Alejandro j-\dem
R. Jan1es Milgram
Cohomology of -roup
i"
Springer -Verlag Berlin Heidelberg N e'vv York
London Paris Tokyo Hong I{ong Barcelona Budapest
Alejandro Adem Department of Mathematics University of Wisconsin Madison, WI 53706, USA
Table of Contents
R. James Milgram Department of Applied Homotopy Stanford University Stanford, CA 94305-9701, USA
Introduction ................................................
1
Chapter 1. Group Extensions, Simple Algebras and Cohomology O. 1.
2.
Mathematics Subject Classification (1991): 20105, 2OJ06, 20Jl 0, 55R35, 55R40, S7S17, I8GIO, 18GIS, 18G20, 18G40
ISBN 3-540-5702S-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-S702S-X Springer-Verlag New York Berlin Heidelberg
3. 4. 5. 6. 7.
8. Library of Congress Cataloging-in-Publication Data Adem, Alejandro. Cohomology of finite groups/ Alejandro Adem, Richard James Milgram. p. em. - (Grundlehren der mathematischen Wissenschaflen; 309) Includes bibliographical references and index. ISBN 0-387-S7025-X 1. Finite groups. 2. Homology theory. I. Milgram, R. James. II. Title. III. Series. QAI77.A34 1995 SI2'.5S-dc20 94-13318 CIP This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereofis permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from SpringerVerlag. Violations are liable for prosecution under the German Copyright Law.
7 8 12 14 16 18 20 23 27 32 34 35 36 36 38 40 43
Chapter II. Classifying Spaces and Group Cohomology O. 1.
2.
© Springer-Verlag Berlin Heidelberg 1994 Printed in Germany Typesetting: Camera-ready copy produced by the authors' output file llsing a Springer TEX macro packagc SPIN 10078665 41/3140-5432 I 0 Printed on acid-free paper
Introduction .............................................. Group Extensions ......................................... Extensions Associated to the Quaternions .................... The Group of Unit Quaternions and SO(3) ................... The Generalized Quaternion Groups and Binary Tetrahedral Group ................................................... Central Extensions and 8 1 Bundles on the Torus T2 ........... The Pull-back Construction and Extensions .................. The Obstruction to Extension When the Center Is Non-Trivial .. Counting the Number of Extensions ......................... The Relation Satisfied by p,(gll g2, g3) ....................... A Certain Universal Extension .............................. Each Element in H~(G; C) Represents an Obstruction ......... Associative Algebras and H~(G; C) .........'................. Basic Structure Theorems for Central Simple IF-Algebras ....... Tensor Products of Central Simple IF-Algebras ................ The Cohomological Interpretation of Central Simple Division Algebras ................................................. Comparing Different Maximal Subfields, the Brauer Group
3. 4.
Introduction .............................................. Preliminaries on Classifying Spaces .......................... Eilenberg-MacLane Spaces and the Steenrod Algebra A(p) Axioms for the Steenrod Algebra A(2) ....................... Axioms for the Steenrod Algebra A(p) ....................... The Cohomology of Eilenberg-MacLane Spaces ................ The Hopf Algebra Structure on A(p) ........................ Group Cohomology ........................................ Cup Products .............................................
45 45
53 55 55 56 57 57 66
VI
Table of Contents
Table of Contents
5. Restriction and Transfer 6. 7. 8.
Transfer and Restriction for Abelian Groups ................. An Alternate Construction of the Transfer ................... The Cartan-Eilenberg Double Coset Formula ................ Tate Cohomology and Applications ......................... The First Cohomology Group and Out(G) ...................
. . . . .
69
Chapter V. G-Complexes and Equivariant Coho1110logy
71 73 76 81 87
O. 1. 2. 3.
Chapter III. Modular Invariant Theory
o. 1. 2. 3. 4. 5. 6.
Introduction ............................................. General Invariants ........................................ The Dickson Algebra ..................................... A Theorem of Serre ...................................... The Invariants in H*((Zjp)n; lFp ) Under the Action of 5 n The Cardenas-Kuhn Theorem .............................. Discussion of Related Topics and Further Results ............. The Dickson Algebras and Topology ........................ The Ring of Invariants for SP2n (IF 2) ........................ The Invariants of Subgroups of GL 4 (lF 2) .....................
. 93 . 93 . 100 . 105 108 . 112 . 115 . 115 . 115 . 116
Chapter IV. Spectral Sequences and Detection Theorems O. 1.
2.
3. 4. 5.
6.
7.
Introduction .............................................. The Lyndon-Hochschild-Serre Spectral Sequence: Geometric Approach ................................................ Wreath Products .......................................... Central Extensions ........................................ A Lemma of Quillen-Venkov ................................ Change of Rings and the Lyndon-Hochschild-Serre Spectral Sequence ................................................. The Dihedral Group D 2n ................................... The Quaternion Group Qs ................................. Chain Approximations in Acyclic Complexes .................. Groups With Cohomology Detected by Abelian Subgroups Structure Theorems for the Ring H* (G; IF p) .................. Evens-Venkov Finite Generation Theorem .................... The Quillen-Venkov Theorem ............................... The Krull Dimension of H* (G; IF p) .......................... The Classification and Cohomology Rings of Periodic Groups ... The Classification of Periodic Groups ........................ The Mod(2) Cohomology of the Periodic Groups .............. The Definition and Properties of Steenrod Squares ............ The Squaring Operations ................................... The P-Power Operations for p Odd ..........................
VII
Introduction to Cohomological Methods ..................... . Restrictions on Group Actions ............................. . General Properties of Po sets Associated to Finite Groups Applications to Cohomology ............................... . 54 ...................................................... . SL 3 (lF 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Sporadic Group Mll .................................. . The Sporadic Group J 1 . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . .
161 165 170 176 178 178 179 179
Chapter VI. The Cohomology of Symmetric Groups
O. 1.
2. 3. 4. 5. 6.
Introduction Detection Theorems for H* (5n ; IF p) and Construction of Generators .............................................. Hopf Algebras ........................................... The Theorems of Borel and Hopf ........................... The Structure of H* (5n ; lF p ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . More Invariant Theory .................................... H*(5n ), n = 6,8,10,12 .................................. The Cohomology of the Alternating Groups .................
181 . 184 . 197 . 201 . 203 . 206 . 211 . 214
117 Chapter VII. Finite Groups of Lie Type 118 119 122 124 125 128 131 134 140 143 143 144 144 146 149 154 156 157 159
1. 2. 3. 4. 5. 6. 7.
Preliminary Remarks ..................................... The Classical Groups of Lie Type .......................... The Orders of the Finite Orthogonal and Symplectic Groups ... The Cohomology of.the Groups GLn(q) ..................... The Cohomology of the Groups O~(q) for q Odd ............. The Cohomology Groups H*(Om(q);lF2) .................... The Groups H*(SP2n(q);lF2) ............................... The Exceptional Chevalley Groups .........................
. 219 . 220 . 227 . 231 . 235 . 240 . 241 . 246
Chapter VIII. Cohomology of Sporadic Simple Groups
O. 1. 2. 3.
4. 5.
Introduction .............................................. The Cohomology of Mll .................................. . The Cohomology of J 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Cohomology of M12 .................................. . The Structure of Mathieu Group M12 ....................... . The Cohomology of M12 .................................. . Discussion of H*(M12;lF2) ................................. . The Cohomology of Other Sporadic Simple Groups ........... . The O'Nan Group 0' N ................................... .
251 252 253 254 254 258 263 267 267
VIII
Table of Contents
The Mathieu Group M22 The Mathieu Group M 23
Introduction
268 271
Chapter IX. The Plus Construction and Applications O. 1. 2. 3.
4.
Preliminaries ............................................. Definitions ............................................... Classification and Construction of Acyclic Maps ............... Examples and Applications ................................. The Infinite Symmetric Group .............................. The General Linear Group Over a Finite Field ................ The Binary Icosahedral Group .............................. The Mathieu Group M12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Group J 1 . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . The Mathieu Group lvI23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Kan-Thurston Theorem ...............................
273 273 275 277 277 278 279 281 281 282 283
Some Historical Background
Chapter X. The Schur Subgroup of the Brauer Group O. 1.
2.
3.
4.
5.
Introduction ............................................. . The Brauer Groups of Complete Local Fields ................ . Valuations and Completions ............................... . The Brauer Groups of Complete Fields with Finite Valuations .. The Brauer Group and the Schur Subgroup for Finite Extensions of 0. .......................................... . The Brauer Group of a Finite Extension of 0. ................ . The Schur Subgroup of the Brauer Group ................... . The Group (0./71..) and Its Aut Group ....................... . The Explicit Generators of the Schur Subgroup .............. . Cyclotomic Algebras and the Brauer-Witt Theorem .......... . The Galois Group of the Maximal Cyclotomic Extension of JF .. . The Cohomological Reformulation of the Schur Subgroup The Groups H;ont(G rr ; 0./71..) and H;ont(G v ;0./71..) ............. . The Cohomology Groups H;ont(G rr ; 0./71..) ................... . The Local Cohomology with Q/Z Coefficients ................ . The Explicit Form of the Evaluation Maps at the Finite Valuations ............................................... . The Explicit Structure of the Schur Subgroup, S(JF) .......... . The Map H;ont(G v ;o./Z)------tH;ont(Gv ;Q;,cycl)' ............... . The Invariants at the Infinite Real Primes ................... . The Remaining Local Maps ................................ .
289 290 290 293 295 295 297 298 299 299 300 301 304 304 307 309 310 311 314 316
References ................................................. . 319 Index ...................................................... . 325
~(
This book deals with the cohomology of groups, particularly finite ones. Historically, the subject has been one of significant interaction between algebra and topology and has directly led to the creatiori. of such important areas of mathematics as homological algebra and algebraic K-theory. It arose primarily in the 1920's and 1930's independently in number theory and topology. In topology the main focus was on the work of H. Hopf, but B. Eckmann, S. Eilenberg, and S. MacLane (among others) made significant contributions. The main thrust of the early work here was to try to understand the meanings of the low dimensional homology groups of a space X. For example, if the universal cover of X was three connected, it was known that H 2 (X; A) depends only on the fundamental group of X. Group cohomology initially appeared to explain this dependence. In number theory, group cohomology arose as a natural device for describing the main theorems of class field theory and, in particular, for describing and analyzing the Brauer group of a field. It also arose naturally in the study of group extensions, N .(l)(n'),g') and this, in turn is (n', g') since >'(1) = id. Likewise, (n, g)(l, 1) = (n>.(g)(l), g) = (n, g) since >.(g) is a homomorphism N -tN. Hence (1,1) is a unit. Also,
(n, 9 )(>.(g-l) (n -1), g-l)
(n)...(g)()...(g-1)(n- 1 )),1) (n>'(l)(n-l),l) (1,1)
while
10
Chapter 1. Group Extensions, Simple Algebras and Cohomology
(A (g -1) (n -1) , 9 -1 ) (n , g)
(A(g-1)(n-1).\(g-1)(n),1) (A(g-1)(n-1n),1)
(A(g -1 ) ( 1), 1) (1,1) . Thus (A(g-l )(n- 1), g-l) is a 2-sided inverse to (n, g). We now show that the product is associative.
(n, g)( (n', g') (nil, gil))
(n, g)(n' A(g')(n"), g' gil) (nA(g)(n' A(g')(n")), g(g'g")) (n(A(g) (n')A(g )A(g') (nil)), g(g' gft)) (nA(g) (n')A(gg') (nil), g(g' gil)) (nA(g)(n'), gg') (nil, gft) (( n, g) (n', g')) (nil, gft).
Hence N x G with the new multiplication is a group. It remains to evaluate the map ¢: G---tOut(N). We have G'---+ N x G as the set of pairs (l,g). Associated to 9 E G we obtain (l,g)(n, 1)(1,g-1) = (A(g)(n), 1). Thus the map ¢: E---tAut(N) restricted to G is our original map (A). 0
Examples 1.4 Aut(Z/nm) = Aut(Z/n) x Aut(Z/m) when nand m are relatively prime. Also, Aut(Z/2 T) = Z/2 x Z/2 T - 2 for r ~ 3 with generators, multiplication by -1 giving the element of order two, and multiplication by 5 or -3 giving the element of order 2T-2. (Note that (1 + 4k)2 = 1 + 8k(1 + 2k), r 2 r 3 so (1 + 4)2 - == 1 mod (2T) while (1 + 4)2 - ¢. 1 mod (2T).) In general, if p is prime, then Aut(Z/pT) is given by multiplication by the elements of Z/pT which are not divisible by p, and there are pT-1(p _ 1) of them, which is thus the order of Aut(Z/pT). If p is odd then 1 + p has order pT-1, and hence generates the p-Sylow subgroup. The projection Aut(Z/pT)---tAut(Z/p) induced from the surjection Z/pT ---tZ/p is onto, but Z/p = IFp is a field so its multiplicative group, being finite, consists of roots of unity, and hence is cyclic and isomorphic to Z/(p-1). From this Aut(Z/pT) = Z/(p - 1) x Z/pT-1 for p an odd prime. As a particular example, Aut(Z/7) = Z/6 with generator, multiplication by 3. Explicitly, the automorphism is given by n I--t 3n mod 7, so its square is n I--t 9n == 2n which has order 3. Hence there is a homomorphism Z/3---tAut(Z/7) sending the generator T to multiplication by 2. Consequently (1.3) gives a group extension Z/7 lR+ .
14
2. Extensions Associated to the Quaternions
Chapter 1. Group Extensions, Simple Algebras and Cohomology
15
Also note that
Proof.
,-1(r,8 + s,8')f
Norm(a,8) = a,8w(a,8) = a,8w(,8)w(a) aNorm(,8)w(a) = Norm(,8)aw(a) =
Norm(,8)Norm(a) = Norm(a)Norm(,8).
o
Norm- 1 { +I} = S3 C lHI - {O}, the subgroup of unit quaternions, which is thus the kernel of Norm. We may embed R+-tlHI - {O} by setting "\(r) = r1. Then the composition is the map .,\-t.,\2 which is an isomorphism lR+-tlR+. Also, the image, im("\) C centerlHI - {O}, and we have a map
=
r,-l,8,
+ s,-l(3',
so the conjugation action of S3 on vV is a map ¢: S3-tGL3(lR), where GL3(lR) is the group of lR-linear isomorphisms of W. But also, Norm({-l(3,) = Norm((3), and, for (3 E W so (3 has the form v
ri
+ 3j + tij
Vii + Vjj + vijij,
we see that Norm( v) = r2 + 3 2 + t 2. Thus, the linear transformation of IV is length preserving in the usual metric on W, and consequently image( ¢) C 0(3), the subgroup of orthogonal transformations.
r: R+ x S3--+lHI - {O} defined by r(r, a) = .,\(r)· a. Thus we obtain the following structure theorem which describes the multiplicative group ]HI - {a}.
Proof. Conjugation by a(3 is conjugation first by a and then by (3. This shows that ¢ is a homomorphism. We now determine the kernel of ¢. Consider a non-zero mEW, w(m) = -m so m 2 = -Norm(m). Also, suppose v.lm, i.e., (V,1/1) = Vimi + vjmj + V'ijmij = 0, then vm E W since the coefficient of 1 in vm is -(v, m), and
Theorem 2.6 r is an isomorphism. Proof. First note that r is a homomorphism
r((r1' ad(r2' a2))
=
r(r1r2' a1 a 2) .,\( rl )A( r2) (al a2) A(rl)alA(r2)a2 r(r1' al)r(r2' a2)'
-vm = w(vm) = w(m)w(v) = m(3.
But the converse is also true. We have
Now note that r(r, 1) and r(I, a) are both injections, hence the kernel of r is determined by the intersection r(r, 1) n r(l, a) which is just the single element {I}! Finally, suppose ,8 E ]HI - {O}, then, setting v = Norm (,8) , we have "\(V)-I,8 E S3. Hence,8 = A(V)("\(V)-I,8). 0
We now study the group S3 of unit quaternions. We will construct a surjective homomorphism S3-tSO(3) with kernel 7L/2, a basic non-split extension which allows us to construct many other non-trivial extensions by specializing to subgroups. Note that S3 can be characterized as those a E JH[ - {a} for which w(a) = a-I.
Let W 3 C lHI be the (-I)-eigenspace of w, the "pure imaginary" subspace of vectors ri + z2j with r real. Thus, for ,8 E W, , E S3 we have
and it follows that
=
wh- 1 )w(/3)w(,)
,/3,-1
E
W if /3
E
W.
=
,(_/3),-1
Proposition 2.8 Let m, v E W be non-zero, then (m, v) = 0 if and only if mv = -vm. Proof of 2.8. It suffices to show w(mv) vm = -mv.
=
-7nv. But w(mv)
== w(v)w(m)
=
o
Corollary 2.9 If m, v E Ware non-zero and orthogonal, then m, v, rnv form a mutually orthogonal basis for W.
The Group of Unit Quaternions and SO(3)
wh/3,-l)
Lemma 2.7 ¢: S3-t0(3) is a homomorphism with kernel7L/2 = {±1} C S3. Moreover, the image of ¢ is exactly the component of 1, 80(3) C 0(3).
=
_,/3,-1
Proof of 2.9. mmv = m 2 v, mvm = -m 2 v, so m.lmv. Also v(mv) = (vm)v = -mv 2 so v.lmv. Moreover since Norm(mv) = Norm(m)Norm(v) f. 0 it follows that mv f- O. 0
Now, consider an element, = A + m E 53 with mEW, A real. Then = A - m, and we have
,-I
(A
+ m)m(A -
m)
(Am - Norm(m))(A - m)
(A2 + Norm(m))m m.
16
Chapter 1. Group Extensions, Simple Algebras and Cohomology
2. Extensions Associated to the Quaternions
N1
=1
.
A space having a non-zero homotopy group in a single dimension is call.ed an Eilenberg-MacLane space. Such a space is unique up to homod~oPY ~qU1~d· t) P G provided the 1menSlon 1S alence and one exists for any ( lscre e grou G· Ab r If the group o or 1 or the· dimension is greater than one but is ~ Ian. 11 d t d is G a~d the dimension n, the Eilenberg-MacLane spac: IS usua y eno ~ K(G, n). We will shortly introduce an ~xplicit constructlOn of the spaces e and Be which will make the constructlOn of the K(G, n) very easy. Corollary 1.4 A model fOT the classifying space for the product G x H of
two groups, Bexf[, can be chosen to be Be x BH·
Chap;,cr II.
Cia~;sifying
Spaces alld
CiOlljJ
1. Preliminaries on Classifying Spaces
CDhOl!lulugy
Proof. A principal G x H bundle over Be x BH is TrG XTrll
Ee x EH ------+ Be x BH but the product of two contractible spaces is itself contractible, so the result follows. 0 The set of homotopy classes of maps X -+Be which is denoted [X, Be] is equal to the set of isomorphism classes of principal G-bundles on X when X is a CW-complex, and the isomorphism is explicit, i'(Ee), the pull-back of the principal G-bundle Ee-+Be is defined as the set of pairs (x, l) E X x Ee for which f(x) = p(l). This is the bundle over X associated to f. In the case of the K(G, n) for n > 1 the set [X, K(G, n)] ~ Hn(x; G) and the isomorphism is again explicit. Note that Hn(K(G, n); G) = Hom(G, G) and if we set I- = {id} then {f} 1-1 J*(I-) E Hn(x; G) gives the correspondence. In particular, K(G,n) x K(G,n) = K(G x G,n) by (1.4), and Hn(K(G x G, n); G) = Hom(G x G, G) for n > 1 so that G is abelian. The element 1-+ E Hn (K (G x G, n); G) is the class corresponding to the sum
We now explicitly construct a model for the classifying space Be. Let O'n be the usual n dimensional simplex. It is given with barycentric coordinates as the set of all n + 1- tu pIes of non-negative real numbers (to, t 1, ... , t n ) with L:~=o h = 1. The n + 1 faces are the sets Fi where ti = 0, 0 :S i :S n. A more convenient coordinatization for our purposes is given by
Proof. Set Ti = to + tl + ... + ti-l' This defines a map from barycentric coordinates to the coordinates in the lemma. Conversely, given 0 :S Tl :::; T2 :::; .. , :::; Tn :::; 1, define to = Tl, ti-l = Ti -Ti-l, 1 < i < n, and tn = I-Tn· This defines the inverse map. 0 Let Ee as follows.
=
U~=o G
x o-n
X
(g, Tl,"" Tn, gl,"" gn)
Gn j(relations) where the relations are given
rv
+
map G x G-+G and, in the correspondence above between [X,K(G,n)] and Hn(x; G) we have that a + (3 is represented by the composition 1.5
X
L1 --+
X xX
c<x/3 --+
if Tn= 1.
Examples 1. G 1. Sl is the Eilenberg-MacLane space K('!L,I), and Ee-+S1 is the usual exp map lR __ Sl where exp(t) = e2Trit . 2. For the group G = '!Ljn we have a free action
S2m-l------+S2m-l
Here the 1\ over the coordinate Ti or gi means that we simply delete that coordinate. This space Ee admits a free G action\~ x Ee--+Ee, which is defined as
(g,(g,T1, ... ,Tn ,gl, ... ,gn))
=
lim cpm m.~oo
(gg,Tl, ... ,Tn ,gl, ... ,gn),
and then, using the relation, to pass to equivalence classes. It is routine to check that this is well defined and G-free in the quotient. Also Ee is contractible. The contraction is given by
Ht((g, Tl,"" Tn, gl, ... ,gn))
defined by T(Zl, ... ,Zm) = (PnZ1, ... Pnzm) where Pn = e / n is a primitive nth-root of 1. The quotient is the Lens space L;m-l. The inclusion s2m-l C s2m+l as the set of points (Zl"'" Zm, 0) commutes with the '!Ljn action and consequently induces an inclusion of quotients L;m-l '---+ L;m+l, so, using the Steenrod Recognition theorem and passing to limits we see that BZ/n = limm~oo L~m-1. 3. We have the principal (Hopf)-fiberings Sl-+S2m+l-+((]p'm, which, as above, under the inclusion s2m+1 '---+ s2m+3 induce inclusions ([]p'm '---+ cpm+l, and passing to limits we have
1-1
= (e, t, Tl + t, ... , Tn + t,g,gl,'"
,gn)
- {t1 tt i then the operations are identically zero, but the same definitions hold for operations in a generalized cohomology theory, and here there need be no such constraint on the operations. An operation a E O{ i, j, A, B} need not be a homomorphism. Indeed, we have the factorization of a(x + y) in terms of the composition of maps
54
2. Eilenberg-MacLane Spaces and the Steenrod Algebra A(]J)
Chapter II. Classifying Spaces and Group Cohomology .:1 xxy.. x X~XxX~B~ xB~ ~Bi .. A A A,
Axioms for the Steenrod Algebra A(2) There are elements 8qi, i 2: 0 in OS( i, Z/2, Z/2) which are uniquely specified by the following axioms. 1. SqO = id, 2 2. If dim(x) = n then Sqn(x) = x , 3. If i > dim(x) then Sqi(x) = 0, j 4. (Cartan formula) Sqi(x U y) = ~~ Sqj (x) U Sqi- (y). As a consequence of these axioms one can show that Sr/ is the Bockstein associated to the exact sequence Z/2-tZ/4-l-Z/2, and [Car] the Adem relations R( a, b)
since, for the fundmental class, we have I- I---' I-®1+1®I- I---' x+y. Consequently a(x + y) = L: A(X) U X(y) + Ext - terms if x *(a) = L: A ® X + Ext - terms. However, there is a circumstance when an operation must be a homomorphism. 2.3 A sequence of opera~ions as E O{s, i+s, A, B}, s = 0, 1, ... , called stable if as(O"(x)) = (-l)tO"(a s_dx)) where 0" is the suspension isomorphism ~efinition
IS
(J"
H*(X;A)-----+H*+1(EX;A) ,
~/~
and EX = 8 1 /\ X = 8 1 x X/(e x X U 8 1 x *). Such a sequence as is said to have degree i, and the set of such operations of degree i is denoted
S qa S qb
=
OS(i,A,B).
