Cohomology of finite groups

0, 2 mod (4), 0 mod (4), odd.

= ~ ~

o

and the result follows .

Milnor, [Mi2] , proved that S3 cannot act freely on any sphere. In fact he proved that a necessary condition for a periodic group G to act freely on a sphere is that any element of order 2 in G must be central.

Remark. The proof of (6.3) illustrates the important fact that G i~ pe~iodic only if its IF p cohomology is periodic for ~ach ~ and then the p:rlod IS the least common multiple of its p-periods. ThIS basIC result follows dIrectly from (6.4) and (6.6) below which determine the possible p-Sylow subgroups of G. Lemma 6.4 G is periodic only 'if the only p-elementary subgroup~ of G are cyclic, which is true only if all the abelian subgroups of GaTe cychc. Proof. Indeed, if (Zjp)n c G for n ~ 2, then IF p[a, b] C H* (G.; IF p) .by the Krull dimension results of (IV.5), for appropriate a, b. But III thIS case dim(Hi(G; IFp)) is unbounded. On the other hand from the coefficient sequence

.p

O--+Z -+ Z-+IF p--+O

we get the long exact sequence of cohomology groups

Chapter 1V. Spectral Sequences and Detection Theorems {),

'p,

{)

--+ H~(Gi Z) --+ H~(Gi Z)--+Hi(Gi lFp) --+ Hi+l(G; Z)--+· .. ,

and it follows that dim(Hi(G; Z) 0lFp) is also unbounded, so G cannot be 0 periodic.

Remark. Using a spectral sequence argument with lFp coefficients in 6.2, we infer that Zip x Zip cannot act freely on a sphere, and hence if a finite group acts freely on a sphere (preserving orientation or not) it must be periodic (see V.0.7). Definition 6.5 A P-group G is any finite group G which satisfies the condition that every abelian subgroup of G is cyclic. The P-groups have been completely classified by Suzuki and Zassenhaus. It turns out, using the classification theorem that every P-group is periodic.

vVe will review the classification shortly. But for now we give further details on the structure of P-groups.

Corollary 6.6 If G is aP-group then Sylp(G) is cyclic for p odd and either cyclic or generalized quaternion faT p = 2.

Proof. We wish to show that if H is any finite p-group where the only abelian subgroups are cyclic, then H is Z/pT if P is odd. This is surely true if IHI = p, so assume the truth of the assertion for IHI < pT. Since H is a p-group it contains a normal subgroup, N, of index p, and by assumption N = Z/pT-l. Hence H is an extension of the form
Z/pn-l --+ H --+Z/p . If r = :2 the action of Zip on N is trivial and H is either Zip x Zip or Z/p2. The first case is impossible, so the second case holds. Otherwise r > 2. Then there are two possible actions of Zip on Z/pT-l, multiplication by 1 + pT-2, or the trivial action. In the first case H2(Z/p; Z/pT-l) = 0, so the extension is the semi-direct product, which 'contains (Z/p)2 as a subgroup, so there remains only the second case. Here H2(Z/Pi Z/pT) = Zip with representatives (p - 1) copies of Z/pT and Z/pT-l x Zip. Thus the case p odd is proved. When p = 2 we once more assume that the result is true for IHI < 2T, where we know the only possibility for IHI = 4 is Z/4. There are four actions of Z/2 on Z/2 T, multiplication by -I, 1 + 2T- 2, 1 - 2T- 2 and the trivial action. We have H2(Z/2j Z/2 T - 1 ) = Z/2 for the first and the fourth, but H2(Z/2; Z/2 T - 1 ) = 0 for the second and third. Thus we are reduced to the first and fourth cases. In the first case the extensions are the dihedral group and the generalized quaternion group. In the fourth case they are the cyclic group and Z/2 x Z/2 T - 1 .

6. The Classification and Cohomology Rings of Periodic Groups

149

When the subgroup of index two is Q2r-1 we are in the situation of H2(Z/2; 7l/2) = Z/2. Here the two extensions are the semi-direct product and the generalized quaternion group of order 2T. 0

Corollary 6.7 Let G be periodic of period n.

1. For each p dividing IGI there is a unique subgroup Zip Zip C Sy12(G) is weakly closed in G. Consequently

C

Sylp(G) and

im (res*: H* (G; lFp)-+H* (Z/p; lFp))

is exactly equal to H*(Z/p;'Wp)Wc(z/p) for p odd. = Z /2 T, the restriction image is IF 2 [e 2 ], the polynomial algebra on a two dimensional generator. 3. VVhen Sy12 (G) = Q2 n we have

2. For Sy12 (G)

im (res*: H* (G; lF2)--+H* (Z/2; lF 2 ))

is W2 [e 4 ], the polynomial algebra on one generator in dimension four. Proof. The f<,Lct of weak closure is clear. Moreover, we have seen in (IV.2.10), (IV.2.12), that the images of the restriction maps are respectively Wp[b2J for p odd or p = 2 and G = Z/2 r , while the image of restriction is lF2[e 4] for 0 G = Q2n. Of course, in these last two cases the Weyl group is trivial. Corollary 6.8 Let G be periodic, then H*(G;Wp) = lFp[bi ]0 E(e2i-l) for i odd, where i divides (p -1). Also, ifSyI2(G) = Z/2 T then H*(Syl2(G)jZ) = H*(G; Zh and H*(G; lF2) = E(e) 0lF2[b 2].

Proof. The integral cohomology of Z/pr is given as Zjpr in each even dimension and is zero in odd dimensions. Consequently, since H*(G; Z)p is a direct summand of H*(Sylp(G); Z) it follows that H*(G; Z)p = Z/pr in each dimension 2im and is zero otherwise. Now apply the coefficient exact sequence in the proof of (6.5). The argument for 2 is similar. 0 The situation which holds when Sy12 (G) = Q2 r is more complex. In order to understand it we review the classification of P-groups as determined by Suzuki and Zassenhaus.

The Classification of Periodic Groups To begin we need Aut(Qs)·

Lemma 6.9 Aut(Qs) ~ 54, the symmetric group on

4 letters.

l.i)U

with Z/3 i -+71.,/3 C Aut( Qs) giving the extension. Each Ti also admits an extension in the evident way by 71.,/2 generalizing the extension above,

Proof. Qs/[Qs, Qs] = 7l/2 x 7l/2. Hence, there is a homomorphism

I---t Tc--tO: ---t 7l /2 - - - t 1 .

Moreover, e is onto since {x t-t y}, {x I--t xy, Y I--t x} both induce automorphisms of Q8, and their images generate S3. The kernel of e is the set of automorphisms

f(x)=xyZa,

fey)

= y1+Zb

The Groups SLz(IFp)

and the inner automorphisms (~ Z/2 x 7l/2) give all such possibilities. Moreover, it is direct to check that the extension

7l/2 x 7l/2--tAut(QS)--tS3 identifies Aut(Qs) with the group of affine transformations Aff(IF z)

rv

S4.

o

0;

The Groups Ti and The action of 7l/3 on Qs gives rise to a semi-direct product, Qs x a 7l/3 = T. T is the binary tetrahedral group. It is also isomorphic to SLz(IF3)' I~deed, we construct an embedding by first embedding Qs C SL z (IF 3) by settmg

x

I--t

(=~ ~1),

Y

I--t

(~1 ~).

N~ extend to T, by sending the element t of order 3 to (~

These groups are clearly all P-groups.

Theorem 6.10 SL2(IF p) is a P-gmup for every prime p.

Proof. ISL 2(IF p )1 = p(p2 - 1) = pep - 1)(p + 1). A subgroup isomorphic to Zip = A subgroup isomorphic to Z/(p - 1) = {( The group IF' 2 embeds in GL2(lFp) when we think of the vector space (JF p)2 as the field' 2, so that JF p-linear isomorphisms are obtained by multiplying by the eleme~ts of lF~2' However, the determinant of {o:} (multiplication by 0: E IF' 2) is N(o:) = o:l+p. Hence, the subgroup KerN n SLz(lF p) which is contai~ed in SLz(IFp) is isomorphic to Ker(N) = 7l/(p + 1). Since p - 1, p, and p + 1 are coprime except for the factor 2 which is common to P - I, P + 1, we are done except for checking the structure of the 2-Sylow subgroup. There are two cases. First, assume that p == 1 mod (4), then the subgroup

{G n}

i ).

A presen-

tation of SL z (IF3) can be given as

where I = -to The subgroup S3 C Aut ( Qs) has two distinct extensions by Qs since H2(S3;Z/2) = Z/2. The first is the semi-direct product but the second 0* is the binary octahedral group. It is obtained from T by ~xtending T by 7l/2: Hence, we have the extension diagram

has order 2(p-l), and is clearly a quaternion group. Second, if p == 3 mod (4), then, if W C 71.,/(p + 1) is the 2-Sylow subgroup, the action of the Galois automorphism 9 is via inversion. However, det(g) = -1. To correct this use w E IF 2 for which w 1+p = -1. Then (gw)Z = -I, and det(gw) = -det(w) = 1. Th~s the subgroup generated by gw, Ker(N) is again a quaternion group and the result follows. 0 Remark. SLz( lF 5) is the binary icosahedral group. It has a presentation

1--tT--tO*--tZ/2--t1.

SL2(F5 )

Specifically, let the new generator be z, and set

z(x) Note that

~ a~')}

-1.

Hence, z(ZX)Z-l zxy-l = (zx)7 = (ZX)-l, and the subgroup generated by x, y, z is isomorphic to Q16. More generally we set

=

s3 = (rs)5} ,

see for example [Co, p. 2]. For the remaining odd primes H. Behr and J. Mennicke, [BM], give presentations as SL2(lF p)

(zx)Z = zxz-1ZZx = xyx = y.

= {r, s I r2

= {A,B IAP=I, (AB)3=B 2, B4=(A2BA~(p+l)B)3=1}.

Indeed, we can map this group to SL2(lFp) by

A

~

GD,

B

~ (~1 ~)

15:2

Chapter IV. Spectral Sequellces and Detection Theorems

6. The Classification and Cohomology Rings of Periodic Groups

The Groups TL2(lFp) With the demonstration that the SL2(lFp) are periodic and the introduction of the generalized binary tetrahedral and octahedral groups Tt and 0; we have almost completed the list of periodic groups. There remains one more family which we construct now. Out(SL 2(1F 2)) = 7l/2 with generator, ~y, acting as conjugation by the

matrix group

(~ ~)

where w is a non-square of

U2 ,p

=

SL2(lFp)

XQ

lF~.

is not a P-group. Moreover, since H2(71/2; 7l/2) = 7l/2, these are the only two extensions of SL2(lFp) by Out(SL2(lFp)). Lemma 6.14 T L2(IFp) is a P-group.

Proof. As usual there are two cases. a. p == 1 mod 4. Then a 2-Sylow subgroup of SL2(JFp) has the form

7l/(p - 1)

(~i ~

where a(xi) is the automorphism conjugation by the subgroup generated by

Consequently we have the

wOw0) .,2.

).

Now Let L C U2 ,p be

(~

-1

(

n(~1 ~) (W~' n (_~-1 ~) =

(~ w~') (~1 ~)

Lemma 6.11 L is central (in particular normal) in U2,p. E

Proof. If T

E

SL2(IFp) then ,2 acts on

( w-Ow0) 1

T

as does

W2 (

0

0) 1 . But

and

(w 0) (w2 0) Ow 01'

=

hence the two actions cancel out. Also, , commutes with

(~1 ~) 'I (~

SL2(lFph,

-;;1)

1

= '1- .

Hence, the 2-Sylow subgroup of T L2 (IF p) is a quaternion group. -1 (

w0

~), so the

action is trivial on 7l/(p - 1) as well, and the lemma follows.

o .

b. p'" 3 mod (4). Here

(-;;1

~) gives the Galois action on 1F~,

Definition 6.12 We write TL2(IFp) for the quotient group U2,p/L. T L2 (IF p) is given as an extension

((~1 ~). v ,v,-l -1 0) (-1 0) v), - -( v)v ( ((-~ 0) v,.) (, 2(-1 ~) ( (-10))2 v (-1 0) vv (- (-1 ~)

I---+SL2(IFp) ---+ T L 2 (lF p)---+71/2-tl

by

,2

=

(~ w~l)"

0

,

acting on SL2(IFp) via conjugation

Remark 6.13 There is a second subgroup L'

0

1

1

-1

_

=

0

,

-

(~ ~).

GL2(lFp),

2

)

generates a 2-Sylow subgroup of SL2(lFp). We have so (f


C

and we can assume that, acts in this way. Let v E lF~2 satisfy v p +1 = -1, then. V,

with extension data

153

-2

1

0

1

0

1

,and

-1

0

-2

v,-2),2

v)v 2

0

=

_v 2.

Also -1

c U2 ,p defined as "

(

0

0) v, 1

-1

=

,(~1 ~){_v-1} -(f

When p == 1 mod (4) the two quotients are isomorphic, but for p == 3 mod (4) we could choose w = -1, in which case L' = (,2), and the resulting quotient

(-1 0) v)v 0

1

-2

=

b

(-1 0) v] 0

1

-1

.

It follows that this group is again a quaternion group and the result follows. 0

The Suzuki-Zassenhaus Classification Theorem In summary we have the Suzuki-Zassenhaus theorem (See tWo], [DM])

Theorem 6.15 A complete table of P -groups is given by Definition

Family

Conditions

I

?l/a

II III

Z/a x{3 (?l/b x Q 2i) Z/a x, (?l/b x T i )

IV

Z/a

V

(?l/a

?l/b) x SL2(lFp)

(a,b)=(ab,p(p2 -1))=1

VI

Z/a xJ1- (?l/b x TL2(lFp))

(a,b)=(ab,p(p2 -1))=1

Xa

X

T

(a,b)=l

?l/b

(a,b)=(ab,2)=1 (a,b)=(ab,6)=1 (a,b)=(ab,6)=1

(?l/b x 0;)

Xa

Thus there is only one non-commutative representation, up to conjugation, of Qs c GL2(lFp). Hence, if we set P(x) =
Proof If Syl2(SL2(lFp)) = Qs we have H*(SL2(lFp);lF2) c H*(Qs;lF 2)T, but this invariant ring is IF 2[e4] @ E (b 3 ). On the other hand by (6.7.3), the cohomology ring cannot be smaller than this. Let Sy12(SL 2(lF p)) = Q2Tt with n > 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
---t

Proof Since p is odd it follows that the group ring IF p( Qs) is semi-simple. On the other hand Qs has four distinct I-dimensional representations, (x f--+ ±1,y f--+ ±1). It also has a non-commutative representation
fx

f--+

(~ ~a)

1~ (~1 y

a+ b 2

2

= -1 in lFp,

~)

for some choice of the pair (a, b). This representation must be irreducible and it follows that '


---t

0;

---t

71../2

TL2(lFp)

---t

71../2

we have a spectral sequence with E 2-term H*(Z/2; lF2) @ E(b3) @ lF 2 [e4]. In each case there must be a differential on b3 since we know that e4 E H 4 (Oi; lF 2) or e4 E H4(SL2(lFp); lF 2) must be in the image of restriction, and we know that the Krull dimension is one for the extension group. But the only 2 3 possible differential on b3 is 84 (b 3 ) = e 4 . Then E3 = lF2[e4](1, e, e , e ), and there can be no further differentials. Thus we have determined the mod(2) cohomology of all the periodic groups.

