Elsl(P)

266
IX Eukr Characteristics
Letting a = L {-IYòj: Cp --+ Cp-l we obtain a chain compie x
(
'..--+C ....:..c 1--+...
p p-
in the category of chain complexes, and hence a dou ble complex. The
acyclicìty of each X à implies that one or the spectral sequences of the doub]e
complex collapses yielding
H .(C)  H ...(5),
where C is the total compJex associated to the double complex. On the
other hand exactly as in VIl.4 the acyclicity of each Se jmplies that the
othec spectraI sequence alsa colla pses and yields
H.(C)  H.(X).
It is easy to check that these isomorphisrns are induced by canonical chain
maps C(X) *- C -+ C(S), cc. exercise 1 of VII.4. D
We wiIl also need the following simple o bservation:
(12.2) Lemma.lfS is (l partially ordered sef with a iargest or smallest element,
then S is contractible.
PROOF. If s e S is the extreme element and S' = S - {s}, then I SI is the cone
over I S' I, wìth s as the cone vertex. O
.PROOF OF LEMMA 11.2. We have Z = UHeS ZH. where S is the set or oon-
trivial finite subgroups of r. For each celI e or Z. the set Se defined in 12.1 is
simply the set or alI non-trivial finite subgroups of the isotropy group re-
Since re is a non-trivial finite group, Se has a largest element and hence is
contractible hy 12.2. The hypotheses of 12.1 are there£ore satisfied and we
obtain a compJex C and weak equivalences C(Z) - C -+ C(S). Since r acts
OD Z and permutes the su bcomplexes ZH according to the conjugation
action of r 00 S, it is clear that the complex C inherits a r -action and that the
weak equivalences above are compatible with the r -action. O
13 Thc p-Fractional Part of X(r)
Theorem 11.1 is not very useful in practice because the computation of Xr<S)
tends to be extremely tediolls. If we fix a prime p, however. and replace S
by thc set ofnon-trivial finite subgroups which are p-groups, then the compu-
tation often simplifies drastically. Moreover, it turns out that this gives the
p-fractional part of X(r). In fact, we can even compute the latter by looking
l
.
13 The p-Fractional Part of x(D
267
only at the elementaryabelian p--subgrou.ps of r, i.e. the subgroups isomorphic
to a finite direct product of copies of ?Lp:
(13.1) Theorem. Let r be a group QI jirzite Jwnwlogical type and let si be the
partiall y ordered set DJ non-trivial elementarJl abelian p-subgroups oJ r,
where p is a fixed prime. Assume that the normalizer N(A) is of finite homo-
logical type for every A E si. Then Xr<d) is defined and
X(r) - Xr< d) mod Z(I1)'
(This is an improvement due to Quillen [1978] of the main resuIt of
Brown [1975b].)
The proor will use:
(13.2) Lemma. Lei r be a group such that vcd r < 00 and let p be a prime. If
r has a torsion-free norma l subgroup r' offinite iptdex such that H .(r', Zp) is
finitely generated, then r has only finitely many p-subgroups, up to conjugacy.
PRODF. Let X be a finite dimensional, contractible, proper, admissible r-
complex such that XH '# 0 for every finite H c r. Let Y = Xjr' and let
G = r{r'. By hypothesis H.(Y; Zp) is finitely generated, so we know that
H.( yP; 7L p) is finitely generated for every p-subgroup p c G (VII.1 O,
Theorem 10.5 and exercise 2). In particular yP has only finitely many con-
nected components.
On the other hand, I claim that
",
yP = U XH /(r' r. N(H».
He
";
where  is a set of representatives for the r'-conjugacy classes or finite
subgroups or r whose image in G is P. To see tbis.. consider the inverse
- - .
image Y p of Y p in X. and let C{j be the set of finite subgroups of r w hase Image
in G is P. For any x E ¥P. we know from IO that the isotropy group rx
maps isomorphically to G c G, where Y E Y is the image or x. Since P ç; Gy,
it follows that thece is a unique H c: rx whose image in G is P. In other
words, there is a unique H e ij such that x E XH. Thus ? = U Heri]:H.
Since r' permutes the XH according to its conjugation action on , it
folIows that
,
yP = (li xH)/r' = il XHj(r' n N(H»,
HEi He
as claimed.
Since yP has only finitely many components and each XH is non-empty.
we conclude that  is finite, i.e., that the finite subgroups of r lying over P
CalI into finitely many conjugacy cIasses. Finally, there are only finitely
many possibilities for p. so the lemma follows. O

268
IX Euler Characteristics
We alsa need:
(13.3) Lemma. Ler S and T be partiall y ordered sets and let /0' Il : S --+ T be
order-preserving maps such that lo(s) < Il (s) for ali s E S. Tlten I fo I 
Ifll: 151--+ I TI.
[Note that this reduces to 12.2 if S = T and either fo or il is ids while
tbe other is a constant map.]
PROOF. Heuristically, the desired homotopy moves fo(s) to fl(S) along the l
simplex {fo{s), fl(S)} of I TI_ Unfortunately, this only defines the homotopy
on vertices of I SI, so we instead proceed as follows. Let J be the ordered set
{O, l} with O < 1. Note that IJ I is the unit interval. Consider S )( J with the
product ordering: (s, i)  (s', i') if and only if s < s and i <i'. Let F: S x J
--+ Tbe defined by F(s, i) = fr.(s). Thc hypothesis thatfo(s) S 11(S) guarantees
that F is order -preserving, so we ha ve an induced map I F I : I S x J I -- I T I.
Now it is immediate from tbe definitions that I SI x 821 = l SII x 1821 for
any two partially ordered sets SI' S2' where the product on the right is the
simplicial product defined in VIII.l1. Thus I F I can be viewed as a map
I S I x I J I -+ I TI t and this is the required homotopy. D
PROOF OF THEOREM 13.1. By Lemma 13.2, r has only finitely many p-
subgroups, up to conjugacy. It now folIows exactly as in the first part of tbe
proof of 11.1 that Xr<d) is defined. We now wish to apply 11.5 to the r-
complex I d J, so we must verify that 1.sI IH is acyc1ic for each non-trivial
p-subgroup H s; r. Now I d 18 = I.JJIH I, where NH is the set of non-trivial
elementary abelian p-gronps normalized by H, and I claim that this is in
fact contractible. F or let C  H be a central subgroup or order p, and con-
sider fOf any A E dH the chain of inequalities
A > AC s C. AC > C
in slH. (Note that AC, C, and C . AC are indeed alI in dH; in tbe case of AC,
this follows from VI.8.1.) This yields, via 13.3, a homotopy from the identity
map or slH to tbe constant map A 1-+ C, since thc interrnediate maps A ...... AC
and A f--+ C . AC are order-preserving maps .si -+ si. Thus 11.5 is applicable
and yields the desired congruence. D
In tbe next section wc will show that Xr<.9I) adrnits an explicit computation
in terrns of tbe normalizers of the elements A Ed. F or the moment, we just
mention one case where this is o bvious. Suppose that every A E .si is or rank
< L [This is equivalent to saying that the finite subgroups or r have p-
periodic cohomology, cf. VI.9.7. It is also equivalent, as we will see in the
next chapter, to saying that r itseIf has p-periodic cohomology in high
dimensions.] In this case Idl is discrete and we obviously have
Xr(d) = L x(N(P»,
PE'
I
, i.
I
I .
..
"
"
.
",
"
t
13 The p-FractionaI Part of X<r)
269
where 9 is a set of representatives for the conjugacy classes or subgroups of
r of order p. Thus
(13.4)
x(r) === I x(N(P» mod Z(P)
Pe9'
in this case.
F or computational purposes. it is often more convenient to work with
elements of order p and their centralizers, rather than subgroups of order p
and their normalizers. It is easy (O rewrite 13.4 in these tenns:
(13.5) Corollary. Le, r be a group offinite homolog;cal type ami assume that
every elementa' y abelian p-subgroup oJ r has rank  l. Suppose further
that lite centralizer C(}') is of finite homological t ype for every y E r oforder p,
and let <il be a set 01 representatives [or the conjugacy classes oJ elements ofr
of order p- Thel'l q: is finite llnd
l
X(r) = l L x(C(y» mod Z(p).
p - re'"
PROOF. F or each P E f!jJ choose a set  p of representatives for the non-triviaJ
elements of P under the conjugation action of N(P)JC(P). Then U PrdJJ 'IJ p
is a set of representatives for the conjugacy classes of elements of r of order p,
so we may assume that ct is equa l to this unioo_ Since C(P) = Cb) for every
nontrivial Y E P, the action of N(P)jC(P) on P - {l} is rree. Thus
card(ct p) . (N(P): C(P)) = p-L
Consequently,
I x(C(y» = L card( p) . xeC(p»)
}'V' PED'
= L card(çc p) - (N(P): C(P» - X(N(P»
Pe
= (p - 1) L x(N(P»).
Pe
The corollary now rollows Crom 13.4.
D
We mention briefly one interesting application or 13.5 to number theory_
(This application is due to Serre, and a detailed treatment is given in Brown
[1974, 1975b].) Fìx an odd prime p and let r be the symplectic group Sp2"flL),
where 2n = p - l.
On the one hand. wc ha ve Harder's formula (8.6) which gives X(r) in
terrns of ( -I), .. +, (1 - Z,z)t and hence in terrns of the Bernoulli numbers
B2,... t B,,-1- Using known facts aboul Bernoulli numbers e'von Staudt's
Theorem n) one concludes that X(r) = O mod Z(I') if and only if p divides the
numerator of one of tbe numbers B2, . . . t B,,_ 3-