L
°
. .
(b - 1 -. a - 2)
J) S qa+b- j S q.i
for 0 < a < 2b. Moreover, A(2) is the graded algebra generated by the Sqi su b ject only to the relations SqO = 1 and the Adem relations R( a, b). The first few relations have the form Sql Sq2 = Sq3, Sql Sql = 0, Sq2 Sq2 = Sq3 Sql = Sq1 Sq2 Sql. One can prove that a minimal generating set for A(2) is {I, Sq1, Sq2, ... , Sq2i , ... }. Moreover, a basis for A(2) as a vector space over Z/2 is the set of monomials SqI = Sqil SqiZ ... Sqi,., I = (il' i 2, ... ,iT) with i j 2: 2i j +1 for all j = I, ... ,r - 1. These are called the admissible monomials. The first few are SqO, Sql, Sq2, Sq3, Sq2,1 = Sq 2Sql, Sq4, Sq3,l = Sq3S q l.
A stable operation is always a homomorphism since LJ: EX -l-EX x EX factors through the wedge EXv EX c EX x EX up to homotopy. Moreover the composition of stable operations is defined in an evident way provided only that the coefficients fit together. In particular the stable operations of type OS(*,A,A) form an algebra. Steenrod used methods from group cohomology [SE] to construct operations 8qi(j) E O{j, j + i, Z/2, Z/2}, i 2: 0 which together form, for each i, a stable operation, denoted 8qi, as well as
Axioms for the Steenrod algebra A(p)
(3(j) E O{~,j + 1,71(p,Z/p}, pi(j) E O{j,j +2i(p-1)/L/p,Z/p} for odd pnmes p. Agam the (3(J) give a stable operation denoted (3 and called operation~
There are elements pi E OS(2(p - l)i, Zip, Zip) for p an odd prime which are uniquely specified by the following axioms. 1. po = id, 2. If dim(x) = 2n then pn(x) = x P , 3. If 2i > dim (x) then pi(X) = 0, 4. (Cartan formula) pi(x U y) = ~~ pj(x) U pi-j(y). Again, as a consequence of these axioms one obtains Adem relations
the mod(p) Bockstein. It is the boundary map in the coefficient sequence associated to the exact sequence of coefficients Z/P-l-Z/p 2-l-Z/p ,
... -----+H*(X; 7l/p2)-----+H*(X; Zip) ~ H*+1(X; Z/p)-----+H*+1(X; Z/p2)-----+ .... Also, the Pi(j) give a stabl~ operation denoted pi. Composing these operations gives rise to an alge.bra of stable operations, the Steenrod algebra A(2) generated by the 8~~'s for p = 2, and the Steenrod algebra A(p) generated by (3 and the p~ when p is odd. The construction of these operations is intimately connected with the cohomology of (finite) groups, and we will give the details in Chapter IV, §7, but for now we record the axiomatic descriptions of the A(p) first given in the book by Steenrod and Epstein [SE].
55
pa pb
1 I I
= [~( _l)aH ((P - l)(b - t) -
°
1)
pa+b-t pt
a - pt
if a < pb. If a :::; pb then
pa{3pb =
[~(_l)aH((P-l)(b-t))j3pa+b-tpt
°
a - pt
[(a-1)/p]
-\-
I
~
°
(_l)aH-l ((P - l)(b - t) a - pt - 1
1)
p a+b- t j3pt.
56
Chapter
n.
Cla,ssifying Spaces and C;roup Cohomology
As above A(p) is the graded algebra generated by {3 and the pi, subject only to the relations po = id, (32 = 0 and the Adem relations above. Again there is a basis for A(p), consisting of admissible monomials. Let
I
=
(EO, iI, E1, i2, . , . , iT, ET )
57
3. Group Cohomology
,
with Ei = 0 or I, then I is admissible if i j 2:: pi j +1 +Ej+1 for each j 2:: 1. Corresponding to an admissible monomial is the element of A(p), /y. o pil ... pi (3f.r , and these monomials, together with pO = 1, form a basis for A(p) over Z/p. r
The Cohomology of Eilenberg-MacLane Spaces We need the following definition in order to give the structure of the cohomology rings of the Eilenberg-MacLane spaces. Definition 2.4 An exterior algebra is a graded algebra of the form
IF (e1, ... , eT , ••• ) subject only to the relations eiej = -ejei for all i, j, and when the characteristic of IF is 2, also the relation = O. Generally the generators are odd dimensional.
e;
The Steenrod operations described above generate O{ i, j, Zip, Z/p} when we add in cup products. Precisely, we have the following theorem ofH. Cartan and J.P. Serre, [Se1]. Theorem 2.5 1. Let p = 2, and suppose that I is an admissible monomial for A(2), then the excess of I, e(I), is i 1 - i2 - ... - iT' In these terms H*(B Zj2 ; 71./2) is the polynomial algebra on generators SqI (/"n) where I runs over all monomials of excess less than n. 2. Let p be an odd prime and suppose that I is an admissible monomial for A(p), then the excess, e(I), is i 1 - (p - 1)(i2 + ... + iT) - L:~ Ej. In these terms H*(B zjp ; Z/p) is the tensor product of a polynomial algebra Z/p[·· . ,pI (in),"'] when dim (pI (in)) is even, and and exterior algebra E(-··, pI (in),"') when dim(pI (/"n)) is odd, as I runs over all admissible monomials with e(I) < n. Examples. We have that H*(K(Z/2, 1);IF/2) is the polynomial algebra on a single one dimensional generator, IF 2 [/.,], while
The Hopf Algebra Structure on A(p) The Cartan formula defines a homomorphism of algebras c: A(p)->A(p) 0 A(p), pi I-t L:~ pj 09 pi- j, {3 I-t {3 09 1 + 1 0 (3. This gives A(p) the structure of a cocommutative, coassociative Hopf algebra. The basic facts in the theory of Hopf algebras are developed in section 2 of Chapter VI, VI.2, and the exposition is independent of the rest of the book. Among the most important elements in a Hopf algebra are the primitives, those B with c( B) = B 091 + 109 B. Such elements are characterized by the property that B( a U b) = B(a) U b ± a U B(b). They correspond to generators of the dual algebra A(p)*. Milnor analyzed the structure of A(p)*. His result [Mi] is Theorem 2.6 1. There are primitives Qi in dimension 2i -1 in A(2), Qi and A(2)* = 71./2[6, ... , ~i,."] where ~i is dual to Qi.
=
[Sq2
i
1
-
,
Qi-l],
2. Let p be an odd prime. There are primitives Qil Si, in dimensions 2pi-1 and 2(pi - 1) respectively. Moreover, A(p)* is the tensor product of a polynomial algebra on generators Xi dual to the Si and and exterior algebra on generators Ti dual to the Qi'
For A(2) the Qi are characterized by the fact that they are primitive 2i so Qi(a U b) = Qi(a) U b + a U Qi(b), and the fact that Qi(e) = e when dime e) = 1. Likewise, the Qi for p odd are characterized by the same formula for cup products and the fact that Qi(e) = (3(e)pi for dim(e) = 1.
3. Group Cohomology Let A be an Abelian group, then we define H*(G; A)
=
H*(Be; A)
and call these groups the cohomology groups of G with (untwisted) coefficients A. If H eGis a subgroup, the inclusion B H ' - t Be induces a map in cohomology (res~)*: H*(G; A)--tH*(H; A) called restriction. Lemma 3.1 Let Ne(H) be the normalizer of H in G, then there is an action of Ne(H)/H
H* (Hi A) Wa(H).
= We(H)
on H*(H;A) and im
[(res~r]
is contained in
58
Chapter II. Classifying Spaces and Group Cohomology
3. Group Cohomology
Proof For 9 E Nc(H) define a map rg: BH-+BH by (*)
rg ((tl, ... ,tn ,h1, ... ,hn ))
Clearly
rg . rgl = rggl
80 :
M, 80
(2:= Bi 1mJ = ~ Bimi .
Next, let our new module be the kernel of 00 and apply the same construction above to construct a surjective map 0~:C1-+ker(00). Then, including ker(oo) c Co = Us Z(G)mi defines 01 as the composition of inclusion with o~. Now, we can repeat this construction to obtain a resolution. More generally, we can require the modules in definition 3.2 to be projective (i.e. direct summands in a free module); all the subsequent constructions will work equally well, as the reader can verify.
so these maps fit together to give an action map
and consequently an action on cohomology rings
r; = id if 9 E H
---+
s
= (tl, ... ,tn ,gh 1g-l, ... ,9hng- 1).
Since rg ~ id for 9 E H, (1.9), it follows that above factors through
Il Z( G)mi
59
and the action
We extend rg to Be for 9 E Ne(H), by the same formula (*), and for each 9 E Ne(H) we have a commutative diagram
Example 3.4 An explicit resolution of Z (with the trivial action) is obtained by taking the cellular chain complex of Ee, since Ee is contractible and G acts freely and cellularly. Explicitly, the cells of Ee have the form 9 x (In X (91, ... , 9n) which we write as 91911921·· ·19nl· Consequently, Cn can be given as UZ(G)1911" ·19nl where the (91, ... ,9n) run over all n-tuples of elements of G with no 9i = e. The boundary map 0 is given by
0(1911·· ·19nl)
= 911921·· '19nl+ n-1
r;
Consequently (res~)* . r; = (res~)*, but rg is obtained from an inner automorphism on Be, so is homotopic to the identity, and the equation above becomes for all
Q:
E
o
H*(G; A).
Definition. The group Ne(H)/H = We(H) is called the Weyl group of H in G. There is an algebraic reformulation of the definition of group cohomology which allows us to generalize to the case where G acts in some non-trivial way on the coefficients A.
with the understanding that when 9igi+1 = e that term in the summand 1 is set equal to O. Thus 0(1911921) = 911921-191921 + 1911, but 8(191191 1) = 1 1 91191 1+ 191 1. This resolution is commonly called the bar construction and is written B(71). It is not hard to extend it to a general resolution of a (left) 7l(G)module M provided M is a free 7l-module. Set B(M) = B(71) 0z M and define the boundary o( 1911 ... 19n 1m) as above, except the last term in the formula above is replaced by
Proposition 3.5 Let ¢: M
-+ N
80
80
81
82
8i
O+--M +--Co +--C 1 +--C 2 +--··· +--Ci +--···
82
81
82
O+--N +--1)0 +--1)1 +--1)2+-- ...
are two resolutions of Mover 7l(G). Then there are 7l(G)-maps ¢i: Ci -+1)i, the dia9ram
o ::; i < 00, so o +--
M
o
N
where each Ci is Z( G)-free. Remark 3.3 For any Z( G)-module M we can construct a resolution. Indeed, take any set S of Z(G)-generators for M, mI, ... , mi,'" and define a map
81
O+-- M +-- Co +-- C1 +-- C2+-- ... 80
Definition 3.2 Let M be a Z( G)-module, then a resolution of Mover G is a long exact sequence of Z(G)-modules and Z(G)-module maps Oi,
be a Z( G) -module map, and suppose
80
1· +--
+--
80 +--
1)0
60
Chapter II. Classifying Spaces and Group Cohomology
commutes. 1I1oreover, given any other choices cPS, cP~, ... making (**) commute there are maps J-Li:Ci-+Vi+l, i = 0,1,··· so that 8i+lf-Li + f-Li-18i = cP~ - ¢i, 0 :::; i < 00. Remark. A map p,: C1 -+Vi+l with 8f-L+p,8
=
i -9 is called a chain homotopy.
PlOof· To begin the definition of cPo choose a basis e I, ... , er , ... for Co. Since 80 is onto, for each e'i we have ¢80 (ei) = Ai is in the image of 80 in the second resolution. Say Ai = 80 (f·i). Then define ¢O(ei) = fi and extend to a Z( G)-module map by freeness. Assume that ¢j defined so as to satisfy 8j ¢j = ¢j-18j for j < k. Choose a basis el, ... , er , ... for Ck and note that 8[cPk-1 (8e r )] = ¢k-2( 8 2er ) = 0 by assumption. Thus, by the exactness of the second resolution, cPk-l (8k (e r )) is contained in the image of 8k :V k -+V k- 1. Specifically, let 8k(fr) = ¢k-1 (8ker), define ¢k(e r) = ir and extend to a map (Pk by Z(G) freeness. To construct the maps f-Lk set f-L-l: M -+Vo to be the O-map, and assume f-Lj defined so that 8 f-Lj + J-Lj -1 8 = cPj - ¢j for all j < k. Then we proceed substantially as above. Let el, ... er , ... be a basis for Ck . We have
so.
3. Group Cohomology
Remark 3. 'l Even if Z(G) acts non-trivially on A we often write H*(G; A) or sometimes H;(G; A) when we want to be explicit about the twisting for Extz(o)(Z, A). These groups are called the cohomology groups of G with (twisted) coefficients A. Similarly we can define the homology groups of G with coefficients in A as H*(G, A) = Tor~o(Z, A) = H*(C*(EG) ®o A). However, it turns out that nothing new is obtained using homology instead of cohomology. In fact, later we will see that homology and cohomology can be glued together to form a Z-graded theory, known as Tate cohomology. Example 3.8 Let G = Z/n, and set I c Z(G) equal to the kernel of the augmentation map E: Z(G)-+Z defined by E(2:niTi) = 2:ni. A Z-basis for I is given by the n - 1 elements T - 1, T2 - 1, T3 - 1, ... , T n- 1 - 1. But Ti_l = (Ti-l+T i - 2+ . . '+I)(T-l), so, as a module over Z(G), I is gener~ted by T - 1. It follows that the map 8 1 : Z( G) -+ I defined by 81 (2: niTt) = 2:niTi(T - 1) is surjective. Note that 8 l (T r- 1 ) = (Tr - 1) - (Tr-l - 1) so that the images of 1, T, ... , T n - 2 are independent over Z. On the other hand 81 (1 +T +T2 + ... +Tn - 1) = O. It follows that the submodule of Z(G) generated by Eo = 2: EO g generates the kernel of 80 , and, moreover, this kernel is a single copy ~f Z. In particular, since gEe = Eo for all 9 E G it follows that the kernel is a copy of the trivial Z( G) module. But this is the situation we started with. Consequently, we can iterate and a resolution of Z over Z( G) is given as T-l
€
f-Lk- 18(er ) Consequently, set f-Lk(e r ) prooE
-
cP~(er)
= - fro
+ cPi(e r ) =
8k+1(fr) .
This completes the inductive step and the 0
Let A be any Z( G)-module. We define
Ext~(G) (!vi; A) as the
Eo
Z-- Z(G) +-Z(G) -- Z(G)
and
ith
61
T-l +- '"
Passing to Ext groups we have, for example Exti(Z/n)(Z, Z/n)
= H*(Z/n; Z/n) = Zjn
for all * ~ 0 since Hom(Z( G), Z/n) = Zjn and (T - 1)* becomes zero here while Eo becomes multiplication by n. On the other hand, if A = Z((n) with action T((}) = en(} where en is a primitive nth root of unity. Then we find i even i odd.
cohomology group of the cochain complex More generally, for any Z(Zjn) module A we have
for any resolution of M. From the proposition above, these groups are independent of the particular choice of resolution.
Remark 3. 6 Ext~(O) (Z, A) = Hi( G; A) if Z, A are both given as trivial Z( G)modules, since, as we have seen the cellular chain complex of Eo is a suitable resolution of 7l while Homz(c) (C*(E c ),A) = Homz(C*(Bc),A) if A is a trivial Z( G)-module. Consequently, these Ext groups are true generalizations of the cohomology of G. They are contravariant as functors of G and M, but covariant in A.
Hi(G; A) =
MO M O / Ez/n(M)' { ker(Ez/n)/(T - I)M
i = 0 i even i odd.
We now consider a method to derive resolutions from a free presentation of a group, i.e. an exact sequence
N
.
72
5. Restriction and Transfer
Chapter II. Cla,ssifying Spaces and Group Cohomology
Proof. We must check two things, first that the cochains b2i and e 2 i+l are co cycles , and second that they evaluate non-trivially on chains representing the generators in homology in these dimensions. We have
8b2i (e.) = b2i (8e.) = {b2~(ej+l - eJ·) = 0 J J b2~(TPel - ep ) = 0
when j < p, for j = p.
Similarly
8e 2i +1 (ej) = e2i+l(8ej) = e 2i + 1 (E}f(el
+ ... + ep ) = e 2i+l(E}fel) =
IHlel ,
and this is 0 mod (p). Also, in odd dimensions el + ... + ep represents a generator in homology, while el represents a generator in even dimensions.
o
Corollary 5.7 Assume G = Z/pi with i > 1 so that H i= {I}. Then we have a. tr: Hj (H; IFp)-tHj (G;rlp) is the zero map for j even and the identity when j is odd. b. res*: Hj (G; IFp)-tHj (H; IFp) is zero when j is odd and the identity when j is even.
Proof. We have LTjb2i(T-jel) = p, j=O p-l
= LTje2i+l(T-jel) = 1.
. res·
.
tr
.
O---+Cc ---+C H---+Cc---+O as long as the action of G on the coefficients A is trivial. Consequently, there is a long exact sequence which turns out to be a special case of the Gysin sequence 5.10 15. res·. tr. 5. res· ... ---+ H~( G; A) ---+ H~(H; A) ---+ H~(G; A) ---+ H~+l(G; A) ---+ '" .
BTl' xid
Be ---+ Be x Be ----+ BZ/2 x Be .
To prove (b) note that the inclusions H C G induces the chain map of minimal complexes
+ T + ... + TP-l )el
in odd degrees, in even degrees.
0
Remark S.B The special case {I} C Z/p is not covered by (5.7). Here, since Hi({l}' Z) = {O ,
More generally, we can use (5.5) and (5.7) to give the transfer and restriction explicitly for G any finite abelian group and H any subgroup. As (5.9) is essentially the only case we need in the remainder of this work we leave this calculation to the reader as an (important) exercise. However, there is a further case where the transfer is very useful. This is when G has an index 2 subgroup H, so we have the extension H <J G-tZ/2. Then the transfer tr: Ck-tCb fits into a short exact sequence
L1
j=O
ec--+(l { ei---+ei
Proof. Since the composition of transfers is the transfer of the composition we can assume H = (Z/p)n-l has index pin G. Moreover, after a change of basis we can assume H = {I} x (Z/p)n-l C (Z/p)n. Then the result follows from (5.5). 0
When the coefficients A = IF 2 the map X turns out to be a 1--+ aUf for all a E Hi(G;IF2) where f E Hl(GiIF2) is B;(e), e E H1(Z/2iIF/2) is the non-zero class, and B7r is the map induced by the projection G-tZ/2. This is actually quite easy, just track back using the chain level definition of 8 and compare it with the cup product associated to the composition
p-l
tr(e 2i+l)(el)
73
Z
when when
~ > 0, ~ = 0,
we see that the transfer is simply multiplication by p in degree O. In particular, with IF p-coefficients it is identically zero.
As a special case note that the Gysin sequence gives a second proof that
H*(Z/2; IF 2 ) = IF2[e] which is the key step in 4.4. An Alternate Construction of the Transfer Geometrically we can view the transfer as follows. First, the map p: BH-tBe induced by the inclusion H "--+ G can be thought of as a covering with fiber the set of right cosets of H in G by simply regarding B H = * X H EG and letting p be the projection onto Be. Then the chain map C#(Be)-tC#(BH) associated to the transfer is given as
!gl!" ·!gn!1--+
L
Hgvlg 1 ! .. ·Ignl
Hgv
Corollary 5.9 Let G = (Z/p)n be an elementary p-group and suppose H S; G is a proper subgroup. Then tr: H* (H; IF p)-tH* (G; IF p) is zero in all degrees.
where the H gv run over the right cosets of H in G. That is to say, one takes the sum of all the cells in the inverse image of a cell of Be.
74
Chapter II. Classifying Spaces and Group Cohomology
We can model this more formally be considering gl) ... ,gn as representatives for the right cosets of H in G so G = Ui= 1 H gi, and define a map
Tr: Ee--1- (Eet by B 1-+ (g1 B, ... ,gnB) as B runs over Ee. (This is the geometric analogue of summing over fibers.) Note that gB 1-+ (g1gB, . .. ,gngB), and writing gig
= h(J'g(i)g(J'g(i)
where a g is the permutation of cosets associated to g, we obtain a homomorphism of G to the wreath product G ~ Sn which is defined in (IV.I),
5. RestrictiOll and Transfer
75
is A-equivariant and (Ee)n x ESn is H2Sn-free and contractible. Consequently h passes to quotients and induces the map
ETr X
tr),: Be--1- (* XH Eet xSn ESn ':::::: Bmsn . tr,\ is, up to homotopy, the map of classifying spaces induced by the homomorphism A. On points this representation of it has the form
where Xl) ... , Xn are the points in the covering * XH Ee-+Be lying over x. In fact, on the level of homology the map tr), is independent of the choice of section gl, ... ,gn used in the definition of ).. since we have
A is called the Frobenius map associated to the section (gl,' .. ,gn)' Thus
Tr· 9 = A(g)Tr, i.e. for all x E Ee, 9 E G we have Tr(xg) = A(g)Tr(x). In particular H (Sn acts on (Ee)n and Tr is A-equivariant. On the other hand (Ee)n is not H 2Sn free though it is Hn-free. However, if we take the symmetric product spn( * X H Ee) then Tr induces a well defined and continuous map
Lemma 5.12 Let g~, ... ,g~ be a second set of right coset representatives of H in G and A' the associated Frobenius homomorphism, )..1: G -+ H (Sn' Then A' and A differ by an inner automorphism of H ( Sn. Proof. We have g~
= higi ,
1 :::; i :::; n, so
g ~g "
hI
I (J'g (i)g(J'!I (i)
higig -
--1
I
hih(J' 9 (i) h(J' 9 (i)g(J' 9 (i) Using tr we can give a geometric construction of the transfer for a E Hm(BH; A) where the A are untwisted coefficients. Let a be represented by a: BH--1-K(A, m)
= BA
.
and conjugation by (hI, ... , hn' 1) takes A to
5.11 and it is a good exercise with chain approximations to verify that (a . tr)*(l.,m) = tr(a) agrees with the previous definition of the transfer. We can actually carry things a bit further, taking advantage of the Frobenius homomorphism A. Let h: Ee---+Esn be any (CW) map which is equivariant with respect to the homomorphism p. A: G--1-H 2 Sn--1-Sn which gives the permutation action of G on the right cosets of H. Then
o
Finally, we note that our second description of the transfer, 5.11, actually factors through the map tr), since the map
From IL1.8 B"A is an abelian topological group, so there is a natural extension of a to spn(B H ), n
)..1.
n
P({Xl,'" ,xn,w})
= LXi
E spn(* XH Ee)
1
gives us a factorization of tr as P . tr),. This last construction of the transfer, using tr,\ is actually very important in applications since it can be generalized substantially. In fact, given any functorial method of associating cohomology classes B(a) E H*(H (Sn; A) to cohomology classes a E H*(H; A) we obtain associated cohomology classes in H*(G; A). It is this principle, first used by Steenrod, which enables us, in Chapter IV, to construct the Steenrod operations. The principle is also very important in homotopy theory where it provides the basis for both the Kahn-Priddy theorem, [KP], and the Snaith splitting theorem [Sn].