156

Chapter IV. Spectral Sequences and Detection Theorems

7. The Definition and Properties of Steenrod Squares

157

7. The Definition and Properties of Steenrod Squares

which is in the kernel of L* and restricts to t, 0·· . 0 t on the fiber. Then, by

We use the notation rp (X) for the space XP x z/ p Ez/ p studied in §1. Recall that rp(X) is given as the total space of a fibering

naturality, r(a) is given as the pullback under the composition

'---v--' p

j

7r

xp-rrp(X)-rBz/p. The space was originally introduced by Steenrod and used in [SE] to explicitly construct the Steenrod P power operations. We review that construction here. There is the map of (4.1),

times

rea) 7r r(X) - r r(K(Z/p, n)) - r Fp(K(Z/p, n)) xZ/ p E z/ p of .t (t). This gives the first two statements. To see the third consider the universal construction for the cup-product of an n dimensional class a and and m dimensional class (3,

K(Z/p, n)

~n0~m

1\

K(Z/p, m) -----+ K(7l.,/p, n + m) .

il P x id: X x Bz/p-rrp(X)

We now pass to the rp-constructions and build the obvious diagram. Define

and these constructions are natural with respect to maps f: X -+ Y as is illustrated, for example in (1.8), so we have the commutative diagram

e: rp(K(7l.,/p, n) x K(7l.,/p, m))-rrp(K(7l/p, n)) x rp(K(7l.,/p, m)) by e{(xl' yd, . .. , (xp, yp), v} diagram i

rp(X)

l

ru

Y x B z/p

-----+

= ({(Xl, ... , Xp), v}, {(Yl,' ", Yp), v}). Then the r(~)

K(Z/p,p(n+m))

)

I

rp(Y) .

Theorem 7.1 Let X be a CW complex and a E Hn(x, Fp), with n

r(,@,)

~

I, then

there is a unique class r(a) E Hnp(rp(X), Fp) so that 1. (j*)* r(a) = a 0 a· .. 0 a in H*(XP; Fp) '-----v----" p times

2. if f: X -+Y is a continuous map r(f)*(r(a)) 3. r(a U (3) = r(a) U r((3).

= r(f*(a))

Proof The existence of r(a) is the argument in the proof of (1.7). Diagram (1.8) shows how to reduce the construction of r(a) to the construction of r(t,) where t E H n (K(7l.,/p, n); Fp). is the fundamental class. But uniqueness is not demonstrated there. 10 see uniqueness, note that K(7l.,/p, n) is connected for n ~ 1. Hence the choice of basepoint in K(7l.,/p, n) is immaterial and we can consider the reduced construction with fiber the smash product Fp(K(7l.,/p, n) ~(7l.,/p, n) 1\ .. . 1\ K(7l.,/p, n).; Here, the fiber is (np - I)-connected with

r p (K(Z/p,n)I\K(Z/p,m))

K(Z/p,np)I\K(Z/p,mp)

/

e

rp(K(Z/p,n))xrpK«Z/p,m)) commutes up to homotopy. From this the third statement follows.

0

The Squaring Operations We now show how to construct the Steenrod squaring operations using the spaces r2(X), Definition 7.2 Let X be a CW-complex and suppose a E Hn(x, F2)' Then

p times

H np(Fp(K(7l.,/p, n); Fp) = F:P ~ Fp .

*

E Fp(K(Z~p,

n)) gives a lifting, L, of B z / p to Fp(K(Z/p, n)) xZ/ p E z / p so we define r(t) as the unique class in

Moreover, the basepoint

Hnp (Fp (K(7l.,/p, n) xZ/ p E z / p; Fp)

defines the Steenmd squares of a. Of course we must show that the classes defined in this way satisfy the axioms of (II.2). In particular Sqn(a) = a 2 follows from the commutative diagram

x

Corollary 7.6 The Sqi maps are homomorphisms, that is to say

1

idxpt

x

X

BZ/2

since (id x pt)* L: f)n+i 0 en - i = f)2n but .1*j*(r(a)) = .1*(a 0 a) = a 2. Since K(Z/2, n) is (n -1)-connected it follows that 5qi(Ln) = for i < 0, so, by naturality, 5qi(a) = 0 for i < 0 as well. Next we verify the Cartan formula,

°

Theorem 1.3

The P-power Operations for p Odd

There are essential differences here from the case of the squares. One starts, as before, with formal operations Di(a) given by Definition. Let X be a CW complex anda

i

5qi(a U (3) =

Proof. r(a + (3) = r(a) + r((3) + E where E is in the image of the transfer (1 + T). But .1 x id(1 + T) ~ 0 so additivity follows. 0

L 5qi(a) U 5qi-r((3).

E

Hn(XjlFp) then we set

r=o

where 8 n - i = b(n(p-1)-i)/2 if i is even and is eb(n(p-1)-i-1)/2 when i is odd.

Proof Since r(a) U r((3) = r(a U (3) 7.2 gives

(L 5qi(a) 0 en-i) U (L 5qr((3) 0 e r ) = L 5qt(a U (3) 0 en+m - t m-

(7.4)

r

q

and since e U e

= ep +q , 7.3 follows on expanding the left-hand side of (7.4).

o Lemma 1.5 Let

phism, then

0-:

j{n(x, lF 2)-tHn+1(EX, lF 2 ) be the suspension isomor-

1. 5qi(o-(x)) = o-(5qi(x)) 2. 5 qO(x) = (x).

Proof. By (3.11) and naturality .1*(r(e 1 )) = e1 0 e 1 + 10 (e 1)2 for e 1 E HI (RpOOj lF2)' Thus, since Rpoo = K(7l../2, 1), 5 qO(L1) = 1,1 as desired. Now take the suspension 0-: EK(7l../2, n)-tK(7l../2, n + 1) so o-*(Ln+1) = o-(Ln). Notice that 0-* is an isomorphism in dimension n + 1. Hence if we knew the first statement then the second assertion would follow for the universal model and hence for all X. To show (1) note EX = 51 1\ X is a factor space of 51 x X. Hence r(o-(a)) = r(e) U r(a) and .1* r(o-(a)) = e 1 0 e U E5qi(a) 0 en - i = Eo-Sqi(a) 0 en+1- i .

o

Then one shows that these Di operations commute with suspension up to a non-zero coefficient determined in (3.11) as above and that they satisfy the Cartan formula, and are linear. However, to show that most of them vanish, and to relate them to the P-power operations and mod (p) Bocksteins requires the extension of the construction from XP x Zip E z/ p to XP x sp Esp. It was the necessity of this extension which was one of the main reasons for Steenrod's interest in the cohomology of the symmetric groups. Finally, the remaining D/s are still not quite the operations pi(a) or f3p i (a), but have the form /-Li,npi(a) and vi,nf3pi(a) where /-Li,n and Vi,n are non-zero constants made necessary again by the coefficient in (3.11). We omit the details here as they are readily available in [SE] and our major interest in the remainder of the book is with the situation at the prime 2.

Chapter V. G-Complexes and Equivariant Cohomology

o.

Introduction to Cohomological Methods

Let G be a finite group and X a space on which G acts. In this chapter we will descri be a cohomological analysis of X which involves H* (G; IF p) in a fundamental way. First developed by Borel and then by Quillen, this approach is the natural generalization of classical Smith Theory. After reviewing the basic constructions and a few examples, we will apply these techniques to certain complexes defined from subgroups of a group G, first introduced by K. Brown and D. Quillen. Using results due to Brown and P. Webb we will show how these complexes provide a systematic method for approaching the cohomology of simple groups which will be discussed later on. One of the aims of this chapter is to expose the reader to the important part played by group cohomology in the theory of finite transformation groups. By no means is this a complete account; in addition it requires a different background than the previous chapters have. We recommend the texts [AP], [BreI] as excellent sources for those wanting to learn more about group actions. For technical reasons we assume that the G-spaces which appear are CWcomplexes with a compatible cellular G-action (G-CW complexes). Note that any compact manifold with a G-action can be given the structure of a finite G-CW complex. We denote the cellular complex of X by C*(X; R) (R the coefficient ring). We have an elementary lemma to begin

Lemma 0.1 Each chain group C*(X; R) is isomorphic to a direct sum of permutation modules, i. e. Ci(X; R) ~

EB

R eTi

@G ui

RG

eTiEX(il/G

where ReT is the sign representation on R defined by the orientation on the cell CJ. Proof. This follows immediately from the decomposition of X into cellular orbits, keeping track of whether elements 9 E G preserve or reverse the orientation of the cells. 0

Note the case in which the G-action is free (i.e. G x = {I} for all x E X); then G*(X; R) will be a chain complex of free RG-modules. Recall

that EG denotes a contractible, free G-CW complex which is always infinite dimensional for G finite. We define Definition 0.2 The Borel construction on a G-space X is the orbit space XXcEG

= *, X

xcEG

= BG; the natural projection X X

<--...t

P

-:f 0 and in fact X

P

has the mod

q -O =? Hp+q(X Xp EP' f ) .

x EG--+EG induces

X XcEG

1

0.3

Theorem. Under the conditions above, X p cohomology of a point.

Choose '!L/p ~ G ~ P central; then x P = (XC)P/c, and so by induction on IPI it suffices to prove the result for P ~ '!L/p. In this case (0.4) becomes

= (XxEG)/G

where G acts diagonally on X x EG. For X a map

Example 0.6 Let X be a finite complex with the mod p homology of a point, and assume that a finite p-group P acts on X. We will prove the following celebrated result due to P. Smith

BG.

Because the action on EG is free, the vertical map is a fibration with fiber X. Hence, associated to {0.3) there is a spectral sequence with E 2 -term

q> 0

'

p

Hence it collapses and so Hp+q(X

Xp

EP;fp) ~ Hp+q(P;f p) .

On the other hand, the inclusion X P

jp: H*(X

Xp

"-----t

X induces a map

EP; fp)----+H*(XP x BP) ~ H*(XP) 18> 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.

<,nd Equivariant Cohomology (;_(~omplcxes ~

Then we have 2 2 If Lemma.

Z

is contractible for each x E

. h 0 X then PI: Z-,X '/,s a om-

x

topy equivalence. Proof. x\PI

=

a normal p-torus in G. In this case 2.6 can be reformulated as the statement that if IAp(G)IG =f. 0 then IAp(G)1 is contractible. We note that the converse of this is still an open problem due to D. Quillen:

. \

, > x}' define
{(x',y) E Z\x -

Conjecture. If Ap(G) is contractible then G has a non-trivial normal psubgroup.

- ' Zx by

W:n::'l


9

then it is easy to venfy tha If I W d \{O < I}\ is a I-SImplex). 11 < l} X WI-' 2 an . t tible for a o { h topy inverses and so x IpI IS con rac 0 determine a map Hence v and
\Y\

are homotopy equ'/,valent.

Proof. By 2.2 \X\ c:::

o

\Z\ c::: \Y\.

" "'naiyze the homotopy theoretic properties of the poset spaces We can nOVY '-'" G). To begin we have S (G) an d A P ( 'bl P . . then A (P) is contract'/, e.

Lemma 2 .4 If

P is a non-trwwl p-group

p

'd t't t 0 f P consisting of the 1 en 1 y Let B C P be the subgroup ~f the c.enh~n B > 1 since P > 1. Hence, the of order E Ap(P), so \idAp(p)\ c::: ct in Ap(G) we have G ~ AB Ap(P) c::: *.

:~~o~ll

ei~ments

>p~c~~~lYalf':

an~

We use this to prove .' 2 5 The inclusion i: A p ( PropOsItIOn . lenee.

G)

'-t

8

p

( G)

is a homotopy equiva-

'bi

t and hence 0

= Ap (P), a contractl e pose ,

Proof. Let P E Sp(G), then i\p

\Ap(G)\ c::: \8 p(G)\.

L1

A (G) is . . l normal p-subgroup then p .' 2 6 If G has a non-tnvw PropOSltIon . contractible. P E 8 (G) 'bl 1£ 1 < K
5 (G) is contractl e. P K' S (G) hence \8 p ( G)\ c::: *. we have P ~ P K '2: m p , f . sub roUp of G then the subgroup E ~ Now if K is a non-trivlal normatl P- f is characteristic in K, hence E 18 er 1 or P in the ten er 0 elements 0 f or d j

i

1---7

stabilizer of Ll

estabishes a contravariant isomorphism between the poset of simplices in II and parabolic subgroups in G. Ap( G) is closely related to this complex.

Theorem 2.7 The poset Ap( G) is homotopy equivalent to the building II. Hence Ap(G) has the homotopy type of a bouquet of (m - I)-dimensional spheres. Proof. We provide the proof for the simple case G = SLn (11(). Let X be the poset of simplices in II and define Z C X x Ap(G) as

Z = {(x,A)lxEXA} = {(x,A)'A~Gx}. For G = SLn(JK), if can be identified with the simplicial complex IT(lKn)1 where T(OC n ) denotes the poset of proper subspaces of lKn. Note that m = n - 1, differs from rkp(G). Now it is clear that an element of G x leaves each point of x fixed, hence x' :::; x implies G x ' :> 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 <

p(n-i)pi so it follows that 7r(M) = {I}. Consequently, M result now follows by checking orbits. Corollary 1.8 The image of the restriction map

is

C

S;"_1 and the 0

Corollary 1.9 1. For all integers n, i

°

> we have that the composite restriction map res:

res·

n

H*(Spn+i; lF p ) ~H*(Spn; IFp) --r H*(Vn(P); IFp) is onto the image of res* . 2. Let v E im(res*; H2*(Spn; lF p)-+H2*(Vn (p); IFp)), then for all i

> 0,

VQ9VQ9"'Q9V "---.r---' pi times

is in the image of the restriction res*: H*(Spn+i;lFp)--rH*(Vn(p)Pi;lFp). (1.9) is a key tool in our final description of the homology structure of the symmetric groups which will be completed in the next 2 sections using the methods of Hopf algebras. We now determine the image of the restriction map in H* (Vn (p); IFp). The Image of res*: H*(Spn;lFp)--rH*(Vn(p);IFp).

We apply the results of (IV.I) on the homology of wreath products, to further understand how the mod (p) homology and cohomology of Sn occurs. Clearly, the critical case is when n = pm. We have a factorization of the inclusion Sylp(Snp) '---+ Snp as follows Sylp(Snp)

'---+

Sylp(Sn) 2Zip

'---+

Sn 2Zip

'---+

Sn 2Sp

'---+

Snp .

In (IV.I) we determined H*(H 2Zip; IFp). It is generated by elements of two types. First there are the (1 + T + ... TP-l )(A1 Q9 ... Ap) E H*(HP; IF p) where the Aj are not all equal and T acts to shift the elements T(AI Q9 ... Q9 Ap) = There are two special cases for the result above. Let

(Ap 0 Al 0 ... 0 Ap_r). These are all in the image of the composite res· tr. Then there are the elements of the form

l::1U

\..;IU:1p\,er Y 1.

1. ne \..;onOIIlOIUl:.Y Ul 0YHlllltLrIC '-..Truup::;

(1.10)

,(A

2. the construction is natural in the sense that if f: X -----+Y is a continuous map and f*(T) = a then UP x id)*(T(T)) = T(a),

0 A 0···0 A), U7r*('Y·) J v p times

where (I'j) E Hj(Zjp; IFp) is a generator and 7r: H2Zlp-----+Zlp is the projection. Precisely, 'Yj is given as follows. Let e E Hl (B'l..lp; Wp) be the fundamental class which corresponds to the identity homomorphism

3. r(a U (3) = ±r(ex) U r((3), 4. r(a + (3) = r(ex) + r((3) + tr(w) for some w E Ht(XP; lF p ). From the appendix to (IV) we have the formulae for the restriction map (,dP

x 1)*: T(a)-----+H*(X x Blip; IFp) :

id E Hom(Hl(B'l..lp; Z); Wp) = HI (B'l..lp; Wp) and suppose b = j3( e) is the Bockstein. Then set 'Y2j = bi, 'Y2j+l = bi e for p odd, and 'Yj = ej when p = 2. For notational convenience we will sometimes write 7r*bj) = ej , 7r*b2j+€) = bief. as well in what follows. There are further restrictions that we can put on the possible classes in (1.10) which are in the image from H*(Snp; Wp) when p is odd. The normalizer of Zip C Sp is Zip XT Z/(p -1). The action of B E Z/(p -1) on (A 0· ··0 A) is trivial if dim(A) is even, but is given by the sign representation if dim(A) is odd. Likewise, we can make Aut(Z/p) ~ Z/(p - 1) correspond to the units p, E Zip and under this correspondence p,bj) = p,[J+ 1/2l 'Yj. Thus, the invariants under this action in H*(Sn ~ Zip; Wp) among the elements of the form (1.10) are the elements (A 0 ... 0 A) U 'Yk with k = 2j(p - 1) or k :;::; 2j(p - 1) - 1 when dim(A) is even and (A 0 ... 0 A) U b(2j+I)(p-l)/2 or (A 0· .. 0 A) U b(2j+I)(p-l)/2-1 e when dim(A) is odd. Lemma 1.11 Let p be a prime. The composite

(1.12) t

L

(,d2x1)*(T(a))

tr· res: H*(S;n_l; Wp)--tH*(Vn(p) n Spn-l; lFp)--tH*(Vn(p); lFp ) and since the index of this intersection in Vn (p) is p the result follows.