270
IX Eukr Charact.eristics
On the other hand, it is not hard to classify the elernents or r of order p
up to conjugacy and to compute their centralizers. ODe finds (i) that the
hypotheses of 13.5 afe satisfied; (ii) that the number of conjugacy classes of
elements of order p is c10sely related to the class number h of tbe p-th cyclo-
tornie number field; and (iii) that the centralizer of any element y of order p
is finite and of order 2p, so that X(C(y» = l/lp.lt then follows from 13.5 that
X(r) 0= O mod Z(P) if and only if h/2p =:: O mod l(p)? i.e., if and only ir p I h.
(Tbe prime p is then said to be irregular.)
Combining tbe results of tbe last two paragrapbs, we ha ve: A prime p is
jrregular if and only if p divides tbe numerator of one or tbe Bemoulli
numbers B 2, . . . , B,,_ 3' This resnlt is oot new; indeed, it is known as Kummer's
criterion for regularity. But our method or proof via Euler characteristics
generalizes to rings of integers other than li. and yields number -theoretic
results which were noi previousl y known.
14 A Formula for Xr(d)
Tbe formula to be given in this section is essentially due to Quillen [private
communication]. It is based on:
(14.1) Proposition. Let V be a vector space oJ dimension r > l over the field
IF q witll q elements, and let S = S( V) be the partially ordered seI of non-trivial
proper subspaces oJ Il: Then
{z.,(')
Hi(S)  O
ifi=r-2
ifi-;;j.r-2,
where n(r) = q,(r- 1)/2.
(This is a special çase of the so-called "Solomon- Tits theorem." The
proofthat we will give is due to Quillen [1973]. It will follow easily from the
proof that I S I in fact has the homotopy type of a bouquet of n(r) spheres of
dimension r - 2, but we will not need to know this.)
PROOf. The proposition is trivially true if r = l, sioce S is then empty. Assume
now that r > l and that the proposition is known for vector spaces of
dimension r - 1. Let L c: V be a fixed l-dimensional subspace and let
S' c:: S be the set of non.trivial subspaces V' c V such that V' + L =I- V (i.e.,
such that V' + L E S). F or an y V' E S', we ha ve relations
V' < V' + L > L
in S', and it follows easily that S' is contcactible (cf. proof of 13.1). Thus
HI..S)  H I..S» S').
I
,.
.
.,
14 A Formula for Xr<:d)
271
I
Let Jff be tbe set of byperplanes H in V which are complementary to L
(i.e,? such that V = L  H). Let S(H) (resp. S(H» be the set of non-trivial
subspaces (resp. non-trivial proper subspaces) of H. Note that every simplex
II of I S I which is not in I S' I has tbe fonn
Vo < .. . < ,
where Jt;, E K. Hence (J is a simplex or IS(H) I but not 18(H) I for a unique
H E.K. Consequently? tbere is an excision isomorphism (already on the
level of simplicial chain complexes)
H.(S, S')::- E8 H.(S(H)? S(H».
He.K
Now S(H) is contractible since it has a largest element, so
-,
_ {lLn{r-l)
Hi(S(H) S(H)  Hi-1(S(H»)  O
i£i=r-2
ifi:#:r-2
by the induction hypothesis. Thus
{ZCard(A").n('- I)
it(S)  O
To compute card(K), note that tbe
spJittings or the exact sequence
o -+ L -t V -t VJ L -+ O,
ifi=r-2
ifir-2.
elements or ff' correspond to
so tbere is a bijection :K  HomF (V / L, L). The latter being a vector space
'l
of dimension r - I, it follows that card(Jf) = q-l. One now completes
the proof by verirying that the function n(r) = qr(;-I)/2 satisfies n(r) =
q;-ln(r - l). D
(14.2) Theorem. Let r be a group and p a prime such that Xr<'w) is defined,
where .!li is the set oJ non-trivial elementary abelian p-subgroups oJ r. Let
do be a sel oJ represenlatives for .91 mod r. Then
xr< = L (-ly(A)....Ip"(A)X(N(A)),
A. edn
where r(A) is the rank oJ A and n(A) = r(A)(r(A) - 1)/2. Consequently,
X(f) =:; L (-1 y(A) - l plt(A)X(N(A» mod Z(p)
Auto
under the hypotheses of 13.1.
PROOF. For each A e .91, let S(A)(resp. S(A» be tbe set of non-trivial subgroups
(resp. non-trivial proper subgroups) of A. Let .9'(A) be a set of representatives
mod N(A) for tbe simplices or I S(A) I which are not in I S(A) I. Then

272
IX Euler CharRClcrislics
UAE'....O 9(A) is a set of representatives ror tbe simplices or 1.911 mod r. lt
therefore follows from the definition of xr(d) that
Xr(d) = L XN(A)(S(A), S(A)),
AE40
where the relative equivariant Euler characteristic on the right is equal, by
definition, to LuE.9'CA) (-l)dimax(N(A)47)'
I c1aim that XN(JI)(S(A) S(A» = X(N(A». X(S(A), S(A». For let C  N(A)
be a torsion-free subgroup or finite index which centralizes A. Then C
acts triviaI1y on S(A). so we bave a spectral sequence (cf. VII.7.2)
E = H p(C, Q) @o Hq(S(A). S(A); Q) =- +q(S(A), S(A); Q).
Therefore idS(A), S(A» = i(C), x(S(A), S(A». The claim now follows at
once from 7.3a and b and from relative versions of7.3b' and c.
Thus
X-rlsFJ = I X(N(A»). x(S(A), S(A»).
A,p/o
Now A can be regarded as a vector space of dimension T = r(A) over IF 11'
so we can apply 14.1. Since S(A) is contractible, we conclude that
x(S(A1 S(A» = (_I)"-Ip'"(r- 0/2, whence the theorem. D
Remark. Let p" be the maximal order of a p-subgroup of r. Then the
denominators of the X(N(A» which occur in the theorem involve p to at
most the a-th power by 9.3. Sinee r(r - 1 )/2 > 1 if r  2, it follows from
14.2 that
x(f) == 1: XeN(p» mod p-(a-l)Z(p),
PE!J
where f?I is a set of representatives for the subgroups of r of order p. Thus
we can compute the "leading term n of the partial fractions decomposition '
of x(r) (i.e.. the term involving pa in the denominator) by considering only
the subgroups of order p and their normalizecs.
EXERcrSl:S
l. Extraet from the proof or 14.2 the following ract; Let S be a partialty ordered r-set
such that Xr<S) is defined. F or each S E S. Jet S <-  (resp. S <- s) be the set or t e S such that
f < s (resp. l < s). Ir each S 5 s is finite then
xS) = L x(rJ. z(S 9' s-< ..),
5
where s ranges over a set or representatives for S rnod r.
2. Show that the hypothesis that each S <: S is finite in exercìse l can be replaced by the
weaker hypothesis that Il.<S<.J is finite]y generated for eaoh s. [Hint: r;s has a
subgroup of finite ìndex which acts trivialJy OD H ...(S :s 8' S<;:s; Zz).]
CHAPTER X
Farrell Cohomology Theory
1 Introduction
.
Let r be a group such that vcd r < 00. If r is torsion-free, then we know
from Chapter VIII that ed r < 00, so that r has no high-dimensional
cohomology. In tbe generai case, one might hope to ., explain" the high-
dimensional cohomology of r in terrns of the torsion in r. (Thìs is analogous
to the situation of Chapter IX, where we tried to explain the non-integrality
of x(r) in terms of the torsion in r.)
For this purpose it is convenient to use modified cohomology groups
R*(r, M), first introduced by Farrell [1971]. Farrell's theory generalizes
the Tate cohomology theory for finite groups (Chapter VI), and it has tbe
following two properties which make it appropriate for our present purposes:
(i) Ri = Hi for i > vcd r and (ii) 8* = O if r is torsion-free.
The next three sections will be devoted to the foundations of the Farrell
cohomology theory. Then in 5 and 6 we will generalize to the Farrell
theory some or tbe results or VI.8 and V19 on cohomological triviality
and periodicity. Finally, 7 will contain tbe main results of the ehapter,
relating R*(r) to the torsion in r.
,
,
2 Complete Resolutions
Let vcd r = n < 00. By a complete resolutiQI1 for r we rnean an acyclic
chain complex F of projective z:r -moduJes, together with an ordinary
projective resolution B: P --. 7L of Z. aver z.r such that F and P coincide
in sufficientIy high dimensions. {As in Chapter VI F is not in generaI
::!:J1

274
X FarreU Cohornology Theory
nonnegative.) If r is finite, for example, then any complete resolution in the
sense of VI.3 is also a complete resolution in the present sense. In VI.3,
however, we required F and P lo agree in all non-negative dimensions, so
the present notion of complete resolution is more generaI. (It might seem
reasonahle. by analogy with VI.3. to require F and P to agree in dimensions
> n = vcd r. Indeed, we wilI see below that complete resolutions of this
type always exist. Our reason for not requiring this is that we want a complete
resolution for r to also be a complete resolution for any subgroup of r.)
If (F, P, E) and (F', P', E:') are complete resolutions, then a chain map
't: F -+ F' wiU be called a m ap of complete resolutions if there is an augmenta-
tion-preserving map p -+ p' which agrees with t in sufficiently high dimen-
SlOns.
(2.1) Proposition (a) There exisr complete resolutions (F, P, E) such that F
coincides with P in dimensiorls >n = vcd r. lf r is of type VFP, then F
CUri be taken to be of finite type.
(b) II (F, p, e) and (F', P', l) are complete resolutions, rhen there is a tmique
homotopy class ofmlJ.psfrom (F, P, E) io (F' P' E'), and these maps are homo-
topyequivalences.
PROOF. Choose a torsion-free subgroup r' c r or finite index. We will
apply tbe relative homological algebra of VI.2 to (r, r'). Thus we bave a
notion of admissible short exact sequence of r-modules and a corresponding
notion of relative injective r -module. Let lJ: P -+ 7L be a projective resolution
of 7L over zr and let K = ker{Pn-l --+ PII-l}. Then (Pi)t?n provides a pro-
jective resolution
. . . -+ PII + 1 -io p -+ K --Jo O
of K. Sincecd r = n, weknow(VIII.2.1)tbat K is lT'.projective. Proposition
VI.2.6 therefore gives us a relative injective resolution
O -+ K -+ QO -+ Q l ---+ . . -
with each Ql projective. Splicing together (Pi)in and (Qj)i o. we obtain the
desired complete resolution. Note that P and Q can be taken 10 be of finite
type if ris oftype VFP. whence (a).
The proof of (b) will req uire the following generalization of VI.3.2 :
(2.2) Lemma. lf ed r < -':I:) then af'lY acyclic chaif'l complex oJ projective
zr -modules is contracrible.
PROOF. Note fust that any r-module M which is Z-free has proj dimzrM < n
= ed r_ For if P is a projective resolution of 7L oflength n, then P (8) M (with
diagonal r-action) is a projective resolution of 7L @ M = M of length f'I.
[Why?] Suppose now that F is an acyclic complex of projectives and let Z"
for any k be the module of k-cycles. Then we have an exàct sequence
0--+ ZA: -+ F1.-. . .. -+ F..-II+ 1 ........ Z"_II --+ O
,
,
,
,
2 Complete Resolutions
275
with each Fi projective. Since proj dimzrZt-1I < n, it follows (VIII. 2. l) that
Zk is projective. Hence F ,,+ 1 "'* Zie splits for ali k and F is contractible. D
..
1
Returning now to the proof of 2.1 b, the lemma shows that any complete
resolution F is r' -contractible, hence it is adrnissible. Also, each projective
module Fi is relatively injective by VI.2.3. Given two complete resolutions as
in (b), we use ordinary homological algebra to construct an augmentation-
preserving map P -io P'; this defines a chain map 1": F -io £1 in high dimen-
sions. Since F is acyclic and admissible and F' is relatively injective relative
homological algebra (Vl2.4) now aUows us to extend T to ali dimensions.
Similarly, any two such maps are homotopic in high dimensions by ordinary
homological algebra applied to P and p', and the homotopy can be extended
to ali dimensions by relative homological algebra. It is clear from tbe unique-
ness that alI maps of complete resolutions are homotopy equivalences. D
Various other chain maps can be constructed by the same technique.
For instance:
J
(2.3) Proposition. If(F. P, E) is a complete resoiution, then there is a unique
homotopy class oJ chain maps T: F -+ P such that t' is the identity in sufficiently
high dimensions.
PROOF. F is acyclic and admissible and each p. is relatively injective. D
.'
In addition, one can construct a .. diagonal map," which will be needed
for the theory of cup products:
,
"
,-
(24) Proposition. Let (F, P, f) he a compIere resolution and let T: P -+ P (8) P
be a diagonal approximation (V.l) with components T.....: P + -+ p IV'. P
 .... p q Il \C) q'
Let F @ F be the completed tensor product as in Vl.S. lf m is an integer such
that F and P agree in dimensions > m - l, then there is a chain map .6.: F -+
,a.. I
F (8) F whose (m, m)-compQnent .6.mm  F 21'1 -+ F Wl @ F J1[ is equa l lo 'tmm.
PROOF. This IS almost identica! to the proof or the corresponding result for
finite groups (end of VI.5). As in that case, it suffices to define .6. in dimension
2m, in such a way that òl! I B2m = O and .6mm = 1''''1'1I. This reduces to the
construction or a family of maps a p: F 2m ........ F m + Il (8) F m  p (p E Z) satisfying
(i) (ò'ap + ò"r:x.p_ ])IB2m = O and (ii) !xo = 1'mm- (Here ò', Ò"t and B2m are as
in VI.5_) Let aD = tIMI and 1 = 1'm+1,,,,-I; this makes sense because F = P
in dimensions > m-L Sinçe 1": P  P @ P is a chain map, condition (i)
(with p = 1) is satisfied. We can therefore define ct.p inductively for ali p,
exactly as in VI.5. O
<
Finally recall that in the finite group case it is possible to construct a
complete resolution by splicing together an ordinary resolution and its duaI.