76
Chapter II. Classifying Spaces and Group Cohomology
6. The Cartan-Eilenberg Double Coset Formula We now describe a useful method for computing the restriction map using double cosets which was first developed by Cartan and Eilenberg [eEl. Let G be a finite group and H, KeG subgroups. For a given Z(G)-module A, we will consider the composition tr~
6. The Cartan-Eilenberg Double Coset Formula a disjoint union of left cosets. Hence [G:K] (1 ).
77
= L:i[H:HnxiKxilj, proving
For (2), let F* be a free resolution of Z over Z(G), and <jJ E HomK(Fn , A):
res~ . tr~¢ = res~ ( L
g<jJg-l)
gEG/K
(res~)*
~ (2t Zj'X'q,x, Zji ,)
H*(K; A) ---+H*(G; A) ---+H*(H; A) .
1
First some notation: cx: H*(H; A)-tH*(xHx-\ A) (x E G) will denote the isomorphism induced by the homomorphism
in HomH (Fn, A)
o
HomH(C; A)---tHomxHx-1 (C; A) given by cx(J)(u) = xf(x-1u). Now take a decomposition of G into double cosets:
Corollary 6.3 If H<JG, thenforb E H*(H;A) we have res~.tr~(b)
for x
Remark 6.1 The double coset decomposition can be understood as follows. Given
c
G the left coset decomposition of Gover f{ defines a homothen the image breaks up int.o separate orbits, or, equivalently, ¢(H) C SkI X ... X Sk,. c S[C:Il.
(~ ~~,). Since
H*(PSL2(lF2n);lF2) ~ lF 2[X1, ... ,xn ]71/(2
0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
-3 -1 3 1 3 -1 -3 -3 -1 2 -1 -1 -3 -1 -3 -3 -1 -1 2 -3 -3 -3 -1 3 1 1 3 -2 3 1 1 3 1 3 -2 3
-3 -2
with normalizer the semi-direct product of (71/2)n with the subgroup N = 7/.,/(2 n - 1) = lF e 2n embedded in PSL2(lF2n) as the diagonal matrices ~ 1-+
= lF 2 [X1, X2, X3]71/7X
T
71/3 .
In Chapter III we discuss the techniques for determining these invariant subrings. An important thing to note is that they are closed under the action of the Steenrod algebra A(2).
7. Tate Cohomology and Applications We now describe a version of cohomology obtained by using a complete resolution instead of an ordinary one, see (11.3.7) for further remarks. By definition, a complete resolution of 7/., (the trivial ZG-module) is an acyclic 7l-graded complex F* of projective 7/.,G-modules which agrees with an ordinary projective resolution of Z in non-negative dimensions. To construct such a complex we proceed as follows. Let F* be a free resolution of Z where Fo = 7/.,G. Now c?ilsider its 7l-dual, F*. This is now a backwards free resolution (a coresolutlOn) of 7/." as 7/.,G is isomorphic to its dual. We now glue the two resolutions where they end (begin) respectively.
82
Chapter II. Classifying Spaces and Group Cohomology t--
C1
t--
~
CO ~
Co
f--
C1
7. Tate Cohomology and Applications t--
/ Z
Let N = :L:: EG 9 be the trace (or norm) map from Fo to F , this clearly factors throu~h Z, as the augmentation followed by the inclusion of invariants. Now we define F* as the chain complex O
i, :F.i = { F F-i-l,
if i ~ 0 if i < 0
endowed with the inherited differentials plus the glueing (norm) map in dimension zero. As for ordinary resolutions, one can show that complete resolutions together with the augmentation map F* -+ Z are unique up to chain homotopy, and hence can be used for homological definitions unambiguously. We can now define Definition 7.1 For any ZG-module M, the Tate Cohomology of G with coefficients in M is defined for all i E Z as
jji(G, M) = Hi(HomG(F*, M)) . Denote by I the augmentation ideal in ZG and by N the norm map as above. Then we can identify Tate Cohomology as follows
jji(G M)~ , -
Hi(G,M), H-i-I(G,M), MG/NM, KerN/1M,
I
if i > 0; if i < -1; if i = 0; ifi=-l.
The above is clear except in dimensions less than -1. For that we use the natural isomorphism HomG (F* ,M) ~ F* 0G M (derived from the fact that F* is ZG-free) and the definition for homology (3.7). Note the following simple fact: if F is a free ZG-module, then jj* (G, F) == O. This follows from the fact that F has a trivial projective resolution (itself) and that FG = NF, KerN = IF. As a consequence the same must be true for any projective ZG-module. An additional advantage of Tate cohomology is that it allows us to consider homology and cohomology at the same time. It will enjoy the same general properties as ordinary cohomology with respect to short exact coefficient sequences, restriction and transfer, maps induced by group homomorphisms etc .. The following is also a very useful device, known as "dimension-shifting": LelTIlTIa 7.2
83
For any finitely generated ZG-module M and integer i, there
exists a finitely generated Z-torsion free ZG-module [2i(M) such that for all nEZ
Proof. Given any ZG-module M, we may map a free module :F onto it, with kernel K. It follows from the long exact sequence associated to this sequence, that jjn ( G, M) ~ jjn+ I ( G, K) for all n E Z. Now K embeds in a free mod ule as Hom(Z, K) p,
then
'!lIp. Proof. If A eGis maximal, normal, abelian and
But:r centralizes B, so in the exact sequence
1---+ B ---+ A ---+ '!l I p---+ 1 r acts trivially on both B and '!lIp. This induces a long exact sequence in cohomology
then we are done. With A as above, if there is an N and C(N) C C(G) then consider the homomorphism 7r
<J
G with GIN = '!lIp
P
¢: G---+GIN ---+C(N) c G By assumption, the terms with coefficient group A are 0 for v coboundary map is an isomorphism .. On the other hand
while, since '!lIp acts trivially on B,
HV('!llpi B) = ('!llpyank(B). Hence, rank(B) must be 1, and we have that
By assumption A is not cyclic so
A = B x '!lIp
> O. Thus, the
where p: '!llp--+C(N) is some injection. We have Ker(¢) = N and so ¢2 is trivial. Define
f..L: G---+G by f..L(g) = 9 . ¢(g). Clearly, f. L is a homomorphism, but since f..L2 = id, f. L is actually an automorphism. Suppose f. L were an inner automorphism. Then there is an h E G and f..L(g) = hgh- 1 for all g E G. Since f..L(n) = n, for all n E N it follows that h E Cc(N). If h does not belong to N then Cc(N) . N = G and (h, C(N)) = Cc(N) so h E C(G), and f. L is the identity. If hEN then h E C(N) c C(G) and once more f. L is the identity. Hence j.J must be an outer automorphism. But all remaining cases are classified by the previous two propositions, and it is easy to show the theorem is true for DB and the semi-dihedral groups. Indeed, for the semi-dihedral groups this is true because we have
n 1 D2 n = 7L/2 - X-I 7L/2 c 7L/2 n
X
-1+2n-1 7L/2 = SD 2n+1 ,
Chapter III. Modular Invariant Theory
and C(D2n) = C(SD2n+l). Thus, the construction above gives an outer automorphism of SD 2 n+l. For
Ds = {x,y!x 2 =y2=(xy)4=1}, the map interchanging x and y is a suitable outer automorphism. The theorem follows. 0
Remark. 8.9 also holds if p is odd and G is abelian since in this case G either has a direct summand 7L/pi with i ~ 2 and hence !Out! is divisible by (p - l)pi-l, or is an elementary p-group with Out(G) = GLn(lFp). When p = 2 and G is abelian 8.9 continues to hold provided IGI > 4.
o.
Introduction
In this chapter we discuss the role of classical invariant theory in determining and analyzing the cohomology of finite groups. Typically, one has a subgroup of the form H = (7L/p)n C G and we note that im(res*: H*( Gj TFp)-+H*(Hj lF p)) is contained in the ring of invariants under the action of N G (H) / H on H*(H;TFp), (II.3.1). In some cases, see e.g. (11.6.8), it is possible to describe the entire cohomology ring of G in this way, but more often they contribute important but incomplete portions which are assembled using restriction maps to give the most important pieces of H*(G; lFp), see (IV.5). In the first section we give some of the basic techniques for determining rings of modular invariants. In §2 we study the Dickson algebras, the rings TFp[X1' ... ,xnJGLn(lFp ). These algebras playa basic role in describing the cohomology rings of the symmetric groups in VI. In §3 we apply some of the results of §2 to prove a very important theorem of Serre. Then in §4 we determine the groups H*((7L/p)n; TFp)Sn which are very useful in understanding the cohomology of the groups of Lie type, VII. Finally, in §5 we prove the Cardenas-Kuhn theorem, one of the most important and effective tools in the theory, and one which underlies most of the calculational results in the remainder of the book. This is by no means an exhaustive exposition of invariant theory, which is a vast subject with many ramifications into algebra and combinatorics. A good reference for further discussions is [StJ.
1. General Invariants We begin by considering the group A 4 , the alternating group on 4 letters. -It can be written as a normal extension
94
Chapter III. Modular Invariant Theory
1. General Invariallts
where K ~ 2/2 x 2/2 is the Klein group with generators x, y and the ~enerator T of 7l/3 acts by T(x) = y, T(y) = x + y. Passing to cohomology
It follows from (II.3.1) and (II.4) that
im(H*(A4;lF2))
'--t
H*(K;lF2) = lF 2[a,b]
(where a is dual to x and b is dual to y) is contained in the subring of invariants
lF 2 [a, bJZ/3 . The action is given by T(a) = b, T(b) = a + b on the generators and it extends multiplicatively to the entire polynomial algebra. Thus a2b) = (T(a))2T(b) = b2 (a + b) = b2 a + b3.
T(
How can we obtain the ring of invariants lF 2 [a b]Z/3? The simplest method d t' fr h th . , , a mg om t e 19 century, is as follows. If we consider the effect of field extension lF 2 [a, bJ ::::;:. lF4 @1F 2 lF 2 [a, bJ
=
lF 4 [a, b] ,
95
2: ni Vi where the Vi run over the irreducible representations and n'i is the dimension of Vi. Consequently, the trivial representation can only appear once, and the result follows. 0 In particular, applied to the situation above, we see that IF 4 is a splitting field for 2/3 and IF 4 @JF 2 I = IF 4 EB IF 4 where the first representation is T I---t (3, a primitive third root of unity, while the second representation is T I---t The claim, (*), follows immediately from this. More generally the argument above shows
(5.
Theorem 1.2 Let G be a finite group, and suppose the order oj G is prime to the characteristic oj a field IF = IF pr, then, given an action of G on the n dimensional vector space IF \Xl, ... , Xn I with 'basis Xl, ... , x n , it extends to an action on the polynomial ring IF[Xl' ... ,xn ] and, for any extension field JK/lF
we have
with the same action of 2/3, we claim first
Indeed, the action of 2/3 extends to a (graded) module structure over the gr?up ring lF2(2/3) by setting (2: AiTi)w = 2: Ai(Ti(W)). But, since 2 is pnme to the order of 7l/3, the group ring is semi-simple by Maschke's Theorem and thus ,splits as a direct sum
On the other hand, after tensoring up, we can split the vector space into the irreducible representations afforded in IK(Xl,' , . ,:rnl, and that usually makes the calculation of the invariants in degree i much easier. In counting the number of invariants we often find the Poincare series convenient. Here the Poincare series of an invariant subalgebra (R*)C ~ R* of a graded IF -algebra is defined as 00
lF2(71/3)
= lF2 EB I where
I ~ lF4 .
Here the action of 7l/3 on the IF 2 summand is trivial, and the action on I is multipli:ation by the units 7l/3 ~ lF4. (Explicitly, the idempotents which gIve the splItting are (1 + T + T2) for the lF2 summand and (T + T2) for the IF4 summand.) Consequently, in each dimension we have a splitting
b~
lF2[a, b]i
= IlF 2 [a, bJi EB lF 2[a, b]~/3 .
pc(R*)(t)
Proof. Und~r these circumstances the regular representation, i.e. the representatIOn VIa left multiplication on the group ring, has the decomposition
L dim
Jl3' ;-2b C onsequen tl y, TJiJJ ;-t+2J fiJ) summand IS I 2 = ':>3 1 2' and the invariants now have a basis of the form
f1f~· with i
This splitting is preserved on tensoring up, i.e. tensoring over IF 2 with a larger field.
Le~ma 1.1 Let IF be any field oj characteristic prime to the order oj the fimte group. G, and let IK/lF be a splitting field Jor G, i.e. a field over which the g~ou~ nng decomposes as a direct sum 2: Mi (IK) where kIi (IK) is the i x i matrtX nng over IK. Then, there is one and only one IK summand in the sum above for which the action map G x JK-+JK is trivial.
=
+ 2j == 0 mod
3.
This equation implies j == i mod 3, so the general solution is hhfrl Jim, Jf Ji Jfl Jim, or ffl Jim. It follows that as a ring lF 4 [a,bjz/3 ~ lF 4 [J{,Ji](hh) with the single (cubic) relation (hh)3 = Ur)(Ji). In particular this ring has 2 Poincare series 1 + x ~3 , and so must t.he ring of invariants IF 2 [a, bl z / 3 . (1 - x ) However, we cannot assert that the relation is that holding above. In fact, calculating, we have
i
96
Chapter Ill. Modular Invariant Theory
1. General Invariants
(a + (3b)(a + (ib) = a2 + ab + b2 3 a + (2a2b + aab 2 + b3 3 2 a + (ia b + (3 ab2 + b3.
Sometimes it is possible to filter V via invariant submodules VI C V2 C '" C Vk = V where we already know the invariants on the quotients Vi/Vi-I and piece together the structure of the invariants on V from that of the quotients. The key technique here involves the consideration of a single extension
Thus, generators in degree three over lF2 are A = ff + f? = a 2b + ab 2, and B = (3ff + (If? = a 3 + a 2b + b3. Reversing the calculation ff = (lA + B, f? = (3 A + B. It follows that for these generators the invariant algebra becomes
j
lFda, b]Z 3 since A2
+ B2 + AB
=
lF2[A, B](C)/(C 3 + A2
is the expansion of
ff f?
+ B2 + AB)
Applying 11.6.8 we obtain
Using II.5.1 we can in fact prove f{
This gives rise to the long exact sequence in cohomology
,
Theorem 1.3 H*(A 4;lF 2) ~ lF 2[a,bl Zj3 ~ lF 2[A,B](C)/(C3+A 2 +B 2 +AB) with deg(A) = deg(B) = 3, and deg(C) = 2.
Theorem 1.4 Let
97
be normal in G and suppose [G: K] is prime to p, then
im (res* (H* (G; lFp)-+H*(K; lFp ))
which is useful because HO (G; A) = A G. From this point of view the most important step in the determination of V G is determining the connecting homomorphism 6. Remember that this is determined by lifting the element in CO(V/VI ) which represents the cohomology class to CO(V), taking the coboundary there, and then lifting this coboundary back to CI(Vl)' Here are some examples of how this process works.
Lemma 1.6 a. Let 7l/2 act on IF 2 [Xl, X2] by interchanging the generators then IF 2 [Xl, X2]Z/2
is exactly
= IF2[XI + X2, XIX2]
.
b. Let 7l/2 act on IF 2 [XI,' " , X4] by interchanging the first and second generators and the third and fourth generators. Then j
IF 2[Xl,X2,X3,X4]Z 2
We now discuss some techniques which enable us to do expicit calculations in special cases. Here is an extension of (1.4) which is very useful in building up invariant subrings for G from the knowledge of the invariants for certain subgroups.
where M2 = XIX3
+ X2X4
=
IFdxI +X2,X3 +X4,XIX2,X3 X4](1,M2)
satisfies the relation
Lemma 1.5 Let p be a prime, G a finite gro'up, and G = UgiH be a coset decomposition with respect to H where Sylp( G) c H. Let V be any IF p( G)mod~lle and VI-! the lFp(H)-invariants. Then ~ gi maps VH to itself and has image exactly VG.
Proof. Let A = IF 2 [XI + X2, XIX2] C IF 2 [XI, X2]' Then we can write IF2[XI, X2] as the direct sum AEBAxl. In particular there is an exact sequence ofIF 2(71/2) modules A - + IF 2 [Xl, X2]-+ AXI,
Proof. Let 9 be an element of G. Then 9 ~ gi = ~ g"-
We now return to the proof of the lemma. The double coset formula gives that res* . tr*: H* (K; IF p)-+H* (G; IF p)-+H* (L; IFp) can be written as the sum of the compositions
•
There are certain important cases in which we can be more precise about im(res*: H*(G;IFp)--+H*(L;IFp)) . 't' 5 2 LCKCG is a closed system, also called a weakly closed sys-r->"• • 1 d . t D e fi III IOn. tern, if every subgroup of K which is conjugate to L m G IS a rea y conJuga e
to L in K.
Proof of 5.4 We can regard the double cosets LgK as corresponding to the orbits of the left co sets of K in G under the action of L. From this point of view the coset gK is fixed if and only if g-l Lg C K, and hence, under our closure assumption and choices of the gi, if and only if gi E Nc(L). Moreover the total variation of gi is LgiNK(L) = giNK(L). 0
Since L is p-elementary and tr* == 0 for any V the gi E Nc(L), and the theorem follows.
s: L we need only sum over 0
From this we obtain the Cardenas-Kuhn theorem,
s:
Theorem 5.5 Let L K ~ G be a closed system with Lap-elementary subgroup of G. Suppose also that IWc(L): WK(L)/ is prime to p, i.e. WK(L)
contains a p-Sylow subgroup of Wa(L), then the image of res*: H*( G; IFp)-+H* (L; JF p)
114
6. Discussion of Rela.ted Topics and Further Results
Chapter III. Modular Invariant Theory
6. Discussion of Related Topics and Further Results
is exactly equal to the intersection
H*(L; IFp) Wc(L) n im (res*: H*(Kj IFp)---'lH*(Lj lF p)) Example 5.6 The 2-Sylow subgroup of the sporadic simple group M12 is given as the semi-direct product (71/4 x 7l/4) XT (71/2 x 7l/2), where if a, b are the generat?r~ of the ~wo. copies of 7l/4 and c, d generate the two copies of 7l/2 then catbJ c = a-tb-], dad = a, dbd = ab- 1 . The subgroup V = (71/2)3 = (a 2 , b2 , c) is normal in this group with WSy12(M12) (V) = D s , and we will see in res·
.
Chapter VIII that the image of restriction H*(SyI2(M12)'lF2)---+H*(V'IF ) . exactIy H* (V; IF2 )D 8. Moreover, it is true that V c SyI2(M12) " IS C M12 is 2a closed system. Thus the Cardenas-Kuhn theorem gives us the image in this group of H*(M12; IF'2)' This group SyI2(M12) ~ccurs in several other contexts as well. It turns out that there is a unique non-split ~xtension
(71/2)3
~ E---'lGL 3(IF 2 )
which has SyI2(M12) as .its 2-Sylow subgroup. (Here the action of GL 3 (IF 2) on (71/2)3 = IF'~ is the usual one.) Consequently, WE(V) = GL 3 (lF'2) contains W Sy12 (M12) (V) as a 2-Sylow subgroup and im (res*: H*(E; IF'2)-----+H*(V; IF2)) is exactly IF'2[el, e2, e3]GL3(JF ) = IF'2[d 4, d6 , d7 ]. This group E is contained in the exceptional (continuous) Lie group G2 as a maximal (finite) subgroup, and the composition 2
res·
115
res·
There are some further special cases where calculations have been made or where there are unexpected connections between the topics in this chapter and other areas of mathematics. The Dickson Algebras and Topology
As we point out in (VIII.5) 1 the cohomology of the classifying space of the fourteen dimensional compact Lie group 02, H*(B02; lF2) = lF 2 [d4, d61 d7] is GL3 2 a copy of the Dickson algebra IFd x 1' X2, x31 (lF ) • It is known that the Dickson algebra lF 2 [X1,'" ,xn ]CLn(lF2) for n ~ 5 cannot be realized as the mod (2) cohomology ring of any topological space, and there were claims in the literature that neither could the Dickson algebra for n = 4, But recently W. Dwyer and C. Wilkerson, [DW] , have actually constructed such a space, V4. Its loop space DV4 has finite mod (2) cohomology, H*( DV4;IF 2) = IF'2[X7l/(X~) ® E(Xll,X13), but is not the mod(2) homotopy type of any Lie group. It is conjectured that there is a connection between the space V4 and the Conway sporadic group C03, in much the same way that BG2 is related to M12 in [M2] (as described in (IX.3)) but the exact connection, if any, awaits a cohomological analysis of the Conway sporadic group C03' There is also a connection between the J 1 and G2, due to F. Cohen, which we discuss in VIII.5, IX.3, which shows that in an appropriate sense, at the prime 2, BlI can be regarded as the total space of a fibration with fiber G2 and base Bc 2 •
H* (B C2 ; IF'2) -----+ H* (E; IF 2) -----+ H* (V)GL 3 (JF 2 ) is an isomorphism in cohomology.
The Ring of Invariants for SP2n (IF 2)
Remark. A brief comment on the history of the Cardenas-Kuhn theorem might be in order. The original idea for this decompostion occurs i~ H. Cardenas' thesis [Card], where it was exploited only for the groups S 2. The senior author of this book in 1965, and H. Mui in approximately 1974 both realiz,ed that this could be extended to give the critical step in studying the rmg structure of the groups H* (Sn; IF'p), but both formulations were restricted to the situation occurring in the symmetric groups so their work should properly be regarded as subsumed in Cardenas'. However, N, Kuhn gave the very useful general formulation of Cardenas' ~dea that we proved above in [Kl and since Kuhn's formulation is applicable m a very large number of cases including most of the groups of Lie type and some of the sporadic groups we felt that calling this basic calculational result the Cardenas-Kuhn theorem was appropriate.
Recently P. Kropholler, motivated by conversations with J.F. Adams, studied the invariants in IF2[X1,.'" X2n] for the groups of Lie type SP2n(lF 2), (see (VII.3) for a discussion of the finite Chevalley groups of symplectic type). He found the following structure for the ring of invariants IF 2 [x 1,,'"
X 2n
lSP2n(JF2)
IF dW3, "/5, ...
-
,"/2n-1 +1,
d2 2 n - l , ••• ,d 2 2 n _2 2n - 2 ] (1, "/2,,+d
where W3 = I':,ih XZXj, Sq2('1.V3) = "/5, and for i ~ 2 we have Sq2i ("(2 i+1) = "/2 i+ 1 +1' As an example there is a classic isomorphism Sp4(lF 2) = S6, and
IF'2[X1, X2, X3, X4]S6 where' Sq2(W3)
=
"/5
= IF 2[W3, "/5, ds, d12](l,"/9)
,
= I':,xtXj, Sq4("(5) = "/9 = I':,x~Xj, and Sq4(d s ) = d 12 ·
11 G
Chapter Ii 1. IVlod ular I IlVariallt Theory
The Invariants for Subgroups of GL 4 (lF 2 )
There are other special isomorphisms like that of 56 with Sp4(lF 2) between some of the smaller finite Chevalley groups in different families. For example (VIL3.8), SLdJF 4 ) = Sp2(JF 4 ) ~ A 5. This and many others are consequences of the classical isomorphism GL4(JF2) ~ As. (See (VI.6.6) for an explicit construction of the matrices which realize this isomorphism.) There is a second A5 in As. This A5 acts on JF 2 [Xl, ... ,X4] by tensoring over IF 2 from the action of 55 on (Z)4 where (Z)4 consists of those elements (nl,' .. ,n5) E (Z)5 with L ni = 0, and 55 acts by permuting coordinates. These two A5 's are both contained in A6 and, in fact, are the two non-conjugate copies of A5 there. These groups occur as parts of maximal subgroups in many of the sporadic groups. For example the semi-direct product (Z/2)4 XT A6 is maximal in the Mathieu group M 22 , while (Z/2)4 XT A7 is maximal in M 23 and is the normalizer of both maximal 2-elementary subgroups (Z/2)4 in the group Me L. Likewise, A5 is important in the Janko groups hand h where one has a maximal group of the form Z/2-tG-t(Z/2)4 XT A 5. Using the techniques sketched in (UI.l) we have been determining the rings of invariants for thE)se actions on IF 2 [Xl, ... ,X4]. In [AMI] the invariants for the first A5 were determined. They have Poincare series with denominator (1 - x 5 )2(1 - x12)2 and numerator 1 + 2x 3
+ 3x 6 +x 8 + 6x 9 + 2xll + 9X l2 + X14 + 10X 15 + X16 + 9X18 + 2x 19 + 6x 21 + x22 + 3x 24 + 2X27 + x 30 .