0

In view of (1.11), to understand the restriction map

H*(Spn-l 2 Zip; lFp)-----+H*(Vn(p); lFp ) it suffices to study it on elements of the form T( ex) where T (ex) is described in the proof of (IV.1.5). From this proof we see that, in fact, T(ex) is defined and non-zero in HPt(X P xr E'l..lp; lF p ) for any cohomology class ex E Ht(X; lF p ) where X is an arbitrary CW complex. T(ex) satisfies the following four properties, all of which follow directly from its construction in (IV.1.5): 1. res(T(ex)) = ~ in H*(S;n_l;lFp), p

times

j

when p = 2,

(,dPxl)*(T(a))

L

[t/2(p-l)]

At

. .

(_l)i[pi(a) 0 bm (t-2i)

+ (_I)t j3p t (a)

0 bm (t-2t)-le]

o where m

= (p - 1)/2 and ( _l)mt(t-l)/2

At =

(m!)t

Note that since (m!)2 = (_1)(P+l)/2, it follows that At is ±1 f~r ~ even. This can be applied inductively to understand the restnctlOn map to H* (Vn (p)) from H* (Spn) since the inclusion Vn (p) <-+ Spn factors as follows

When p

Proof. We use the double coset formula for S;n-l and Vn(p). There is only one double coset in the wreath product. Hence the composite factors as

et -

o

Vn(p) is identically zero.

Sqj (a) 0

= Vn-I(p)

.,:1Pxl

X

Z/p----+Spn-l (Z/p--+Spn .

= 2 and n = 2 this gives res*(T(e))

= e2 01 +

e 0 e E H*(V2(2)).

This element is not invariant under GL2(2) but res*(T(e))+10 e2 = d 2 is invariant, and since res* (e) = 10e, it is also in the image of res. Consequentl~, d is in the image of H* (S4; IF 2)' and since d3 = S ql (d 2 ) it follows that d3 IS afso in the image. We thus have a proof that lF 2[Xl,X2]L2(2) is in the image of restriction from H*(S4; IF2)'

Example 1.13 H*(S4;lF2) 2 In this case we need only 2 detecting subgroups, VI (2) and V2(2). Then we have

= S2

x S2 C 54,

192

Chapter VI. The Cohomology of Symmetric Groups

where ¢ = (res1' res2), and cp = res~~ g\~nv2 + res~~ (2)2nv2' Note that im ¢ ~ kercp; now Vl(2)2 = ((12), (34)), V2(2) = ((12)(34), (14)(23)), hence VI (2)2 n V2(2) = ((12) (34)) = .d(V2 (1)). The invariant subrings are given as follows:

1. Detection Theorems for H* (Sn; IF p) and Construction of Generators

II

193

(x-v)

vEVn(p)

H* (VI (2)2)Z2 ~ IF 2[CTI' CT2], H*(V2 (2))GL 2(lF 2) ~ lF 2 [d2 , d3 ], H* (VI (2)2 n V2 ) ~ lF 2[e], and

From the above we deduce the existence of classes CTI, CT2, c3 E H*(S4), with

and so n

L AjXPj ((dpn-Lpi-1)P + BjeP- I ). This approach can be iterated. First, we need an inductive formula for constructing d(p_l)pn-l E H*(Vn(p))GLn(P). We fix the generators dpn_ pj for the Dickson algebra discussed in (III.2) as follows. Write Vn(p) = (Zlp)n-1 x Zip where the elements in the summand Zip are written Ae, 0 :::; A ~ p - 1. Then the restriction of dpn_ pi to the subalgebra for (Zlp)n-1 is (dpn-l_pj-l)P for j > 1, and when j = 1 we have dpn_1 = dpn_l_lepn-l(p-l) where dpn-l_* is the Dickson invariant for (Zlp)n-l.

Lemma 1.14 The following expansion is valid for all primes p. .-r

pn

n-I

-

L AjX

j=O for each integer n, 1 ~ n <

pj

dpn_pj

=

II

j=O In particular, by restricting to IF P (Vn - I (p) 1 we see that when j f. 0 the restriction of the coefficient of Xpi is -d~n-l_pi-l' Thus, since there is only a one dimensional set of invariants in IFp (Vn (p) 1 with degree pn - pi we obtain the result for all the lower coefficients. But the top coefficient is ± TI v where v E Vn(p) and v =f=. O. The inductive step follows. 0 Corollary 1.15 Write Vn(p) = Vn-I(p) X Zip, then. the lowest dimensional ., ' d pn_pn-l = 'l\"n-I GLn(P) 'mvanant can b e wntten 0j=0 ep1 dpn-l_pi + dPpn-L pn-2' (Just expand out the coefficient of x

pn

1

- in the expression above.)

(x - v) ,

vEVn(p) 00.

Proof. To begin consider the case n = 1. We have TIf:Ol(X -ie) = xP -xeP-1 ~o the result is true here. Assume the result true for n, we show this implies It for n + 1. Write Vn(p) = Vn-I(p) X Zip, so

Corollary 1.16 dpn_pn-l is contained in the image of the restriction map, im(res*), from H*(SYlp(Spn)). n n-l 2n-1 Proof. Indeed, d pn_pn-l = res*(T(d p n-Lpn-2) ± 1 0 bP -P ,(or 1 0 e when p = 2) but if 7r: SYlp(Spn-l) 2 Zlp-+Zlp is the projection onto the new Zip and e = 7r* (e) then res* (e) = e in the expression above for p = 2 and similarly for b when p is odd. 0

This shows, since the dpn_pi are all Steenrod pth powers of dpn_pn-l, that when p = 2 the entire invariant sub algebra is in the image of the restriction map and when p is odd, the entire invariant polynomial sub algebra is also in the image of restriction, but it leaves open the question of the part which involves the exterior terms. Here we follow the discussion in [Maj. Define

- ._ .....

bf. (1.17)

Li=

.....

....,

_- .... _----

i-I

b1!t

i-I

Example 1.19 H*(Sp2;'iFp). The following calculation is due to H. Cardenas. We have

pr

lJ( 1

bi

h

bi

i.e. the k, j entry of Li Ie

is b~ , (::; r ::; i-I).

Similarly, set i-I

bP1

Mj,i =

b1!t

---

bpi 1

ll]j

h

bi ei

t

el "

i-I

We have pr? (br?) =

i.e. the bPi row is omitted (1 :::; j ::; i-I).

'+1'

If

.

but prY (b T ) = 0 for r < pJ. From this we find

while

---

ll]i

and we see that the GL2(p)~invariant class Nh,2Mo,2L~-3 is in the image of restriction from H*(SYlp(Spz);lFp). Thus, applying j3 we have that MI,2L'{-2 is in the restriction image, and applying pI j3 we have Nfo,2L"{-2 is also in this image. The polynomial algebra 'iFp[d p2_p,d p2_I = L~-I] is also in the restriction image, and, from (III.2.9), we see that this is the entire GL2(p) invariant subalgebra of H*(V2(P); 'iF p). The groups H*(Sp2;lFp) are detected by restriction to H*(V2(p);lFp), H*(Vl(p)P; lFp ). From (III.4.2) and (1.8) above it follows that the image of this restriction for VI (p)P is

t

lFp[Sl,"" sp]

Note that dim(Mi,i+l) q = (p - 3)/2 = m - 1, (1.18) res*(T(Mi_1,i))

= ±

p(dim(Mi-1,i)) - p

+3

so we have, setting

(~(-l)j Mj,ib~iq - Libf+le) = M ,i+l br+l i

@

E(h, ... , jp)

where dim(si) = 2i(p -1) and dim(fi) = 2i(p - 1) - 1. Also, the only gener~ ating class which restricts to H*(V2(P); lF p ) non-trivially and also restricts to H*(Vl(p)P; lFp) gives the pair (O'p, dp 2_ p). The remarks preceeding (1.19) show that more generally

T(Mi-l,iNfi-2,iLf-3) = ±Mi,i+lMi--l,i+ILf;{

)=0

up to a sign. Here the restriction is from Spi l Zip to Vi+ 1 (p) = L\P (Vi (p)) x Zip and the classes bi+l' e are associated to the rightmost Zip. Moreover, PP'Mi,i+lbr+l = Mi-1,i+lbr+l' A similar calculation shows that res*(T(Li))U bi+l = Li+l. Also we should note that each of the matrices above is invariant under SLn(P) but that gJ = det(g)J for 9 E GLn(p), so any product of p-l of them is invariant under G Ln (p). .

and so this element is in the image from the restriction' of H* (Spi; IF p) for each i. Applying Steenrod p-power operations and Bocksteins we obtain the following table of further elements:

196

Chapter VI. The Cohomology of Symmetric Groups

2. Hopf Algebras

197

where Li,T is the (j, r) minor of L given as (-ly+jDet(Vj,T) where vj,r is the matrix obtained from the matrix, £, with determinant L by deleting the (i - j) th row and the rth column. But then it is easy to verify that P1'

Mi -

i-2

~

Mi_2 M i_3 Lp - 3

1 ...

ele2'"

eiDet (Lr,j) .

On the other hand

1

p"-'

, C·

pp i - 2

~

Mo =

W")

Mi_2Mi_4Lp-3

=

(

L

o

0

L

~

o

o

so taking determinants of both sides (1.20) follows.

Ip,,'-'

Finally, to complete the demonstration that the elements above generate the entire image of res* we proceed by induction using the factorization pp2

--7

NI 2 M ILp-3

Ip, PP

i-3

~

111,£_1 L1'-2

LlPxid -----* Vi-l(P)

(res}/id

lZ/p----+Spi-l lZ/p

'-+

Spi

for the restriction map. The details are direct. Remark 1.21 The entire ring of invariants has been discussed in (III.2.9), and for p > 3 and n > 3 also, [Mal shows that the classes above generate a proper subalgebra of this ring.

2. Hopf Algebras

Further, one can verify that MJ = 0,0 ::; j ::; i-I but M i - 1 M i - 2 ··· Mo -# O. Thus, there are further products that we can write down and Mann shows using a counting argument depending on the homology of symmetric product~ of spheres, that the image of the restriction map is exactly the free module over the Dickson algebra of GLn(p)-polynomial invariants generated by 1 and the elements together with their distinct products above. It is possible to give an algebraic proof of these results thus avoiding any reference to topology. Here are details for these last two steps. Lemma 1.20 For each i

Vi(p)

> 0 and every odd prime p we have

An augmented, graded, algebra over a field JF is an associative, graded, unitary JF-module together with a ring homomorphism €: A-+JF, where JF is thought of as concentrated in degree O. Here, to be precise the multiplication /-L: A 0lF A -~ A is given and the unit corresponds to a graded homo€01

A

10€

0lF JF ---+ A 0lF

~

A - - - t A are both the identity maps, as is the composition

d: JF -+ A -+ JF. The associative condition can be written diagramatically as saying that the diagram

Proof. One has an expansion for each j,

A

i

lvIj

=

±

L 7'=0

eTLj,T

~'

morphism 1: JF -+ A so that the compositions JF 0lF A ---+ A 0lF A ---+ A and

commutes.

198

Chapter VI. The Cohomology of ~ymmetric Groups

A graded coassociative, counitary, coaugmented, coalgebra over a field IF is a graded IF-module, B, together with a coproduct map

¢ :B

----+ B 0lF

B,

counit, Le. a graded IF-module morphism 1* : B----+lF

so that (id 01 *)¢

= (1* 0

id)¢

= id,

Example 2.3 Let G be a group and IF( G) its group ring, then

and coaugmentation map

¢: IF(G)-tlF(G) 0lF(G)

which is also graded and satisfies the condition that the composition (€* 0lF €*)id : IF -----t IF 0lF IF -----t B 0lF B is just ¢€*. As before the coaugmentation and counit are related by 1*€* = id: IF -----tlF. The coassociative condition is that the diagram below commutes B


}-

(xi)' . (xk)' =

e; k)

(:d H

)'

It follows that if IF has characteristic 0 then A * is also isomorphic to the polynomial algebra on one variable x* in dimension 2i, but if char(lF) = p then a direct exercise with binomial coefficients mod (p) shows that Ip i

, •••

l/(x*P

z=

In the sequel all the Hopf algebras we consider will be connected, commutative, and even co commutative of finite type. So we assume these conditions in the remainder .of the section. Hopf algebras arise for us as follows. vVe are given a sequence of groups and injective homomorphisms

:s: n < 00,

Go

= {1},

for which the f-L(n, m) are associative up to congugation, i.e. for any triple (n, m, l) we have that there is an element g(n, m, l) E G n + m+ l so that

Example 2.2 Let A be the polynomial algebra on an single variable x of dimension 2i over the field IF. Then the coproduct structure is determined by ¢(x) = x01 + 10x so ¢(xi) = L:~i=O G)x j 0x i - j • Thus, in the dual algebra the product is given by the rule

= JF[x*, IP"'"

z=

is defined by ¢(g) = (g 0 g) for all 9 E G, so ¢ figi = fi(gi 0 gi)' This is also a unitary, co connected Hopf algebra, but it is not connected unless G=l.

Gn , 0

Definition 2.1 A Hop] algebra (A, f-L, ¢, €, 1) over a field IF is a graded, associative, unitary, augmented algebra, together with a counitary, coaugmented, coassociative, coalgebra map ¢ so that ¢ is a homomorphism of graded algebras for which the counit is € and the coaugmentation is 1.

A*

co augmented , counitary coalgebra. In particular, (A *, ¢*, IL*, 1 *, E*) is also a Hopf algebra and (A **, f-L**, ¢**, E**, 1 **) = (A, f-L, ¢, E, 1). , The Hopf algebra (A, f-L, ¢, E, 1) is said to be connected if A ~ = 0 for i < 0 and 1: IF -----t Ao is an isomorphism. A is said to be commutative in case f-L:A0A-----tA satisfies f-L(a0b) = (-1)lallblf-L(b0a) for all a E Alai, bE Albl'

= '" = I;i = ... = 0),

dimhp-i)

=

2ipi .

g(n, m, l)-l(f-L(n + m, l)(f-L(n, m)(a, b), c)g(n, m, l)

= f-L(n, m + l)(a, f-L(m, l)(b, c)) for all (a, b, c) E G n x G m x G l . We also assume the f-L(n, m) are commutative up to conjugation. This means that for each pair (n, m) there is g(n, m) E G n +m so that

g(n, m)-l f-L(n, m)(a, b)g(n, m) = f-L(m, n)(b, a)

V(a, b)

E

Gn x G m

.

For example if G n = Sn, the symmetric group and the pairing f-L(n, m) is the usual inclusion of Sn x Sm C Sn+m, with Sn acting on the first n elements and Sm acting on the last m, then these conditions are satisfied. Here the g(n, m, l) = id but the g(n, m) are non-trivial. Of course U~ H* (G n ; IFp) is only an associative, commutative, graded algebra, but not a Hopf algebra since the diagonal map .d: Be-----tB e x Be which, on passing to cohomology, is supposed to give the coproduct, ¢, has the form n n .d*(an) = an 0 t + t 0 an + a~ 0 a~'

L i

In case A is of finite type, Le. Ai is finite dimensional over IF for all i these notions are self dual, At = HomF (Ai, IF), (A *, ¢* , €* , 1*) becomes an associative, unitary, augmented algebra and (A*, f-L*, €*,l *) becomes a coassociative,'

where an E H*(G n ; lFp ) and t n E Ho(G n ; lF p ) is the element given as the image of a chosen generator t E Ho(G 1 ; lFp ) under the iterate of the multiplication map,

200

Chapter VI. The Cohomology of Symmetric Groups

n times

Next, adjoin a unit 1 to the algebra and note that the quotient of this algebra by the ideal (1 - t) is a Hopf algebra. In dimension i we have, clearly, that this quotient is given as limn (Hi (G n ; JF p)).