276
X Fanell Cohomology Th.eory
It turns out tbat there is an analogous (but more complicated) construction
whenever r is of type V FP. For simplicity we will treat here only the case
where r is a virtual duality group. referring lo the exercìses below for tbc
generaI case.
Let r be a virtual duality group wìth dualizing module" D = Hft(r, .lT)
(cf. VIII. 1 1.3). I claim that D is or type F P  as a zr -module. F or let r' c r
be a torsion-free subgroup of finite index and reca11 that D ìs of type F P as a
Zr'-module (VIIII0. exercise 1); it now follows as in the proof of VIII. 5. l
that D is of type F P 00 over zr.
(2.5) Proposition. LeI r be a virtual duality group with dualizing module D,
let B: P -+ Z be a finite type projective resolution o[ 7L aver zr, and let
ti: Q -+ D be afinite type projective resolution oJ Dover zr. Then there is a
complete resolution F such thaf fhe dual complex F = :J'fomzr<F, Zr) is the
mapping cone oJ a chain map 1: -"Q --+ P. In particular, Fi = P; for i > n and
Fi = (QIt-l-i)* for i  -1.
PROOF Consider the n-th suspension E" P or the dual P or P. Then
HlI:"P) = Hi-tlP) = HIt-i(r, .zT), which is o for i #- o and D for i = O. It
follows easily that there is a weak equivalence Q -+ 1:11 P. [Start by lifting
  Qo ......,. D to a map to the zero-cycles of n P; now extend this to a chain map
by the fundamentallemma 1.7.4.] This map can also be viewed as a weak
equivalencef: I:-lIQ -+ P, and the mapping cone C orfis an acyclic complex
of finiteIy generated projectives. Let F be the dual complex C. Thus
Fi = (C _.)* = (p - i $ (1: - lIQ)_ i -1)* . Pt* ffiI (QII-:i :-1)* = Pi  (Qrz- ì-:-l);
in particular" Fi = Pi Cor i  n. Checkmg the definlt10n of the ddferentlal In
F, one finds that it agrees with that in P in dimensions > n, up to sign. The
proposition will be proved, then, if we verify that F is acyclic. Let r' ç r
be a torsion-free subgroup of finite index. Then we bave F = Jlfi»nzrtC, Zr)
 lr'(C, Zr'). [This follows from the theory of induced and coinduced
mod ules, exactly as in the proof ofVI.3.4 ; altematively" use Lemma VIII. 7.4.J
But C is r -contractible (by 2.2 for instance), so its r' -dual omzr'( C, Zr')
is also r' -contractible and hence acyclic. D
Remark.. Consider the complex C = F of this proof. It coincides with
1:1-"Q in dimensions > l, so rI-1C coincides with Q in dimensions
> n Thus 1:11- l C = ,£"-1 F ìs what one would reasonably cali a complete
- .
resolution of D.
EXERCISES
1. Let r = r I x r 2' where ed r l < DJ al1d vcd r 2 < 00. Show tht one can costct
a complete resolution for r by taking the tensor product of a firnte length proJect]ve
resolution of lL. aver zr 1 and a complete resolution for r  .
I
.

3 Definition and Properties or Il-Cf)
.
277
.2. A non-negative chaìn complex C is said to be or type F Pro if it admits a wea.k
equivalence P --4 C. where P is a non-negative chain complex or finiteJy generated
projec:tives. (If C consists of a single module M concentrated in dimension O. for
example. this just means that M ig of type F P "" .) If r js a group and r' is a subgroup or
finite index. prove that a chain complex of lT-rnodules is of type F P eD over .zr
if and only if it is of type FP «J over £'T'. [This is proved in Brown [1982]. Ali
the essenlìal ìdeas can be found in Brown [197Sa], and one can even deduce the
result from Theorem 2 ofthe latter by a "'Shapiro's lemma" argument.]
.-
.3. Suppose r is oftype V,FP and let P be as in 2.5. Let!7J be the subcomplex
/
.I 0-+ 1'0 --+... _ pn-l -. Z" __ O
I
,
of P. where ZII i8 the moduJe of II-cocycles. Note (hat the inclusion !2 c., P is a weak
equivalence. Show that !2t is of type FP  and deduce that 2.5 extends to groups of
type VFP. with Q now taken to be a finite type "projective resolution" of", i.e..
a finite type complex of projectives such that there is a weak equivalence Q _ I:1f9.
",
.
A
3 Definition and Properties of H*(r)
We continue to assume that r is a group such that vcd r = n < 00. We
can then choose a complete resolution (F, P, t) and set
fJ*(r, M) = H.(.tthmr<F, M))
for any r -module M. By 2.1, fJ* is well-defined up to canonical isomorphism.
If r is finite, the present definition is o bviously consistent with that of
Chapter VI. For simplicity we will often write B*(r) or even B* instead of
R*(r" M). I t will always be understood however, that there is an arbitrary
r -module or coefficients.
Tbe cohomology theory R*(r, -) will be called the F arrell cohomology
theor y. I t has formaI properties analogous to those of the Tate theory for
finite groups (in which case n = O):
(3.1) R*(r) = O if r is torsion-free.
For in this case we can take F = O.
(3.2) fl*(r" -) has ali the "usua]' cohomological properties: long exact
sequences, Shapiro's lemma, restriction and transfer maps, and cup products.
This is proved exactly as in ordinary cohomology theory, except for tbc
construction of cup products. Wc will discuss the latter below, after stating
two more properties.
(3.3) n-cr. M) = O if M is an induced module zr @zr' M', where r' is a
torsion-free su bgroup or finite index and M' is an arbitrary r' -module.
Consequent1y. tbe functors e(r. -) are effaceable and coeffaceable.

278
X Farrell Cohomology Theory
This follows from 3.1 and 3.2 (Shapiros lemma). Note that 3.2 and 3.3
allow us to shift dimensions in botb directions as in VI.5.4. In view or the
next property this means that questions about Farrell cohomology can often
be reduced to questions about (high-dimensional) ordinary cohomology.
(3.4) There is a canonical map llì -+ fJi which is an isomorphism for i > M
and an epimorphism for i = n. Moreover, the sequence
H"(r') Ir . HlI{f) , R"{r) -+ O
is exact for any torsion-free subgroup r' c:: r of finite index. where tr is the
transfer map.
Tbe map Hi -+ Eli is induced by the chain map F -+ P or 2.3. Since the
latter can be taken to be the iden tity in dimensions  n (cf. 2.1)t the first
assertion of 3.4 is c1ear. To prove the second assertion. one could simply
directly examine tbe n-coboundaries in r<F M), where F is assumed
to be constructed from P as in the proof of 2.1. It is easier, however to proceed
as follows: Consider the 'dimension-shifting" exact sequence
O  K -+ zr 0zr' M -+ M -+ O.
This gives rise to a long exact sequence in ordinary cohomology which maps
to the corresponding sequence in Farrell cohomology. Using Shapiro's
lemma and definition (A) of tbe transfer map (III.9), we deduce a diagram
H"(r'. M) tr. H"(r. M)  H"+ l(r, K) · O
l l.
O .lì"(r, M) ,. R"+ l(rt K) t O
with exact rows. (The zero in the top row is H" + l(r', M) which vanishes
because ed rE = n.) The second assertion of 3.4 follows at once.
Remarks
l. If r is finite, 3.4 reduces to the statement that Hi = fli for i > O and
that HO is the cokernel of the norm map M -+ MO.
2. It foJlows from 3.4 that the image of tr: Hn(r') -+ H"(r) is independen t
oftbe choice ofr'. This should not surprise the reader who bas done exercise
6 of VIII.2.
We can now explain tbe construction or cup products. Let (F. P, e) be a
complete resolution and choose : F -+ F @ F as in 2.4. Using fi we can
construct cup products
tip(r. M) (8) flq(r N) ..... flp+'l(r, M @ N)
in the usual way, and these will be compatible .with coboundary maps in
long ex.act sequences. Moreover, wc have by 2.4 an integer m, such that
I
,
.
1
3 Definition and Properties of fI.(f)
279
Hi  fli for i > m and such that the cup product Rm 0 .Bm -+ fl2m agrees
with the ordinary cup product Hm Q9 Hm -+ H2m. The standard dimension-
shifting arguments (cf. proof or VI.5.S) now show that the cup product in fJ*
is compatible with that in H*, in the sense that the diagram
HP (8) H'l
l
lìF <8> flq
Hp+q
I
1
u ) flp+q
v
.'
commutes for ali p, q. [Assuming this for (p q), use the coeffaceability of H*
to deduce it for (p + l, q) and (p q + I) and use the effaceability of lì. to
deduce it for (p - I q) and (p, q - I).]
It is now c1ear that our eup product is independent of the choice of F
and.d. For any two choices would give the same product in high dimensions
by w hat we ha ve just done. and hence they would agree in ali dimensions by
dimension-shifting. Finally, one uses dimension-shifting once again lo estab-
lish the usual properties of the eup product (associativity commutativity
unit); for these properties are known to hold in high dimensions. [Note: The
unit for the cup product is the image in flo(r, Z) of l E HO(f, Z) = Z.]
Recall that tbe Tate group lìi for i > O could be interpreted in terms of
homology. This resulted from the duality theory whieh allowed us to inter-
pret the negative part or a complete resolution as the dual of an ordinary
resolution. We can generalize this to Farrell cohomology provided r is of
type VFP. For simplicity we will assume that r is a virtual duality group.
but the reader who has done exercises 2 and 3 of 2 should be able to treat the
generaI case. -fSee also - exercise 5 of VIII.I O.]
(3.5) Suppose that f is a virtual duality group witb dualizing module D.
and set fl.(r. M) = H .(f. D (g) M). Then there is an isomorphism f/ì
 HII-I-i ror i < -1 and a monomorphism lì-l -+ Hn. Mo reo ver. the
seq uence
o
 iì-l(r)
) R ncr) tI I il ner')
is exact for any torsion.free subgroup f' c r of finite index.
F or let F be as in 2.5. and let E = 1:n - 1 F (cf remark at the end or 2).
Then we have
Ri(f. -) = Hi(Jfi..mr(F -» = H -lF @r -) = Hn-l-lE @r -).
Now the construetion of E shows that there is a projective resolution Q or D
which receives a chain map E -+ Q which is the identity in dimensions >Pr.
Hence we have a map HJ{E @r )  Torf(D. -) which is an isomorphism
for j > n nd a monomorphism or j = 11. Since Tor.f(D, -) = HJ{r. -) by
111.2.2, thls pro ves the first assertlon of 3.5. Tbe second assertion follows by a