The second A5 has invariant sub ring of the form
The invariants under A6 are a degree two integral extension of the invariants for Sp4(lF 2),
and finally
lF 2 [Xl,
...
,X,d A7 = JF 2 [d s , d 12 , d 14 , d 15 ](1, blS, hl, b22 , b23 , h4' d27, b45 ) .
Further details can be found in [AM2].
Chapter IV. Spectral Sequences and Detection Theorems
o. Introduction In this chapter we introduce the main computational techniques for determining the cohomology of finite groups. We start with the various forms of the Lyndon-Hochschild-Serre spectral sequence, first from a geometric point of view in §1 and. then from a purely algebraic point of view, following work of A. Liulevicius and C.T.C. Wall, in §2. The work in §1 is particularly well adapted to wreath products, and we determine the cohomology ring of G 2Z / P in terms purely of H* (G; IF p). Then, in §2 we determine the rings H*(G; lF 2 ) where G is the dihedral group D2n or the generalized quaternion group Q2 n • Section 3 concentrates on chain level techniques. In particular we use these techniques to get some information on cup products and the ring structure in H*(G;JF p ) induced from the map BG x BG-tBG defined in (11.1.8). Section 4 considers detection theorems for the cohomology of wreath products and we prove some theorems of D. Quillen which are very important in understanding the cohomology of symmetric groups and finite groups of Lie type. Then §5 proves most of the general structure theorems for the ring H*(G; lFp), Evens' finite generation theorem, and many of Quillen's results on the role of the p-elementary subgroups of G in determining H*(G; lFp). In particular we describe Quillen's proof, [Ql], of the Atiyah-Evens-Swan conjecture on the Krull dimension of H*(G; lF p ). The groups of Krull dimension at most one at all primes are the periodic groups and they are studied and their cohomology rings determined in §6. We also review the results of Suzuki-Zassenhaus which completely classify such groups there. Finally, in §7 we use the ideas and techniques developed in this chapter to sketch a modification of Steenrod's construction of the Steenrod Squares and p-power operations. These results summarize most of the techniques available for calculations in group cohomology until the more recent work of Peter Webb which we discuss in Chapter V.
118
Chapter IV. Spectral Sequences and Detection Theorems
1. The Lyndon-Hochschild-Serre Spectral Sequence:
Geometric Approach Let H
cG
be a normal subgroup, and consider the induced surjection Bp: Bo-+Bo/H.
Note that B;I (*) consists of those points of the form {tl' ... ,tr , 91, ... ,9r} with each 9i E H and this is just BH C Bo. Every continuous map f: X-.Y can be turned into a (Serre) fibration by replacing X by the space of paths in the mapping cylinder of f which start at X, E':,~~f). (Projecting a path to its initial point gives the equivalence to X and projecting to its endpoint gives the map to Y c:::: M (j).) The fiber over y E M (j) is the set of paths which end at y, E':.~f). Consequently there is a natural inclusion of BH into the homotopy fiber of the map B p , the space of paths E;~/:I, in the mapping cylinder of Bp by sending (b, t) to (b, t) in the mapping cylinder for b E BH. Lemma 1.1 The inclusion BH C E;~/: sending B E Bo to ,(t) = (t, B) in the mapping cylinder is a homotopy equivalence.
1. The Lyndon-Hochschild--Serre Spectral Sequcilce: (;collwtric Approach
liD
it can actually be analyzed using homotopy theory or geometric constructions. More often, in order to study the differentials we need an algebraic reformulation which will be given in the next section. Here, though, we will consider a special case where geometric methods prove the spectral sequence collapses and E2 = Eoo for a small, but very important class of groups. Wreath Products Let Zip act on
P x G xv· .. x ~ by cyclic shifting, i.e. p-times
T(911 92,· .. ,9p) =
(9p, 91, 92,· .. ,9p-l)
where T is a selected generator for '!lIp. Then, with respect to this twisting, define the wreath product G ('!lIp as the semi-direct product GP XT '!lIp· A classifying space for G ( Zip can be explicitly given as (Bo)P xZ/ p E z / p
where '!lIp also acts on (Bo)P by shifting coordinates cyclically. In this case, since (Ba)P = Bop, the homotopy fibering above becomes an actual fibering
Proof First, from the homotopy exact sequence of the fibration ... .'-+ 7ri+l (Bo/ H )-+7ri(E;~/: )-+7ri(Ba )-+7ri(B a / H )-+ ...
we see that E;~ /: has the homotopy type of B H, since, by a theorem of Milnor, the path space E;~/: has the homotopy type of a CW-complex. Next, from the factorization of the inclusion BH C Ba through E;~/: constructed above
l
EBe/H Be *
---t
Ba
we see that A induces an isomorphism in homotopy. Consequently, by Whitehead's theorem, it is a homotopy equivalence. 0 Corollary 1.2 There is a spectral sequence converging to H* (G; A) for untwisted coefficients A with E~,j term Hi (G I H; HI (H; A)). This is just the Serre spectral sequence for the fibering above. [eE], [Brown] and [Sp] provide good descriptions of spectral sequences for readers interested in more background material. The spectral sequence above is called the Lyndon-Hochschild-Serre spectral sequence ([HSj, [L]). Sometimes
(1.3) where the normal subgroup H is (G)p. We now consid.er the Lyndon-Hochschild-Serre spectral sequence in this case, with A = IF p. First we study the E 2 -term, so we study the action of Zip on H*(Ba; lF p) 0···0 H*(Ba; fp) = H*((Bo)P; fp) . \.
V'
."
./
p-times
Lemma 1.4 Let A be a 9raded IF p -vector space, A = It Ai. Then as a lFp(Zlp)-module the p-fold tensor product AP = A 0 ···0 A wher~ the. a.ction of Zip is by cyclic shifting, is a direct sum of free modules w~th tnvwl modules, the trivial ones generated by elements of the form (Ai 0 Ai 0· . ·0 Ai) as the Ai run over a lFp-basis for A. Proof. A IF P basis for A0· . ·0A is given by the elements A1:10· . ·0 Ai p as the Aij run over a ZIp-basis for A. Moreover, the action of Zip preserves (possibly up to sign) this basis. Suppose, then, that Ti(Ai1 0· . ·0 Ai p) = Ai! 0· . ·0Aip. Since p is prime, Ti generates Zip unless i == 0 mod (p), and we can assume T fixes this element as well. But this implies Ai! = Ai2 = ... = Ai p. Consequently, except in this case, the Ti on a basis element give a total of p distinct basis 0 elements, and consequently a copy of the free module fpClllp).
,-;llap1.(,J" 1V. ::11)('('11':11
SCqlIClll"CS
and
U('\.CCt.lOll
Theorems
1. The Lyndon-Hochschild-Serre Spectral Sequence: Geometric Approach
T'his lemma makes the calculation of the E 2 -term routine for G ~ Zip. However it remains to study the differentials. In order to do this we enlarge the dom~in of discussion. The fibration (1.3) above is a special case of a fibering associated to an arbitrary space X,
XP xZ/ p E z / p
(>,)pxid ----+
B z/ p
id --+
where, as above, the action on XP is by cyclically shifting coordinates. In this context the lemma above gives directly
The next result is of basic importance in topology as well as in studying the cohomology of groups. Theorem 1.7 Let X be a CW -complex. Then the Serre s.p~ctral.s.equence for the fibering (1.5) above collapses, i. e. for that fibering E~,J = E:fl for all i,j.
Proof We start by observing that any element in Eg,j corresponding to an invariant for a free IFp(Zlp) module is an infinite cycle. The map Xli ':::::' XP
X
Ez/p--+XP xZ/ p E z / p
is a ZIp-covering. Consequently we have the transfer map tr: H* (XP; IFp)--+H*(XP xZ/ p E z / p; IFp) and (res· tr)* = (1 + T + ... + TP-l):H*(XP;IFp)~H*(XP;IFp). But the image of ET is exactly the subset of H* (XP; IFp)z/p associated to the free IFp(Z/p)-modules. Thus, these classes must survive toEoo. It remains to consider the classes A 0 ... 0 A, "----v----"'
Since X is a C1V-complex there is a map (A): X ~K(Z/p, n) so that (A)*(bn) = A where K(Z/p, n) is an Eilenberg-MacLane space and bn E Hn(K(Z/p, n); IFp) is the fundamental class. Consequently we have maps of fibrations
1 1
K(Z/p, n)P xZ/ p E z / p
Bz/p
and passing pto Serre spectral sequences, at E 2, the class (bn)P maps back to (A)P in Eg,n . Consequently, if (bn)P is an infinite cycle, then (A)P must be also. We now turn our attention to the right hand fibering in (1.8). Note that K(Z/p, n) is (n - I)-connected, so, below dimension np all the terms on the fiber (the vertical line Eg,*) come from invariants of IFp(Z/p)-free modules, and we have E;,j ;: : : 0 for 0 < j < np if i ~ 1. Consequently, if (bn)P has any non-trivial differential it must be dnp +1 , the transgression, taking EO,np to Enp+l,O. The terms on the line E~o are identified with the image of H*(Bz/p; IFp) in the cohomology of the total space of the fibration, and the fibration (1.8) has a section, y f-7 (x, x, ... ,x) xz/ p y where y projects to y in B z/ p and (x, x, . .. ,x) is some point on the p-fold diagonal in XP. Consequently, the cohomology of the base injects into that of the total space, and it is not possible that any term along the base line (E*'O) can be in the image of any differential.
It follows that (bn)P is, indeed, an infinite cycle, and the proof is complete. D Remark 1.9 The argument above using (1.8) shows more generally that the class A 0"·0 A survives to Eoo for H*(Z/p (Sn; IFp), n < 00. '---v---' n
Thus we have a complete calculation of H* (Z/p (Z/p; IFp) and more generally,
H*(Z/p2Z/p2'" 2Zlp; IFp) '-
A E H n (X; IF p) .
p
(K(Z/p, n))P
---+
1 1
(1.5)
Corollary 1.6 Let Aij be a IFp-basis for H*(X; IFp), then, for the fib~rfng (1.5) above, the E2-term of the Serre spectral sequence has the form E.2,J = Hj (XP; IFp)z/p, while, for i > 0, E~,j equals 0 unless j = pr, an~ then E~'P7' ~ H7'(X; IFp) with explicit generators (Aij 0·· ·0AiJUBi where Bi 'lS a generator for H'i(Bz/p;IFp) = E~'o.
CA)!'
XP (1.8)
121
V
I
i times
for any i. Later we will study these groups much more carefully, since, for example, Z/p2' .. 2Z/p is the p-Sylow subgroup of the symmetric group Spi. ~ i times
We now provide an explicit calculation using this result.
Example. The dihedral group of order 8, denoted DB, can be identified with Z2 2Z2. First we note that
122
1. The Lyndon-Hochschild-Serre Spectral Sequence: GeollleLric Approach
Chapter IV. Spectral Sequences and Detection Theorems
where the Z2 acts by exchanging generators, and the Ui are the usual symmetric classes. Observe that 0"1 = Xl + YI is a trace class and hence will multiply trivially with elements from the cohomology of the base, generated by a I-dimensional polynomial class e. T~erefore we obtain
H*(D8)
~
IF 2[e,u1,u2]j eu1 .
The action of the Steenrod Algebra is determined by Sql(U2) a fact which will be verified in (2.7).
= (U1 + e)u2,
Remark 1.10 This construction and 1.7 provide the basis for the construction of the Steenrod operations. We will give details in IV. 7. Also, a different proof of 1. 7, working at the chain level, is implicit in [SE]. In some ways that proof is more general than the one given here, but this one provides us with better control of cup products.
123
Ig11g211--+ Ill(g2)-1[(g2)-1l(g1g2)11 E B(Bz/p) lifts to give a map of principal fibrations BZjp-+{B;;1(BG,2)-+BG,2 to the fibration over the two skeleton of B(B z / p ). After this one extends the map by using the principal action and 0 extending over lifts of the individual cells of B G . This next result is very useful theoretically, and provides a nice application of our result on the cohomology of wreath products and the Frobenius map discussed in (II.5). Lemma 1.13 Let '!ljp-+E-+G be a central extension, then there 'is a finite n so that En = Eoo in the spectral sequence of the fibration (l.ll).
Proof. The Frobenius map associated to the inclusion ?ljp '------+ E gives a homomorphism rp: E-+'!ljp (SIGI, and, from (1.9), rp*(b 0··· 0 b) is non-zero '---v---"
IGI and restricts to blGI in H* ('!ljp; IF p) since ?ljp is central in E. Thus the class G I· ' EO,2I . fi 111'te cyc le 'm t 118 spectra1 sequence. Now write biGI m 2 IS an m
Central Extensions Cn ---l> ... ---l>C 1 ---l>Co---l>Z bP -
1
Iblbp -
1
be a resolution of Z over Z(H). Then we have •
bJJ - 2
~.
&2
0
b
Iblbp -
Lemma 2.1 1. Z( G) 0ZeH) Z = Z( G j H) as a left Z( G) -module 2. '" Z( G) 0Z(H) Cn --+ ... --+Z( G) 0Z(H) Co --+Z( G/ H) is a Z( G) resolution ofZ(GjH) .
2
~ ~
&
IblbP-11
Ibl
Proof. Let H U g2 H U g3H U ... U gmH U '" be a coset decomposition of G. Then Z(G) ~Z(H)Z as a (free) 'll-module has a basis (1), (g2), (g3), "', (gm), and the actlOn of G on these basis elements is given by g(gi) = grh(gi) = (grhgi) = (grgi) when we write g = grh in the decomposition above. This gives the first claim. Now the second is clear. 0
The form of the higher differentials and the Kudo differential. ?orol.lary 2.2 Let A be a Z(G)-module regarded as a Z(H)-module via the mcluswn Z(H) C Z(G), then there is an isomorphism
Ext~(G)(Z(GjH),A) ~ Ext~eH)(Z,A).
A Lemma of Quillen-Venkov <J
7r
Consider an extension of the form G--+E--+Z/p. Let u E H2(E;Yi p ) be 7r*(b) where b E H2(Z/p; lF p ) is the Bockstein of the fundamental class. Then we have the following useful result of [QV].
Proof. Homz(G) (Z(G), A) = HOmZ(H) (Z(H), A) = A. Thus the map Z(G)0ZeH): C---l>Z(G) @Z(H) C on passing to co chain complexes yields an isomorphism
126
2. Change of Rings and the Lyndon-Hochschild-Serre Spectral Sequence!
Chapter IV. Spectral Sequences and Detection Theorems
127
HOmZ(H)(C, A)--+HomZ(G) (Z(G) 0Z(H) C, A)
1
o
and the result follows.
8
Example 2.3 Let D 2n be the dihedral group D2n Let H
= (T)
COl
= {T, T I Tn = T2 = 1, TTT = T- 1} .
8
1
= Z / n. There is then an isomorphism
Ext~(D2n) (Z(T), A) ~ Ext~(z/n) (Z, A)
= Hf(Z/n; A)
coo
.
1 '0
The equation above is a typical example of the change of rings principle, i.e. the specification of Ext terms for a module over A as Ext terms for a simpler module over B where B c A is a subring. This change of rings principle allows us to give an algebraic construction of the Lyndon-Hochschild-Serre spectral sequence. The idea is to first resolve Z over Z( G / H), then resolve each term of the resolution over Z( G), lift the boundary map in the resolution of Z and then make a guess that the boundary in the resulting doubly graded complex is of the form 8 + (-l)€d l where d l is the lifted boundary and 8 is the boundary in the resolutions of the terms in the original resolution of Z, and then start adding correction terms to make the resulting guess into an actual differential. Thus, to begin we obtain a diagram
t
Z
f---
Bo
If G = H X G / H is a Cartesian product the diagram above could be achieved by tensoring a resolution of Z over H with one of Z over G / H so we would have explicitly Cij = R j 0 Bi where R j is the jth term in a resolution of Z
over H, and
d
=
3N 01
+ (-1)j10 3E
would be the total differential. This motivates attempting to use the same formula with the diagram above. The difficulty is that, in general
d2
(3 82
+ (-l)j dlj) 2 + (-l)j 3d 1j + (-1)j- 1 dl ,j_ 1 3 + (-1)d 1 ,j- I d 1j
(-1 )d 1 ,j- 1 d1j
1a
1a
1a
COl
Cl1
C2l
1
1 '0
Z
+---
do
Bo
8
ClO
coo
€
1
8
1a
+---
1"
B1
is not necessarily zero. However, Ed1,j-ldl,j = d j -1 djE = 0, so the chain map dl,j-ld j is homotopic to zero, and we can construct a family of maps d 2{ Cji ----+C j -2,i+ 1 so that 3d 2j + d 2j 3 = dl,j-.ld lj . We add these in, maki~g a new (prospective) differential d = 3 + (-1 )J d Ij + d2j . Next we square thIS operator to find the deviation from being a differential, correct that deviation and iterate the procedure. We obtain a result due to A. Liulevicius, [Li], and C.T.C. Wall, tWa],
C20
dl
+---
1" B2
d2
+---
where the vertical sequences consist of Z( G) free resolutions of the B· 'so Next, as indicated, we lift the di's to obtain chain maps between the verti~al columns, so the diagram above now takes the form
Theorem 2.4 In the situation above there exist a series of Z( G) -linear maps drs:Csj----+Cs-r,j+r-1, so that if we setDm = U~=mCt,m-t' d = 3+(-l)sd 1s + 2:;:2 dz*, then d2 = 0 and the resulting complex is a resolution of Z over Z( G). Specifically, each dr* can be chosen so that (1) do = 3 is the vertical differential, (2) diEi = Ei-ldli, (3) 2:~=o di,*dk-i,* = 0 for each k; conversely, any map with properties (1), (2), (3) is a differential which makes the complex above acyclic.
2. Change of Rings and the Lyndon-Hochschild-Serre Spectral Sequence
Chapen IV. Speclral Seqnences and Detection Theorems
Proof. We show first that any d with properties (1), (2), and (3) above makes the total complex acyclic. Filter Cij by FPC = L,i:;p Cij . The differential d preserves filtration, and the associated spectral sequence converges to H* (C). The differential in EO is precisely do = O. Hence E1 = 5, the resolution of Z by Z( G / H) free modules with which we started. Moreover, (2) implies that d 1 is exactly the differential d*, the differential in B. Since 5 is a resolution of Z it is acyclic, and E2 = Eoo = EGO = Z, hence C is acyclic. To complete the proof we must construct the maps dr * for r 2: 3. To begin we set drs = 0 if S < T. Now, suppose that drs has been defined on all Cij with i + j < v. Let f = - L,~=1 did r - i · We claim there is a map d r so that dod r = f. To prove this, it suffices to show that dof = 0 and E*f = 0, but this is direct
i=1
i=1 r
B3
j=1
1
r-j
I:: dj L di-jdrj=l
1
L ~ djdi-jdr-i
- ~dodidr-i
i
= O.
B2
i=j
1 1 1
0
which completes the proof.
We now illustrate the explicit resolution techniques above, by constructing resolutions for the dihedral groups D 2n and the quaternion group Q s ·
B1
The Dihedral Group D 2n
Bo
We will apply the Lyndon-Hochschild-Serre spectral sequence to the situation obtained by using the index two subgroup Z/n <J D2n with quotient Z/2. The associated resolution of Z over Z(Z/2) has differentials which are alternately T - 1 and T + 1. Consequently, to begin we need to lift these maps to chain maps of resolutions of Z over Z(Z/n). Proposition 2.5 Let D2n = {T, To I T2 = Tn = 1, TTT = T-1}. Let E7 = 1 + T + T2 + ... + Tn-I, and C = Z(D 2n ), then the following is a comm1dative diagram of chain maps C
1 C
7-1 +--
T-1 r-1 +--
C
1 C
ET +--
-(rT+1)
ET +--
C
1 C
7-1 +--
-(T+1) 7-1 +--
ET
C
+--
1
rT-1
C
+--
Er
Proof. Note that TTT = TT-1T = T so 1. - (TT + 1) (T - 1) = - T - T + TT + 1, (T - 1) (T - 1)
•
o
Remark 2.6 The first non-trivial dihedral group Ds = Z/2 ( Z/2. Hence, 1.6 gives the structure of H*(Ds;JF 2 ). In particular, the Lyndon-HochschildSerre spectral sequence collapses in this case. But for the general case we still need to make an explicit calculation. We turn to this now. From the proposition we obtain the following diagram for D 2n which we have turned on its side for convenience.
i
r
r
dof
+ 1) = -E7(T + 1) = -(T + l)ET 3. (TT - 1) (T - 1) = T - T - TT + 1 -(T - l)(T + 1)= -TT - T + T + l. 4. (T - 1)E7 = Er(T - 1) = Er(TT - 1).
2. -Er(TT
129
€3
+--
T-l €2
+-T+1 t1
+--
T-1 to
+--
1 1 1 1 C
C
C
C
r-1 +-T-1 T-1 +-T+l r-1 f--
T-1 r-1 +--
1 C
1 C
1 1 C
C
Er f--
-(rT+1)
Er +--
-(rT-l)
Er +--(7'1'+1)
Er f--
1 1 1 1
T-1
C
f--
-(T+1) T-1
C
f--
-(T-1) r-1
C
f--
-(T+1)
C
T-1 +--
1 1 1 1 C
C
C
C
Er f--
rT-1
Er +-TT+1
Er f--
7T-1
Er f--
'0
Z
Note that (TT -l)(TT+ 1) = (TT)2 -1 = 0 = (TT+ 1)(TT-1) so d2 ::::: 0 and the construction of the higher differentials stops. It follows that alternating the signs on the columns above defines the total differential for the complex above as OT + (-l)j d 1 . For example, if n is even and we take our coefficients as JF 2 all the differentials are zero and Hi(D4m; lF 2 ) = (Z/2)i+1. It follows, in particular, that this resolution is minimal in the sense that, in each dimension, the number of free summands cannot be reduced.
C
We now apply the above result using induction and the central extensions
IT-l C
-.) . '----v---" P times
Then we can construct an explicit chain approximation
J.L(h): C(Z(Z)) 0z C(Z(Z/p))----C(Z(Z))P 0 C(Z(Zlp))
Now, assume the formula of the theorem is true for ei 0 ej with i Then in dimension n + 1 we have 1. If n + 1 is even, then
+j
:::; n.