Lemma 2.4 Let G =

then

lim G where the inclusion Gn C Gn+1 is p.,(n, 0)) n->oo n

(Any chain in the bar construction for G appears in the bar construction of G m for some sufficiently large m.) In particular, while the maps p.,(n, m) need not pass to limits to define a "reasonable" homomorphism G x G-tG, in homology the maps do pass to limits to define a pairing

which, together with the diagonal map

gives H*(G;71lp) the structure of a commutative, cocommutative, connected Hopf algebra. Of course, it is not always going to be true that the Hopf algebra is of finite type but this does turn out to be the case for the symmetric groups above and the other cases we consider, finite groups of Lie type which are studied in Chapter VII. Thus, associated to USn is the Hopf algebra H*(Soo; JFp). In §3 we will show that the restriction map res*: H*(Sn; IFp)-tH*(Soo; IFp) is injective for all n, and we will determine the Hopf algebra structure on H * (Soo; JF p) completely. As a result we will obtain the groups H*(Sn; JF p) by simply describing the image of the restriction map in the resulting Hopf algebra. Before we can do this we need to discuss some basic structure theorems for commutative and cocommutative Hopf algebras and for this we need to discuss sub-Hopf algebras and quotient Hopf algebras. The notion of sub-Hopf algebra is evident, as is the notion of quotient Hopf algebra and Hopf algebra map ,\: A - t B of Hopf algebras . Duality connects these notions together very nicely. Given a sub-Hopf algebra AI C A, note that AS = Ao = IF by our assumptions and let lA' = Ker(E\A'). (fA is called the augmentation ideal in A.) Then AI A' A is a two sided ideal in A and B = AI AI A' A is written AI I AI. B inherits a unit, augmentation, product and coproduct from A, and so becomes a Hopf algebra. It is the quotient of A by the Hopf subalgebra AI. Dually, B* C A* is a Hopf subalgebra of A* and A* I jB* = AI*.

2. Hopf Algebras

201

Proposition 2.5 Let B C A be a sub-Hopf algebra of the commutative Hopf algebra A, and 0 = AI j B be the quotient. Let e: A - t A 0JF 0 be the compo-

sition

4> 10p A-tA0A~A0JFO

where p: A - t o is the projection. Then e( (J)

= (J 0

1 if and only if (J E B.

(The kernel of p has the form I(B)A and can be written I: I(B)wj where the Wj map onto an JF-basis for 0 since A is commutative. But from this the result is direct.) An element a E Ai is said to be primitive if ¢(a) = a 0 1 + 1 0 a. The set of primitives, 1'(A) is a graded IF-submodule of A and is closed under the pth_power operation, ~: a f--+ a P , if char(JF) = p. It is always closed under the Lie bracket operation [a,,B] = a,B - (-l)la ll .Bl,Ba. Thus the set P(A) is a (restricted) Lie algebra where "restricted" means the existence of the operation ~. A very important property of P(A) is that P(A 0JF B) = P(A) EB 1'(B) where A and Bare Hopf algebras. A dual notion to being primitive is being indecomposable. The set of indecomposables Q(A) for A is the quotient IAj(fA)2 of IA by the square of its augmentation ideal. The lifts of a basis for Q(A) to A form a set of algebra generators for A.

Proposition 2.6 Let v: A-tB be a surjection of Hopf algebras with A connected. Let /l E Ker(v) have minimal dimension then /l is primitive.

(v 0 v(¢(/l)) = 0 since v is a map of Hopf algebras, but ¢(p.,) = /l0 1 + 1 0/l + I: /ll 0 p.," and the connectedness of A implies that dim(/l/), dim(/l") are both less than dim(/l), thus the sum term is identically zero.)

The Theorems of Borel and Hopf A Hopf algebra is said to be primitively generated if the lifts of its indecomposables can be chosen to be primitives. For example, this is the case with JFp[x] where ¢(x) = x 01 + 10 x. But note that it is not the case for lFp[x]* where only x*, among the indecomposables, can be taken to be primitive. We have

Proposition 2.7 Let A be a connected Hopf algebra of finite type which is commutative and primitively generated. Suppose also that IF is perfect if

char(JF) < 00. Then as an algebra, A = Q9i M(i) where each M(i) has the form IF[xJ/ (x T ) and r is constrained by the following conditions: 1. r = 2 if dim(x) is odd and char(IF) =1= 2, 2. when dim(x) is even r = 0 if char(lF) = 0 and has the form pI (or zero) if char(IF) = p.

202

Chapter VI. The Cohomology of Symmetric Groups

iI, ... , in, ...

so that if i < j then dim(fi) ~ dim(fJJ We prove the result by induction. Set A( i) to be the subalgebra of A generated by iI, ... , fi. Since the fi are primitive A(i) is also a Hopf subalgebra of A. Clearly, the proposition holds for A(l). We assume it for A( i) and consider ~he surjection

Proof. Order the generators of A,

A(i) 0JF IF[IH1]-tA(i + 1) where IH1 1-+ fH1' The primitives in A(i) 0lF[h+1] are P(A(i)) EEl (le1 I j = 0, 1, ... ). Let 8 be an element of minimal degree in the kernel. Then 8 is primitive and so can be written as (Ir~l + Et
that

fit

i:~l

=

=

At

for each t. Replace iH1 by if+1 = ii+1 -

E pdf

s(t)-j

. Then

0, A(i+1) is the surjective image of A(i)0lFlF[h+l]j(lf~1) and there is no longer any primitive in the kernel so the surjection is an isomorphism. 0 This last result generalizes to the important theorem of Borel and Hopf. Theorem 2.8 Let A be a connected Hopf algebra of finite type over the perfect field IF. If the multiplication in A is commutative then, as an algebra A is a

tensor product A = 0iMi where each Mi is of the type above. Proof. Define A( i) C A to be the sub algebra of A generated by all the elements of dimension::; i in A. It is a sub-Hopf algebra. Since A(l) is primitively generated we can begin an induction. Assume the result proved for A(i). Set W C AHl to be a complementary IF-subspace to A(i) n A H1 , and choose a basis Xl, ... ,X r for W. We have the sequence of Hopf-subalgebras

c (A(i),Xl) c (A(i),X1,X2) c ... c A(i + 1)

A(i)

and we assume the result for (A(i), Xl,"" Xj). Thus we are reduced to considering the situation (A, b) where the indecomposables of A all have dimension ::; dim (b) and there is an exact sequence Q(A)-tQ( (A, b) )-t(b). We have that (A, b) I I A is primitively generated and thus has the form I IF[b]/(bP ) (or IF[b]/(b 2) when dim(b) is odd, but"since this case is easy we assume for the rest of the proof that dim(b) is even). Consider, as before the -

e

-

A 0 A we have that ¢(e (6)) = ~I (6) 01 + 1 0 ~I (6) in C. We need to show that this implies ~ 1(6) = 0 in C and this follows from

Proposition 2.9 If A is a connected Hopf algebra over a field of character-

istic p > 0 and the multiplication in A is commutative then if 1. Q(A)r = 0 for r > n, and 2. ~ I (I A) = 0 for some in.teger f, it follows that P(A)r = 0 for r > pi-ln. Proof of 2. g Suppose that the assertion is proved for Hopf algebras with q or less generators as an algebra and A has q + 1 generators. Choose a set of q + 1 generators for A and let X be one having highest degree. Let AI be the subHopf algebra generated by the remaining generators, and let A" = AI I AI, so A" is a Hopf algebra with 1 generator that satisfies (1) and (2). We have an exact sequence P(A/)-tP(A)-tP(A"). Now we can apply the inductive hypothesis. 0 Returning to the proof of the theorem, we have that ~ I (b) = bpI = 0 under J projection onto C. Thus (id 0 p)¢(b pJ ) = ~I (b) 0 1. Consequently bP E B pJ PJ l and there is an element v E A so b = v and if we set b = b - v we are in the previous case and the result follows. 0

The element in H* (Spi-l I Zip; IF p) which restricts to Li (defined in (1.17)), which we write as L when there is no possibility of confusion, has the form r(Li-d U 7r*(b). Consequently it restricts to 0 in H*(S;i_l;JF p). It follows that all the elements described in (1.14) - (1.20) which restrict to the ideal

(3.1)

Ii = lFp[dpi_pi-l, ... , dpi_l](dpi_l' M oLp-2,"" M j Lp-2, . .. , Mo'" M i _ 1 Lr)

all restrict to 0 in H*(Spi-l;IF p). Also, by (1.9(1)) they survive non-trivially to H * ( Soo ; IF p) as do the classes

r

surjection A0JFIF[b] -t (A, b) and let bP +w be the least dimensional element in the kernel. If w is zero we are finished since the composite map

A 0lF[b]j (bPI) --=---. (A, b) ~ (A, b) 0 (A, b) I I A is clearly injective so e is an isomorphism of algebras. In case w i=- 0 we proceed as follows. Let B C A be the sub-Hopf algebra of A consisti~g of pi -powers. (Thi~ is where we use the fact that IF is perfect.) Consider (A, b)1 IB = C. Since ¢(b) = b01+10b+ I::vi0v~ with I::vi0v~ E

(3.2)

8080"'08

'------v-----'

pi-times

for 8 E Ii and dim(8) even (by (l.9(2))). In particular, if 8* is the dual of any such even dimensional element in Ii it has image an indecomposable in the Hopf algebra H*(Soo;lF p), ~j((j*) i=- 0 for j 2: 1, and is independent of ~1(A*) for any other A E I k . Thus the dual groups Hi = It generate a polynomial algebra on even dimensional generators tensored with an exterior algebra on

204

Chapter VI. The Cohomology of Symmetric Groups

205

odd dimensional elements which is a split summand of H*(S<X);lF p)' Letting A(M) denote this algebra we have from (1.8), the discussion after (1.19), and the Borel-Hopf theorem (2.8), that <X)

H*(S<X);lFp) ~ @A(M).

(3.3)

o Moreover, applying (l.8) again we see that the map

2. For each odd prime p H*(S<X); lFp ) is a tensor product of a polynomial algebra on classes indexed by r-tuples (iI, i 2,· .. , ir) with i l :S i2 :S ... ir and dimension 2(p-l)(il +pi2+" '+pr-li r ) together with an exterior algebra on odd generators dual to the odd generators for H.(Soo; IFp) .together with a divided power algebra on generators indexed by the remaining elements in the union of the Nj. Remark F or p = 2 the cohomology generator which corresponds to D 2r where D = d~Ll ... d~:_l E'Ii is actually the element which restricts to

res*: H*(Sn; lFp)-tH*(S<X); IFp) must be injective since the Weyl group of a group of the form appearing in (1.3) is a product of the Weyl groups for the Vr(p) x .. , \..

X

Vr(p)

v

2 r times

and a similar discription is given for the elements corresponding to of Dickson elements at odd primes.

I

ir separately and this Weyl group is GLr(P) I Sir' Thus the homology images of tensor products of elements in Mr surject onto H*(Sn;IF p) and, since these elements last to H *(S<X); IF p) subject only to the same symmetrizing conditions imposed in H* (Sn; IF p) - namely symmetry under Sir - the injectivity follows. We specify the image of H*(Sn; lFp) more precisely by giving a second degree to each monomial generator of N i , deg2 (8) = pi. Then to each monomial in (3.2), 81 0··· 08 l E H*(S<X); lFp), we assign the second degree

pth

powers

By divided power algebra on an even dimensional generator " we mean the tensor product

where dimbr)

= pr . dimb). Then classes of the form e.g., , = { dhpi_pi-l . " djipi_l M r M LP-3} 8

deg2(8 1 0 ... 0 81)

=

deg2(8 l )

+ ... + deg2(8l)

,

and it follows that H*(Sn;IF p) C H*(S<X);lFp) is the additive subgroup spanned by the tensor product monomials having second degree :S n. When p = 2 we have a very convenient method for indexing these classes. Let I = (i 1 :S i2 :S ... :S iT) be a sequence of non-negative integers, not all of them zero. Then there is a unique generator Q1 = Qil Qi'2 ... Qir E N r having dimension i 1 + 2i2 +. + 2r - 1 i r corresponding to it. It is the image of the class 11 1 0· .. 0 :,.. There are similar descriptions for the generators of M for odd primes, but the conditions on the sequence J = (i l :S i2 :S ... :S ir) are more complex. For notational convenience, when p = 2 we write the product in homology as a * (3. thus a typical monomial in H*(S<X);lF 2 ) will have the form

e:

e:

r

-

Q II

* Q 1'2 * ... * Q ir

.

It should also be pointed out that there are significant differences between the dual algebras. The following result is direct when we check the pth power operation on the su brings of the H* (Vi (p); IF p) given as the images of res* . Theorem 3.4 1. H* (S<X); IF 2) is a polynomial algebra on generators in one to one correspondence with the generators for H*(S<X); IFp).

together with the classes restricting to

generate the divided power algebras. The structure of H*(Sn; IFp) was first discussed by Nakaoka [Na]. Later H. Cardenas [Card] determined the cohomology rings H* (Sp2 ; IF p). N akaoka's idea was to determine H. (Sn; IF p) using the cohomology of symmetric products of spheres which he had determined previously, but which were also determined by Steenrod (unpublished) at about the same time using a key result of Dold and Thorn, [DT]. Cardenas' ideas were very much in the spirit of the development in §1. Perhaps the interest in these groups was prompted during the 1950's by J. Adem's thesis [Adem], where he used preliminary information about H* (Sp2 ; IF p) to obtain the Adem relations in the Steenrod algebra. From another point of view Bsoo occurs in an essential way in the study of infinite loop spaces. Recall that the n-fold loop space fln X is the space of based continuous maps f: (sn, oo)-t(X, *) with the compact open topology. If X is a space with basepoint, the natural inclusions fln sn(X)-tfln+l sn+l(x)

'2Uo

Ghapter v 1. Tne vonornolOE!,y

Vl uYllUlltLll<.- ' ..HVUY'"

on passing to the limit define the space Q(X) = limn -+ oo [2nsn(x). Q(X) depends functorially on X and 1fi ( Q (X)) = 1ft( X), the i-th stable homotopy group of X. Now Q(80) has a "loop sum product", which is given as the limit of the .an- 1 (*): .ansn x .ansn-+.ansn where *: [2sn x .asn -+[2sn is the loop sum. We have the following fundamental result of E. Dyer and R. Lashof Theorem 3.5 There is a map e: Z x Bsoo -+Q(8°) which induces isomorphisms in homology with all (untwisted) coefficients and which is additive in the sense that the diagram

i: (V2)k

'-t

(S4)k

'-t

Sn .

In cohomology, the image of i* lies in (0~~1 P[xo, x d) Sk, where Sk acts by permuting the polynomial generators. We will now analyze this ring of invariants, which is, as we have seen in §3, a homomorphic image of H*(Sn)' We start with the case k = 2: Theorem 4.1 The Poincare series of (P[xo, Xl]® P[xo, Xl])S2 is 1 + t5 t2)(1 _ t 3 )(1 - t 4 )(1 - t 6 )

= (1 _

P2(t)

.

Proof. As we are taking two-fold symmetric tensors, there are two types of X 0 Y + y 0 x and X 0 x. The latter are counted invariants, by the Poincare series (1-t 4 )\1-t 6)' Consequently, the former are counted by

the series homotopy commutes where the product on the left is addition in Z times the product discussed in §2 on Bsoo'

Although the language was different, this result first appeared in the original 1960 preprint of [D-L] which was widely circulated at the time but not published. It later was independently rediscovered by Barratt-Priddy, [BP], and Quillen, also unpublished, and has been of fundamental importance in algebraic topology since.