280
X FarreU CohomoJog)' Theory
dimensìon-shifting argument analogous to that used in 3.4; details are left to
the reader,
We have not yet said anything abaut Hi for O .:s;; j < n - L This range of
dimensions which is vacuous if r is finite, is harder to understand than
thase treated in 3.4 and 3.5. We can say something about il, however in
view of the mapping cone construction in 2.5. F or .Jrmnr<F, M) = F @r M
is the mapping cone of a map I:-nQ @rM - P @rM = .Jt'omr<P, M). We
therefore ha ve a long exact homology sequence, which takes tbe fono
il . Ri il i+ l
. 1 . -+ 11_ i -+ H' -+ -+ n-l - ì -+ H - . . . .
Since H-l = O and R -I = O we conclude:
(3.6) With tbe hypotheses and notation of 3.5, there is an exact seq uence
0-+ 8-1 -+ HI'I -+ ftJ -+ fJD -+ Hn-I -+ HI -+... -+ Ho -+ H" -+ R" -+ o.
We can summarÌze 3.4-3.6 as follows. If r is a virtual duality group then
the Farrell cohomology functors fli include: the cohomology functors Hi
for i > n; the homology functors Ri for i > n; a certain quotient of HR; a
certain subfunctor of RR; and n additional functors flo, ..., H,,-I, which
are some sort of mixture of {Hì}osisn and {R;}o Si:Sl'I:
. .. R-3 li-2
HO . . . Hn-1 HR Hn+1
l l l
fl-l flO . . . fll'l-l fln fll'l+ 1
I- l l
R" il ,,-1 ... Ho
. . .
Hn+ 2 h.
fl" + 2
...RI'I+2 HII+l
EXERCISES
1. Let r = ";£ x 7L2. Construct an explicit complete resolution and use it (O compute
R*(r. Z). [Cf. exercise 1 or2.]
2. Prove that tbe groups 8"(1) aretorsion-groups. More precisely. they are annihilated
by the integer d of IX.9.1. [Hint: T ransrer.]
Remark. I t is not known whether fl*(r) is annihilated by the integer m of IX.9,J.
In view of the cup product structure on n"(r), it wou[d be enough to show thal m
annihilates the identity element l E RO(r. Z).
.
3. (a) Define Farrell homology groups fl.(r).
(b) If r is a virtual duaJity group with dualizing module D. show thaf fli(r. M)
 Il.._ 1- i(r, D (81 .M) for ali i. More preciseJy. show there is an element
I

4 Equivariant Farrell CohomoJogy
281
z E HII-1(r. D) such that cap product with z is an isomorphism. [Hint: To construct
an isomorphism ljJ: Ri(r, M) -) il PI- I _ i (r, D @ M), use 3.5 and dimcl1si on-shiftjng ;
alternatively. take F and E as in the proof of 3.5 and show by homoJogical a[gebra
tbat tbere is a map E - F <8> D which induces the desired isomorphism. Now let
2 E RII -l(r, D) be the image of 1 e jìO(r. Z) under t/t. Show that '" is given by cap
product with z (up IO sign) as foJlows. The two composites
Hi(r, M) - Ri(r, M) .:: HII-1-lf", D @ M)
agree when i = O by naturality, hence Ihey agree for ali i by dimension-shifting. But
then the two maps iÌ1(r. M) =: HII_] _r. D (8) M) agree for large i and hence for
ali i.]
*( c) If D is Z -Free in (b). show that cap product with the same Z E fili _ 1 (r, D) gives
isomorphisrns Dir, M)  f/n-l-i(r, Hom(D. M».
*(d) State and prove analogues of (b) and (c) for arbitrary groups r of type JI F P.
.4. Let r = r 1 x r 2 as in exercise 1 of 2. Derive, under suitable hypotheses. Kiinneth
formulas relating fl.(r) to H *(r 1) and il ,.,( r 2) and simiJarJy for cohomology.
5. Let 1 - r' - r - r' - 1 be a short exact sequence of groups or finite virtual
cohomological dimension,
(a) If r" is torsion-free, show that there is a Hochschild-Serre spectral sequence
(derived as in ordinary cohomology) ofthe form
E';' = HP(r". Rq(r'» =- JjP+'}(r).
"'(b) If r' is torsion-free, show that there is a spectral sequcnce
E = {Jp(r". H'l(r'» ;;;:;lo fJp+q(r).
"
.....
\
(Method l: U se tbe ordinary Hochschild-Serre spectral sequence and dimension-
shifting. Method 2\ Use a complete resoJution of the forrn F = F"  C. where
C is a fini(e dimensional complex of r-modules which is a r'-projective resolution
of Z.]
"'6. Define and study composition products fl* @ fl. - 8* anaJogous IO tbose of
VI.6. [The only trìcky thing here is to prove the analogue or VI.6.1a. This is tme.
but the proof requires more work than in the case or finite groups.]
4 Equivariant Farrel1 Cohomology
Let r be a group of finite virtual cohomological dimension and let X be a
finite dimensioMal r -CW -complex. As in Chapter VII, we then choose a
complete resolution F for r and we set
flr(X; M) = H*(,.(FJ C*(X; M»)

282
X FarreIJ CohomoJogy Theory
for any r-module M. These equivariant Farrell cohomology groups have
properties analogous to those of the ordinary equivariant cohomology groups
(spectral sequences, etc.).
An important special case is that where X is taken to be proper and
contractible (cf. VII!.!!.I). We tben obtain as in VII. 7. l O a spectral sequence
(4.1) Efl = n flQ(r(" Ma);;;;> f/p+'l(r M)
11£1:"
where I:" is a set of represen tatives for the p-cells of X mod r. This spectral
sequence relates tbe Farrell cohomology of r to the Tate cohomology of its
finite subgroups. Note that the spectral sequence lives in the first and fourth
quadrants, but there is no problem with con vergence because dim X < CI:;.
Indeed. the spectral seq uence is concentrated in the vertical strip O < p
< dim X, so Er = Ea) as soon as r > dim X.
The rest of tbis section will be devoted to establishing some properties of
tbe spectral sequence 4.1 which will be needed in 6. F or simplicity we will
assume that X is an ordered simplicial complex with order preserving sim-
plicial r-action. In addition we will assume that XH Is non-empty and
connected for every finite subgroup H c r; we saw in VIII.ll that such an
X exists.
Note that the order preserving' assumption implies that Mf1 = M, i.e.,
that Zr; = 7L with trivial rO'-action, for each simplex u. We may therefore
suppress M from the notation in what follows.
Our first goal is to give a purely algebraic description of the left-hand
edge Eg,q. Far this we will need to compute the differential dy,q. Recall
that for any simplex (j of X and any rE r we bave a conjugation isomorphism
c(y - 1)*: H*(r u) .; fl*(r ra),
denoted u  yu (cf. II18).
(4.2) Lemma. Let X" be the set oJ p-simplices oJ X. Then Efl cali be identified
with the subgroup oI naexp {Jq(rt;r) consisting oJ those jamilies (Utt)tJEXp
such thar '{UIf = u}'tJ for alI}' E r, (T E X r The differential dr is the restriction
to this subgroup oJ tlle map
d: n flq(ru)  n fIq(rt;)
ue;Xp teX.... 
defined as follows. For QI1Y T = (VO ...  Vp+ l) E x p+ l with Va < ... < vp+ l'
let ti = (vo,.. " vh ..., vp+J), i = O ...,p + I, and let Pi: {Jq(r,) - Rq(r,)
be the restriction map. Then d is given b y
(P+l )
(ua) .-. T .-. i Lo (  I)i Pr(Ui) .
(Note that r-c 5 reI because ofthe assumption that the r-action is order-
preserving. Thus the definition or d makes sense.)
I
.,
.'
,
.
.
,
.
,

4 Equivariat'Jt Farrell Cohomology
283
PROOF. Note first that
n 11"(ru) = il n {Jq(r }14)'
tleX p al; l:p )'E" rfr t1I
Since the conjugation action of r tF OD fltl(r o) is trivial (cf. 111.8.3), it follows
that tbere is a well-defined injection
ti: Er = n Rq(ra) -+ n f/Il(rq)
EI aXp
given by (U"")E1:p 1--+ (vUu)ae1:p, yerJI'a' Tbe image of ex is clearly equal to the
set of families (U)06:Xp satisfying yu = U"17 for ali u E X p and }' E r. whence
the first part of the proposition. The descrìption of di fol1ows from the
cohomology analogue of VI I. 8. 1. F or the con venience or the reader, however,
we will give a direct proof.
Let F be a complete resolution far r. Then the injection
a: Efl -+ O f/4(r a)
r7Xp
is given in terrns of cochains as follows: We bave
Efl = fl4(r, C"(X; M»
= H4(Jt'unzrl.F, Hom(X p' M»)
= H'l(Hornr<:X p J't'omz{F. M»),
where the second equality comes from the ordering on X. An element
U E £1 is therefore represented by a family (ct'J)o£xp such that Ca E.7thmz(F. M)
and rCO' = c}'t'J for ali )' E r and (J E X p. Taking Y E r u' it follows that
Cu E Jt"'.omzr ,,(F, M). Hence Ca represents an element Ua E {Jq(r (J  AI), and one
checks that o:(u) = (ua),.,'eXp' The differential di is simply the map
Rfl(r, b): flq(r, CP(X; M}) -+ flq(r, Cp+ I(X; M)),
where h: CP(; M) -fo CP+1(X; M) is the coboundary operator. The lemma
now foIlows 4t once from the definition or b in tenns or the face rnaps
X p+ 1 -+ X" an from tbe fact that the restriction map Pi: flq(r rl) -fo fltl(r r) is
induced by the indusion .JFomZI'TI(F, M) 4 Komzr.{F M). D
In particular, we can now easily calculate E,q = ker d?q, and we obtain:
43) Lemma. E,q can be identified with tlte subgroup ofOvlOxo{Jq(rv) consist-
mg DJ those families (Utr)VI;XO satisfying the following t\Vo conditions:
(i)
YUt) = u}'t) lor any Y E C V E Xo.
(ìi) if e is a l-simplex oJ X with vertices vo, VI' tllen uvo and UV! restrict lO the
same element oJ R4J(r.J.
Using now our hypothesis that XH is non-empty and connected for H
finite, we will deduce the desired algebraic description of E' q:

284
X Facrell Cohomology Theory
"
(4.4) Proposition. Let «J be the set of finire subgroups DJ r. Then E,q is iso-
morphic to the subgroup f)tl(r) ç; OHt?iJ flll(H) consistilg of those families
(UH)Heft satisfying the following two conditions:
(i) )/UH = U)'Hr--1 for ali 'V E r H E iJ.
(iì) lf H' c H thel1 resJJ,uH = UH"
(Note: Conditions (i) and (ii) could be combined into tbe single condition
that the Un are compatible with respect to "restriction maps' flll(H 2) -+
iiq(H l) induced by embeddings H I  H 2 given by conjugation by elements
ofr.)
PROOF. Using the description of E.q given in 4.3. we have an obvious map
qJ: i)q(r) -+ E.q given by (UH)Ue81--+ (urJveXo. Sìnce every H E iY is contained
in some rv [because XH '# 0]. it follows from condition (ii) above that tp is
injective. To prove that rp is surjective suppose (u1,)ve Xo satisfies conditions
(i) and (ii) of 4.3. Given H E 5. choose a vertex v such that H c r v and set
WH = res.uv' Sinçe XH is connecled. condition (ii) of 4.3 shows that WH
is independent of the çhoice of v. I t is easy to check that the resulting family
(WH)Heif is in b(r) and that its image under tp is (uv), D
Next we wish to introduce a multiplicative structure in to tbe spectral
sequence 4.1. T o simpliry the notation we will assume that the coefficient
module M is a commutative ring R with trivial r -actioD. The simplicial
cochain complex C*(X) = C*(X; R) then has a cup product
C*(X) @ C*(X) -7 C*(X)
which is strictl y associati ve and is commutati ve (in the graded sense) up to
homotopy. Moreover the product and the homotopy afe defined canonicall y
in terrns of the structure or X as an ordered simplicial complex- [See the
exercise below ror a review or these facts.] They are t herefore compatible
with the r -action.
Now choose a complete resolution F with a diagonal map tJ.: F -+ F @ F.
We then have a product
.Jtbm:r(F, C*(X» (8I.Jf'omr<F, C*(X» -+ r<F, C*(X) @ C*(X»
-i> r<F, C*(X)
where the first map is induced by !1 and the second map is induced by the
cochain cup product C*(X) @ C*(X) - C*(X). As we explained at the end
of VII.5, this product is compatible with the filtration definìng the spectral
sequence 4.1. hence there 18 an induced product
E,. @ EF' -+ E,
.,
,
, .
4 Equivaciant Fanell Cohomology
285
for r > 1. The following proposition sumrnarizes the properties of this
producI which we will need:
(4.5) Proposition. (i) The differential dr is a derivatiol'l with respect to the
product on £1" i.e.,
dr(uv) = d,.(u) . v + (_l)degu U . drCv)-
(ii) The product on E1'+ I = R(E,.) is obtained from the product 01'1 Er by
passage to homology.
(iii) The product on E l is the composite
flq(r CP(X» @ fl4'(r, CP'(X» -+ flq+q'(r, CP(X) @ CP'(X»
-+ Rq+q>(r, CP+P'(X)
where the first map is the usual cup product in fì*(r, -) and the second map is
induced by the cup product in C*(X).
(iv) The product on E, is associative for r > 1 and commutative for r  2.
(v) The product on E is compatible with the usual product on R*(r, R)
under the identification oJ E with Gr fì*(r, R).
(vi) The isomorphism E'.  {)*(r) oJ 4.4 is a ring isomorphism.
PROOF- (i) and (ii) follow from the fact that the product OD Er is induced by
the product OD .Jt"omr<F, C*(X». (iii) is immediate from the definitions. In
view or (iii) and properties of the ordinary cup product, the product on E l is
associative and is commutativeup to homotopy (with respect to thedifferential
di)' (iv) now follows from (ii). The product on EC(J is obviously compatible
with the product OD the abutment H*(.1e<»nr<F, C*(X»). But the identifica-
tion of the abutment with R*(r) is induced by a chain map Kmnrf,F, R) -i>
.w'ewtr(F, C*(X» which is easily seen to be a Ting homomorphism, whence (v).
Finally, to prove (vi) it suffices to show that the map a: E'. -+ IlCJexo fl*(rv)
defined in the proof of 4.2 is a ring homomorphism. Consider the v-compo-
nent of a, \,
\ E'. = R*(r, Co(X» -+ fì.(r (1)'
I
I t is given on th cochain leve] by a map
.w'6'fflrlF, CO(X» -+ Jf()?»,rJF, R).
which in turn ìs induced by tbe ring homornorphism CO(X) = Hom(Xo, R)
-+ R given by evaluation at v. This cochain map is easi1y seen to be a ring
homomorphism, so (vi) is proved. D
As a simple application of this multiplicative structure we will prove a
result due to Quillen [1971]. Fix a prime p and consider the ring t)*(r, 7L )
defined in 4.4. There is an obvious homomorphism p
p: R*(r, Zp) ..... f;*(r, Zp)
given by the restriction maps Iì*(r) ..... fl*(H) (H E fj).

286
X FarreU Cohomology Theory
(46) Proposition. Tlle map p has lhe following two properties:
(i) Everyelement ofker p is nilpotent.
(ii) For any U E f)*(r. Zp) there is an integer k > O such thac uYC E im p.
(Thus p. is an isomorphism 'up to p-th powers.' Following Quillen, WC
say that p 15 an F-isomorphism.)
PROOF. Note first that p is simply the edge homomorphism
R*(D -... E*  E>*  s*(r)
associated to the spectral sequence 4.1. This is easily seen by direct definition-
checking (see also exercise 1 of VII.7). Recall that the construction of the
spectraI sequence gives us a filtration
R*(r) = FOH*(f') => F1 fl.(r) => ...
which is compatible with the cup product (Le., FS FC c EH r). Since the kernel
of the edge homomorphism above is precisely F1 fl*{f). it follows that any
Z E ker p satisfies z" E F" B*(r). But pkft.(r) = O for k > dim X so z is
nilpotent. Next Jet U E E¥' *. I claim that d2(IiP) = O. This follows C;om anti-
ommutativity if p and deg u are both odd (in which case u2 = O); otherwise,
It follows from (i) and (iv) of 4.5, which give
d2(uP) = puP- ld2(u) = O
since the coefficient module is 7Lp- Thus uP E E' *. Iterating this argument
we find that urJ< E E+ , hence upl< E E * = im p as soon as k + 2 > dim X.
D
Remark. Quillen actually proved a much stronger resu)t. Namely, he proved
the analogue of 4.6 with t)*( f) defined in terrns of the finite su bgroups or r
which afe elementary abelian p-groups.
EXERCISE
Let K be an ordered sìmplicial complex and let C(K) be its chain compIex. with one
bis elemen for every (n + l )-t u pIe (lìo. . . . , VII) of vertices such that {vo. . . . . v,J is an
n-slmplex WJth v(] < ... < v". The Alexander-Wlzitney diagonar map A: C{K)-+
C( K) @ C( K} is defined by
"
A(vo,..., v,,) = I {vo..... r"J @ (z:;p..... v,,).
1'=0
Verify that A is a chain map and that the ìnduced cochain cup product is associative
and is commutative up to (canonicaI) homotopy. [Hinr: The homotopy commutativity
carI be proved by an acydic models argument, which is most easily carried out as. follows:
Because or thc ordering 011 the vertices of K. a simplex of K can be regarded as a singular
simpJeJl. in the geometric realization or K. Thus C(K) is a subcomple)( of the singular
I
5 CohomoJogìcalJy TriviaJ Modules
287
chain compJex cin8(K). Now look at the standard proof vìa acyc1ic models that the
Alexandcr- Whitney map on Cin.l(K) js naturally hornotopy commutative. and observe
that the homotopy can be taken to map C(K) into C(K) @ C(K).]
5 Cohomological1y Trivial Modules
We continue to assume that vcd r = " < 00.
(5.1) Lemma. Il M is a r-module such thac fjll(G, M) = O for every finite
subgroup G c r, then R*(r, M) = O.
PROOF. Consider the spectral sequence 4.1, and recall that X can be chosen
so that M(J = M for aH (l. Our hypothesis implies that E1 = O, hence the
abutment n*(r, M) is also zero. D
We will say that M is cohonwlogically trivial if R*(r' M) = O for every
subgroup r' £ r. Applying 5.1 to each r' we obtain:
(52) Proposition A rmodule is cohomologically trivial if and only if il is
coJwmologically trivial as a G-module lor every finite subgroup G c: r.
Thus the theory of cohomologically trivial moduJes is reduced to the case
where r is finite. In particular. we can use this to prove the following general-
ization of Rim's theorems VI.8.10 and VI.8.12:
(.5.3) TheoreDL ThefollowiPlg conditions on a r-module Mare equivalent:
_.-_....
(i) M is cohomologiéiIlly trivial.
(ii) proj dimzr M < n + 1.
(iii) proj dimzr M < (I).
lf these conditions hold and M is Z {ree. then proj dimzr M < M.
PROOF. We have (ii) ==-- (iii) trivially, and (iii) => (i) exactly as in the proof of
V18.12. To prove (i) =- (ii), assume first that M is .l-free, in which case we
wiIl prove Ihat proj dim.zr M < 11. According lO VIII.2.1, it will suffice to
show that Extr+1(M, N) = O for every r-module N. Moreover a glance at
the proof of VIII.l.I shows that we only need to consider rnodules N which
are lf-free. In this case we know from VI.8.11 that the r-module Hom{M, N)
s cohomologically trivial over every finite subgroup ofr hcnce Hom(M. N)
IS cohomological1y trivial by 5.2. We therefore have
Extr+ I(M. N) = Hl1+ l(r. Hom(M, N»
= fln+ l(r. Hom(M, N»
_O
- .
by 111.2.2
by 3.4