.by these techniques using s@ in the complex for the wreath product. . To begin we may assume C(Z(Z)) is the resolution discussed above
'"
T-l
Z+-Z(Z)[eoJ +-Z(Z)[el] ,
s[
(w: ~ ~ 1) + (w +~ - 1) 1Ten
(w: l)e + n
J.L(h)(el 0 eo) =
(2::: 1j
particular class which is of most interest in the sequel is the image J.L(h)(el 0 ep-l). When p = 2 we obtain, explicitly
1'
J.L(h)(el 0 er)
while
e2w+l 0 2. When n
e2l+1
+ 1 is odd,
~
1
S [ (w : l) - (w: l) Ten
0 el 0 TP-j-l) 0 eo, J.L(h)(l 0 ej) = 1P 0 ej. The
0.
then
e2w 0 e21+1 ~ SJ.-l( Te 2w-l 0 e21+1 + e2w 0 T e 21) s(w: l)Te 2 (w+l)
=
s@((T x T - 1 x l)eo 0 eo 0 el - (T - l)(el 0 T + 10 el) @ eo) (el 0 Teo + eo @ el) @ el - s@T @ el @ Teo (e 1 @ Teo + eo @ e 1) @ e 1 - e 1 @ e 1 0 Teo·
The most important class here turns out to be the last term, -(e10er)0Teo. More generally, for odd p, we have Theorem 3.11 In the chain approximation constructed above fOT Z x Zip with p odd, we have that, afteT Teducing mod the action of Z l Zip,
(w: l) e2(w+I)+1 . Moreover, the assumed commutativity of the formula in degree n implies the same in degree n + 1. 0
Remark 3.9 Passing to homology, the map above gives a map
This is called the Pontrjagin product. For the case when P = 2 the result above shows that H*(7L/2; Ylz) = E(el' ez, e4, ... , ezi, .. .), is an exterior algebra. More generally, the same arguments can be extended to the case of P odd to show that H.(Z/Pi Yl p ) = E(el) 0 p [r2], where p [r2] is the divided power algebra on the two dimensional generator /2. (The divided power algebra has generators /2i in each even dimension 2i and multiplication given by the rule
r
/2i . /2j =
(t+.) / 12(i+j)')
r
Proof. The proof is a close study of the way in which the iterate operator S@(T -l)s@ Er acts. It turns out that the only term at any stage, which can reach the last class under these operations has the form of a tensor product of el's and T's in the first p positions - no ones - tensored with Tl eo in the rightmost position. Moreover, the effect of looking at s@ (T) s@ ET shows that it produces classes with two adjacent el's in the first two positions. After iterating this operator (p - 1) 12 times we achieve terms of the form
j >-'(l,j)ei 0 T 0
ei- 2j - 1 0
T1eo .
140
Chapter IV. Spectral Sequences and Detection Theorems
For each j, the sum
'£l A(l, j)
is
Cp;
1) -
4. Groups With Cohomology Detected by Abelian Subgroups
I)! corresponding to the order in
S1
8°
.
l,J
A(l, j) e 1 0 ... 0 e 1 07 1+2'J eo .
X
sn)p
p times
(The signs are all positive since we are always .permuting an even number of el's.) But now, summing the A(l,j) over both l, j, gives (~)! as asserted.
o
\;Ve will see later that this formula is the essential step in defining and proving the basic properties for the Steenrod pth power operations. It is also the basic step in proving a key lemma which is crucial in our study of the cohomology of the symmetric groups and many of the groups of Lie type. We turn to some of this in the next section.
X
B z/ p
shuff·L).
---+
(Sl x B z / p )
XZ/ p
E z/p
(S1)p
XZ/ p
E z/p
We begin with a topological application of the last result in the previous section. The following result is due to N. Steenrod, and is found in [SEl· Theorem 4.1 Let X be a CW -complex, and consider the map
E
X
1
B z /p )
X
(sn)p
XZ/ p
E z/ p.
K(Zjp, n) x B z / p
1
"'xid
XZ/ p
E z/p
and (e n )*: Hn(K(Zjp, n); lFp)-+Hn(sn; Fp) is an isomorphism. Consequently, the truth of the result for en implies it for /'n- But since /'n is universal, the theorem follows. 0 This result has very strong consequences for the cohomology of finite groups. Definition 4.2 Let G be a finite group. We say that H*(G; lF p ) is detected by abelian subgroups if there is a family of abelian subgroups Hi C G so that
L1P x id: X x Bz/p---+XP xZ/ p E z/ p .
Then, for all 0:
(sn
From the naturality of the construction of IP (0:) we can assume p* (')'p (e 1 ) 0 'P(e n )) = Ip(e 1 0 en), and from this, the inductive assumption and the commutativity of the diagram above, the result holds for sn. Next, consider the map en: sn-+K(Zjp, n) where (en)*(/'n) = en. The following diagram commutes
K(Zjp, n)P
4. Groups With Cohomology Detected by Abelian Subgroups
X
("'Xid)'
"'xid
(S1
'----v-"
sn
1
which the pairs of adjacent e1 0 el's appear, except for the leftmost, which we know just appeared. Then applying the final ET we obtain a sum "L.t
X
141
H* (X; IF p), we have is an injection.
where b is a generator for H2(Zjp; lFp), and dim(O:j)
=
dim(o:)
+j
.
Proof. For X = S1 this is a restatement of the last result in the previous section. Suppose the result is true for a generator, en, of Hn(sn; Fp) = Fp. We wish to show it for a generator of Hn+ 1 (sn+ \ IF p). To do this consider S1 X sn and the projection S1 X sn-+sn+1 which induces an isomorphism of cohomology groups in dimension n + 1. Consequently, if the result is true for S1 X sn it is true for sn+l. But the following diagram commutes
Note that if G l , G 2 both have mod p cohomology detected by abelian subgroups then the subgroups H i (G 1) x H j (G 2) detect H*(G 1 x G 2; Fp), so the family of such groups is closed under products. Moreover, if G is such that a subgroup H contains a Sylow p-subgroup of G and H has mod p cohomology detected by abelian subgroups, then the same is true for G, . since (res5})*: H*(G; lFp ) L-+ H*(H; lFp ) must be an injection. The following result, due to D. Quillen, [Q6], but anticipated in large measure by M. Nakaoka, [Na], will show that the set of such groups is really rather large. Theorem 4.3 Let H*(G; lFp ) be detected by abelian subgroups, then the same property holds for H*(G ~ Zjp; lF p).
Proof. Let {Hi} detect H*(G;Fp). Then, our previous calculation of H*(G ( Zlp;Fp) shows that the collection of subgroups {Hi l Zip} detect H*(G (
5. Structure Theorems for the Ring
Zip; lFp ). Thus, we are reduced to proving the result for an abelian group C.
There are a number of general things which can be said about the structure of the ring H* (G; F p) when G is a finite group. Among them is the result of L. Evens [Evl] that it is Noetherian, in particular finitely generated, with only a finite number of relations. Also, Quillen, in his landmark paper [Ql]' proved a series of beautiful theorems relating the structure of H* (G; F p) to the set of conjugacy classes of elementary p-groups in G. He showed l. that the Krull dimension, (the number of generators of the largest polynomial subring contained in H*(G;Fp), an alternate definition is given in 5.4), of H*(G;Tf!p) is equal to the largest rank of any elementary psubgroup of q, 2. that the intersection of the kernels of the restriction maps
The theorem will, thus, follow directly from the following result. Lemma 4.4 Let G be abelian, then GP and G x Zip embedded as a subgroup of G (Zip via the inclusion L1P x id, detect H* (G l Zip; F p). Proof 4.4 We have already seen that
where
ei
comes from Hi (Zip; Zip), and ,(7) restricts to 70···07 in '----v----' p times
H* (GP; F p). Consequently, it is only necessary to check the result on the elements ,(7) U ()i. In the group L1P x id(G x Zip) this element has image (L1P x id)*(f(7)) U (10e i ). Thus, the lemma follows from Steenrod's theorem
above since it implies that under the assumptions above (L1P
X
id)*(f(7))
= ±,0 bdim (')')9 + :LDj(f) 0 Aj j>O
where dim(Dj(f)) = dim(f) + j for p an odd prime, and it is equal to , 0 (ed dimb ) + I:j>o Dj(f) 0 e~im(')')-j for p = 2. 0
o
This finishes the proof of the theorem.
In Chapter VI one of the crucial facts which makes the determination of the cohomology of the symmetric groups possible will be proved: that Sylp(Sn) is a product of wreath products of the form !-lp2Zlpl' .. lZ/~ with v j times
pi < n. Thus the cohomology of Sn is detected by its elementary abelian p-subgroups for every n. But even more is true. Let G = GLn(Fq) for some finite field F q. We will see in chapter VII that for p odd the Sylow p-subgroup of G is a product of wreath products having the form Zlpl l Zlpl'" (l.lp for appropriate l, j, '--v---' j times
depending on q and n as long as p does not divide q. Moreover, a similar result is true for many other groups of Lie type. Consequently, (4.3) is a very powerful tool in analyzing the cohomology of many of the most important finite groups.
H*(G;Fp)
as A runs over the set of p-elementary subgroups of G is contained in the Radical of H*(G; lFp), and 3. that the set of minimal prime ideals of H* (G; Tf! p) is in one to one correspondence with the set of conjugacy classes of maximal p-elementary subgroups of G. In this section we prove Evens' theorem and the first two of Quillen's results. (Note: Venkov [V] obtained an independent proof of the finite generation of group cohomology; this is described in [Ql]' pp. 554-555.) Evens-Venkov Finite Generation Theorem Recall the definition of Noetherian ring: a ring is (left or right ) Noetherian if and only if, for each ascending sequence of (left or right) ideals h C h c ... , there is an n so that I j = I j + 1 if j 2: n. We use the same definition for the graded rings which occur in cohomology, and a similar definition for Noetherian modules. In particular quotients and ideals are Noetherian for a Noetherian ring. Also, a finitely generated exterior algebra over a Noetherian ring is Noetherian, as is a polynomial ring on a finite number of generators over a (graded) commutative Noetherian ring. (This last is just Hilbert's theorem.) Lernma 5.1 Let E be a finite p-group, then H*(E; lF p ) is Noetherian. Proof. It is clear that H*(Zlp; Fp) is Noetherian. Now we proceed by induc n - 1 and E~m = 0 for l + m ~ n. ~om this .Eg,J ~ HJ (S~-l IG; Z) for J < n - 1, an~ otherwise dn must be an Isomorplusm for each ~.
Since 8 3 is a continuous group any subgroup acts freely, so the generalized quaternion groups, the binary tetrahedral group T, the bi~ary octahedr~l group 0, and the binary icosahedral group all act freely on 8 and hence ~le periodic. (We discuss them more precisely after 6.10.) But there also eXIst periodiC groups which do not act freely on any sphere.
Lemma 6.3 S3 is periodic with period four.
6. The Classification and Cohomology Rings of Periodic Groups i i i i
The results of (IV.5) on the Krull dimension imply that it is a good invariant for organizing groups. In this section we study the groups G which have Krull dimension one at all primes p which divide IGI.
Definition 6.1 A finite group G is periodic and of period n > 0 if and only . Hi(G; Z) ~ HHn(G; Z) for all i ~ 1 where the G action on Z is trivial. We have already seen that if G is cyclic then it is periodic and if G is the quaternion group Q2n, (IV.2.10- IV.2.12) show that it is periodic as well. Periodic groups come up naturally in topology when one considers finite groups acting on spheres.
Lemma 6.2 If G acts freely and orientation preserving on a sphere then G must be periodic of period n.
sn-l
Proof. If G is a finite group and acts freely on X there is a standard fibration which is used to study the quotient, cp
X ----tXjG ----t Ba where
3. Then, if x and y generate this copy 2 of Q2Tt and x 2n - = y2, y4 = 1 there are two copies of Qs, QS,l =
The Mod(2) Cohomology of the Periodic Groups
(X
2n
3
-
,YI,
and QS,2 = (xy, YI
.
It is direct that the two restriction maps
res';: EB res;: H* (Q2Tt; IF 2) ---tH* (QS,I; IF 2) EB H* (QS,2; lF2) From the classification result above the only cases which need to be considered now are the cases IV, V, VI. We will first determine the situation in case V and then an easy spectral sequence argument will give the remaining two cases. To begin we need a lemma. Lemma 6.11 Let Qs C SL2(lFp) where p is an odd prime. Then the normalizer of Qs in SL2(lFp) contains an elmen(;nt T with T3 = 1 which acts as the non-trivial outer automorphism of Qs of order 3.
together are injective, so QS,l and QS,2 detect mod (2) cohomology. Now the remainder of the argument is clear. 0 It remains to discuss the groups H*(Oi; lF 2) and H*(TL2(lFp); lF2)' In both cases, corresponding to the extension data H*(BP) .
Here 1-£* (X; JR.) is a twisted coefficient system as G = 7f1 (BG) may act nontrivially on it. The term E~,q may be identified with the group cohomology with coefficients, HP(G; Hq(X; R)).
In sufficiently high dimensions jp will be an isomorphism because Pacts freely off X p . X is finite and so H*(X Xp EP,X P x BP) = 0 for *» o. So P p *. for large * we have H*(X P ) 18> H*(BP) ~ H*(BP), which implies x This theorem also admits a very interesting extension due to T. Chang and T. Skjelbred, [CS] which is not needed in the sequel but is well worth pointing out.
Remark. This spectral sequence is originally due to J. Leray, H. Cartan, and R. Lyndon in various forms provided that G is discrete and the action is sufficiently reasonable. Many of the applications before Borel's work are discussed in chapters XV and XVI of [CE].
Theorem. Let G = '!L/p and J{ = fp 01' G = S1 and J{ = Q. Suppose that G acts on the compact Poincare duality space X of formal dimension n. Then each connected component of the fixed set satisfies Poincare d1wlity over J{ and, if G i- '!L/2, has formal dimension congruent to n mod (2).
f"V
Definition 0.5 The G-equivaTiant cohomology of X is Ha(X;R)
=
H*(X Xc EG;R).
Note that if X is a free G-complex, the map X Xc EG--+X/G is also a fibration with contractible fiber, hence X Xc EG ~ X/G. Also note that the arguments given in Chapter V §5 can be adapted to show that if H*(X; R) is afinitely generated R-module, then Ha(X; R) is a finitely generated Ralgebra. ,The analysis of the spectral sequence (0.4) was first undertaken by A. Borel. It yields numerous restrictions on finite group actions on familiar spaces such as spheres, projective spaces, etc., (see (Bol]). We give two simple applications of these techniques to illustrate their utility.
Example 0.7 Now let us assume G is a finite group acting freely on X = STL (the n-sphere). In this framework we can also recall the classical result due to P. Smith proved in Chapter IV:
Theorem. If a finite group G acts freely on a sphere subgroups are cyclic.
sn,
then all its abelian
Remark. Recently this result has been extended to products of spheres of the same dimension, namely, if ('!L/pt (p odd, or if p = 2 then n i- 1,3,7) acts freely on (sn then r :s; k. (See the papers by G. Carlsson [Cal and Adem-Browder, [AB].)
l,
164
Chapter V. G-Complexes and Equivariant Cohomology
At this point it is worthwhile to point out that all of the preceding constructions can be derived algebraically from C*(X). Let F* be a ZG-free resolution of the trivial G-module Z. Consider the double co-complex
Then it is not hard to show that
Hc(X; R)
~
H*(HomG(F*, C*(X; R))) .
1. Restrictions on Group Actions
165
where rCG is the sheaf on X/G associated to the presheaf V ....... Hc(-rr- 1 V), 1f: X XG EG-+X/G. In the case of a finite G-CW complex (assuming constant isotropy on the cells) the d1-differential is just the map induced by the coboundary operator on C*(X) and Ei,q = Hq(G, CP(X)). Recall that in Chapter II we defined Tate cohomology of ZG-modules; instead of an ordinary projective resolution we used a complete resolution i.e. an acyclic Z-graded complex of projective ZG-modules which in non-negative degrees coincides with an ordinary projective resolution of Z. Taking such a complete resolution F*, we can define Tate hypercohomology groups
Note that we endow the double complex with the usual mixed differential:
H*(G; C*) ~ H*(Hom(F*; C*)) . In case C* = C* (X), we obtain the equivariant Tate cohomology of X, first introduced by R.C. Swan [S3],
which is given by
(0.9) For completeness we recall the Cartan-Eilenberg terminology Definition 0.8 The G-hypercohomology of a G-cochain complex C* is defined as
H*(G; C*(X; Z))
Hc(X).
The main technical advantage is the disappearance of the orbit space for free actions: Theorem 0.10 If X is a finite dimensional free G-CW complex then
H*(G; C*) = H*(HomG(F*, C*)) where F* is a free (projective) resolution of the trivial ZG-module Z.
=
Hc(X) ==
o.
Hence we may say that the equivariant cohomology of a G-CW complex is isomorphic to the hypercohomology of its cellular co chain complex. This has certain technical advantages, above all if a specific cellular decomposition is available. In addition we may filter the double complex in (0.8) using either of two filtrations associated to the "vertical" and "horizontal" directions respectively. This yields two spectral sequence which applied to C* (X; R) become (I) E~,q = HP(G; Hq(X; R))} p+q. (II) Ei,q = Hq(G; CP(X, R)) =* HG (X, R) .
Note how the analogous spectral sequence (1) will not involve the orbit space, hence strengthening cohomological arguments.
The spectral sequence (I) is canonically isomorphic to (0.4). As for (II), this can be identified with the E1-term of the Leray spectral sequence associated to
1. Restrictions on Group Actions
Proof. There is a spectral sequence analogous to (II), say
This converges because X is finite dimensional. Now each CP(X) is G-free, hence G-acyclic, so E~,q == 0 and the result follows. 0
X xGEG
1
X/G.
In the general situation the E1-term is not so easy to handle, but the E 2 -term can be identified with E~,q = HP(X/Gj
nb)
=* H[/q(X)
In this section we will outline some instances of how the machinery described in §O can be applied to transformation groups. As this is not our main topic of interest, we urge the reader to consult the original sources for the foundational results [Bol], [QI]. The first result to take note cf is the localization theorem. This result shows that for certain groups the equivariant cohomology contains substan-
tial information about the fixed point set after inverting certain cohomology classes, and hence this makes the entire spectral sequence approach
quite powerful. More precisely we have the localization theorem of Borel and Quillen
Theorem 1.1 Let G = (Z/PY (p prime) act on a finite complex X. Then the inclusion of the fixed point set XC ~ X induces an isomorphism
H*(X Xc EG; lFp)[e A l ] ~ H*(XG
X
BG; lFp)[e Al ]
where eA E H2pr -2 (G; IF p) is the product of the Bocksteins of the non-zero elements in Hl(G,lFp) ~ Hom(G,lFp), and [eAl] denotes localization by the multiplicative system of powers of eA. Note that here we are considering equivariant cohomology as a graded H* (G; IFp)-module.) The next theorem, again due to D. Quillen, is an important structural result for equivariant cohomology, which can be proved using finite generation arguments as in IV.5 (see [Ql]).
ii. Given x
E
A we define
exp(x)
exp((x)).
Using restriction and transfer it is elementary to see that JGJ· fIe (X) == a for any finite dimensional G-complex X; hence Tate cohomology provides an interesting sequence of Z-graded torsion modules. For free actions we have a theorem of W. Browder, [Brow],
Theorem 1.4 Let X be a free, connected G-CW complex. Then
'G'Ill
i 1 exp(iI- - (G; Hi(X; 2))) .
Proof. The proof,we give is from [A2]. Consider the spectral sequence E~,q
Theorem 1.2 Let X be a finite dimensional G-complex and denote
=
= fIp(G; Hq(X; Z))
=}
fI~/q(X) ==
a.
Look at the E~,O-terms; we have exact sequences
00
PG(X)(t) =
L dim(Hi(X
XG
EG; lFp))t
i
-r-l,r EO,o EO,o 0 E r+l - - r+l - - r+2--
.
i=O
Then PG(X)(t) is a rational function of the form p(t)/ I17=l (1- t 2n ) and the order of the pole at t = 1 is equal to max{nJ(Z/p)n
~
G fixes a point x E X} .
= 1,2, ... ,dim(X).
:From this we obtain r
+1 exp (00)1 exp(Er00)/ Er+2 exp (E-r+lMultiplying all these out we obtain
This is the version of the result for G-spaces which we discussed previously for the case X = pt. This result was the starting point for the idea of introducing varieties associated to cohomology rings. As we have seen, this may be considered as a special case of H* (G; M) where M is the cellular complex associated to a G-space. This theorem however, has its natural extension to any coefficient module M. In other words, the asymptotic growth rate of H* (G; M) (known as the complexity of M) is determined on the pelementary abelian subgroups of G. The proof of this (due to Alperin and Evens [AE]) is an algebraic formulation of Quillen's result, and has important applications in modular representation theory. The above results are not too interesting in the case of free actions. For this situation Tate cohomology comes in handy because we obtain a spectral sequence which abuts to a zero term. First a
Definition 1.3 i. Let A be a finite abelian group. We define its exponent, exp(A) N/n·A = a}
for r
= min{n E
exp(iIo(
G; 2)) = 'G'lrr exp(E;;j"C)
III
1r
' )
C 1 exp(iI- - (G; W(X; 2))) D
completing the proof.
Remark. Tensoring C* (X; Z) with a torsion-free ZG-module M and applying the same proof yields exp(ftJ(G; M))
III
exp(iI- c - 1 (G; W(X) 0 M)) .
This result has a few interesting consequences which we now briefly describe.
Corollary 1.5 If (Z/Py acts freely on a connected complex X '}!Jith homologically trivial action, then at least r of the cohomology groups H*(X; Z(p)) must be non-zero.
168
1. Restrictions on Group Actions
Chapter V. G-Complexes and Equivariant Cohomology
169
n2
Proof. Under the above hypotheses, if G = (7L/pt then
II exp(iI- r- k- 1(G; M))
exp(iI-k(G; M))
p. iI*(G; H*(X; 7L(p))) = 0 .
for all k E 7L .
r=l
o
(K unneth formula.)
In particular this shows that (7L/pt+ 1 cannot act freely, homologically trivially on (sn) r.
Dimension shifting yields n2
exp(iIk+n2+2(G; M))
II exp(iIk+r(G; M)) r=l
= min{nlG acts fee ely, homologically trivially on an n-dimensional connected complex}. Then n(G) :2: maxpIIGI{p-rank(G)} + 1.
~s before we get iI* (Gj M) = 0 for H*(G; M) == O.
Corollary 1.7 Let M be a finitely generated torsion free 7LG-module, G a finite group. Then there exists an integer N (depending only on G) such that
Remark A different proof of this result appeared in [BCR] using purely algebraic methods. Notice that the restrictions on M are not important as any finitely generated 7LG-module is co homologous to a torsion free one.
Corollary 1.6 For a finite group G, let n(G)
* :2: k + n 2 + 1 and
we conclude that 0
N
EBiI*+i(G;M) ~ 0
Recall (see [MS]) that
i=l
for all
*E
H*(BU(n), 7L) ~ 7L[Xl' X2,"" x n ]
7L, or else iI* (G; M) == O.
where each Xi is 2i-dimensional. Now given a representation ¢: G '-+ U(n) as before, the polynomial generators Xi E H2i(BU(n);7L) can be pulled back under ¢ to define
Proof. Take a faithful unitary representation ¢: G
'-+
U(n) .
i
Then G will act freely and cohomologically trivially on U(n), which has the homology type of Sl x S3 X ... X s2n-l. From the remark after (1.4) we see that
exp(iIO(G; M))
II exp(iI- r- 1(G; M))
the ith Chern class of ¢. Under this map, the cohomology of G is a finitely generated H* (B U (n); 7L)-module. These classes naturally carry some torsion; the following quantifies this.