~

[(I_t 2 )1(1_t 3 )

(1-t 4

-

)(1-t 6)], and doing an easy manipulation we

obtain:

P2(t) =

~ [ Cl _t2)1(1 - t

=

~ ((1 - t )1(1 _ t

=

~ ((1 - t

3)

r

2

3 ))

2

)1(1 _ t 3 ) ) 1+t

+

(1 - t 4 :(1- t 6 )]

((1 - t 2 )\1 - t 3 ) + (1 + t 2 )1(1 + t 3 ))

((1 _2t~~t~ t

6 ))

5

o

4. More Invariant Theory As we saw in §3, the additive structure of H* (Sn) has been completely understood for over 20 years. In (1.13) the ring structure of H*(S4) was determined to be H*(S4) ~ P[0'1,0'2,C3]/(0'1C3 = 0). It was assumed that the same type of simple relation would hold for symmetric groups of higher degree. In this section we will use invariant theory to show otherwise. In fact we will exhibit numerical evidence that indicates the presence of very complicated relations, rich with symmetry, that build up successively. We will use these to give complete descriptions of the rings H*(Sn) for n = 6,8,10,12 in §5. As in §1, let j: Vn (2) = (Z/2t ~ Z/2 2 ... 2 Z/2 ~ S2 n denote the embedding where we consider S2n as the automorphism group of the set ffifZ/2 and Vn (2) as the set of translations. Its normalizer is the group of affine transformations, Affn(Z/2). Therefore N (Vn(2)) /Vn (2) = GL n (1F 2 ) and we have seen that in fact imj* = H*(Vn(2))GL n (F2), the ring of invariants. For n = 2, we obtain that imj* ~ P[Xo, Xl], deg Xo = 2, deg Xl = 3. For n = 4k, we have a natural inclusion

Examining this series, we deduce the existence of a five-dimensional class, which satisfies a quadratic relation. This is the first indication of the occurrence of relations which are not of the form seen before, the products of two generators being zero. Theorem 4.2

(P[xo, xl]0P[xo, Xl])S2 ~ P [dOl, d02 , d ll , d22 ] (X5)/ (x~ + X5 dOl dll + d22 d61 + d02 dil where

dOl d02

= Xo = Xo

0 1 + 1 0 Xo, d ll 0 Xo, d22

X5 = Xo 0

Xl

+ Xl

= Xl

0

0 Xo .

= Xl 0 1 + 1 0 Xl

Xl

= 0)

4. More Invariant Theory

Chapter VI. Thc Cohomology of Symmctric Groups

208

dOl

Proof. The elements dOl, d 02 , d ll , d22 are independent, generating a subpolynomial algebra. Verifying the relation, we conclude that together with X5 they generate a sub algebra with the same Poincare series as

o The relation in 4.2 will pull back to H* (58), as will be shown in §5. N ext we analyze the case k = 3, denote

d ll X5

+ t S + t lO + tIS

P3(t) = (1 _ t2)(1 _ t3)(1 - t 4 )(1 - t 6 )2(1 - t 9 )

= (1/ (1 - t ) (1 - t ) a b = (1/(1 - t 4 )(1 _t 6 ) a 3 = (1/(1 - t 6 )(1 - t 9 ) 2

abc 2

3

--

-

-

1)

X~

+ d02X~ + d02 dOl Xs + (d 03 d ll + d02doldll)X5 + d62 d il + d61 d22 do2 + d03dold22

(2)

X~

+ d22X~ + d22 d ll X7 + (d 33 dol + d22dlldol) X5 + d~2d61 + dil d02 d22 + d33dlldo2

(3)

XSX7

+ d01X~ + [d61 + d02 ]xs + d61 dllXs + d~l d22 + d03 dii + d03 d 22 + dOl do2 dii

(4)

XS X8

+ dllX~ + [dil + d22 ]X7 + dil dOIXs + drl d02 + d33 d61 + d33 do2 + dll d22d61

(5)

=

3

+-

cubes

el2 = a 2 b -

e3 +-

a

products of type a 2 b which are not cubes

elll = abc - 3el2 - e3

+-

X~ + dlld01X~

+ X7 XS + d ll d6l X S + dOl di l x7 d + (d61 22 +d02 dil) X5+ +d~l d 33

The letters denote the type of products they represent. Let e3

0 Xl 01),

(1)

1) 3

1) (1/(1 - t 2 )(1 - t 3 ) 1) .

0 1 01), d 22

.

Proof. Denote

= S(xo 0 Xo 0 1), d03 = Xo 0 Xo 0 Xo = S(XI 0 Xl 01), d33 = Xl 0 Xl 0 Xl X7 = S(xo 0 Xo 0 xI), Xs = S(xo 0 Xl 0 Xl)

0 1 0 1), d02

Relations:

Theorem 4.3 The Poincare series for P 3 is

1 + t S + t7

= S(xo = S(XI = S(xo

209

+ drl d03 + d03 d33 .

Proof. The elements d ij i = 0,1, j = 1,2,3 are algebraically independent, and generate a polynomial sub algebra. We examine the numerator of the Poincare series for P3:

products of type abc not of two other types.

As we are taking symmetric invariants, the formula for the desired Poincare series is 1 P'3(t) = '6ell1

3) . + el2 + e3 + ( 1/(12 - t )(1 - t)

The term on the extreme right records the 3-fold symmetrization of the generators of Plxo, xIl· Using macsyma, this simplifies to the asserted Poincare series. 0 As before, we use the numerator to deduce the relations. Let S( UI 0 U3) denote symmetrization.

Theorem 4.4

where

U2

0

The quintic term must come from Xs = S(xo 0 Xl 0 1) and similarly X7 = S(xo0xo0xt), Xs = S(X00XI0xI) account for t 7 , tS. Likewise xg explains the occurrence of t lO . (One checks that Xs does not satisfY a quadratic equation). There must be relations involving x~, x?, x§, XSX7 and X5XS. Finding the exact relation is a rather tedious task, which involves taking all possible products of the right dimension involving Xs, X7, Xs, xg and expressing their spill over in terms of the dij . Knowing the exact formulas, 'their verification is a lengthy but straight forward calculation, which we leave as a horrible exercise to the reader. Note that relation (2) is obtained from (1) by exchanging Xo and Xl; likewise we get (4) from (3) in this manner. Relation (5) is invariant under this exchange. To complete the proof we just compare Poincare series. 0

For the cases k = 4,5, 6 we do not have the algebra structure, only the Poincare series. These can be obtained in a combinatorial way extending the methods used in 4.3; we used macsyma to perform the simplifications. As the denominators of these series are clear, we only give the numerators: Theorem 4.5 The invariant algebras P4, P5 , P6 have the following polynomials as numerators of their Poincare series:

(1)

N 4(t)

= t 30 + t 25 + t 23 + t 22 + t 21 + 2t 20 + t 19 + + t 18 + t 17 + t 16 + 2t 15 + t 14 + t 13 + t 12 + t l l

+ 2tlO + t 9 + t 8 + t 7 + t 5 + 1 (2)

(3) N6(t)

N 5(t)

= t 50 + t 45 + t 43 + t 42 + t 41 + 2t 40 + 2t 39 + 2t 38 + 2t37 + 3t 36 + 3t 35 + 3t34 + 3t33 + 4t 32 + 4t 31 + 4t 30 + 5t 29 + 5t 28 + 5t 27 + 5t 26 + 6t 25 + 5t 24 + 5t 23 + 5t 22 + 5t 21 + 4t 20 + 4t 19 + 4t 18 + 3t 17 + 3t 16 + 3t 15 + 3t 14 + 2t 13 + 2t12 + 2t11 + 2t 10 + t 9 + i8 + t 7 + tS + 1

= es + eo'+ t 68 + t 67 + t 66 + 2t 6S + 2t 64 + 2t 63 + 3t 62 + 4t 61 + 4t 60 + 5t S9 + 5t 58 + 6t S7 + 7t 56 + 8t 5S + 10t 54 + 10tS3 + llt S2 + 13t51 + 14t 50 + 15t49 + 17t48 + 18t 47 + 19t46 + 20t 45 + 21t44 + 22t43 + 23t 42 + 23t41 + 24t40 + 23t 39 + 24t 38 + 24t 37 + 23t 36 + 24t35 + 23t 34 + 23t33 + 22t 32 + 2lt 31 + 20t30 + 19t 29 + 18t 28 + 17t27 + 15t 26 + 14t 25 + 13t24 + llt 23 + 10t 22 + 10t 21 + 8t 20 + 7t 19 + 6t 18 + 5t 17 + 5t 16 + 4t 15 + 4t14 + 3t 13 + 2t12 + 2tll + 2t 10 + t 9 + t 8 + e + t 5 + 1 .

Let P[X1,' .. , xnl be a polynomial algebra which has an Sn action defined by permuting the n generators. A fundamental result in Galois Theory is that the ring of invariants of this action is also a polynomial ring, on the symmetric generators 0"1, ... , CT n . We have shown that this result does not extend to invariants ?f the form (01 P[xo, xd) Sk. As more variables are introduced, the complexity of this ring of invariants increases and this will be reflected by interesting relations in the cohomology of the symmetric groups.

5. H* (Sn),

n

=

6,8,10,12

In this section we will combine the well-known additive structure of H*(5n) with the invariant theory of §4 to obtain precise descriptions of the cohomology rings H*(Sn) n = 6,8,10,12, as well as the action of the Steenrod algebra on them. We start by describing H*(S6): Theorem 5.1

deg O"i = i, deg C3 = 3 where

Sq 10"2 = CT1 CT2 + CT3 + C3, Sq1c3 = 0, Sq 2c3 = CT2C3 2 Sq 1CT3 ~ (C3 + CT3) CTl, Sq CT3 = CT3CT2 + c30"i .

Proof. From §1, we see that H* (56) has a basis given by elements

* Qj * Qk, Qr,s * Qd = 2s +r and Qo,s = Qs * Qs·

{Qi where 1 ::; r::; s, dimQr,s sequence of inclusions

Using a basis dual to the one above, we have that if k k* ((Q1

k* ((Ql k* ((Ql

We now consider the

=j

. i,

* 1 * 1)*) = 0"1

* Ql * 1)*) = CT2

* Q1 * Qd*) = 0"3

.

Using the symbol O"i to denote the cohomology classes identified to symmetric classes, the above implies that

* 1 * 1)*) = 0"1 0 1 + 1 00"1 j* ((Ql * Q1 * 1)*) = CTI ® CT1 + 0"2 ® 1 j* ((Ql * Q1 * Qd*) = 0"2 0 CTI . j* ((Ql

Note that the 3-dimensional generator C3 in H* (S4) restricts to zero. As j*(Q11 ® 1) = Q11 * 1, we conclude that j* ((Q11 * 1)*) = Q11 ~ 1 = C3 ® 1. The elements CT1 ® 1 + 1 ® 0"1, CTI ® CTI + CT2 01 and CT2 ® 0"1 are mdependent, as they map to the symmetric generators under i*. Recalling that in H* (S4) C30"1 = 0 is the only relation, we obtain a unique relation in the su balgebra generated by the above elements together with C3 ® 1:

(C3 ® 1) [(0"2 ® 0"1)

+ (0"1

® 1 + 1 ® CT1)(CTI ®

0"1

+ CT2 ® 1)] =

0.

212

5. H*(Sn), n=6,8,10,12

Chapter VI. The Cohomology of Symmetric Groups

Hence observing that j* is a monomorphism and that this subalgebra has the same Poincare series as H* (S6) (which is 1 + x3 /(1- x)(l- x2)(1- x3) ) , we conclude

Ci2 Sq1

Ci1Ci2

Sq3 0"1

01

+ 1 00"1""---;'0"1,

0"1

00"1

+ 0"2 0

Sq1

1 i*(Sq 0"2 01

i* (Sq1 (C3 0 1)) = 0 i*(0"2 c301)) = 0"2 C3 i*(Sq 10"2 0 0"1 + 0"20 O"f) i*(C3 00"1 + 0"10"200"1 + 0"20 O"r) (C3 Ci3)Cil

+

2 Sq 0"3

+ 0"1 Ci 4

3

= 'i*(Sq 2Ci2 0 Ci1 + Sq 1Ci2 0 Sq 1Cir) = i*(u~ 00"1 + C3 0 O"i + 0"1 Ci 2 0 Cii)

d6

d7

0"1 Ci 4

O"lxS

d7

0

+ d 6 + XSO"l

0"2 X S

0"2 d 6

0

C3 d 6

0,

Ci4d6

Ci4d7

+

Ci2Ci 4

+ Xs

c2

X5

Xs

Sq2

= + 0"10 O"r) = + O"i 0 0"1 + 0"10 O"n = i*(C3 0 1 + 0"10"201 + O"f 0 0"1 + 0"10 O"i) = i* ((0"1 0 1 + 1 0 0"1) (0"1 0 0"1 + 0"2 0 1) + 0"2 00"1 + C3 0 1) = 0"10"2 + 0"3 + C3 Sqli*(C3 0 1) i*(Sq2(C3 (1)) Sqli*(0"2 0 O"r)

0 Ci2C3

Ci~

Ci4

For the action of the Steenrod Algebra, we apply the Sqi in H* (S4) 0

Sq 0"2

r+

0"2 Ci 3

1""---;'0"2, 0"200"1""---;'0"3,

H* (S2) and use the relations there: 1 Sq i*(0"2 01

' C3

0"1(C3+0"3) C3 Ci

0

c301....---;.c3·

1

+ 0"3 + C3 O"~

Sq2

Here we identify

Ci3

+ Ci3)X5

Sq3

m7

Sq4

Ci~

ng

SqS

0

x2

Sq6

0

0

(C3

S

213

x Sd6

+ Ci4d7 d~

0

d6 d7

Here m7 = d7 + C3Ci4 + Ci30"4 + Ci2XS, and ng = Ci4XS + c3(d6 + Ci1X5). From this a relatively direct calculation gives the cohomology of SlO. Indeed, going from S4i to S4i+2 only has the effect of freeing up products with Cil, and adding a new generator 0"2i+1' We omit the A(2) action as it can easily be obtained using the embedding H* (SlO)....---;.H* (Ss x S2). Theorem 5.3

= 'i*((0"1 0 0"1)(Ci2 0 1 + 0"10 Cir) + (C3 (1)(0"101 + 100"1)2) o = 0"30"2 + c30"i .

H*(SlO) ~ P[Ci1' Ci2, Ci3, Ci4, CiS, C3, X5, d6, d 7 ]/ (R) where R is the set of relations 0"2d7 = 0"3d7 = C3d7 = x S d 7 = 0 (CiS Ci10"4)d7 = 0 (Ci3 + Ci1Ci2)d6 = (Ci5 Ci1Ci4)d6 = 0 = 0"2C3XS + C~Ci 4 Ci~d6 . (Ci3 + CilCi2)X5 C3(CiS + 0'l Ci 4 + Ci2Ci3 (0"5 + 0"10"4)X5 = C3( Ci 3 + Cil0"2)Ci4 .

+

Next we consider S8: From this point forward we will omit details and just describe the results. Full details can be found in [AMMl]. Theorem 5.2

xg

+

=

+

2

+ CilCi2)

Theorem 5.4 where deg O"'i = i, deg C3 = 3, deg d i = i, deg Xs set of relations: d 6Cil = d 6 0"3 = 0 d 7 Cil = d 7 Ci2 = d 7Ci3 = d7C3 = d7Xs XSCi3 C3Ci4Cil = 0 C3(Ci3 + 0"10"2) + O"lXS = 0 x§ + XS0"2C3 + d6Ci~ + 0"4C§ = O.

= 5, =

and R is the following

0

+

The following tables describe the action of the Steenrod algebra

, H* (S12) ~ P[Ci1, Ci2, 0"3, C3, 0"4, CiS, Ci6, d 6 , d 7 , d g ] (X5, X7, XS) /(R) where deg Cii = i, deg C3 relations

X~ X~

= 3, deg

di

= i, deg

Xi 2

=i 2 C3

and

R

is the set of

2

+ Ci4X§ + 0"2Ci4XS + (Ci6C3 + Ci2Ci4C3) X5 + Ci4 + + 0'2 Ci4 d 6 + Ci6Ci2d61 + d 6 x§ + C3d6X7 + (d gCi2 + C3d6Ci2)X5 + d~Ci~ + C§0'4d6 + d9C30'2,

+ Ci2X§ + [Ci~ + Ci4] Xs + Ci~C3XS + Ci~d6 + 0'6 C§ + Ci6 d 6 + 0'2Ci4C~, X5XS + C3xg + [C§ + d 6 ] X7 + C§Ci2XS + C~0'4 + d gCi 5+ dgCi4 + C3 d 60'5, X~ + C3Ci2Xg + X7X8 + C30'5X8 + 0'2C§X7 + (0'5d6 + 0' 4C~) Xs +Ci~dg + C~Ci6 + 0"6d9, XSX7

vl1tl}J\"tl

V.i. .