288
X FarreU Cohomology Theory
If M is now cohomologically trivial but noi Z-free. choose a short exact
sequc O -+ M' -+ P -+ M -+ O with P projective. Then M' is cohomologic-
ally Inviai and Z-free so proj dirn M' < n by what we just proved. Therefore
proj rum M < n + 1. D
EXERCISE
Show that the following conditions are equivaJent:
(i) M is cohomologically triviaL
(ii) For every subgroup r c [there is an integer m such that Hi(r', M) = O for ì > m.
{iii} fJ .([', M) = O for every subgroup r c r.
(i v) For every subgroup r £: r tbere is an integer m sucb that RAr. M) = O for i > m.
6 Groups with Periodic Cohomology
A group r or finite virtual cohomologicaJ dimension is said to bave periodic
cohomology if for some d * O there is an element U E Bd(r, Z) which is
invertible in the ring 1J*(r Z). Cup product with u then gives a periodicity
isomorphism
Ri(r, M)  Jìi+d(r, M)
for any r -module M and any i E lL. Similarly, we say that r has p-perìodic
colwmology (where p is a prime) if the p-primary component fl*(r, Z)( ).
which is itself a ring (cf. exercise 2a of VI.5), contains an invertible elemet
ofnon-zero degree d. We then have
Ri(r, M)(p)  Jìi+d(r, M)(p).
As in exercise 2b or VI.5 (see also exercise 2 of 3), we have
B*(r, Z) :::;: n B*(r, lL)(p),
p
where p ranges aver tbe primes such that r has p-torsion. Since this is a
finite direet product or rings it is cIear that r has periodic cohomology if
and only if r has p-periodic cohomology for every prime p. We will therefore
restrict our attention to p-periodicity. Our goal, as in the previous section,
will be to reduce to the case where r is finite, which we already understand
by VI.9. First of all, it is sometimes convenient to use il*(r, 7l..p) instead of
fJ*(r, Z)(P)' This is justified by:
(6.1) Proposition r has p-periodic cohomology if a,ul orJly if il*(r, lLp)
conrains an invertible element oJ non-zero degree.
..,.........-...
,
,
..
,.
, .
6 Gro\lpS with Periodic Cohomology
289
Tbe proof wiU be based 00 ,3 'Bockstein" argument. LeI H(k) =
R*(r Z"..). For any k, l there is a canonical shorr exact sequence 0-. Zpl -+
71.,;.+[ -+ lt.pI< -+ O, which yields a cohomology exact sequence
(6.2) . . . -+ H(l) -+ H(k + l) :;O;k+j.k H(k) fJkol, H(f) -+ . . , .
(63) Lemma. lf k > i then {3"., is a derivation, i.e.,
PJc.,(uv) = f3", ,(u) . v + ( -1 )de8 "u . PII., J(v).
(The right-hand side makes sense and is in H(l) because we bave a cup
product H(k) (8ì H(l)  H{l) for k > l.)
PROOF. Choose a complete resolution F and a diagonal map F --+ F @ F,
It is easy to see that PII, l can be computed as follows: Gi ven U E H(k), choose
a cochain c E.J'fl)fflrlF, Z) whose reduction mod p" is a cocyc1e representing
u; then oc is divisible by p", and ocjpt. is a cocycle whose reduction mod pl
represents /31;.'( u). Tbe derivation property or /3",1 now foIlows from the
corresponding property of (). D
(6.4) Lemma Eor any U E H(1) and any k > 1, up'< E im a,,+ 1,1 = ker P1.k.
PROOF. I c1aim first that for any U E H(k), we bave uP E ker f1/(, l = im C(k+ 1.11;.
This follows from the fact that f1" l is a derivation whose target R(I) is
,
annihilated by p (cf. proof of 4.6), Thus C'Xk+ 1.1 is the composite of ring
homomorphisms
H(k + l) l<+ l,'\ H(k) -+ ' .. -+ H(I),
each or which has the property that p-th powers are in tbe irnage. The lemma
follows at once. D
Let H(oo) = ([Ll)(P)' Note that we stili have an exact sequence or
the form 6.2 if r = 00 (and k < co); this is obtained by taking p-primary
com ponents in the cohomology exact sequence associated to
O -+ Z  li. -+ ?Lpk -+ o.
(65) Lemma. F or sufficiently large k, irn ':;(k. 1 = im lXoo. 1 .
PROOF. We know (exercise 2 of 3) that H( 00) is annihilated by pie for some k.
The sequence 6.2 witb l = 00 therefore yields an injection O:co,l;: H( 00) 4- H(k)
for large k. N ow it is easy to see that p l, k = aoo. kfJ l. ro, so ker /h,,, = ker {j.. UJ
for large k. But ker {31.1 = im aJ.c+ l,l and ker (h, IX> = im a, 1. D
(6.6) Lemma. The map ex = aC'C, 1 : H( 00) -+ H(l) has the following two
properties:
(i) Every element ofker a is nilpotent.
(H) For any U E H(I) there is arz integer k such that u"" E im ex.

290
X FarreU Cohomo]ogy Theory
(Thus a is an F9jsomorphism in the sense of .)
PROOF. If U E ker a. then 6.2 with l = 00 and k = 1 shows that u = pv for
some v E H( co). Hence u" = Jfv'" = O for large k. since H( (0) is annihilated
by some r'. This proves (i), and (ii) foHows from 6.4 and 6.5. O
PROOF OF 6.1. The map 0:: H(oo)- H(l) is a ring homomorphism (taking 1
to l). This foJ1ows from the fact that thc map R*(r, Z) _ R.(r 7L. ) is a
ring homomorphism which maps alI primary components to zero xcept
R*(r Z)(PJ' Thus a takes invertible elements to invertible elements. whence
the ., only if" part of the proposition. Conversely, suppose R(l) contains an
invertible element U of non-zero degree. Raising u to a power if necessary, we
can assume (6.6(ii») thar u, Il - I E im IX. Let u = o:(ii) and u -l = IX{V). Then
a(ùv) = l, so 6.6(i) implies that ùf; = l - x with x n ilpotent. But then
I - x i8 in vertible with inverse l + x + x 2 + . . . , so il is invertible and r
has p-periodic cohomology. D
We now give the main result of this section ;
(6.7) Theorem4 The following conditions are equivalenc:
(i) r has p-periodic cohomology_
(ii) There exist integers i und d with d #- O such that Rì(r, M)(p) 
fJ'+d
l (r. M)(p) Jor ali r -modules M.
(iii) Every finite subgroup oJ r has p-periodic cohonwlogy.
(iv) Every elementary abelian p-subgroup oJ r has ra"k < l.
PROOF. (i) "* (ii) trivial1y. If (H) holds, then (H) holds for every finite subgroup
or r by Shapiro's lemma. But this implies (Hi) by the analogue of VI.9.1 for
p-periodicity. Since (iii)<:> (iv) by VI.9.7, it remains to prove (iii) =- (i).
Let fj and *(r) be as in 4.4 with Zp as coefficient module. If (iii) holds,
then I c1aim that we can find a family u = (UH)HI:1J or invertible elements
UH E R4(H) = Rd(H Zp) such that U E .t;d(r). To see this, note fust that the
elements of W fall into finiteJy many isomorphisrn c1asses; for if r' £ r
is a torsion-free normal su bgroup of finite index then every H E fj is iso-
morphic to a subgroup of the finite group rjr'. Now choose for each H E iY
an invertible element VH E fJ.(H) of degree dH  O. By what we have just
observed. we can certainly assume that only finitely many distinct degrees
dH OCCUf. Raising each VH to a suitabJe power if necessary we can then
arrange that ali the d H afe equal to a single integer d.
Now let c: H t c,. H 2 be an embedding of finite subgroups given by
conjugation by some element of r. Tben VB, and C*VH2 are two generators
of fJ4{H l)  RD(H 1). hence c*VH"l = AVHI for some non-zero scalar A E Zp.
Since ÀP- J = 1 it follows that vp-I = (c*v )P- 1 = c*(vP-1) Thus if we set
' HJ - ff2 Hl .
Un = vJl- I, we have (u B) E s*(r). Clearly (UH I) is alsa in V*(r) so 5*(r) con-
tains an invertjble element ofnon-zero degree. In view ofthe F-jsomorphism
fJ*(r)  t;*(r) of 4.6 it follows exactly as in the proof of 6.1 that R*(r)
= R*(r 7L p) contains an invertible element of non-zero degree. O
,
.-
.
1 Jì.(r) and the Ordered Set ofFinite Subgroups or r
291
Remark A weaker version of the implication (iii)::::;.. (i) was proved by
Venkov [1965], but not in the language of Farrell cohomology-he spoke
instead of the ordinary cohornology being periodic in sufficiently high
dimensions. Restated in terrns of Farrell cohomology. his result said: if
J}d(r) contains an element u whose restriction to R*(H) is invertible for every
H E tv then u is in vertible. (His proof, lik:e ours, was based OD the multi-
plicative structure in the spectral sequence 4.1.) The missing ingredient
which was needed to go from Venkovs result to Theorem 6.7 was Quillen's
observation 4.6.
EXERCISE
lf6. 7(ij)holds and the isomorphism is naturaJ,show that jj*(r.Z)cp,containsan invertibJe
elernent or degree d. [Hint: By dimension-shifting. 6. 7{ ji) hoJds for aJJ i, wìth isomorph-
isrns that are compatible wilh connecting homomorphisrns. Show that the composite
Hi -+ Rp) -- R:d is necessarily given (up to sign) by cup produci with the image or
l E HO.]
.'
A
7 H*(r) and the Ordered Set of Finite
Subgroups of r
In this section we will prove analogues for F aneli cohomology of Theorerns
IX.I1.1 and IX.13.l.
If r operates on a partially ordered set S such that I S I is finite dimensional,
then we set
R(S) = Rt(!SI).

-
As usual, it is undeÌòod here that there is an arbitrary r -module or co-
eflicients. Recall that for any finite dimensional r -complex X there is a
canonical map
B*(r) = fì(pt.) -+ Rt(X),
induced by the map X -+ pt.
(7.1) TheorelO. Le! r be a group such that vcd r < 00. and let S be the seI oJ
non-trivilll finite subgroups oJ r. Then the canonical map
B*(r)  81(5)
is an isomorphism.

292
X Farrell Cohomology Tbeory
PROOF. As in the proof of IX.II.I, let X be a finite dimensional, contractible,
admissible, proper r -complex with contractible fixed-point sets X H (H E S),
and let X o = UReS XH, Then we ha ve
fJ.(r) :::;:: fl(X) because X is contractible
:::;:: fl(X o) because r acts freely in X - X o (cf. VII.IO.I)
 R(S) by IXlI.2.
To see that the composi te isomorphism is given by the canonical map
Jì.(r) -> R(S), consider the diagram.
Jì.(I)
e.) ;/ j 
fl:ex) 4 flt(X o)  Jìt(S),
where all maps coming from Jì.(I) are tbe canonical ones. The left-hand
triangle obviously commutes. For the right-hand triangle, we must go back
to IX.I 1.2 and note tOOt, in the notation of the latter, the square
C - C(S)
j j
C(Z) - C(pt.)
commutes. This is easily verified. Thus (.) commutes, whence the theorem.
D
As in Chapter IX, there is a p-Iocal version of7,l which is more usefuL Its
proofwill be based on the following analogue ofIX1l.5:
(7.2) Proposition. Let Z be a finire dimensional admissible r-complex and
let p be a prime such thar ZR is acyclic for every non-trivial finite p-subgroup oJ
r, Then the canonical map
Jj.(r)(p) -> Rt(Z)(p)
is Wl isonwrphism.
PROOF. Assume fust tbat every finite subgroup of r is a p-group. We can
then apply the method of proof of 7.1 10 r acting on X x Z (with X as
above), and we obtain an isomorphism fl(X x Z) :::;:: Rt{S) such that tbe
lriangle
fl.(r)
/ \
Jj(X x Z)  flt(S)
7 e.(T) ano\! the Ordered Set or Finite SUbgTOUpS of r
293
commutes. On the other band, the projection X x Z -> Z yields
R.(r)
/ \
flteZ)  flt(X x Z).
We therefore have
R.cr)  flt(Z)
in this case.
In tbe generaI case, choose a subgroup r' ç:;; r of finite index such that
(r: r') is relatively prime to p and every finite subgroup of r' is a p-group
(cf. proof ofIX.9.2). Using restriction and transfer maps in tbe usual wa (cf.
III.lO), we ha ve a naturaI embedding of fJteW)(p) (far any W) as a dlrect
summand of flt.(W). In particular, taking W = Z and W = pt.. we see that
the canonical map
fl*(r),P) -> fl(Z)(p)
is a direct summand of the canonical map
fl*(r') -> Rt.(Z).
Tbe Iatter being an isomorphism by the previous paragraph, it follows
tOOt the former is also an isomorphism. D
We can now prove the main result of this section:
(7.3) Theorem. Let r be a group such that vcd r < oo,let p be a prime, and
let si be the ordered set oJ non-trivial elementary abelian p-subgroups oJ r.
Then -------..........
.... '"
if*(I)(,,) - Ht(sI)(PI'
PROOF. Apply 7,2 with Z = I si I (cf. proof of IXI3.1).
D
One specia) case where Resl) can be easily understood is that where si
is discrete. In that case 7.3 yields:
(7.4) Corollary. Suppose that every elementary abelian Jrsubgroup oJ r has
rank :s; L Then
fl.(r)(,,)  n fl*(N(P»(,,),
p,,;¥
where {JiJ is a set oJ representatives Jor the conjugac y classes of subgroups oJ r
oJ order p.