.
r=l
Using 1\11* instead of M and identifying iIi(G; M) ~ iI-i(G; M*) (Tate duality) yields
exp(iIO(G; M))
= 1, ... ,n
Theorem 1.8 Let G be a finite group and ¢: G '-+ U(n) a faithful unitary representation of G with Chern clases Cl (¢), ... , en (¢); then
IGII!!
II exp(iIr+1(G; M))
exp(ci(¢)) .
r=l
Proof. Consider the Borel construction on U(n) and its associated spectral sequence
By dimension shifting this can be generalized to n
exp(iI k (G; M))
E~,q =
2
II exp(iIr+k+1(G; M))
for all k E 7L .
r=l
Suppose now that for some k E 7L, iIr+k+1(G; M) = 0 for r = 0,1, ... , n 2. Then iIk(G; M) = 0 and consequently iI*(G; M) = 0 for * ::; k + n 2 + 1. On the other hand, using the dual again yields
HP(Gj Hq(U(n))) => Hp+q(U(n)/G) .
Now H*(U(n); 7L) ~ AZ(Xl,"" x2n-d and by construction these classes transgress down to Ci (¢) E H2i (Gj 7L) for i = 1, ... , n. This implies that [exp(ci(¢))] . X2i-l E Eg,2i-l is a permanent co cycle for i = 1, ... ,n, which implies that [TI exp(ci(¢))]' J-LU(n) E im(i*) where J-LU(n) is the top cohomology class on the fiber and i: U(n)-+U(n)jG is the projection. However, i is a
covering map of oriented manifolds, hence it has degree IGI on the top class 0 from which the result follows.
Example 1.9 We now compute the cohomology of Qs using the action on S3 as a subgroup. The spectral sequence (I) with integral coefficients for this action degenerates into the long exact sequence (for i ~ 0) ... --t
H i - 4(QS)
-t
Hi(QS)
-t
Hi(S3/QS)
-t
were first introduced by K. Brown, [Brown], and D. Quillen, [Q5]. The objective was to construct a natural complex which distilled the p-Iocal structure of the group G as well as to provide analogues of Tits buildings for general finite groups. Given a map f: X -+Y of posets and y E Y we define
fly = {x ylf = {x
H i - 3(QS) ...........
From this we deduce that H2(S3 /Qs) ~ H2(QS) ~ Z/2 x Z/2, and as the action is free, H4(QS) ~ Z/8. Hence we have that
E
E
Xlf(x) :S; y} Xlf(x) 2: y} .
The following result is important for determining the homotopy type of posets. Let f: X -+ Y as above:
Iflyl is con:tractible for all y E Y (respectively Y). Then If I is ahomotopy equivalence.
PrQposition 2.1 Assume
where
n is the set of relations 2a, 2(3,8"
a(3, a 2 , (32.
is contractible for all y
E
Iylfl
Proof. We sketch the proof in the simply connected case; fundamental groups must be dealt with using twisted coefficients. For more on this, see [Q]. Con-
A similar analysis yields
f
sider the Leray spectral sequence associated to and the relations in this case are x 2 + xy + y2, x 2y + xy2. These are obtained easily using the additive structure (computed in Ch.IV) and symmetry considerations.
2. General Properties of Posets Associated to Finite Groups For the remainder of this chapter we will specialize to certain G-complexes which are defined from the lattice of subgroups in G. The point of this is that their equivariant structure is an important device for tackling the plocal structure of G and its mod p cohomology. As before let G be a finite group and consider the collection Sp(G) which is the set of all finite p-subgroups of G which are non-trivial. Under inclusion this becomes a partially ordered set (poset for short) endowed with a natural G-action induced by conjugation. In the usual way we can associate a Gsimplicial complex to it denoted by ISp(G)I. We recall how this is constructed: its vertices are the elements of Sp(G) and its simplices are the non-empty -finite chains in Sp (G). Note that the isotropy subgroups of the vertices are the normalizers of the corresponding p-subgroups of G. Similarly we denote by Ap( G) the poset of p-elementary abelian subgroups of G which are not trivial, and its associated space by IAp( G). . These two· functors {Finite groups}
IApOl.lSpOI ) {G-simplicial sets}
E2p,q = Hp(IYI; Hq)
=}
IXI-+ IYI
:
Hp+q(IXI; Z)
where Hq is the sheaf which comes from the pre-sheaf associating to any open subset U c IYI the group
As f is a map of posets If I is a map of simplicial complexes and ~he s~eaf can be computed combinatorially. In fact we have that Hq can be identified with the local coefficient system y J---+ Hq(lfyl). By hypothesis we have that fly is contractible for all y E Y and hence the E 2 -term becomes
q=O otherwise and it follows that f induces a homology isomorphism of simply connected CW-complexes. By Whitehead's theorem this implies that If I is a homotopy equivalence. In the case y If contractible the above result is proved using xap, yap instead. 0 A subset K of a poset X is said to be closed if x' ::; x E K implies K. Let Z be a closed subset of a product of posets X x Y; and PI: Z -+X, P2: Z-+Y the projections. We can identify the fiber of Plover x E X with the subposet
x'
E
Zx
= {y
E YI(x,
y)
E
Z};
similarly for P2 we have
Zy = {x E XI(x,y) E Z}.
172
Chapter V.
G x and so Z is closed. Hence we need only show that Zx = Ap(Gx ) and ZA = XA are contractible. Consider a simplex x; it is a flag
o < VI
N ext we have
Proof. We show
The cases rkp(G) = 1,2 have been settled affirmatively since then Ap(G) is a finite set of points and a graph (respectively) in which case contractibility always implies a G-fixed point. Now suppose that G is the finite group of rational points of a semisimple algebraic group defined over a finite field lK of characteristic p. Then one may associate a "building" II to G in the sense of Tits [T]. It is a simplicial complex of dimension m -1 with a G-action where m is the rank of the underlying algebraic group over lK. Let L1 c II be a simplex; the correspondence
< ... < Vr < lKn ,
r 2: 1
with stablizer G x . The elements of G x which induce the identity on each quotient of the flag form a non-trivial normal p-subgroup of G x , hence IAp( Gx)1 c::: *. Now let JIH be the simplicial complex associated to the poset T(lKn)H of proper H-invariant subspaces of lKn. If H is a p-subgroup of G, WH > 0 for W E T(1I(n) H, hence this poset is contractible; W ~ W H ~ (JFn) H. Thus the X A are contractible and we conclude jAp(G)1 ~ II. 0
174
Chapter V. \..:i-Complexes ana r-qUlvanam IJonomOlogy
We now introduce the following notation for a G-complex X: the singular set of the action is, by definition, the sUbcomplex .
Examples 2.11
G
Sc(X) = {xEXIGx#{l}}. We have a key lemma,
= 54,
the Symmetric Group on 4-Letters,
A 2 (G)/G:
Lemma 2.8 Let H ~ G be a non-trivial p-subgroup. Then IAp(G)IH is contractible.
7l/2 x 7l/
D8
Proof. Let A be a non-trivial p-torus in G normalized by a p-group H ~ G; then A H =I {I}. Denote by E the p-torus of central elements of order dividing p in H. We have
7l 2
D8
7l/
7l/2 x 7l/2
o
whence the result follows.
We can now prove the following proposition which will be vital for our later study of the cohomology of simple groups. Theorem 2.9 Let P
7l/2 x 7l/2
D
In this case IT(1F~)1 ~ IA2(G)1 c:::: v~ Sl is a trivalent graph. The parabolics are
= Sylp(G); then Sp(IAp(G)I) is contractible.
Proof. Replace IAp(G)1 by its barycentric subdivision lXI, where X is the poset of simplices in IAp(G)I. Then Sp(IXI) is the sub complex
U IXI H HSP
=
UIXHI
and the Borel subgroup is
= ISp(Ap(G))I·
H
As before, let Z C Ap(P) x Sp(Ap(G)) be the closed subset consisting of pairs (H, x) where x E X H or equivalently H ~ Px ' By definition of the singular set Px # {I} for all XES p (Ap (G) ); hence Zy = Ap(Px) is contractible by 2.4. Also, if A E Ap(P) then ZA = X A, the poset of simplices in IAp(G)AI. However, we have seen in 2.8 that this is contractible. Hence we have shown
o Corollary 2.10 X(IAp(G)I) == 1 mod ISylp(G)I. Proof. For any finite G-complex we have
x(X) == X(Sc(X)) mod IGI . The result follows from applying this to X = IAp(G)I, G = Sylp(G). To conclude this section we provide some examples of posets Ap(G).
0
G acts edge transitively on
n with quotient B
It is known that H1 (T; 7l/2) ~ St the Steinberg module, which is projective of rank 8 as an 1F 2 G-module.
G=Ml1 ,p=2,
176
3. Applications to Cohomology
Chapter V. G-Complexes and Equivariant Cohomology
177
However, using restriction-transfer we have
G=J1 , p=2,
Ha(6(x); lFp) '-+ Hp(6(X); lFp)
A 2 (J1 )/ J 1 :
res
Z/2
X
A5
A4
(Z/2)3
71,/2~-----'--'---...:.-----"'(71,/2)2
X
Z/2
([G: P] prime to p). Hence H*(G; 6*) == 0 and so
tc is an isomorphism.
0
What this illustrates is that the equivariant structure of Ap( G) determines the cohomology of G for p-torsion coefficients. The next result, a theorem of P. Webb" [We], makes this more precise.
(71,/2)3
Nh ((Z/2)3)
Theorem 3.2 In the Leray spectral sequence associated to IAp(G)1 Xc EG-+IAp(G)I/G with lFp coefficients we have
We point out here that from the above one can verify that this poset has negative Euler characteristic, hence its I-dimensional homology is non-trivial. It is an example of a non-spherical poset space.
3. Applications to Cohomology We have seen in §2 that we can associate a finite G-CW-complex X = IAp(G)1 to any finite group which satisfies the condition that the fixed point set IAp( G) IH is contractible for any p-subgroup H ~ G. In this section we will show how this very strong condition allows one to extract important and useful cohomological information about G as long as we concentrate only on the prime p. To begin we have a result of K. Brown" [Brown, pg. 293, Theorem X.7.3], Theorem 3.1 The map H*(G;lFp)-+Hc(IAp(G)I) induced by IAp(G)I-+pt is an isomorphism for all * E Z. Proof. We have a short exact sequence of G-cochain complexes
EP,q 2
C::'.
-
{OHq(G; lF
p)
0,
P> p = 0.
Proof. Our proof of this result requires standard techniques in equivariant topology; a good general reference for this material is [Bre2]. Recall that IAp(G)1 is a finite G-CW complex with constant isotropy on each cell. Under these conditions the E1-term of the above spectral sequence can easily be identified with
E9 and d 1 is the differential induced by the coboundary map on C*. Define for any H ~ G This is a finite co chain complex and (3.2) is equivalent to proving that for all q 2
°
Hi(D* (q)) = {Hq(G; lFp )
°
C
i = 0, . otherwIse.
Now we have (for all q > 0) a split monomorphism of co chain complexes
Dc(q)
This induces a long exact sequence .,... .
f.~
"'.
........-
'"
~
Dp(q)
and so Dp (q) :::: Dc (q) EEl D'. However, if we define
... -+H1(G; lF p ) -+ Hc(X; lFp)-+Hc( C*)-+Hi+l (G; lF p )-+ ... , where X = IAp(G)I. Now, if P = Sylp(G) recall that Sp(X) :::: * and furthermore, as Tate cohomology is identically zero on free complexes we have
Hp(X;lF p) ~ Hp(Sp(X);lF p) ~ H*(P;lFp).
we see that Ep(q) ~ Dp(q), hence Dc(q) is a direct summand in Ep(q). It is now clear that it suffices to prove the claim for the cochain complex Ep (q). For this note that
As this isomorphism is clearly induced by the augmentation we ded uce that
ilp(C(X); lF p ) == 0
for all
*E Z .
induces homotopy equivalences
The Sporadic Group Mll
for all H ~ P. Hence this complex is equivariantly contractible and so we have an isomorphism of spectral sequences at the E 2 -level
E 2P,q
-
-
HP(SP(IAPFI)/ P; 1-{q)
For G = Mll we get
H*(lVIll) EEl H*(Ds) ~ H*(GL2( lF 3)) EEl H*(S4) . The Sporadic Group J 1
E 2P,q
-
-
Hq(*;1t q)
For G = h we get
which completes the proof for q > O. However one observes that the argument above holds if we use Tate Cohomology and the corresponding spectral sequence, for all q 2: O. Using the fact that for any finite group G with order divisible by p we have HO(G,JF p) ~ jjO(G,JF p) completes the proof of (3.2).
o
Corollary 3.3
H*(G;JFp) EB (
Using the fact that H* (A4) ~ H* (A5) at p = 2 we recover the result of II.6.9,
H*(Jd ~ H*(Nh ((Z/2)3)) ) where NJ 1 ((Z/2)3) is a group of order 168.
E9
H*(Gai;JFp))
~
UiEAp(G)/G i odd
for all
H*(h) EB H*((Z/2)3) EB H*(Z/2 x A4) EB H*(Z/2 x A4) ~ H* ((Z/2)3) EB H* (Z/2 x A5) EB H* (A4 x Z/2) EEl H* (Nh ((Z/2)3)).
E9
H*(Ga;;JFp)
u;EAp(G)/G i
even
* 2: O.
Remark. This extends easily to arbitrary twisted coefficients M and all using Tate cohomology instead in the proof. We may apply this to the examples in §2.
Examples 3.4 (all for p=2)
For S4 we obtain nothing new
H*(S4) EB H*(Ds) EB H*(Ds) EB H*((Z/2)2) ~ H*(S4) EBH*(Ds) EB H*(Ds) EB H*((Z/2)2).
For G = SL 3 (JF 2 ) we get
*E Z
Chapter VI. The Cohomology of Symmetric Groups
o. Introduction There are intimate connections between the homology and cohomology of the symmetric groups and algebraic topology. The first of these is their connection with the structure of cohomology operations. This arises through Steenrod's defini~ion of the pth power operations in terms of properties of certain elements in the groups H * (Sp; IF p). Indeed, the original calculation of H*(Sn; IFp) by Nakaoka was motivated by this connection. These connections were exploited and developed from about 1952 - 1964 in work of J. Adem, N. Steenrod, A. Dold, M. Nakaoka and others. In particular, Adem used properties of the groups H* (Sp2; IF p) to determine the relations in the Steenrod algebras. The complete cohomology rings, H*(Sp2; IFp), were then determined by H. Cardenas, [Card], and it is here that the CardenasKuhn theorem first appears. Then, in the period from 1959 - 1961 E. Dyer and R. Lashof discovered a second connection of the symmetric groups with topology: a fundamental relationshi p between H * (Sn; IF p) and the structure of the homology of infinite loop spaces. The particular relation that expresses the spirit of their results best is a remarkable map of spaces f: Z x Bs ~ lim ,nnsn = QSo 00
n-+oo
which induces isomorphism in homology, but it is not a homotopy equivalence. In the late 1960's and early 1970's this isomorphism formed the starting point for much of Quillen's work relating the classifying spaces of finite groups of Lie type to stable homotopy theory.· We discuss some aspects of this in Chapter VII. The original calculations of the groups H*(Sn; IFp) was based on an important connection between these groups and the cohomology of symmetric products of spheres spn(S2m), .
H2mk-s(spn(s2m); IFp)
~
Hs(Sn; IFp)
for s < 2m (recall from Chapter II that spn(x) = xn /Sn where Sn acts on xn by permuting coordinates). In turn, the spaces 8 pn (8 2m ) were identified
with subspaces of Eilenberg-MacLane spaces by the fundamental theorem of Dold and Thom, [DT] ,
H*(Sn; lF 2 ) c H*(Soo; lF 2 ) is generated by a.ll the monomial products of the generators with bidegree :S n. Precisely, these monomials have the form
00
SPoo(X) ~
IT K(Hi(X; Z), i)
for all connected OW complexes X. From this, Steenrod, in unpublished work, Milgram, [M3l, and Dold, [Do], determined the homology groups of the spn(x). This work has led to recent connections between the homology of symmetric groups, certain moduli spaces of holomorphic maps from the Riemann sphere to symmetric spaces, the geometry of spaces of instantons and monopoles, and even results on linear control theory. Some of this work is described in the papers [C 2M2l, [MM1], [MM2J, [BHMMl, [Segal]. In this chapter we give a fairly complete and self-contained exposition of the structure of the homology and cohomology of the symmetric groups, Sn, for all n. The connection with infinite loop space theory is summarized in Theorem (3.5), but we dQ not discuss the other applications. . We now d~scribe the form of the answer. The homology of Sn injects WIth any untwIsted coefficients, A, onto a direct summand in the homology of Soo for each n via the homology map induced from the usual inclusion of groups. In ~articular th~s implies that H*(Soo;A) ~ Ll~ H*(B sn ,BSn _ 1 ; A) . for all untwIsted coeffiCIents A. On the other hand, if A is a ring there is a ring structure induced on H*(Soo; A) from fitting together the maps Sn x Sm -+Sn+m' This induces pairings
which makes H*(Soo; A) into a bigraded ring. Then the main structure theorems have the form
Theorem. H*(SooiIFp) is a tensor product of exterior algebras on odd dimensional generators and polynomial algebras on even generators where each generator has a known bidegree when p is any odd prime. When p = 2 it has the form of a polynomial algebra on generators of known bidegrees. As an example, here is the answer for p = 2. An admissible sequence, of length n, I = (iI, i 2 , . .. , in), is any non-decreasing sequence of positive ~ntege:s 1 ~ i1 ::; i2 ::; '" ::; in. The dimension of the sequence, d(I) = 1,1 + 21,2 + 41,3 + ... + 2n - l i n , and the bidegree of I, b(I) = 2n. Then
H*(Soo; IF2) ~ IF 2[XI, X2,
..• ,XI, ... J
as I runs over all admissible sequences. These admissible sequences are give~ by certain constructions called (iterated) Dyer-Lashof operations, [DL], applIed to H*(Z/2;1F 2 ). (We will discuss this further in §3.) In any case
Such a monomial has bidegree I:::~ ijb(Ij) and dimension I:::~ ijd(Ij ). In particular H*(S2; IF 2 ) has generators Xi, 1 :S i < 00, where Xi has dimension i and bidegree 2. Similarly H*(S4;lF 2 ) has these generators, their products XiXj, i ::; j, and further generators corresponding to the admissible sequences (iI, i2) of dimension i 1 + 2i 2. The Xi'S are used by Steenrod to construct the basic Steenrod squaring operations, Sqj, while the XI with I = (i 1,i2) are used by J. Adem to construct the iterates Sqi Sqj and determine the relations between them, such as the basic relations Sq2n- 1 qn = O. The first section of this chapter is primarily algebraic. We determine the groups Sylp(Sn) and show that H*(Sn; lF p) is detected by restriction to elementary abelian p-groups. Then we determine the conjugacy classes of these groups in Sn and show that certain key conjugacy classes satisfy weak closure conditions in Sprn. This allows us to determine the image of the restriction homomorphism for these groups and at this point we are able to write down the groups H*(Sn;Fp) for n::; p2. The hard work in §1 is the determination of the image of restriction
s
where Vn(p) ~ (Z/p)n and the inclusion is the regular representation. It turns out that the classes detected by these groups construct every cohomology class in H* (Sn; IF p) via certain composition pairings. Here, for the most part, we follow the exposition of [Mal, reporting on work done in the early 1970's. We extend his ideas in a few places to make things more self-contained. To understand these groups and the composition pairing which builds them for all n we must introduce a more global point of view. This is accomplished in the second section where we introduce the techniques of Hopf , algebras. We first introduce the notions of Hopf algebras, then prove the basic theorems of Borel and Hopf on the structure of commutative and cocommutative Hopf algebras. In §3 we apply the Borel-Hopf theorems to the work of §1 to quickly obtain the Hopf algebras H* (Soo; lF p) for all primes p. In §4 we discuss the structure of some rings of invariants. This is in preparation for §5 where we give complete calculations of H*(Sn; lF2) for n = 6,8,10,12. Finally, in §6 we discuss the cohomology groups H*(An; F 2). The work in §4 is largely incomplete and the complete answers should be of considerable interest when they are finally understood.
184
Chapter VI. The Cohomology of Symmetric Groups
1. Detection Theorems for H* (Sn; JFp) and Construction of Generators
1. Detection Theorems for H*(Sn; JF p ) and Construction of Generators
so, if n
= L~ arpr then [~] = L~ arpr-t and the sum above is
'L., " ar (r-l p In this section we concentrate on the structure of the groups Spn. In particular, we construct a great many cohomology classes which are non-trivial in these groups using the Cardenas-Kuhn theorem and restriction to some of the more important maximal elementary p-subgroups of Spn. These subgroups are classified in (1.3). Then, after we have discussed Hopf algebras in §2 we will show that the elements constructed here generate H* (Sn; JF p) for all n.
185
+ pr-2 + ... + 1) =
-1) =
'L., " ar (pr p_ 1
P _1 1 (n -
Q p (n)).
0
It follows that J embeds the wreath product as the 2-Sylow subgroup of
S2 n
•
Now let m be an arbitrary integer. Write it out in terms of its dyadic expansion Then
The Sylow p-Subgroups of Sn
with odd index. hence Recall that for any G ~ Sn and any other group H, the wreath product H l G is defined as the product H n x G with multiplication
(17,1, ... , hn, 9 )(h~, ... , h~, g') = (hlh~_l (1)' ... ,hnhg-l(n)' gg') . In particular, 8 m I Sn may be thought of as the set of permutations of pairs (i,j), 1 ::; i ::; m and 1 ::; j ::; n by defining
Syl2 (Sn)
Note that, in particular, we have
X .•. X
Syl2 (S2ir) ,
from which we conclude that the Sylow 2-subgroup of any finite symmetric group is a product of iterated wreath products of 2j2. A similar analysis applies for p odd. One checks, as before that
(Zjp) 2 (Zjp) \.
( hI, . . . , hn, g) (i, j) = (h j ( i ) , 9 - 1 (j)) . Then, using lexicographic ordering, this provides an embedding
= Sy12 (S2 il)
2 '" l
(Zjp)
'V
Spr
C
J
r-times
is Sylp(Spr) and a product, depending on the p-adic expansion of n, of these wreath products is Sylp(Sn) for general n. From (IV.4.3) we obtain an important detection theorem for symmetric groups. Theorem 1.2 The mod p cohomology of Sn is detected by its elementary abelian p-subgroups.
and, iterating this, we obtain J: Z2 l ... l 2;2
'->
S2
The Conjugacy Classes of Elementary p-Subgroups in Sn Tt
•
~ n times
Lemma 1.1 1. The power of 2 which divides n! is n - a( n) where a( n) is the number of 1 's in the dyadic expansion of n. I ' n-ap(n) 2, Let p be an odd prime, then t he power 0 f p w17"'lC h d"d 'tV'l es n. 'lS p-l where ap(n) is the sum of the coefficients in the p-adic expansion of n.
Proof. The power of p which divides n! is
Let Vn(p) = (Zjp)n '-> Spn be the regular representation, (the permutation representation on the cosets of the identity). Theorem 1.3 Write n = a + i1P + i2p2 + ... + irpr with 0 < a < pi, >0 , J for 1 ::; j ::; r. (Note that the i j can be greater than p in this decomposition.) Then there is a maximal p-elementary subgroup of Sn corresponding to this decomposition
V1 (p) x ... x V1 (p) x ... x Vr(p) x ... ....
v
ii
I
\"
yo
ir
X
Vr(p) ./
'J.1l.CltP\.rCl.