..l...11~ VUl.l.Vl..1.1V.1V

d9(J'1, d 9(J'3, d9(J5 d7(J'3, d7X5, d7(J'I C3, d7(X7

d7 (X8

o

J

V.1

U'y.1J.J..11l.vll.L.l\,,;

'-....l.LVU'pQ

+ (J'4C3) , d7((J'5 + (J'4(J'1) , d7(0'6 + (J'40'2) ,

+ d6 (J2) ,d7 (dg + d6C3) ,

X7(J'1 + X5 (0"10"2 + 0"3) + C3 (0"20"3 + 0"50"1 + 0"10"4 + 0"5) , X70"3 + X5 (0"5 + (J'1(J'4) + C3 (0"10"6 + CT3(J'4 + CTI0'2(J'4) , X7(J'S + XS(J'1(J'6 + C3 ((J'3(J'6 + (J'10'20'6) , XS CT I + d 6 (CT3 + CTICT2) , XS(J'3 + d 6 ((J'5 + 0'1 CT4) , XSCT5 + d 6CTl (J'6 .

with the transfer. This can also be proved algebraically using the short exact sequence of G-modules

o

The action on the Steenrod Algebra on H* (S12) is determined by the following values: Sq1(J'2 = (J'1(J'2 + (J'3 + C3, Sq1c3 = 0, Sq2c3 = (J'2C3 + X5, Sq2(J'3 = C3(J'r + CT2CT3 + CT1(J'4 + CT5, Sq1(J'4 = X5 + 0'1(J'4 + (J'5, Sq2(J'4 = (J'2CT4 + CT6 + d6 + C3 (0'3 + 0'1(J'2) , S q2 (J'5 = d6(J'1 + X50'r + C3 (CTI CT3 + (J'r (J'2) + 0'2(J'5 + 0'1 CT6, Sq4(J'5 = C3(J'~ + (J'4(J'5 + CT1X50'3 + CT3CT6 + c3CT1(J'2(J'3 + C30'1(J'5 + d7(J'r, Sq2X5 = (J'2 X5, Sq 1CT6 = X7 + 0'1(J'6, Sq2(J'6 = Xs + X7(J'1 + (J'2(J'6, Sq4(J'6 = (J'4(J'6 + X7(J'3 + d60'~ + X~ + X50'2C3 + C~(J'4 + C3X7 + C3(J'10'6, Sq 1d6 = d7 , Sq 2d6 = X8 + (J'2 d6, Sq 4d6 = (J'4CT6 + X5(J'2C3 + d6(J'~ + CT4C~ + xg + X7C3 + X80'2, Sq 1d 7 = Sq 2d7 = 0, Sq 4d7 = (J'4d7, Sq 6 d7 = d6d7, Sq1X8 = dg + 0'2d7 + (J'3d6 + (J'1(J'2d6, Sq4X8 = X8 ((J'4 + C30'r) Sq 1 d g = d7C3, Sq 2d g = (J'2d9, Sq 8d g = d7X7C3 + X8 d 9 .

6. The Cohomology of the Alternating Groups An important class of simple groups whose cohomology rings have not been computed are the alternating groups An. The absence of proper normal subgroups makes the cohomology of simple groups particularly inaccessible, but in our situation we can make good use of H*(Sn). The Gysin sequence, as in 11.5, allows one to relate the cohomology of a group to that of an index 2 subgroup. For convenience we recall its statement and give an independent proof. Lemma 6.1 Let v E Hom(G, Z/2) non-trivial, with H a long exact sequence

= kerv.

Proof. (Compare [DM2J, Appendix 1). The fibration 7l/2 - ) BH - ) Be is the sphere fibration of a line bundle. Consequently MC(p) = Bel BH is the Thorn space of the line bundle with first Stiefel- vVhitney class v. Now apply the Gysin sequence. Transversality identifies

We apply this lemma to the situation An ~ Sn, with [Sn: An] = 2. We know Hl(Sn) ~ lF2' generated by the symmetric class 0'1; hence Theorem 6.2 T!;,ere is a short exact sequence of H*(Sn)-modules:

o -) H*(Sn)/(CTl)~H*(An)~Ann(CTd

-)

o.

From the considerations in §1, it is not hard to see that the ideal I n = Ann((J'I) is generated by "pure" Dickson cla.sses. These can be described as follows: let n = 2j1 + ... + 2j ,·, where 1 ~ j1 ~ ... ~ jr (the case jl = 0 can be reduced to this situation); the pure Dickson classes are those of the form (Qft * ... * Q1,.)*' where 2 ~ length(h) = jk for all k = 1, ... ,T and each Ik = (1, ... ,1). An immediate consequence of this is the following Corollary 6.3 If n is congruent to 2 or 3 mod H*(Sn) -+ H*(An) is onto, and

4,

then the restriction map

We remark that the results in §1 together with the preceding discussion can be used to completely determine the additive structure of H* (An), any n. The ring structure is of course much harder, but we can recover the rings H*(A6), H*(As), H*(AlO) and H*(A12) in what follows. From 5.1 and 6.3 we have:

Then there is Corollary 6.4 with Sql(J'2

H*(A6) ~ P[CT2,0'3,c3]/(c3CT3

= 0'3 + C3, Sq1c3

= 0)

= 0, Sq2c3 = CT2C3, Sql(J'3

= 0, Sq 20'3 = 0'30'2·

216

6. The Cohomology of the Alternating Groups

Chapter VI. The Cohomology of Symmetric Groups

Proof. By 5.1, Ann(O"l)

=

0, hence

H*(A6)

~

o

H*(56)/(0"1)'

Applying 6.2 to As, we obtain the following

Corollary 6.5

H*(As) ~ P[0"2' C3, 0"3, 0"4, d6, e6, d7, e7](xs)/(R) where deg O"i = i, deg C3 = 3, deg di = i, deg ei = i deg Xi = 5, and R is the following set of relations: d60"3 = 0, d6d7 + d6e7 + e6e7 = 0, dg + d6e6 + eg = 0, d70"2 = d70"3 = d7C3 = d7Xs = 0, d6d7 + d7e6 + e6e7 = 0, d? + d7e7 + e~ = 0, e60"3 = e70"2 = e70"3 = e7 C3 = e7 x S = XS0"3 = 0,C30"3 = xg + XS0"2C3 + (d6 + e6)0"~ + 0"4 C§ = 0.

°

°

°

The action of the Steenrod algebra on the generators different from e6, e7 can be read from 5.2 using the restriction map; for e6, e7 the action is identical to that on d6 and d7 substituting e6, e7 for d6, d7 throughout.

217

The maps j* and I!* are onto: as noted before their images will be subpolyn
+ (ei + di)ei)

°and eidi

=

E im res

U sing the restriction to V3 and 9V3 g -1, it follows that

yielding the relation

e; + eidi + d; = °.

The same procedure yields the relations

e6e7 e6e7

+ d6e7 + d6d7 = + d7e6 + d6d7 =

0 0 .

The other relations, involving e6, e7 and the other generators, follow from the corresponding relations for d6 , d7 . The remaining relations follow from .0 applying res to relations in H* (Ss).

Proof. Using 5.2 and 6.2, we have a short exact sequence

°

-+

H*(SS)/(0"1)~H*(As)~(d6,d7)

-+

0.

Here (d 6, d7) is the ideal generated by d6 and d7 in H* (58), and res is an isomorphism until degree 6, where a new class e6 appears in H6(As); similarly there is a new class e7 E H7(As), and we have tr(e6) = d6, tr(e7) = d7. Note that tr(xy) = tr(x)· y for y E H*(5 s )/(0"1)' hence we need only adjoin e6, e7 and their products with elements in im res to obtain H*(As). It only remains to determine the multiplicative relations involving these two classes. Let N S8 (V3), N As (V3) be the normalizers of V3 in 5 s and As respectively. Then [58: N S8 (V3) 1= 2 [As: N As (V3) 1as there are twice as many conjugates of V3 obtained by using elements in 5 s as those obtained by using only elements in As. Hence there exists 1 =I=- 9 E 5 s /As such that V3 and gV3g-1 are not conjugate in As. The classes e6, e7 will be detected on gV g-l. For this note 1 that N As (V3) = NSs(V3) = Aff 3 (Z/2) ; similarly NAs(gV3g- ) = NSs(V3)' We have a diagram of restriction maps i'

H*(Ss)

---+

The explicit calculation of H*(As) is particularly interesting in light of the fact that As ~ GL4(lF2). This isomorphism, due to Jordan and found in Dickson [D], can be described by the following correspondence of generators:

(23)(12)

(45) (12)

(67)(12)

~ G~ ~ D

, (34)(12)

0 1 1 0~1)

~ (~ ~ ~

(56)(12)

~ G~ ~ D

, (78)(12)

~ (~ ~

D

~ ( ~1 ° ~ ~1

~~)

~ (~ ~ ~

J)

[

r

---+

A calculation for H*(GL4(IF2)) was first attempted in [T-Y]; there was an error which was corrected later. However, at best their approach led to a listing of elements without giving any insight as to their significance or explicit multiplicative relations.

218

Chapter VI. The Cohomology of Symmetric Groups

Corollary 6.6

The Poincare Series for

= (t+l)(t

P(t)

15

H*(GL4(lF2))

Chapter VII. Finite Groups of Lie Type

is

_t14+t13_t12+tll_t9+t8+t1 +2t 6+2t 4-t 3+2t 2-t+l) (t 7 -1)(t 6 -1)(t 3 -1)(t 4 -1)

Using an analogous approach, we obtain

Corollary 6.7 H*(AIO) ~ set of relations O"sd 7 0"3 d 7 0"3 d 6 xg

= =

O"sd6

O"SXS

C3d7

=

x Sd 7

O"s, C3,

xs,d 6 ,d7 l/(R) where

R

is the

=0 1. Preliminary Remarks

+ C~0"4 + 0"~d6

= 0"2C3XS

0"3 X S

= 0"3 d 7 = =0

P[0"2, 0"3, 0"4,

= C30"30" 4 = C3(O"S +

0"20"3) .

(As for A6 the A(2)-module structure follows directly from that of H*(51O).) Finally, we have

Corollary 6.8

H* (A 12 ) ~ where

deg

P[0"2' 0"3, C3, 0"4,

= i, deg C3

O"i

O"s, 0"6, d 6 , d 7 , d g , eg, elO] (XS, X7,

= 3, deg

di

= i, deg

X8) /

= i, deg ei = i

Xi

(R) and

R

is

the set of relations

x~ + 0"4X~ +

0"20"4X8 + (0"6C3 + 0"20"4C3) Xs + O"~ + c~ + 0"~0"4d6 + 0"60"2 d 6, + d 6 X s + C3 d 6 X 7 + ([eg + d 9 ]a2 + C3d60"2)XS + d~O"~ +c~0"4d6 + [eg + d 9 ]C30"2' XSX7 + 0"2 x g + [O"i + 0"4] X8 + 0"~C3XS + 0"~d6 + 0"6C~ + 0"6d6 + 0"20"4C~, XSX8 + C3 x g + [C~ + d 6] X7 + C~0"2XS + C~0"4 + [eg + dglO"~ +[eg + d 9j0"4 + c3d60"~, X~ + C30"2xg + X7XS +C30"~XS + 0"2C~X7 + (O"id 6 + 0"4C~) XS+ O"~[eg + dgJ + C~0"6 + 0"6[e9 + d g ], X8

d90"3,d gO"s,e90"3,egO"s elO0"4 + d 7 (X7 + 0"4 C3) , d 7 0"s, d 7 (0"6 + d 7 (eg + d g ) + d6(d7C3 + elO), d 70"3, d 7 X s,

XS0"3

+ C3

(0"20"3

d60"3, X80"3 d§

+

dgeg

d g d 7 C3

+

+

+

0"40"2), d 7

(Xs +

d 6 0"2) ,

+ O"S), X70"3 + XsO"s + C3 0"3 0"4 , X70"S + C3 0"3 0"6 , XsO"s,

d 6 0"s,

+ d7C3elO + eto, + egelO, dgd7C3 + d7C3e9 + egelO .

e§, (d7C3)2

dgelO

The action of the Steenrod Algebra on generators different from eg, elO can be obtained from 5.3 using the restriction map; for eg, elO the action is identical to that on dg and d7C3 doing the appropriate substitutions.

One of the most remarkable results of this century in mathematics has been the classification - completed in 1980 - of all the finite simple groups. This took over 20 years and occupies almost 5000 pages in the literature. It is conceivable that there are some errors there, so the details of classification are not really available to us, but the main results can be summarized. There are 17 families of simple groups, the alternating groups and 16 families of "Lie type". These in turn are broken up into subfamilies in several different ways. There are, first, the historical breakdowns, 6 families of "classical groups", the projective special linear groups over finite fields, the orthogonal groups, (three types), unitary groups, and the symplectic groups. Then there are the 5 families of groups of "exceptional types", the groups C 2 (q), F 4 (q), E6(q), E7(q), and Es(q), the q denoting the order of the finite field over which they are defined. Finally, there are five special families, the groups of "twisted type" , the Suzuki groups Sz[22n+l], the Ree groups 2C 2[3 2n +1 ] and 2F2 [2 2n +1 ], the twisted exceptional groups 2E6[22n+1], and the groups 3D 4 (q), the "triality twisted D 4 (q) 's". Of course, the most natural classification of these groups is via Lie theory and Dynkin diagrams. A complete discussion can be found in R.W. Carter's book [Carter]. In each family the first few groups are somewhat exceptional: they may not be simple, for example A4 or GL2(lF2) = 53, SP4(lF2) = 56, or they are not distinct, PSL 2 (lF 4 ) = PSL 2(lFs) = As, PSL 4 (lF 2 ) = A 8 . Additionally, the situation for some of these groups at the prime 2 seems to be much more involved than at other primes. For these reasons and probably others there arise 26 further simple groups, the sporadic groups. The first five of these, the Mathieu groups M n , M 12 , M 22 , M 23 , and M 24 , have been known and explored for about 100 years. The remaining ones have been known for less than thirty years. The first of them J 1 - the first Janko group - has already been discussed in Chapters II, III and V. A good discussion of these groups, their construction and history, can be found in Gorenstein's book [Gor]. In this chapter we describe the main features of the classical groups of Lie type and discuss some aspects of Quillen's calculation of the cohomology of these groups. The most striking of his results appears in his paper in the Proceedings of the International Congress of Mathematics, Nice, 1970 [Q2]'

220

Chapter VII. Finite Groups of Lie Type

where he indicates the existence of a spectral sequence for groups sufficiently closely related to the continuous Lie groups G with E 2 -term

2. The Classical Groups of Lie Type

221

B is equally well determined by the bilinear form { , ) B: VK x VK - t K,

( , )B(V, v') = {v, V')B = B(v)(v'). We have Lemma 2.1

where G(q) is the group defined by the same (integral) equations over the finite field IF q and q is prime to p. He also indicates there how to use this result to get information about the groups of twisted type. However, the indicated proof, using Etale techniques from algebraic geometry is far removed from our discussion. Rather, we give proofs of most of his results here but use the more elementary techniques discussed here. An alternate source for much of this discussion is [FP l. It should be emphasized that the groups we are actually studying are not the simple groups, generally quotients of the groups here by their centers, but the full groups, described as the groups of automorphisms of a finite dimensional vector space over a finite field, in the case of the general linear groups, or which preserve a bilinear form in the case of the orthogonal and symplectic groups, or which preserve a Hermitian form in the case of the unitary groups. In brilliant work of Quillen in the early 1970's these groups and their cohomology played an essential role in vital questions in homotopy theory some of which we will also outline in the introduction to Chapter Vln and in Chapter IX.

a. Bis symmetric if and only if (v, v') B = (v', v) B for all v, v' in V'K. b. B is skew symmetric if and only if if (V,V')B = -(V',V)B for all v, v' in V K· c. B is non-singular if and only if, given any v E V K there is a v' E V K so that (v, v') =I- O.