294
X Farrell Cohomology Theory
. Far a concrete ex ampIe where 7.4leads to explicit cohomology calcula-
hons, see Brown [1976]. (These calculations. however, which involve
r = SL3(Z), ha ve been superseded by the caIculations of Soule [1978].)
Far an example of 7.3 with dim d > O. see Brown [1979]. This time
r = SL3(Z[lj2]). p = 2, and dim ,r./ = l.
Remark. Since Rxr) = Hi(I) for i > vcd r, the canonical map
H'(r)cp) --+ n Hi(N(P»(p)
P",9
is an isomorphism for i > vcd r under the hypotheses or 7.4. It is natural
to ask, by analogy with similar situations in algebraic topology. whether the
map is surjective for i = vcd r. It can be shown that this question, whose
answer is not knowll, Is closely related to the question asked in VIII.ll as
to whether one can always find a proper. contractible r -complex of di-
mension equal to vcd r.
EXERCISE
Show that the components or the map in 7.4 are restriction maps
il.( q(PI --+ fl.( N (P) )(P)'
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N otation Index
( -, -] (commutators] 2
 (homotopy] 5
< -, -) (evaluation] 9, J 13
(go,. . . , g.), [YIIY2I' dlg.] [basis or
standard resolution] 18- 19. 35- 36
0,0G [tensor product or C-modules]
55
><J [semi-direct product] 87
)( [cross-product] 109
u (eup product] 109
n (eap product] t 13
o (eompleted tensor product] 137
 (eonvergenee of a speclral sequenee]
167-168
A' [dual or an abelian group A] 145
Aut N [automorphism group of N]
102
B(C) [boundaries or C] 4
B2k [Bemoulli number] 255
ed r [cohomological dimension or f]
184-185
C.(C, M), C.(C, M), C.(C. M) [ehains.
cochains, and normalized eochains
or C with coefficienls in M] 59
Crt}') [eentralizer of}' in f] 241
CoindM [coinduced module] 67
cor [eorestriction map] 80
D2.. [dihedral group] 98
Der (C, A) [derivations] 89
E )( GC' (pull-baek] 94
£(P),6rtP) 230
Epq. E:" [spectrdl sequence] 163, 164
E7 [exceptional group] 254.256
Ext (M, N) 60
e(G, A) (extensions or C by A] 93
F (dual ofa el1ain complex f] 133
FP. FP., FL, etc. see "finileness
conditions" in the index or
tcrminology below
[y] (eonjugacy dass or}'] 239
nN) (eongruence subgroup] 40
ny), r(yj), ro-) [divided polynomial
algebra] Il 9, 125
Cl. AC2 [amalgamation] 49
Cnb [abelianization] 36
G/H, H\C (eosets of H in C] vi
GL. [generallinear group] 38
geom dim r [geometric dimension
ofT] J 85
Gr M [associated graded module]
162
H(C) [homology of a chain complex
C] 4
H .C, H .(C) [homology of a group C]
1,35
RiG, M), H.(C, ,\-1), H.(C, M),
fi.(G, Al) [homologyand
cohomology or G with coefficienls
in a module M] 56-57, 134-135,
277
H ,"(C. C), H'"(C, C), fl*(c, C)
[homology and cohomology with
coefficients in a chain complex]
168. 170
1m

302
H"(G)cp" B*(G)IPI (p-primary
L'Omponent J 84, 141
ll(X, .1'11), H(X, M), iìù(X, 1'1'1)
[equivarient homology and
cohomology J 172, 281
fI · (r) 284
H:'(X) [cohomology with compitct
supportsJ 209
fI.( Z) [red uced homology J 21I
H .( S), l11{S), X r< S), r;:tc. [5 an ordcrcd
seI] 262
Hom [homomorphism module wilh
diagonal actionJ 56
Home 208
£'- [homomorphism (or funelion)
complex J 5
li.. A (H NN extensionJ 179
Ind M [induced module] 67
K(G, l) [Eilenberg-MacLane spaceJ
15
,,2, "., "(X), I\*(V), ,,(V)
(exterior powers or algebraJ 97.
119, 121
1'01* [dualofamoduleMJ 28
MG [invariants] 27
MG [eo-invariants] 34
N, N (normJ 20,58
IV(H) [normaIizerJ 259
Ou t N [ouler aUlomorphism group
of IVJ 104
P(G, A) [prineipal derivationsJ 89
proj dim M, proj dim RM [projeetive
dimension J 152
Q4'" [quaternion groupJ 98
p(P), pr<P) 230
NOlation Index
R(P), R..(P) (Hallori Stallings rankJ
233
Rr<P), Rr<P)(y) [Hattori-Slallings
rank over a group ring] 239
rk.o(P) 235,236
rkz(A) 242
iJtl.o(P) 236
ResM [restriclion of scalarsJ 69
rcs [restriction ma p] 80
ISI 2(i1-262
reA) [symmetric group] 48
I:C [suspension J 5
SL" [speciallinear groupJ 39
SPZ" [symplcctic group] 255
Spec A 235
Tor (M, IV) 60
T(A) 231
tre 1:). tr .0(1:), tr B;.o [trace] 23] - 232
tr, tr [transfer mapsJ 81 82
vcd [virtual cohomological dimensionJ
226
V FP, VFL WFL scc "fìniteness
condilions" in the index of
tcrminology bdow
X(C),X(X} (Euler charactenstie] 243
i(r) 245
Z(r) [Euler eharaclerislic of a group n
247, 249
Zr(X), Xr(S) [equivarient Euler
characteristicJ 249-250,262
'(5) 254
l" [integers mod nJ vi
Z(C) [cycles] 4
lG, kG [group ring] 12, 14
lX, Z[X] [permutation module] 13
Subject Index
abetian groups (calculation of homology
oI) 121ft'
abetianization 36
abutmcnt 163
acyelic chain complex 5
contractibility of 24, 30, 132, 274
acyclic cover 168,265
acyclic models theorem 25
acyclic module (with respect to a
homology theory) 72
acyclic space 15
additive category 25
additive functor 33
admissiblc acyclic chain complex 129
admissiblc cxact sequence 129
admissible r -complex 258
admissible injection 129
arlmissible product 118
Alexander-Whitney map 108,286
amalgamation (or amalgamaled frcc
product) 49ff
tree associated to 53
anti-commutalivity 111, 118
arithmetic groups 217218
Euler characteristic of 254ft'
aspherical space I, 15 (see also
.. Eilenberg - MacLane space")
associated graded module 162
augmentation idea1 and map 12
augmemation-preserving chain map
23, 132
automorphism group 102
ouler 104
bar resolution 19
Bernoutti numbers 255
binary icosahedral group 47, 155
binary octahedral group 155
binary tetrahedral group 155
boundaries 4
boundary operator 4
cap produet 112- 1I3, 141
Cartan-Leray specttal sequence 173
chain complex 4
chain map 5
Chevallr;:y group 254
coboundaries 4
co-compact 217
cocycles 4
co-eft'aceable 73
coefficient system I (i7
co-extension of scalars 63
co-free module bb
cohomological dimension 184
virtual 226
cohomological functor 78
cohomologieatly Irivial module 148,
287
cohomology of a cochain compIex 4
cohomology of a group 57, 134, 170,
277
equivarianl 172, 180ft', 28lff
cohomology with compact supports
209
co-induced module 67


304
co-invariants 34
combinatorial path 42
commutator pairing 97
compact supports 209
complete diagonal approximation 138,
275
complete resolulion 132, 273
complete tensor product 137
composition product Il 5, 143
congruencesubgroup 40
conjugation action 48, 79, 80
connecting homomorphism 6,71,75
interpretation in terms of group
extensions 95
contraetible, contracting homotopy 5
(sce also acyclic chain complex ")
convergence of a spectral sequence 163
corestrietion map 80, 136
crossed homomorphism 45, 88
(see also derivation")
crossed module 102
cross product 109
cup product 109. 130ff,278
in cyclic groups 114
cycles 4
cyclic group (resolution, homology, etc.)
16,21,35, 58, 114
deck transformation 31
degree 4
derivation 45, 88
principal 60,89
ò-functor 75
lì-functor 78
diagonal action 55, 56
diagonal approximation (or map) 108,
138, 275, 286
differential 4
dihedral group 98
dimension
cohomological 184
geometric 185, 205
homoIogical 204
projective 152, 184,287
virtual cohomological 226
dimension-shifting 74,136
dircet limits and homology 121,195
discrete subgroups of Lie groups 38ff,
217ff
divided polynomial algebra 119, 125
divided powers 124
divisible module 66
doub1e complex 164
Subject Index
double coset fornmla 69,82
dual 28, 133, 145
dualilY group 221
Poincaré 222
virtual 229
duality pairing 145
dumity theorem 28, l44ff
dualizing module 221,276
effaceable 72
Eilenberg-Ganea theorem 205
Eilenberg-MacLane complex (or space)
15,40
exislence of 18, 19,205
Eilenberg trick 187-188
elementary abelian p-group 156,267,
271
ends 223
equivariant homology and cohomology
172, 180ff, 281 ff
equivariant Euler characteristic
249-250
Euler charaeteristic 164, 243ff
of a group 247,249
eva1uation maps and pairings 9, In,
145,234
exact functor 8
Ext 60
extension (group) 47, 86ff, 171,248,
257,281
extension of scalars 62
exterior algebra, exterior power 97,
119, 121
factor set 91
Farrell cohomology 277ff
filtration 162
finite chain complex, resolution 243
finiteness conditions l84ff
FL 199
FP, FPn. FP", 193. 195, 197
VFL 226
V F P 226
WFL 226
finite homological type 246
finitely dominated space 200
finitely presented module 193
F-Isomorphism 286
five-term exaet sequence 47, 171, 175
fil[ed-point-free representation 154
fixed-point theorems 11,181,246,261
flat module 29, 56
Subject Indel[
free derivative 45, 90
free G-set, G-complex 13
free group (homology, resolution, etc,)
16-17,37,247
Cunction complex 5
fundamental class 145,204
fundamenlal group of a graph of
groups 179
fundamentallemma of homological
algebra 22, 130
Gauss-Bonnel measure 253
G-complex (or G-CW-complex) 14
generallinear group 38
gencralized quaternion group 98, 155
geometric dimension 185,205
G-module 13
Goulieb's theorem 252
graded module 4
graph 52
graph of groups 179
groupexlension 47, 86ff, 171, 248,
257, 281
group ring (or algebra) 12, 14
G-set 13
Hattori-Stallings rank 233
Hirsch numbcr 186
H N N extension 179
Hochschild-Serre spectral sequence
171,281
Hom 56
Jt'6", 5
homological dimension 204
homological Cunctor 75
homology oC a chain compieI[ 4
homology oC a group 1, 35, 56, 135,
168, 172
equivariant 172. 180ff, 281ff
homotopy, homotopy equivalence 5ff,
19
Hopf's theorems 1,41
el[plicit formula Cor 46
idcmpotent 27, 234, 238
indceomposable 231,235,236
induced module 6O,67/f
injeclive module 26,65/f
invariants 27
invariant cohomology class 84
invertible ideai 27
305
K(G,I)-complex 15 (see also
.. Eilenberg- MacLane complex ")
knot groups 212
Krull dimension of H"'(G) 159
Kummer's criterion 270
KOnneth formula 7, 109, 120
Lefschetz fixed-polOt theorem 21,246,
261
length oC a resolution t I
Lie groups (discrete subgroups 01) 38ft;
217ff
long exact homology sequence 6, 71, 75
L yndon 's theorem 37, 44, 185, 228
mapping cone 6
Mayer-Vietoris sequence 51, 74, 178ff
Nakayama's lemma 150,236
nerve of a covering 166
nilpotcnt group (cohomological
dimension 01) 186,213
norm element 20, 58
norm map 58
normalized bar resolution 19
one-rclator group 37,44,185,228
ordered simplicial complex 227
outer aUlomorphIsm group 104
partially ordered sel (topological
coliccpts applied to) 261,262,291
perfcct group 46,96, 198
periodic cohomology and resolution 20,
133,153, 158,180,288
permutation module 13
p-groups
calculation of 110mology via 84
witl1 a cyclic subgroup ofindex p
97ft
with a unique subgroup of order p 99
with every abelian normal subgrou.p
cyclic 101
Poincaré duality group 222
Pontryagin producI 117
primary dceomposilion of homology
groups 84, 141
principal derivation 60,89