Y..L •
'-'Vl..lV.l.lL'-..,fJ..'-..I'bJ
.l... J.l.'-'
..........
-'J ............................. ". .... '-'
_
.... ...., ....... t-' .......
and as we run over distinct decompositions (iI, ... , ir) these give the distinct conjugacy classes of maximal elementary p-subgroups of Sn. Proof Let H = (Zjp)i C Sn be any p-elementary subgroup. The action of H on (1, ... ,n) breaks this set into orbits, each of length a power of p. Now restrict H to its action on a single orbit. This gives a homomorphism H -+Spt with image conjugate to vt (p). It follows that H is contained in one of the groups of the theorem. 0
H* (~pn-l
X ... X
Spn-l) IF p) .
p
¢
Proof Every elementary p-subgroup of Spn is conjugate to one contained in either Spn-l X ... X Spn-l or in Vn(p) and H*(Sylp(Sn); IFp) is detected by . ,
v
'
p
p-elementary
~mbgroups.
group itself. Thus there is h E Vn(p) so that Ihl- 1 = e and there is an element .\ E Aut(Vn(p)) ~ NSpn (Vn(P)) so that f· Ae(f . A)-l = e and, of course, f . )..Vn(p)(f· )..)-1 = fVn(p)f-l. But this implies that f· A E C(e) and the claim is verified. Now we are ready to prove the theorem. The proof is by induction, so we assume it is true for n-l. Suppose that we have V' = gVn (p)g-l C Sylp(Spn). We wish to show that there is an h E Sylp(Spn) so that hVn (p)h- 1 = gVn (p)g-l. To begin we assume that 9 E Zjp (Spn-l by the remarks above. Then we project onto Spn-l. Note that, if 7r denotes the projection, then 7r(Vn (p)) = Vn-1(p). Thus 7r(g)Vn _ 1(p)7r(g)-1 C Sylp(Spn-l) and, by the inductive assumption, there is some A E Sylp( Spn-l) so A7r(g) Vn - 1 (p) (A7r(g)) -1_ = Vn- 1(p) and A7r(g) is an automorphism of Vn- 1(p). Hence, there is some
o
Weak Closure Properties for Vn(p) C Sylp(Spn) and pi (Vn-i (p)) C Spn-l 2 Zjp
E
Spn-l C Zjp 2 Spn-l n NS pn (Vn(P))
so that A7r(g¢) commutes with Vn - 1 (p). But then, since Vn-1(p) is its own centralizer in Spn-l, it follows that A7r(g¢) E Sylp(Spn-l), so 7r(g¢) E Sylp(Spn-l) as well, and g¢ E 7r-l(Sylp(Spn-l) = SYlp(Spn) and the induction is complete. 0 Using this result we have a precise determination of the image of the restriction homomorphism from H* (Spn ; IF p) to H* (Vn (p); IF p). Corollary 1.6 The image of the restriction homomorphism
Recall that N C H eGis said to be weakly closed in H if every subgroup of H which is conjugate to N in G is already conjugate to N in H. Theorem 1.5 For all n > 0 and each prime p the subgroup Vn(p) C Spn obtained via the regular representation of (Zjp)n is weakly closed in Sylp(Spn).
Proof The regular representation G '-+ SIGI is defined by regarding the poi~ts of G as the elements being permuted and the embedding as permutations is 9 ({ h }) = {g h }. Then the centralizer of G '-+ SI GI is a second copy of G acting from the right, c(g)( {h}) = {hg-1}. (This is well known, but the proof is easy: if xg = gx for all 9 E G, we have xg{h} = gx{h} = g{h'\} for some .\, and xg{h} = x( {gh}) = {gh'\}.) In particular, in the case where G is abelian, it follows that G is its own centralizer in SIGI when G is embedded via the regular representation. Next, choose e E Vn(p), e =1= 0, and let C(e) ~ Zjp2Spn-1 be the centralizer of e in Spn. In particular, Vn(p) C C(e). Then we also have C(Zjp 2Spn-l) = (e) ~ Zjp, and we suppose! E Spn given so that !Vn (p)!-l C C(e). We claim that there is agE C(e) so that fVn(p)f- 1 = gVn (p)g-l so that Vn(p) is weakly closed in C(e). To see this note that e E C(jVn(p)f-l) = fVn(p)!-l since, as we have seen, the centralizer of the regular representation of an abelian group is the
is precisely equal to
n-times
Proof. First, the normalizer of the regular representation of H C SIHI is always Aut(H) so the Weyl group is Out(H). In the case of (71jp)n this is GLn{lF p). Now (1.5) implies that we can apply the Cardenas-Kuhn theorem to Vn(p) C Sylp(Spn) and (1.6) follows. 0 We also have further weak closure properties which are very useful in understanding the groups H* (Spn; IF p). Theorem 1.7 For allprimesp and 1 ~ i ~ n-l, the subgroup (Vn_i(p))pi C Spn-l 2 Zjp is weakly closed in Spn..
188
Chapter VI. The Cohomology of Symmetric Groups
1. Detection Theorems for H" (Sn; IF p) and Construction of Generators
Proof. Let T = (1,1, ... , 1, T) E H 2Zip where T acts by
189
then
v0I0···0I+I0v0·"0I+···+I0"·010v '-----v----' "---.r---' pi times pi times pi times
T(h 1, ... ,hp,1)T- 1 = (h 2 ,h3 , ... ,hp,h 1,1),
'-----v----'
and suppose that T E H 2Zip and T have the same image under the projection 7r: H 2?llp-+?llp. Then, we can write T = (Tl, ... )Tp, T), and a direct calculation shows that
is in H*(Vn(p)pi;IFp)N. Also, when dim(v) is even
v 0 v 0··· Q9 v pi times
E
H*(Vn(p)pi; IFp)N .
'-----v----'
so TP = 1 if and only if T1 ... Tp
Consequently we have
= 1.
Proposition. Suppose that T and T have the same image 1mder then, if TP = 1 it follows that T and T are conjugate in H 2 Zip.
7r
in Zip,
(Write e = (T1, T2T1, ... , Tp-1 ... T1, 1, 1) E H2?l I p. Then directly eTe- 1 = T and the proposition follows.) Now we turn to the proof .of the theorem. We introduce the following notation. Let M = g(Vn _ i (p))p'g-1 C Spn-i 2?lIp for some g E Spn. Suppose that the projection 7r on M is not {I}. Then the proposition shows that we may assume T E M. The centralizer of (T) in Spn-l 2Zip is L1P(Spn-l) x (T) and consequently, M = (MnL1P(Spn-l)) x (T). Thus, the intersection with L1(Spn-1), now regarded as the permutation group on pn-1 points, must have pi orbits, each of length pn-i-1 to give pi orbits each of length pn-i under under the action of the entire subgroup in Spn-l 2 ?lIp. It follows that
and the order of the resulting group (M n S;n_l, T) is at most p(n-i-1)pi+ 1
IF 2 ) H*(O~(q) x (O;-(q)) ;IF2) { H*(Of(q) x (O;-(q))m-l;IF2)
when q == 1 mod (4) q == 3 mod (4), m odd, q == 3 mod (4), m even,
orH*(02m+l(q);IF2) to H*(O~(q)m x Z/2; lF 2 ) { H*(O;-(q)m x Z/2;IF 2 )
for q == 1 mod (4), for q == 3 mod (4)
is injective in cohomology. In particular there are exactly two conjugacy classes of elementary 2groups (Z/2)2 C O~ (q). Indeed, we have
n 1 Lemma 5.4 The dihedral group D 2n = {x, y I x 2 = (xy)2 = y2 - = I} has precisely 2 conjugacy classes of maximal elementary 2-groups (Z/2)2, the n 2 n 2 first conjugate to (x, y2 - ) and th~ second conjugate to (xy, y2 - ). Proof. The elements of order 2 are precisely the classes xyi, a :::; i :::; 2n - 1 - 1 n 2 and y2n-2. y2 - is central and xxyix = xy-i, y-j xyiyj = xyi+2j. Thus there are precisely 2 conjugacy classes of non-central elements of order two, the first represented by x and the second by xv. Also, xyixyj = yj-i and we see that xyi, xyj commute if and only if j = i ± 2n-2. Hence there are two n 2 n 2 possibilties for conjugacy classes of (Z/2)2, (x, y2 - ) = {I, x, xy2n-2 ,y2 - } n 2 and (xy,y2 - ) {I,xy,xy2n-2+1,y2n-2} and these are clearly not conjugate. 0 As an example of how 5.3 works we have
We now wish to precisely determine these images. The result will be direct from the Cardenas-Kuhn theorem once we have proved Lemma 5.8 Let G = (Z/2)2m C D 2n 2Sm. Then G C (D2n)m C D2n 2Sm· In particular G c O~(q) 2 Sm c O~m(q) when q == 1 mod (4) and ± is + or q == 3 mod (4), ± = - and m is odd, or ± = + and m is even.
Proof. Consider the projection
and look at 7r(G). It is clear that any 2-elementary subgroup (Z/2) C Sm must have r :::; [~]. Thus K~r(7r) n G = (Z/2)V with v ~ m,: [~]. In particu~ar G n (D2n)m must contam a subgroup of the form (Z/2) = (tl,"" t m) WIth ti contained in the ith copy of D2n in (D2n)m and Jr(G) must centralize this group. But this is impossible unless 7r( G) = {I} and the first statement of (5.8) follows.
238
Chapter VII. Finite Groups of Lie Type
O.
To see the second statement we use (5.3), (5.4), (5.5) to assert that G has the form VI x ... X Vm C (D2n)m where each Vi is either conjugate in D 2n
to (x, y2
n
2
-
)
or (xy, y2
n
2
-
).
L.):)
a. H*(O(q);lF 2) = U~=l H*(On(q),On-l(q);lB'2) so the map taking H*(On(q); lB'2)--tH*(O(q); lF 2 )
conjugate to
for a unique s, and (5.3) shows these groups are not conjugate in O~(q) for distinct s. Thus the closure conditions are satisfied. 0
UUU
Theorem 5.10
Moreover, within (D2n) l Sm each such group is
m-s
Lne vonornulOgy Ul LIte uruups un\q) lUr (}
is an inclusion for all n. b. The inclusions On(q) X Om(q)--tOn+m(q) induced by orthogonal sums of vector spaces fit together on passing to limits to define a unitary, commutative and associative multiplication
Corollary 5.9 Under the conditions of (5.8)
and gives H*(O(q); lF 2 ) the structure of a biassociative, bicommutative Hopf algebra. c. As an algebra
H*(O~m(q);lF2) C H*(D2fi;JF 2)Sm
consists of exactly those elements () which satisfy res;(() E H*((VJ)S x (VII)m-s;lF 2)s2 8XS2(m-s),
O~s~m.
Example. The case q == 1 mod (4) We now assume that q == 1 mod (4) and (3 will represent the a non-square in JF q • The inclusions j: otm(q) i(1), i({3): 02m+1
--+
02m+l(q)
--+Otm+2(q)
induce injections in homology and surjections in cohomology. Here i(l) is the inclusion induced by regarding 02m+ 1 ( q) as the group of the form (1)-1 ... -1(1) and
----------
where the subscripts denote the dimensions of the generators and ei is the non-zero element of dimension i in H i (Ol(q); lB'2) = lF2 while hi E H i (02(q); lB'2) is the sum ei + i({3)(ei)' Proof. (a) has already been discussed in part. To see that the homology of the limit is the limit of homology groups is standard since homology is carried by chains with compact support. To see (b) note that the inclusion /-Ln,m: On(q) X Om(q)--tOn+m(q) has images all orthogonal matrices of the form m x m. But the element
2m+1
(1)~ (,(1)-1.~.-1(1)/) = ~ 2m+l
2m+2
while i({3) is the inclusion obtained by regarding 02m+I(1) as the group of the form ((3) -1 ... -1 ((3) and noting '-v-'
2m+l
((3)-1 ... -1((3) ' -v -'
2m+2
=
(~n
1'0)
E
(~ ~)
with A n x nand B
On+m(q) now acts to congugate /-Ln,m
and /-Lm,nT where
is the interchange. Consequently, this pairing is commutative. Also, a similar conjugation shows that the multiplication fits correctly with inclusion so that, on the level of homology groups it passes to the limit groups and defines the desired algebra structure on H*(O(q);lF2)' Finally we verify (c). Here, note that from our description of
(1)-1 ... -1(1) . __________
2m+l
i({3)j is conjugate to i(l)j but ji({3) need not be conjugate to ji(I)' Hence, to be definite we define O(q) = lim--t(On(q)) where On(q) = {o;t(q) n even On(q) n odd, and the inclusion On(q)--tO n+l(q) is j if n is odd and i(1) if n is even. We have
we have res*(w) is the same in both conjugacy classes of subgroups ('£/2)2 C D21. From this, if we set Ii = i({3) (ei), then ai is dual to ei, bi is dual to Ii and both e~ and fl are dual to Wi. Consequently we have h~ = O. On the other hand it is clear that the ei, hj generate H*(O(q);lF 2 ) and are independent. Hence by Hopf's theorem that commutative associative Hopf algebras over a field are tensor products of monogenic algebras it (c) follows. 0
240
241
Chapter VII. Finite Groups of Lie Type
Examples 5.11 H*(02(q);1F2) = H*(D21;1F2) = 1F2[a + b,w](b) where b2 = (a + b) b gives the extension data and the Poincare series is
u L W2j X2k-1
l+x P02 (q)(x) = (1 _ x)(l - x 2) .
u U
+ -(1---x)-:"2-(1---x2:-:-")
(1 - x)(l - x 2 )(1 - x 3 )' Finally, we consider 04(q). Here there are three conjugacy classes of subgroups (71/2)4 with contributions respectively IF2 [0"1 , 0"2, 0"3, 0"4], elements dual to fiJj ® ere s with 0 :::; i < j and duals to elements of the form fdjfkfl with o :::; i < j < k < l. Consequently the Poincare series is 1 + x6
(1 - x)(l - x2)(1 - x3 )(1 - x 4 ) (1 + x)(l + x 2)(1 + x 3)
(1 - x)2(1
_X
)2
The Cohomology Groups H*(Om(q);IF2) The restriction map
Il
H*(Om(q);lF2)--t
H*((71/2)2i)S2i ®H*((71/2)m-2i)Sm-2i
°Si l. c. For n = 1 there are two conjugacy classes of subgroups (Z/2) C SP2(q) ( 7l/2. The first VI is the center of Sp2(q)2 and is normal in Sp2(q) 171/2 ,1xid
Corollary 6.3 In homology the product map n
H*(SL 2(q)i lF2)0 --+H*(SP2n(q); lF 2)
is surjective. In particular H*(SP2n(q); lF 2 ) is a quotient of the subset of elements in lF 2[b4, b8 , ... , J ® lF2[e3, e7,"" e4i-l, ... J consisting of products of n or less terms where the subscripts denote the dimensions of the generators. (The mapping H*(SL 2(q);lF 2)n-+H*(SP2n(q)jlF 2) factors through the 5 n invariants and they have the description given in (6.3).)
Lemma 6.4 The elements e~i-l are zero in H*(Sp4(q)jlF 2), so it follows that there is a surjection of the products of n or less terms in the algebra lF 2[b 4, b8 ,.·., b4i ,· .. J ® E(e3,"" e4i-l, ... ) onto H*(SP2n(q)j lF2)'
Proof. (6.1) shows that the two inclusions p,: SL 2(q) x Z/2-+SL 2(q) lZ/2 and Ll x id: SL 2(q) x Z/2-+SL 2 (q) l Z/2 are conjugate in Sp4(q). Consequently,
e E im(H*(Sp4(q); Z/2) C
H*(SL2(q) 2 Z/2; Z/2)
only if p,* (e) = (Ll x 1)* (e). In particular consider the elements r(e3bi). We have p,*(r(e3bi) J.L*(e3bi ® e3bi) = 0, but (Ll x id)*(r(e3bi)) = e3bi ® e4i+3 + higher terms which is certainly not zero. It follows that r(e3bi) is not in the image of H*(Sp4(q);lF2) so dually e4i+3 ® e4i+3 maps to zero in H*(Sp4(q);lF'2) and e~i+3 = 0 as asserted. 0
while the second VII C Sp2(q) x 7l/2--+Sp2(q) 171/2. Proof (aj Given (71j2)W C SP2n(q) we argue as in (5.3) using the ±1 eigenspaces for generators of (71/2)W to show that if w < n then there is a subspace V C lF~n with the restriction of the symplectic form non-singular on V and V is a -1 eignenspace for t 1, but a + 1 eigenspace for the other generators, and dim(V) 2: 4. This allows us to split V non-trivially as the orthogonal sum
V
= (S)~
.. · ~(S)
and extend (Z/2)W to a subgroup (71/2)w+l C SP2n(q). This process stops only when w = n. In particular, every (71j2)n C SP2n(q) is associated uniquely with a splitting of the symplectic form
and from this it is direct that any two such subgroups are conjugate in SP2n(q)· (b). Let W = (71/2)W C SP2n(Q) 171/2 and consider the projection 1f: SP2n(Q) I Z/2-+Z/2. If im(1fIW) -I {1} then there is atE W with t of the form (e, e' ,T). Since 1 = t2 =
(e, e' ,T) (e, e' ,T) = (ee', e' e, 1)
it follows that t = (e, e- 1 , T) for some e E SP2n(Q). Thus, when we conjugate W by (e- 1 , 1, 1), t is conjugated to (1,1, T), so we can assume (1,1, T) E W. On the other hand, the centralizer of (1,1, T) in SP2n(Q) 171/2 is
244
Chapter Vll. Finite Groups of Lie Type
245
which, by (a), has exactly one conjugacy class of elementary 2-groups (2/2)n+1. Since n ;:::: 2 it follows that n + 1 < 2n and we assumed w = 2n. Thus, W c ker(7T) = SP2n(q) X SP2n(q) and (b) follows from (a). (c). The proof of (b) actually shows that for all n, SP2n(q) 22/2 contains exactly 2 conjugacy classes of elementary 2-subgroups, the first of the form (2/2)2n C ker(7T) and the second of the form (2/2)n+1 which is not in the kernel of 7T and which is conjugate to a subgroup of il(SP2n(q)) X 2/2. (c) is a special case of this. 0 Note that we can regard Sp2(q) 22/2 as the normalizer of a 2/2 x 7l/2 C SP4(q) and the (2/2)2 is contained in Sp2(q) 2 7l/2 as the normal subgroup VI· Lemma 6.7 N SP4 (q) (VII) n N Sq4 (q) (VI) is a split extension of the centralizer oj VII, il(Sp2(q)) x 7l/2 in N SP4 (q) (VI) by a copy oj71/2, N SP4 (q) (VII)
n NSp'l(q) (VI) = (il(Sp4(q)) x 7l/2)
xT
((1,-1,1)) .
Proof. The Weyl group WS P4 (q) (VI ) = 7l/2 C GL 2(2) = 53. (1, -1,1) normalizes VI I and acts non-trivially on it so is contained in the intersection. Hence the intersection above is the extension of the centralizer of VI I in SP2 (q) 27l /2 by (1, - 1, 1) as asserted. 0 We now consider the double coset decomposition
Sp2(q) 22/2
Ii Sp2(Q) 22 /2 n 9;1S P2 (q) 271/29i
where Li
= 9iSp2(q) 22/29;1 n Sp2(Q) 22/2.
1'"
9'-'SPZ(f, I7l/29 i
¢i,j: H* (L~; JF 2 )
res
Sp2(q) 271/2
and the resulting image of VI I will be a copy of VI I, hence by modifying 9i it will be VII, except in the one case covered in (6.1) where it will be VI. Let 91 = 1 and 92 give the coset where this interchange happens. Consider the composite map
res ----t
res
H* (Sp2 (q) ( 7l/2; JF 2) ----t H* (VI; JF 2)'
tr
Cj
----t
H* (L~,j; JF 2) ----t H* (L~:j; JF 2 ) - - t H* (VI; lF 2 )
s:
and L~,j = 9;1 L~9j n VI. In particula,r ¢i,j == 0 if L~,j VI, so the only terms which can enter non-trivially into the sum above are ¢l,! = res and the terms ¢2J' . In the case of ¢2,j we can choose the 9j giving the double coset decomposition Sp2(q) 22/2 = UVI9jL~ = U9jL~ so that cj = id for each j since L~ is given by (6.7) and this group surjects onto im(NSq4(Q) (VII )~Aut(VII). It follows that the sum L:: j ¢2,j = \Sp2(q) 22/2, L~\res = 0 since the index of L~ in Sp2(q) 22/2 is even. Thus we have proved that E is just the restriction map
= res: H*(Sp2(Q) 22/2; lF2)--tH*(V[; lF 2)
and the image of H*(Sp4(q); lF2) in H*(1i]; lF2) is lF 2[b 4 , bar Now, to complete the proof of (6.5) we can proceed by induction, (6.4) and the result above providing the theorem for Sp4(Q). By Hopf's theorem on commutative Hopf algebras the only possible truncations on the polynomial parts would be relations of the form b~~ = 0 for some l. So it suffices to show that no such truncation can occur. Thus, assume there are no such truncations for H*(Sp21 (q); lF 2), l ;:::: 2. But (2/2)2
1'"
Again, each of the composites
can be rewritten using the double coset formula as a sum of composites
E
For certain 9i we have the composite
VII
By applying the double coset formula to the first two maps E can be rewritten as a sum of maps of the form
1
X
(2/2)21 C SP21 2Z/2 C SP21+1 (q)
is a closed system satisfying the hypothesis of the Cardenas-Kuhn theorem for l ;:::: 2 and the inductive step is direct. 0
7. The Exceptional Chevalley Groups
where v is 1 or 3 and n acts on each SL2 as conjugation by (
The exceptional Chevellay groups are of two types. First there are the groups G 2(q) C P4(q) C E6(q) C E7(q) C Es(q) which are analogues for finite field of the exceptional classical Lie groups. Then there are the groups corresponding to ~he "graph automorphisms" of the Dynkin diagrams. Classically, only the umtary groups have this type of description, but for finite fields (and other fields with similar automorphisms) there are further types. In particular the groups 02n are associated to graph automorphisms. There are two further families of this type which were discovered by Steinberg, the first associated to the rotation through 27r /3 of the Dynkin diagram D4,
written 3 D 4(q), and called the "triality twisted" D4(q). The second is associated to the flip of the Dynkin diagram for E6(q), f--
•
flip
----4-
•
•
"
r
•
Then there are three further exceptional families, associated to graph automorphisms which are only valid at certain primes, the Suzuki groups Su(22m+l), and the two Ree families, 2G 2(3 2m +1 ), 2P4(22m+l). The groups 2G 2(3 2m +1 ) have Syl2(ZG2(32m+l)) = (Z/2)3 with Weyl group the semidirect product (Z/7) XT Z/3. Consequently .
H*eG 2(3 2m +1 ); JF 2 ) = JF 2 [x, y, z]Z/7X T Z/3 . Aside from this and Quillen's general result mentioned in the introduction to this chapter not too much is known about most of these groups. However recently, the situation for the groups 3D4(q) and 2G 2(q) has begun to becom~ much clearer, and there is related work by T. Hewett which promises to also clarify the situation for F4(q). The groups 3D4(q) and G 2(q) are simple for odd q and are characterized by a result of P. Fong and W.J. Wong, [FW], ~heor~m
1.1 Let G be a finite simple group with one conjugacy class of mvolutwns. If the normalizer of an involution is the central product
J.L a non-square in
~J.L ~)
with
IF q, then G = G 2 (q) if v = 1 and 3 D4(q) if v = 3.