Proof. (a), (b) are direct. To see (c) note that B is a linear map of finite dimensional vector spaces over K having the same dimension. Consequently B is invertible if and only if Ker(B) = O. But v E Ker(B) if and only if (V,V')B = B(v)(v') = 0 for all v' E V K . D Two non-singular bilinear forms Band B' are said to be isomorphic if there is an isomorphism g: V K - t V1 so that the diagram

2. The Classical Groups of Lie Type Given a field K and a vector space Vj{ of dimension n over K we have the group GLn(K) of invertible linear transformations 0:: V K - t V 1{. GLn(K) has a center C(GLn(K)) = K· and there is a determinant homomorphism Det: GL n (I{) --r KO --rOo The kernel of Det is the special linear group SLn (K) and SLn(K)/SLn(I<) n C(GLn(I<)) = PSLn(I{) is called the special projective linear group of rank n over K. It is simple except for K = IF 2 ,IF 3 , and n = 2. As K runs over the finite fields IF'q (where q = pn, p a prime in Z+) we obtain in this way the finite simple groups Ln(q) = PSLnOFq). This is the first family of finite simple groups of Lie type. Next, suppose Vj{ is given together with a non-singular bilinear form b. A bilinear form can be thought of as a K-linear map B: VK - t (VK)* where (VK)* = Hom I< (Vj{, K). There are natural isomorphisms e: (Vj{ )** - t V K (for n =I=- 00, e(f) is given by w(e(f)) = f(w) for w E V1), and given B there is the induced map B*: (VK)** - t (VK)* defined by B*(f)(w) = f(B(w)). Consequently B induces a map Be-I: (VK)* - t V K which we again write as B* using a mild abuse of the language. B is symmetric or anti-symmetric according to whether B* = B or B* = -B respectively. B is said to be non-singular if B is an isomorphism.

commutes. Isomorphism is an equivalence relation. Moreover, to each nonsingular bilinear form B there is associated its group of isometries G B:

G B = {g: V K - t V11 9

E

GLn(K), (g(v),g(V'))B = (V,V')B}

forallv,v' E V K . (g,h E G B implies that (gh(v),gh(v'))B = (h(v),h(v'))B (v, v') B for all v, v' in V K so G B is closed under comp~sition. Also

=

so G B is closed under taking inverses and G B is a subgroup of GLn(K).)

Remark 2.2 If B is isomorphic to B' then G B ~ G B/, since if 9 takes B to B' in the sense of (*) then gGBIg- 1 c GB, and g-lGBg C GB,. Examples 2.3 Over K any bilinear form B is given by (k, k') B = kk'b for some b E K using the usual identification K* = K (b E K* is the map k I--t kb.) Then (*) says that two forms x b and x b' are equivalent if and only if there is a k E Ke so that b' = k 2 b and the equivalence classes of non-singular forms on J{ (they are all symmetric) are the elements of K e/(K e where (Ke)2 denotes the multiplicative subgroup of squares in K'. The group GB in each of these cases is K e .

?

222

Chapter VII. Finite Groups or

Lle

.type

In particular, if K = IF q , (q =I 2r) there are 2 equivalence classes of forms, the first represented by (k, k')I =. kk' and the second represented by (k, k')a = kk'a where a E K e is any non-square. Next, passing to based vector spaces, given Vf;: together with a basis

(el, ... ,en ), there is a dual basis for (IFf;:)*, (el, ... ,en ) where ei is defined as the homomorphism ei(ej)

=

{01

~ =I= J.cr ~With respect to these choices

for 9 E GLn(K). It follows that 9 E G B if and only if B = gBgt. In particular for B = (1) ~

Det(g)

(~ ~)

so 9 E G B if and only if ggt

= ±l. In general, for

=

I. Consequently

L- - (ac b)-l __ (d-c -b) a d

l

1 - Det(L)

2

(J E

SO~(K)

if and only if

(J =

(~b ~)

with Det(tI) =

+ b2 =

l. If B: Vf;: - t VI<, B': Vl{ - t (Vl{)* are given non-singular bilinear forms then B EB B': VK EB VI( - t (VK)* EB (Vl{)* = (VK EB Vl{)* is a non-singular bilinear form on VK+ m . It is symmetric or skew-symmetric according as B, B' are, and it has the property of orthogonality, (v, v') BffiB' == 0 if v E VI< while v' E Vl{' This direct sum form is usually written B ~ B'. If B is isomorphic to B then B ~ B' is isomorphic to 13 ~ B', and similarly for 13 isomorphic to B', B EB B' is isomorphic to B ~ B.

a

Theorem 2.4 Let K = V ql (q =l2r) then there are exactly two equivalence classes of non-singular symmetric bilinear forms on V K, the first equivalent to (1) ~ (1) ~ ... ~ (1) and the second equivalent to (1) ~ ... (1) ~ (a) with a any non-square in K. Proof. We need a splitting principle.

W if and only if (w, v) B

v) + (l,~)

B

= O.

v

and l - ¥v is contained in (v)..L. Consequently Vf;: = (v) b =I 0 it is clear that (v) n W = 0 so the sum is direct.

+ Wand,

since 0

(More generally, if L C Vf;: is a sub-vector space of VI< then the same proof shows that if the restriction of B t~ L is non-singular there is a splitting IFf;: = L EB L..L and E splits as (ElL) ~ E.) Lemma 2.6 Let V K be given with K = IF pr, p odd, and B symmetric and non-singular. Then there is a vector v E Vf;: so that (v, v) B =I o.

(v so L- l = Lt if and only if c = -b, a = d if Det(L) = 1, and c = b, a = -d if Det(L). = -1. Denote by SO~ ~K) those elements of G B with determinant +1 so, if Char(K) is not 2 S02 (K) is normal in G B with quotient 'lL/2. Summarizing,

= (l _ (l,~) B

W E

Proof of 2.6 Choose v =I 0, v E VI< and suppose that (v, v) B = O. Then, by the non-singularity of E, there is a v' E V1{ with (V,V')B =I O. If (v',v') -I- 0 we are done. If not, then

2 x 2 matrices L,

1 _

Proof of 2.5 Let W = (v)..L. That is to say, Let l E Vf;: be any non-zero vector. Then

~

for bases B is represented by a matrix (which we again denote B) where the entries Bi,j satisfy Bi,j = B)',i if B is symmetric or Bi,j = -Bj,i if B is skew symmetric. Also, if 9 E GL(Vf;:) is represented with respect to (el,"" en) by the matrix (gi,j) then g* is represented by the transpose of g with respect to the basis (e l , ... , en) for (VI<) *. It follows that in matrix notation our equivalence relation becomes

(1) the matrix is

Lemma 2.5 Let v E VK and let B be a non-singular symmetr-ic form on V[{. Suppose (v, v) B = b =I 0, then there is an n - 1 dimensional subspace W c VK so that V'K = (v) EB W~-l as K -vector spaces, and with respect to this splitting B = (b) 1. fJ.

+ v' , v + v') B

(v, v) B + 2(v, v') B + (v', v') B 2(V,V')B

=I 0 so the proof is complete.

o

At this point we have proved that B is isomorphic to (b 1 ) ~ ... 1. (b n ). It remains to show that all but one of the bi can be chosen equal to 1. To do this we study non-singular forms on V}. Lemma 2.7 Let E be a non-singular symmetric form on V 1{ where K = lFpr with p odd, then E is isomorphic to (1) 1. (1) or (1) 1. (a) where a is a nonsquare in KO. Proof of 2.7 After diagonalization E rv (1) 1. (1), (1) 1. (a) or (a) 1. (a). We now verify that (a) 1. (a) is isomorphic to (1) 1. (1). By the pigeon hole principle there are k, k' in K so that k2 + k,2 is a non-square in KO. Then (k 2 + k,2)a is a square in KO, and the two vectors (k, k'), (k', - k) are mutually orthogonal in (a) 1. (a) and provide an explicit equivalence with (1) 1. (1). 0

224

2. The Classical Groups of Lie Type

Chapter VII. Finite Groups of Lie Type

Iterating the lemma above shows that there are at most two isomorphism classes, (1) l- .,. l- (1), and (1) l- ... l- (1) l- (a). It remains to show that these two forms are not isomorphic. Suppose then that we have an isomorphism between them for some n 2:: 2, and suppose that 9 is a map which effects the equivalence. Then

o 0

100 010 001

o

o

0

0

00010 o 0 0 0 b and Det(ggt)

= b = Det(g )Det(gt) = (Det(g))2 which is impossible. The 0

theorem follows.

vk

In a similar way we can analyze skew-symmetric forms. On we see from our previous discussion that there are none. On V'k we have (v, v) B = 0 for every v E V'k, but by the non-singularity of B given v =I=- 0 in V'k there is a Vi so (v, Vi) B = 1. It follows that, with respect to the basis v, Vi the form

B is given by the matrix (

~1 ~). Iterating this argument we have

k 2+k,2. Moreover, since Norm is an onto homomorphism IFe 2 = 'll/(q2 -1) -+ IF~ = 'll/(q - 1), its kernel is 7l/(q + 1). Finally, for this ~ase, if N(B) = 1 where e = k + k'ye:r, then

k kl) k E sot (K) ( -k' and conversely, so SOt(K) = 'll/(q + 1). To prove (b) note first that now -1 = k 2 for some k E K so there are two lines (1, k) = h, (1, -k) = l2 and only two so that (l, l)A = 0 for l Eli. Also, since (1) J.- (1) = (a) l- (a) it follows that there is a one to one correspondence between those lines lj for which A restricts as (1) and those for which it restricts as (a). Consequently, since V'k contains exactly q + 1 lines there are (q - 1)/2 lines, l+,1,"" l+,(q-1)/2 where A restricts to (1). Each such line has exactly two vectors a and -a so that ((3,(3)A = 1, and each vector can be written with respect to the original basis as (k 1 , k2) with + k~ = ~. This gi:es that IS01(K)) = q - ~. It remams to verIfy that S02 (K) IS as clalmed. To see this consider

kr

9

We have .

Theorem 2.8 Let B be a non-singular skew-symmetric form on IFi<- where the characteristic of K is not 2, then n is even and

B

(V

(S) l- ... l- (S)

Remark 2.9 Note that when n = 2m + 1 is odd and K is a finite field of odd

~

n

times

so 9 gives an explicit isomorphism (1) J.- (1)

J{

(0 1)

109

2ab ( bc+ad

. '----v---'"

times

= 1Fq be a finite field of odd characteristic.

a. If q == 3 mod 4 then SO~(I() = 7l/(q + 1). b. If q == 1 mod 4 then SO~(I{) = 7l/(q - 1).

Proof. Suppose that q == 3 mod 4, then -1 E K is a non-square and IF q2 IFq(yCT). Write a E IFq2 as k + k'yCT. Then we have that the norm of a is

\

(~ ~)). Hence SOt(K) is

having determinant 1. However,

(~~)(~ ~)(~~)

t

(1) l- ... l- (1) l- (a) . n-1

(~ ~)

(V

here we have 9

But this implies that for such n, K, the groups for the two forms are isomorphic since, if A represents the form with all ones, while B represents the form with all a's, we have (v, Vi) B = a(v, Vi) A for all v, v' in Vi<- so g E G B if and only if g EGA· Lemma 2.10 Let

(kJ:.. 2k

characteristic we have that

(a) l- ... l- (a)

gg t =

(kA ~)

=

isomorphic to the subgroup of Aut

where S is the fomL on VJ< given above.

225

bc+ ad) 2cd

(~ ~) and this implies that 9 E Aut

a

E

K· or 9 =

(l~a ~)

=

(~ ~)

if and only if 9

=

for

(~ l~a) (~ ~). But the latter terms have

determinant -1 so they do not occur and the result follows. Corollary 2.11 Ot(K)

(~ l~a)

= {D 2(q+l) D 2(q-l)

q == 3 mod 4 q == 1 mod 4.

0

226

3. The Orders of the l<'inite UrtllOgonal and 2iympleCtIc Groups

Chapter VII. Finite Groups of Lie Type

Proof. For q == 1 mod 4 this follows from the calculation above with the

group Aut (

~1 ~)

(~ ~). For q '" 3 mod 4 the element (~ ~l)

and fixes

n

(~

(1o 0) -1

acts to invert

Thus

(k

+ k' F-l)

(1 0) 0

-1

= (k - k' yCl)

0

Next we consider the case B = (1) ..L (a).

=

{Z/(q+l) Z/(q-1)

k') ( ' " k -a",'

E

SO~(K) .

",' ) (k'" - k' ",' a '" -a(k"" + k' "')

{D 2 (q+1)

D 2(q-1)

0

which is

(~l ~)

if and only if Det( (:

~)) =

1

is the speciallineaT group

ad -0 be)

o

L

3. The Orders of the Finite Orthogonal and Symplectic

Definition 3.1 A quadratic form on Kn is hyperbolic if and only if n = 2l A -

k",' + k' '" ) k", - ak' ",' .

\(~ ~))1-"\(~ ~))

\,

v

I

when q == 1 mod 4, when q == 3 mod 4.

k ( ak'

-k') (k k -ak'

,/

times

In what follows we denote this form as (H21)'

(I) 1- (1) Remark 3.2 (H2) = (1) ..L (-1) = { (1)..L (a)

(-1o 0) (k k') (-1 0) k

(~ ~l)} = Sp2(K)

0 ( be - ad

and

for q == 1 mod 4 for q == 1 mod 4.

Assume that K is finite of odd characteristic. When n is odd we know there is only one orthogonal group so we may as well study it in the case of the form H 2n ..L (1) = (A).

(In the case q == 1 mod 4 we have

-ak'

Proposition 2.14 AutK {

Groups

Similar considerations show

1

~)}

(=~ :) (~ ~)

The first statement follows. The proof for q == 3 mod 4 follows the lines of proof of the previous lemma (for SO~ (K) in the case q == 1 mod 4. 0

Corollary 2.13 Of (K) =

{C

when q == 1 mod 4, when q == 3 mod 4.

Moreover, the correspondence above is a .homomorphism since k ( -ak'

Aut

Proof. We calculate

Proof. If q == 1 mod 4 then (1) ..L (a) has no isotropy vectors and on every line the restriction of the forin is either (1) or (a). But the interchange mapping (k, k') f--1o (k', k) has the effect of exchanging such lines so there are exactly (q+1)/2lines l with Bll = (1). Consequently BllJ.. = (a), and as before there are 2 transformations /3, -/3 taking (1,0) I---Jo Lo E l, (0,1) I---Jo L1 E lJ.. and preserving the form. It follows that ISO~ (K) I = q + 1 if q == 1 mod 4. On the other hand if J = K (Va) and k + k'Va nas norm 1, then

k k') k ( -ak'

=

SL2(K).

Lemma 2.12 Let K have odd characteristic. Then 2

of (K)

and now the calculation follows.) This completes the discussion for the orthogonal groups of rank::; 2. We now turn to the symplectic groups.

and this is just inversion on the elements of norm 1. The corollary follows.

SOB(K)

When q == 3 mod 4 we have that B = (1) 1- (-1) and

k')-l k

. .

Proposition 3.3 Let IKI = q < 00 be of odd characteristic. Then we have a. The set ofv E K 2n so (V,V)(H 2n ) = a has cardinality

228

when 0: = 0, otherwise. b.

The set of v E K2n+l so that (v, v) (A) when when when

= 0:

E

0:

E

0:

= 0.

We now turn to the calculation of these numbers for the other even dimensional form

has cardinality

where a E Ke is a non-square.

Proof. We begin with the case n = 1. Then (a, a) (a, -a) are the only vectors of length zero (called isotropy vectors). Hence there are 2q-1 isotropy vectors. Next, since

(0:)

~

(-0:)

rv

(1)

~

o

so the induction is complete.