306
producI
cap 112--113,141
composition 115, 143
cross 109
cup 109, 130ft", 278
of ordere<! simplicial complexes 227
shuffie 118
tensor 7,)0, 55, 107, 137
projective dimension 152, 184,287
projective module 21, 26ft", 56
over a group ring 27, 149, 152,201
over a local ring 235
rank of 230ff
stably free 201
proper action 39
pro per r -comp1ex 226
pull-back 94
quaternion group 98, 155
Quillen 's theorem 159
rank
or a finitely generated abelian group
242
ofa nilporeO! group 186
or a projective module 230ff
reduced homology 211
regular cover 31
spectral sequence of 173
relabon module 43,44,90, 198
relative homological algebra 25, 129ft"
relative injective module 129
relative in jective rcsolulion 131
representable faclor 25
resolutiotl 10
bar 19
complete 132, 273
finite '99
offinite type 193
periodic 20, 133, 153
standard 18
uniqueness of 24
via lopology 14ft"
rcstriction map 80, 136
restriction of scalars 62, 69
Rim's theorem 152,287
Schanuel's lemma 192, 193
semi-direct producI 87
Serre's Iheorem 190
Shapiro's lemma 73,136
.
Subje<;;t Index
shufflc product 118
simple module 149
simplida1 complex (ordered) 227
assodaled to a partially ordered set
261"-262
simplidal product 227
Solomon- Tits Iheorem 270
Spcc A 235
spcciallincar group 39, I 57/f. 213/f,
229, 255
spectral sequence 162/f
split eXlension 87
stably free 201
Stallings-Swan theorem 185,223
standard resolution 18
strict anti-commutativity 118
slricl upper triangular group 38, 185,
213
suspension 5
Swan's theorem 240-241
Sylow subgroups (cakulaIion of
homology via) 84
symmetric group 48,85, 114
symplectic group 255,269
Tate homology and cOhomology 134ft",
170, 180/f
tensor product 7, 10,55, 107, 137
Tor 60
trace 231-232
transfer map 80, 83/f
tree 53
type FL, FP, elc.-see ""finiler1CSS
conditions ..
universal centrai extension %
universal coefficient theorem 8, 127,
170,198,202
upper triangular group 38, 185,213
virtual notions 225/f
weak cquivalence 5ff, 29
Yoncda's lemma 25
Zariski topology 236-237
zeta function 254
.
Graduate Texts in Mathematics
I T A KE un/ZA RIN G. Introduction to Axiomalic Sei Theory. 2nd ed.
2 OXTOBY. Measure alld Category. 2nd ed.
3 SCHAEI'I'ER. Topological Vector Spaees.
4 HILTON/STAMMBACH. A Course in Homological Algebra. 3rd printing.
5 MACI.ANE. Categories for Ihe Working Mathemaliciall.
6 HUGHES/PIPER. Projeclive Planes.
7 SERRE. A Course in Arithmetic_
R T AKEt:TI/ZARING. AxiomClie Sct Theory.
9 H U M PH REYS. 1 ntrodul"' ion lo Lie A Ige bras and Representation Thcol)'. 3rd priming,
revised.
lO COHEN. A CouIse io Simple Homotopy Thcory.
Il CONWAY. FunClions or One CompIe" Variable. 2nd ed.
12 BEALS_ Advanccd Mathematieal Analysis.
13 ANDERSON/FulLER. Rings and Catcgories of Modules.
14 GOI.UBITSKy/Gu[LlEM[. Slable Mappings :md Their Singularities.
15 BER RI' R[A N. Lcclure s i n FUnel i(1nal Anal ysi s and Operator Theory.
16 W[TI'R. Thc Slructure of Fields.
17 ROSI'NBI.....n. Raodom Processes. 2nd cd
18 H A L MOS. Mcasure Theory.
19 HAI.MOS_ A Hilbcn Space Prnblem Book_ 2nd ed.. revi!oed.
20 HUSEMOLLER. Fil1re Bundles. 2nd ed.
21 HUMPHRHS. Linear Algebraic Grnups_
22 BARNl's/MACK. An Algel1rak Inlroductinn In Mathematieal Logic.
23 GRI'lJF\. Lincar AIgel1ra. 41h ed.
24 HOL \1I'S. Gcomctric Funclillnal Anal ys i s a od ib Appl ications .
25 HEWITT/STROMBIOKG_ Real and AbSlraCI Aoalysb. 5[h printing.
26 MANFS. Algcbraic Theories.
27 K E L LE 'i. Gencral T opology .
28 ZAR[SKI/SAMUI'L_ CommUlalive Algebra. VoI. I.
29 ZAR[sK[/SAMul'L_ Commulative Algcbra. VoI. Il.
30 JACOHSON. Lcqures in Abstrac[ Algebra I: Basie Concepts.
3 I J ACOIJSON. Ledu res i n A b,traci Algehra Il: ti ncar Algebra.
32 JACOIJSUN. Ledures io Ab,traci Algehra 11I: Theory or Fields and Galois Theory
33 HIRSUI. Differemial Topolo!!y.
34 S PI"! ZER. Princi pie!> of Random Wal k _ 2nd ed.
35 WERMER. Banach Algebra, and Sevcral Complex Variables. 2nd ed.
36 KELI.I'Y/N....M[OK,\ et al. Linear Topnlogical Spaces.
37 MO:>;K. Ma1hemalical LogiL
38 G RA U ERI! F RnZSCHE. Several CompieI( V ariables.
39 ARnso"_ An Invilation tI' C*-Algebras.
40 KEMEI'iY/SNEluKNAPP. Dcnumerahle Markov Chains. 2nd ed.
41 APOSTOL_ Modular Functions and Dirichle[ Serie!> in Number Thenry_
42 StoRRI'. Linear Represenlations of Finile Groups.
43 GILLMAN/JERISON. Rings of Cominuous Funclions.
44 Kf.NDlG. Elementary Algebraie Geomelry.
45 .LoÈVE. Probability Theory I. 41h ed.
46 LoÈVE. Probability Theory Il. 4th ed.

47 MOISE. Geometric Topology in Dimensions 2 and 3.
48 SAC Hs!W u. Genend Relali vily for Mall1emalicians.
49 GRUENBERdWEIR. Linear Geomelry. 2nd ed.
50 EDWARDS. Fcrmat's Lasl Tl1eorern. 2nd prinring.
51 KUNGDIBERG. A Course in Differcmia] Geomelry.
52 HARTSHOR
E. Algebraic Geometry.
53 MANIN. A Course in Mathematical Logic.
54 GRAVER/WATKINS. Combinatorics witl1 Emphasis on the Theory of Graphs.
55 BRowN/PEARCY. Inlroduelion lo Open1lOr Theory I: Elemenls of Functional
Analysis.
56 MASSEY. Algebraic Topology: An Introduction.
57 CROWEL LI Fo x. lntmdUClion IO Knol Theory.
58 KOBLlTZ. p-ad ic N umbers. p-ad ie Anal ysis. and Zew- FUnel ions.
59 LANG. Cyclolomic Ficlds.
60 ARNOLD. Mathematical Methods in Classical Meehanics. 3rd priming.
61 WHITEHEAD. Elemenls or Homolopy Theory.
62 KARG;\POLOv/MIOKZLJAKov. Fundamentals of the Theory of Groups.
63 BOLLOBAS. Gmph Theory.
b4 EDWARDS. Fnuricr Series. VoI. l. 2nd ed.
65 WElLS. Differenlial Analysis 00 Complex Manifolds. 2nd ed.
66 WATERHOUSE. hnroduclioo IO Affine Gmup Schemcs.
67 SEKRE. Local Fields.
68 WEIDMANN. Linear Operalors in Hilbert Spaces.
69 LANG. Cyclntomic Fields Il.
70 M A SSEY. Si ng ular Homnlogy Theory.
71 FARKAslKRA. Riemann Surf
Kes.
72 S TI Ll WEll. CI assical T opology a od Combinatorial Gmup Theory.
73 HUNGERFORD. Algebrd.
74 D A VE
 PORT. Muhipl icali ve N umber Theory. 2nd ed_
75 HOOISl'H[LD. Basic Theory of Algebraic Groups and Lie Algebras.
76 ItTAK.'\_ Algebraic Geomelry.
77 HRKE_ Leclures on the Theory or Algebraic Numbers.
78 BURRIS/SANKAPPANAVAR. A Course in Universal Algebra.
79 W A II ERS. An Inlroduetion lO Ergod ic Theory.
80 ROIUNSON. A Course in the Theory of Groups.
81 FORSTER_ Leclures on Riemann Surfaces.
82 BOTT/Tc. Differential Forms io Algcbraic Topology.
83 WASHINGI-0N. Introduclion [(J Cyclolomic Fields.
84 I RE LA N DI ROSE N. A CI assical IntroduCI ion Modero N uml1er Theory.
85 EDWARDS. Fouricr Serie!>. VoI. Il. 2nd ed.
86 VAN LINT. Introduclion lo Coding Theory.
87 BROWN. Cohomology or Groups.
88 PIERrE. Associalive Algcl1ras.


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