XT (n). From this Syl2(G) = (n), and this group, in turn, is isomorphic to
It follows that SyI2(G) C SL 2 (q) * SL2(q1J)
Q2 m
* Q2
m
XT
where y acts to invert the elements of (71,/2 m- 1 ) while v interchanges them. In particular, this representation shows that G has 2-rank 3, so the largest 2-elementary subgroups are of the form (Z/2)3.2 For example, 2m 2 2m if a, b generate the two copies of 71,/2 m- 1 then I = (a - , b - , y) and II = (a 2m - 2 , b2m - 2 , aby,) give two distinct copies of (Z/2)3 C Syl2(G). For simplicity; in the rest of this section we assume that q == 1 mod (4). The involution j which is centralized by SL 2(q)*SL 2(q1J) xT(n) is (ab)2=-2. With this notation results of [FW), [FM) , show that G has 2 conjugacy classes of (71,/2)2's, the first represented by (j, a 2=-2), and the second by (j, aby) and 2 conjugacy classes of (71,/2)3's, the first represented by I and t,he second by II. In particular, every (Z/2)2 is contained in a (Z/2)3. Further, in [FW) it is proved that the normalizer modulo the centralizer for each (Z/2)2 is G L2 (2) = 53' Then in [FM) it is proved that the normalizer of I is given as a non-split extension I o BEn, using the inclusion En X Em "-7 En+m to give the multiplication. There is a natural map
BEoo---+Q(So)o , which induces an isomorphism in homology. Now 7rl(Q(SO)) ~ Zj2 ~ 7r1 (BEoo) / Aoo· Hence, taking the plus construction with respect to the maximal, normal, perfect subgroup Aoo we deduce that
BE! ~ Q(80)0 , i.e. 7ri(BE;t) ~ 7[}(SO), the ith stable homotopy group of the spheres. Now, if G y Eoo is a perfect group, we have a natural map BG+ ~BE+. In {3:'
00
particular we have for 7ri(BAt) ~7ri(BE~J that [H] l.·,6i is an isomorphism for 2 :S i < (n - 1)/3 or for 2 :S i < (n + 1)/2 and n == 2 mod (3). 2. When we invert 3, ,6i is an isomorphism for 2 :S i < (n + 1)/2 except if i = 3 and n = 6. 3. ,6i is an epimorphism with kernel isomorphic to 7l/3 if n = 3i or n = 3i+ 1.
both 7 and 13 are adopted to 3, while 11 is only adopted to 5 and 23 is only adopted to 11. It is well known that given q there is some p which is adopted to q. We have
Theorem 3.2 Let Pl and P2 be adapted to q) then the BGL(W pJt) i = 1,2) are homotopy equivalent) and each is a factor of Q(50). That is to say
Q(SO) ~ Vq x BGL(lFpJt. (The idea of the proof is to consider the injections
The General Linear Group Over a Finite Field Let IF q denote the field with q elements and take G = G L (IF q), N commutator subgroup. Then by definition
= E, the
the higher K-groups of lF q. Using the homology calculations we described for the general linear groups Quillen proved that, in fact, there is a homotopy equivalence BGL(lF q)+ ~ F¢q where F¢q is described as the homotopy fiber· of a map BU~BU. Precisely, let ¢q denote the element in [BU, BU] which represents the corresponding Adams operation in K -theory. Then F ¢q is the homotopy fiber corresponding to the operation 1 - ¢q: B U ~ BU. From this Quillen [Q3] calculated the higher Ii-groups of lFq as
Kn(lF q) ~
{71/(qi o
1)
n = 2~ - 1, n = 22.
This leads to information about Q(SO) and the stabl~ homotopy groups of spJ:ieres as follows. Since the homotopy groups of BGL(lF p)+ are all finite a standard theorem in homotopy theory tells us that, up to homotopy type we have a product decomposition
BGL(lFp)+ ~
II
BGL(lFp)t
I prime
where 7ri(BGL(lFp)t) = Syll(7ri(BGL(lFp)+). It turns out that various of these BGL(lFp)t are factors of Q(SO). In order to describe which ones we need a definition.
Definition 3.1 Given an odd prime q we say that p is adopted to q if p == 1 mod (q) but p i= 1 mod (q2). Note that p is adopted to only a finite number of primes since p adopted to q implies that p > q.For example 19 is not adopted to any prime but
where p is the inclusion as permutations of coordinates. From the determination in (VIl.4) of H* (G Ln (W p); Wq) we see that the cohomology calculation for the composition (p. reg)* is determined by restricting to the maximal torus. A direct calculation then shows that the map surjects through a range which increases with n. Consequently it is an isomorphism through that range and thus a homotopy equivalence through that range when restricted to the the q piece. It follows on passing to limits that Q(SO)q splits in the desired way.) Remark. This process does not work to obtain a splitting at 2. The relevant space here is BSO(Wp)t, for p == 3 mod (8), and the proof of splitting is quite a bit more complex.
The Binary Icosahedral Group Let G denote the binary icosahedral group. It is a group of order 120 which can be thought of as a double cover of A 5 . Classically it has been known to act 3 freely on S3 since it is a finite subgroup of the group 8U(2) ~ 8 which can also be thought of as the unit quaternions 8p(l) or 8pin(3). Indeed 1 thinking of it as 8pin(3) it double covers 80(3) and the conjugacy classes of finite subgroups of 80(3) are well known. In particular A 5 , the symmetry group of the icosahedron is a subgroup. Then the binary icosahedral group in 8pin(3) is the inverse image of A5 under the double covering map. We have that H 1 (A 5; Z) = 0 since A5 is simple. Also, we have _that H2(A5; Z) = 7l/2 so it has a unique maximal central extension Z/2~A5 ~A5 which is non-split. On the other hand it is not hard to show that the Sylow 2-subgoup of the binary icosahedral group is the quaternion group Qs, so the extension above describes G. In particular it follows that H 2 ( G; Z) = Hl (G; 7l) = O. Now, by construction G acts freely on the unit sphere 8 3 , since it is a subgroup. Consequently the quotient manifold S3/G = M3 has HI (M 3;7l) = Hl (G; 7l), H2(M3; 7l) = H2(G; 7l) and H3(M3; 7l) = 7l since the action preserves orientation and the quotient is a compact oriented manifold. Thus M3 is an example of a homology 3-sphere. It was originally discovered by Poincare
280
3. Examples and Applications
Chapter lX. The Flus Construction and Applications
and is known as the Poincare sphere. Taking a cellular decomposition of M3 and lifting to the universal cover 8 3 we obtain a G-free cellular decomposition of 8 3 which gives us an exact sequence ~
281
that multiplication by the integer 120 on the 2, 3, and 5 torsion of 7r*(S3) is trivial. Thus the homotopy groups of BG+ split into two copies of 7r*(S3)120 with a dimension shift. (We thank one of the referees for pointing this out to us.)
f
Z--+ C3(83)--+C2(83)--+C1(83)--+Co(83) --+ Z , where the Ci (8 3 ) are finitely generated free Z(G) modules. When we paste copies of this exact sequence together using fh . E we obtain a long exact resolution of Z over Z(G). From this (since the map I-h(8 3)-+H3(M3) is just multiplication by deg(G) = 120), it follows that
Hj(G;Z)
=
lVI
j
-+
otherwise.
BG and the induced map on plus BG
--+
1
f2(BG+) ::: Fiber(j) ::: F120 , where F 120 is the homotopy fiber of the map of degree 120 from 8 3 to itself. Thus we have an exact sequence 3 x 120
which is now a 2-local homotopy equivalence. On the other hand, from [FM], [M2], there is also a homomorphism W *H W'-+G 2(q) - which is injective on H and an isomorphism to Syl2(G2(q)) if q s::! 3,5 mod (8) - for any q ';p o mod (2), and these homomorphisms fit together to give a homomorphism W *H W'-+G 2(pOO). For p s::! 3,5 mod (8) this gives a map
1
F As the maps between the plus constructions induce homology isomorphisms and G acts trivially on H*(8 3 ; Z), we can apply the comparison theorem to conclude that X is a homology isomorphism. Since Hi(G; Z) = 0, i = 1,2, we see that BG+ is 2-connected, hence F is simply connected and X is a homotopy equivalence. Clearly M+ ::: 8 3 and it follows that
8
We have the map from the amalgamated product W *H W', to M12 given in VIII.4.1, which, from VIII.4.2 induces isomorphisms in mod (2) homology. Taking plus constructions gives us a map
j
o
Now consider the classifying map M3 constructions 3
M12
= 0, j == 3 mod (4),
Z { Z/(120) f
The Mathieu Group
3
8
--+7ri(F120)--+7ri(8 ) --+7ri(8 ) --+ 7ri-l(F120 )--+ ... It follows that there is an exact sequence
VII.7.6 shows that H*(B~2(pOO);lF2) s::! lF 2[d 4 ,d6 ,d7 ] and that
e;: IF 2[d4 , d6 , d7 ]-+H* (M12 ;lF 2 ) is an injection onto the same subalgebra in H*(M12; lF 2 ), (the subalgebra which restricts to the Dickson algebra in each of the three conjugacy classes of (Z/2)3,s). Consequently, when we pass to the homotopy fibration, the fiber at the prime 2 is 14 dimensional with Poincare series the numerator in the Poincare series for H*(M12; lF 2 ), 1 + t2
+ 3t3 + t 4 + 3t5 + 4t6 + 2t1 + 4t S + 3t 9 + t lO + 3tll + t 12 + t 14 •
. It would be very interesting to have a good geometric :realization of this fiber. In particular, if it were the homotopy type of a closed parallelizable manifold of dimension 14, this would be very useful.
0--+7ri(8 3) / (120· 7ri(83))--+7ri(BG+)--+7ri-l (8 h20--+ 0 3
for each i 2. 2, while 7r1(BG+) = O. Here 7ri(83)r20 denotes the subgroup of 7ri(8 3) consisting of elements whose order divides 120. This analysis is due to J.C. Hausmann [H]. In fact Hausmann has proved that if H is a perfect group with H 2(H; Z) = 0 then 7rn (BG+) for n 2. 5 is in one to one correspondence with the set of topological homology spheres with fundamental group H up to an appropriate notion of cobordism.
Remark. From the work of P. Selick, [Sel], F.R. Cohen, J.C. Moore, and J.A. Neisendorfer, [CMN], I.M. James, [Jam], and F.R. Cohen, [Coh], it follows
The Group J 1 There is also a close connection between Bt and B G2 . Here, if
with the action induced by regarding (Z/2)3 as the additive subgroup of the field ]Fs, then LJ c E c G 2 where E is the non-split extension 23 . L3(2) discussed in VII.7 (see VII.7.6 in particular).
If we could pass to plus constructions (with would be a 2-equivalence lij:
7rl
]\1[22
(B) equal to '!L/3) , there
1 1
B~/2)3XT(Z/7XTZ/3) ---?Bt
induced from the inclusion of the left hand subgroup as the normalizer of Syl2(J1) in J 1. Of course, since ('!L/2)2 xT '!L/7 is not perfect there are difficulties with this step, but what we can do is to kill the fundamental group by adding a two dimensional cell to kill the element of order three. A direct check shows that the resulting space has H 2(B(,1,/2)3 XT(Z/7XTZ/3) U e2;'!L) = '!L and this is the second homotopy group, so we can kill this '!L by adding a single three cell and we have a space with homology unchanged at 2 and 7, but the '!L/3 in dimension 1 is gone. The inclusion of (7L/2)3 XT (7L/7 XT '!L/3) into O 2 induces a map of classifying spaces eJ: B(Z/2)3 xT(zi7XTZ/3) ---+B02
Jvh3
"-------+
S22
5 23
"-------+
where the embedding given by the left hand vertical arrow is given in VIII.5. Since H* (Sn; IF 2) is determined by restriction to 2-elementaries, we can gain a great deal of information about the map on classifying spaces by restricting to the 2-elementaries E and F. Using the particular Sylow subgroup given in VIII.5.3 we find that E has four generators: (1,17)(3,18) (1,18)(3,17)
(2,15)(9,12) (2,12)(9,15)
(5, 19) (16,20) (5,20) (16, 19)
(7,10)(13,14) (7,13)(10,14)
together with (1,2)(3,9)(15,17)(12,18)
(5,10(7,19)(13,16)(14,20)
(1,16)(17,20(3,5)(18,19)(2,13)(7,12)(9,10)(14,15).
which injects H* (B02; ill' 2) as the Dickson algebra in
H* (B(Z/2)3 xT(Z/7XTZ/3); IF2)' Since B02 is three connected eJ lifts uniquely to a map of the space above with a two and a three cell added. Similarly, since J 1 is simple and the multiplier is {I}, [Co], so H 2(J1 ;'!L) = 0 we can lift 7rJ to the space with cells attached and this lifted map is an equivalence at the prime 2. Consequently, at 2 we can identify the two spaces and we have a map
eJ 0 7rJ"1: (BjJ2---+(Bo2h, and the fiber in this composition, at 2, and in cohomology looks like the fiber of eJ which is just the fourteen dimensional closed compact manifold O 2/ ((7L/2)3 XT ('!L/7 XT '!L/3)). This manifold, from VIII.2, has cohomology ring H*(Jl;IF2)/(d4,d6,d7) which is isomorphic to H*(02;IF2) as a module of the Steenrod algebra A(2). The lift of 7r J restricted to this fiber induces a surjection in cohomology which explains the numerator in the Poincare series for H* (J1 ; IF 2) given in VIII.2.1. Also, John Harper has shown that any simply connected OW complex with the mod (2) cohomology of O2 as a module over A(2) must, in fact be homotopic (at 2) to O 2. Thus, Bjl' at 2, fibers over B02 with fiber having the homotopy type of O 2 . This fibering is exotic, and we thank F. Cohen for describing it to us. The Mathieu Group
M23
The embedding M 23 C 5 23 given by letting it act as permutations on the 23 cosets of M22 is quite explicit. In particular, we have the commutative diagram
These last two elements generate the intersection V2 of E and F, and E is clearly contained in (K 2 '!L/2) ( '!L/2 where K c 54 is the Klein group. The group F is generated by the last two elements together with two others, where the elements can now be written as follows. (1,13)(2,16) (1,16)(2,13) (1,2)(13,16) (1,16)(2,13)
(6,11)(8,21) 6,21)(8,11)
(3,5)(9,10) (3,9)(5,10) (3,5) (9, 10)
(14,20)(15,17) (14,20)(15,17) (14,20)(15,17) (14,15)(17,20)
(7,18)(12,19) (7,19)(12,18) (18,19)(7,12)
and we see that F c 1< x K x K x K x K. From the results of VI.l, VI.2, we see that the symmetric sum Sd 3 ® d3 ® 1 0 1 0 1 in H*( K 5;IF 2) is in the image of restriction from H* (522 ; IF2), and it is a direct calculation to check that the image of this class under the map H*(K 5 ; IF 2)-tH*(F; IF 2 ) described above is non-zero. Consequently, using the splitting of 3.2 and the following remark, we can project Bt23 to V2 = coker( J), which, from our knowledge of the stable homotopy of spheres, we know is 5-connected with 7r6(coker(J)) = '!L/2, and it must be the case that the induced map 7r6(Bt23)-+7r6(coker(J)) is an isomorphism.
4. The Kan-Thurston Theorem From the results outlined in the previous section we can deduce that there are certain interesting topological spaces which have the homology of a K ( 7r, 1). Among them are [200 2)00, FlJr q , and certain spaces of homeomorphisms we have not discussed. This very naturally leads to the question of whether or not this is true for a large class of spaces.
284
4. The Kan-Thurston Theorem
Chapter IX. The Plus Construction and Applications
This was settled in the affirmative by Kan and Thurston [KT]; in fact they proved that every path connected space has the homology of a K(7r, 1). In this section we will outline a proof of their result (due to Maunder [MauD, which can be stated more precisely as follows: Theorem 4.1 For every path-connected space X with basepoint there exists a space TX) and a map tX TX ----"7 X
which is natural with respect to X and has the following properties: 1. the map tX induces an isomorphism on (singular) homology and cohomology H*(X,A) ~ H*(TX,A) for every local coefficient system A on X) 7ri (T X) is trivial for i =I- 1 and 7r1 tX is onto) 3. the homotopy type of X is completely determined by the pair of groups G x = 7r1TX) and Px = ker7r l tX (note Px C Gx perfect). In fact) we have that X -::::= J( (G x, 1) +) where the plus construction is taken with respect to Px·
2.
To prove this result, we will need (given any group G) to construct an acyclic group CG into which the original group embeds. To do this we will use ideas due to Baumslag, Dyer and Heller [BDH]. First we need Definition 4.2 A supergroup !vI of a group B is called a mitosis of B if there exist elements s, d in M such that 1. Ai = (B, s, d) 2. bd = bb s for all b E B, and t 3. [b t , bS ] = 1 for all b, b E B. Definition 4.3 A group M is mitotic if it contains a mitosis of everyone of its finitely generated subgroups.' Our goal will be to show that every mitotic group is acyclic, and that every group embeds in a mitotic group. We introduce some notation. Let t K,: B x B --+ M be the homomorphism K,(b , b) = b'bS and )..: B --+ B x B defined by /\(b) = (b, 1). Then if J-L: B -+ M is the injection, clearly J-L = K,)... Lemma 4.4 Let ¢: A --+ B be a homomorphism) !vI a mitosis of B) and J-L: B -+ M the given injection. Let IF be any field such that, ¢*: Hi (A, IF) --+ Hi(B, IF) is 0 for i = 1,2, ... , n - 1. Then (jJ,¢)*: Hi(A, IF) -+ Hi(M, IF) is 0 for i = 1,2, ... ,n.
285
Proof. Clearly we only need to verify the claim for i = n. By the Kunneth formula, ' Hn(B x B, IF) ~ Hi(B,lF) 0 Hj(B,lF) .
L
i+j=n Now let N: A --+ A x A, N(a) = (a,l) and p': A -+ A x A, p'(a) Then we have that Wp = /'i,(¢ x ¢)N, hence if a E Hn(A, IF),
= (1, a). , (a)
We also have that CdJ-L¢ = /'i,(¢ x ¢)L1A, where Cd is conjugation by d. Using the fact that inner automorphisms are trivial in homology, we obtain
(J-L¢)*(a) Similarly, csf-L¢
= /'i,(¢ x
=
/'i,*(¢*a 01)
+ /'i,*(1 0
¢*a).
(b)
¢)p' and hence (f-L¢)*(a)
=
/'i,*(1 0 ¢*a).
Combining (a), (b) and (c) yields (f-L¢)*a = 0, proving the lemma.
(C)
o
We now prove Theorem 4.5 Mitotic groups are acyclic.
Proof. Assume that G is mitotic. If KeG is a finitely generated subgroup, then K = Ko C KI C K2 C ... c G, where each injection Ki C Ki+l is a mitosis. Now note that given any mitosis B --+ M, the induced map H 1 (B, Z) -+ HI(M, Z) is zero. Hence we obtain, using (4.4), that Hi(K,F)-+ Hi(Kn , F) is zero for any field F, and i = 1, ... , n - 1. From this we deduce that Hi(K,lF) --+ Hi(G,lF) is zero for all i > O. Now G is the directed coli mit of its finitely generated subgroups, and the colimit of their inclusion maps is the identity Ie. Homology commutes with directed co limits , from which we deduce that Hi(G,F) = 0 for any field, i > 0 and hence G is acyclic. 0 We introduce the notion of algebraically closed groups. Definition 4.6 A group G is said to be algebraically closed if every finite set of equations
in the variables Xl, ... ,X m and constants g1, ... ,gn E G which has a solution in some supergroup of G, already has a solution in G. Theorem 4.7 Algebraically closed groups are mitotic.
Proof. Suppose that G is algebraically closed and denote A = (g1,"" gn) a finitely generated subgroup of G. Let D = G x G and let G = G x I ,
H = Ll(G), K = 1 x G be the corresponding copies of G embedded in D. We construct the extensions 1
E = (D, t; t- (g, l)t = (g,g), g E G) m(G)
Then G embeds as
= (E,u;u-l(g, l)u = (l,g),
G in m( G),
9 E G) .
and the finitely many equations
g~i(gigf2)-1
=
1, [gi,gj2]
isomorphisms of homology and cohomology for any coefficient system. Now note that
f TL K(C1f' 1) · I' an d so t h e Inc uSlOns 0 , " of 1f'l-hence TK (the universal cover) universal covers of all three. Using a implies TK is acyclic, hence that T K
=1
with i,j = 1, ... , n have a solution Xl = t,X2 = u in m(G). Thus they have a solution Xl = d, X2 = s in G itself. As a consequence of this the group (A, d, s) is a mitosis of A in G, and so G is mitotic. D Using the fact that any infinite group embeds in an algebraically closed group of the same cardinality, we obtain Theorem 4.8 Every infinite group can be embedded in an acyclic group of the· same cardinality.
T(f)(J) in T J( induce monomorphisms . contains multip.le cOP.ies of the acycl~c lifted Mayer-Vlaetons sequence, thIS is aspherical. Also note that as
1f'l(K) = 1f'1(L)
*7\1(80-)
1f'I(o-) ,
the map t*: 1f'l(TK) -r 1f'1(J() is onto. ' . . Using induction on N, one can construct T J( for all fi~lt: (order ed) SImplicial complexes K. This can be extended to infinite simplICIal complexes by taking the direct limit over finite subcomplexes. 'f X is a path-connected space, let SX be its singular complex, N ISXI ~~ ~eometric realization. Denote by ISXI" the secon~1 deriv:d c?mplex (considered as a i1-set). Then we can take TX = T(ISXI ). T~lS WIll be a natural construction satisfying the desired properties, as a contmuous n~ap of X gives rise to a simplicial map of ISXI" that is strictly order-preservmg on each simplex. Then tX is the map
We have therefore proved that given any group G there exists a group CG containing G such that CG is acyclic. We will now give the proof of Theorem 4.1, following Maunder. Proof The first step is to prove the existence of T X satisfying (i) and (ii) when X = L, a connected simplicial complex with ordered vertices. We proceed inductively: suppose that for each such L with at most N -1 simplexes, t: T L -r L has been constructed satisfying (i) and (ii) and that this construction isnatural for simplicial maps of L that are strictly orderpreserving on each simplex. Assume also that, for each connected subcomplex MeL, TM = t- l M, ~nd that 1f'l(TM) -r 1f'l(TL) is 1-1. Note that because every connected 1-dimensional complex is a K(1f', 1), we may start the induction by taking t to be the identity. Let K be obtained from L by attaching an n-simplex (n ~ 2) 0- to 80- C L. Then T(8o-) C T(L) and if j: 0- -r Lln is the (unique) order-preserving simplicial homeomorphism to the standard n-simplex, the corresponding map Tj:T(8o-) -r T(8Lln) is a homeomorphism, and T(8Lln) is a K(1f', 1). Now let g: T(8Lln) -r K(C1f', l) be a map realizing the embedding 1f' ~ C1f', C1f' acyclic. We take the mapping cylinder of the composition gT(f): T(8o-) -r K(C1f'; 1), and attach it to T(L) along T(8o-) C TL; this will be TK. To extend t to T K -r K, .we do it as usual on mapping cylinder coordinates (x, t) and by mapping K(C1f', 1) to the barycenter fj of 0-. The construction can be verified to be natural for simplicial maps that are strictly order-preserving on each simplex. Using the Mayer-Vietoris sequences for K, T K and the 5-1emma, it follows that t: T K -r K induces
1f'l(TI