K e is a non-square, K e is a square,

0:

229

3. The Orders of the Finite Orthogonal and Symplectic Groups

Chapter VII. Finite Groups of Lie Type

(-1)

Proposition 3.4 The form (B) satisfies the condition that the cardinality of 2n+ 1 n+ 1 + n h \ the set of v so that (v v) (B) = ), is q - q q w en /\ = , { qn(qn+l + 1) , when), i= 0.

°

Proof. Clearly \

the cardinality of the set of v so that (v, V)(H 2 ) = ), is the same as that for (v, v) (H2) = T provided )" T are both non-zero, it follows that this cardinality is q2 _ 2q + 1 -'-'-----=--- = q - 1 . q-1

(~ ~o:))

has only the O-vecto; as an isotropy vector.

Consequently, the remaining cases, all having the same cardinality, are q + l. Now consider the, general case. Here the number of isotropy vectors is q2n-l

+ qn _

qn-l

+ qn-l(qn _

l)(q

Thus (a) is verified in the first case. Now we proceed by induction. First we show (a) for H j with j ~ n implies (a) for Ihn+2. Since (H 2n +2) = (H2n) ~ (Ih) we can count isotropy vectors as

+ l)(q - 1) q2n-l + qn _ qn-l + qn-l(qn _1l)(q2 q2n-l

+ qn

1)

_ qn-l

+ qn-l(qn+2 _ qn _ + qn.

q2

+ 1)

q2n+l _ qn+l

which directly expands out to q2n+l + qn+l - qn. As before the .cardinality of the sets so (V,V)(H2,,+2) =), with), i= are all equal, so this cardinality is

°

qn) --=-__~__+~-.:.. = q2n+l

(q2n+2 _ q2n+l _ qn+l

..:..=.-_ _

From this the number when),

#-

°

is

q2n+2 _ (q2n+l _ qn+l

_ qn

=

and this case is complete. We now verify that (a) for (H2n ) implies (b) for (A) = (H2n) ~ (1). First we obtain the count of isotropy vectors as

+ q2n-l + qn

+ qn-l

2

°

o

and the result follows.

Now write sotn (K) for the group associated to the hyperbolic form (H2n ) and S02n (K) for the group associated to the form (H 2n - 2) 1.. \ Then we have

_ qn-l

Suppose next that 0: is a non-square, then 0: = (0: - k ) + k and 0: - k #- 0, so as k runs over K this gives q X qn-l (qn - 1) = qn (qn - 1) solutions. Finally, suppose that 0: is a square. Then 0: - k 2 = for two values of k so the count is 2

2

+ qn

q-1

q-1

q2n _ q2n-l _ qn

+ qn)

___. . .:. : . __--=-__-.:....!... = q2n+l

qn(qn+l -1) ,

Theorem 3.5 a. ISOtn(K)1 = qn(n-l)(q2 - 1)··· (q2n-2 - l)(qn - 1), b. IS02n(K)1 = qn(n-l)(q2 - 1).·. (q2n-2 -l)(qn + 1), 2 c. IS02n+l(K)1 = qn (q2 -1)(q4 -1)··· (q2n -1). Proof. We prove (a), (c), together. Clearly

(~

_00: ) ) .

Chapter V11. l<'lmte Groups 01 LIe .lype

.L .............. ___

since it suffices to map the generator i of (1) to any element v of length 1 in K2n+l = (H2n) J.- (1). Then (v)-.l ~ (H2n ).Nextnotethat(H2) = (1) J.- (-1) SO

~"""'J'-"''''''''''''.''''''''''''''''bJ

............................

---" '-'

-~I-~

---,t.

\~1/

Proof. Given 9 E SP2n(K), 9 can take the non-singular subspace (el, h) to a new non-singular subspace (in a form preserving manner) in exactly

(H2n ) = (-1) 1 ((1) 1 (H2n - 2 )) so

= qn-l(qn -1)IS02n-l(K)1 .

ISOtn(K)1

ways. Hence,since it also maps (el' h)-.l to the perpendicular complement of g((el' h)) we have

From these two formulae (a) and (c) follow. Finally we prove (b). Since

!SP2n(K)1 = q2n-l(q2n - 1)!SP2n_2(K)! . Also, as we have already seen, Sp2(K) = SL2(K) which has q(q2_1) elements. Now, note that

we have IS02n(K)! = qn(qn-l

+ 1)!S02n-l(K)!

and the theorem follows.

0 (2n - 1)

+ (2n -

3)

+ ... + 1

Let K be any finite field (here characteristic 2 is allowed). We consider

_2n_(_2_n_+_1_) _ 2 n( n + 1) . 2 2

K 2 n with the skew form

(~l ~).

where 8 = In particular, we have a basis el,···, en, 2 K n so (el,ej)B = (fi,fj)B = 0, (el,fj) = -(jj,ei) = Oij. Lemma 3.6 Let v E K2n with v =I- O. Then (v, v) B precisely q2n-l vectors w E K2n so that (v, w) B = 1.

(v, V)B (v, W)B

h,··., fn for

o and there are

2::)ViAi - AiVi) = 0 L v/J L i -

o

so the corollary follows.

B = (8) J.- ... J.- (8)

AiWi

and since v =I- 0 there is a Vi or Ai which is not O. Say Vi =I- 0, then (v, ~ fi) B = 1 so the set of w with (v, w) = 1 is non-empty. On the other hand if (v, w) B = (v, w') B = 1 then (v, w - w') = 0 and w - w' is contained in the hyperplane

Remark 3.8 Let K jJF be an extension of degree d, then

by forgetting the K-structure. As an example

and we note that the index of this inclusion is 12. But the normalizer of A5 in SP4 (JF 2 ) is generated by the Galois automorphism of IF 4 over JF 2, and this gives a copy of 55 c Sp4(JF 2) having index 6. In turn, this gives, via the action of 8p4(JFi) on cosets, a homomorphism Sp4(IF 2) --+ 56. Moreover, the kernel is contained in 55 and hence is either As or 1. But since the normalizer of 8p2(JF r ) in 8p4(JF 2) is 55 it follows that the kernel is a proper subgroup of As and is thus 1 (since it is normal). Finally 1Sp 4 (IF 2) 1 = !S61 so the coset representation constructs an isomorphism 8p4(IF 2) ~ 56'

4. The Cohomology of the Groups GLn(q) Conversely, if, is contained in this hyperplane, then (v, w any hyperplane has q2n-l elements the lemma follows.

+ ,) B = 1.

Since 0

The group GLn(q) is a model for our calculations, so we study this case in detail now.

Corollary 3.7 Let SP2n(K) be the group of B, those 9 E GL 2n (K) so that

(gv, gw) B = (v, w) B for all v, w, in K2n. Then

Theorem 4.1

a. Suppose that p divides q -1. Then 8ylp(GLn(q)) is the p-Sylow subgroup of 'Zjpr 25n C GLn(q) where q - 1 = pr() with ((),p)

=

1 and r 2: 1.

232

4. The Cohomology of the Groups GLn(q)

Chapter VII. Finite Groups of Lie Type

233

permutation matrices and Z/(q - 1) embeds as the diagonal matrices. Now the power of p which divides the order of this group is

b. If plq then Sylp(GLn(q)) = UTn(q)

is the subgroup of upper triangular matrices with ones along the diagonal in GLn(q). c. If (q, p) = 1 but p does not divide q - 1 then there is a unique non-negative integer s, s :s: p - 1 so that qS =:: 1 mod (p) and

pn1'+([n/pJ+[n/p2J ... J) but from the lemma this is the p-order of

qn(n-l)/2(qn _ l)qn-l - 1) ... (q - 1) .

o

(b) and (c) are similar.

Moreover, if qS - 1 = p1' () then Corollary 4.3 Let q - 1

Sylp(GLn(q)) C (Zlp1'

XT

= p1'w,

l'

~

1 if p is odd or l' ~ 2 if P = 2. Then

(Zls)) (S[:;]

where ZI s acts on Zlp1' by T(x) = x q for T a generator of ZI s. Proof. We need a lemma.

where ai is the coefficient of pi in the p-adic expansion of ni n = ao + alP +

Lemma 4.2 Let q - 1 = p1'wJ l' ~ 1 with (w,p) = 1 and suppose either p is odd or p = 2 and l' ~ 2. Then qj -1 = p1'+sw' where (w',p) = 1 and j = pSv

... + aipi + ....

with (v,p) = 1.

The cohomology of Zlp1' (Zip 2·· . Zip is detected by groups of the fOfm (Zlp1')S x (Zlp)1 as we see from the discussion in Chapter IV. On the other hand we have

Proof of 4.2 Write (qJl"v - 1) = (qP" - 1)(1 + qp8 + .. ' + qP"Cv-l»). Since q =:: 1 mod (p) the second term i~ congruent to v mod (p) and so is prime. to p. Thus we need only analyze (qP" - 1), and this, in turn, reduces to studymg qP - 1. Write q = 1 + p1' A

Theorem 4.4 Let E TIi(Z/~i) C GLn(q) with q - 1 = p1'() and each ji r. Then E is conjugate to a subgroup of the diagonal (ZI (q - l))n. In

:s:

particular, H* (G Ln (q); IF p) is detected by restriction to the diagonal subgroup (Z/p1')n.

with (A,p) = 1. Then 1 + p . p1' A

Proof. Since p1'lq - 1, a primitive p1'th root of unity is present in IFq so each element x E E is conjugate to a diagonal matrix. Next, since E is commutative, it follows that we can simultaneously diagonalize all the elements of E, and the result follows. 0

+ (~) p21' A 2 + ... + p1'P AP

1 + p1'+l A

+ p21'+l B 1 + p1'+l(A + p1' B) provided p is odd. If p = 2 then 1 + 21'+1 A

(i) =

+ 221' A2

1 + 21'+1(A

=

and if l' = 1 this is 1 + 21' +2 W while The lemma follows.

Corollary 4.5 Let q - 1 = p() with p an odd prime and () prime to p, then 1 and

l'

+ 21'-1 A2)

> 1 implies that (A + 21'-1 A2) is odd. 0

We now prove (a) of the theorem. Note that Z/p·r

c

Z/q - 1

= GLl(q)

so (ZlpTt c GLn(q) as a subgroup of the diagonal matrices. The normalizer of this group in GLn(q) is Z/(q - 1) I Sn where Sn is embedded as the

Proof Note first that the subgroup of the diagonal (Zlp)n C Syl2(GLn(q)) is normal in U Z/(q -1) 2Sn , and, indeed, it is the only elementary subgroup of this order in this wreath product. (This depends on the fact that p is odd.) Consequently (Z/p)n c (ZIp) 2 Sn C GLn(q) is a closed system. From our discussion of the cohomology of wreath products with Sn it follows that the restriction map from H*(Zj(q - 1) l 5 n ; IFp) to H*((Zjp)n; Fp) is surjective onto the ring of Sn invariants. The Cardenas-Kuhn theorem now gives the r~uli.

0

234

Chapter VII. Finite Croups of Lie Type

O~ (q)

for q Odd

Remark. The condition that p2 not divide q - 1 in (4.5) is, in fact, satisfied in the major application of this calculation in Quillen's original proof of the

5. The Cohomology of the Groups

Adams conjecture [Q6]. However, it is clearly desirable to have the general

Lemma 5.1 For the orthogonal groups O~n(q) and 02n+l(q) with q odd a copy of Syl2( G) is always contained in the normalizer of a maximal torus. These tori normalizers are given as follows. 1. For q == 1 mod (4) the maximal tori normalizers are

result. In fact we have the following theorem of Quillen,

Theorem 4.6 Let p be an odd prime and suppose q - 1

= pT() with r 2:: 1

ot (q) l Sm

and () prime to p. Then the restriction map

0;-

S; otm(q)

x (ot l Sm-I) S; O;-m(q)

Z/2x (OtlSm) S; 02m+l(q).

is an isomorphism onto the entire ring of invariants. (The proof of this result does not appear to be a formal consequence of the Cardenas-Kuhn theorem when r is greater than 1. Quillen's original method involved the use of the Adams operations in K-theory and is described in considerable detail in [FP]. However, it is also possible, using 4.4, III.4.3, and the results on Hopf algebras of Vl.2, to give another proof, starting with GL I (q), and using the Hopf algebra structure of the disjoint union of the H*(GLn(q); lFp ).)

2. For q == 3 mod (4) and m odd the maximal tori normalizers are ot(q) x (O;-(q) 2Sm-l) S; otm(q) 0;- l Sm S; O;-m(q)· 3. For q == 3 mod (4) and m even the maximal tori n07'malizers are

O;-(q)/Sm S; otm(q) ot(q) x (O;-(q) l Sm-l) S; O;-m(q)· 4. For q == 3 mod (4) the maximal tori normalizers are 0;- / Sm x 7l/2 S;

Remark 4.7 The situation where p does not divide q-l is not much different. Here the action of (71/ s) described in (4.1) gives that the invariants are in

02m+l(q). (This is a direct counting argument.) Corollary 5.2 The groups H*(0~m(q);lF2) and H*(02m+l(q);lF2) are de-

tected by restriction to elementary 2-subgroups for q odd. But

Proof. From 2.13 and 5.1 we see that so the rings are formally the same as those before and, except for the evident change in dimensions of generators the arguments go as before. Thus we have

Corollary 4.8 Suppose p does not divide q and qS -1 =pT() as in (4.1{c)).

Then H*(GLn(q);lFp) ~H*((Z/pT)[n/sl;lFp)(z/S)IS[n/sJ and this invariant algebra is the tensor product of a polynomial algebra on generators in dimensions 2si and an exterior algebra on generators in dimensions 2si - 1, 1 ~ i ~ [n/s]. (Of course, we have only given a proof in case p2 does not divide qS - 1, but (4.6) gives the proof in the general case.)

SyI2(0~m(q)) ~

II D 2 l ~/2l71/21 ... l71/~ j

I

l

where the I run over the powers of 2 appearing in the dyadic expansion of m or such a group times (Z/2)2. Similarly, Syl2(02m+l(q)) ~ 7l/2 X SyI2(0~m(q)). But we have already seen in Chapter IV that the cohomology of such groups is detected by restriction to elementary 2-subgroups. 0 Now we want to understand the conjugacy classes of elementary 2subgroups in O~m(q) and 02m+l(q). Consider an elementary 2-subgroup (71/2Y = (tl,"" t T) contained in one of these orthogonal groups. For ease of notation we assume for the moment that we are dealing with one of the groups O~m(q). Since q is odd we can choose an orthogonal basis el,"" e2m so that the ti become diagonal with only ±I's along the diagonal. Suppose t 1 (el) = -el' Then we can choose new basis elements for (71/2)n,

Chapter VII. Finite Groups of Lie Type

236

tl, t~, . ..

,t~, so that t~(el) = el, 2 :::; i :::; n. Then V = (e2,"" e2m) is taken to itself by (Z/2)n. Write V = V+ EB V- where x E V is contained in V± if and only if h(x) = ±x. Then (t;, ... , t~)(V±) = V± and if we define T by el if x = el, T(X) = X ifxEV+, { -x if x E V-, T E O~m(q) and T commutes with (Z/2)n. Consequently either T E (Z/2)n or we can adjoin it obtaining an extension (Z/2)n+l C O~n(q). By

then

repeating this construction we obtain Lemma 5.3 Let G = (Z/2)n C O~m(q) or 02m+l(q), then G is contained 'in a maximal subgro'U,p (Z/2)2m or (Z/2)2m+l. Such a group splits the form uniquely as a sum of 1-forms (I)S ~(,B)2m+E-s and s is a complete characterization of the conjugacy class of a maximal 2-elementary subgroup of O~m (q) or 02m+l(q).

Proof. The argument above shows that (Z/2)n is contained in a subgroup (Z/2)2m+c = (t 1 , ... , t2m+c) and there is an orthogonal basis el, ... ,e2m+c for the vector space so that ti(ej) = ej if j I- i, and ti(ei) = -ei' Since the ( -1) eigenspace for t'i is orthogonal to the (+ 1) eigenspace it follows that ei is not an isotropy vector for the form and so the ei give an orthogonal splitting of the form into a sum of I-forms. On the other hand it is direct to see that among the 22m +c elements of this group exactly 2m + t of them have I-dimensional (-1) eigenspaces. Hence, this splitting is intrinsic to the group and the result follows. 0

5. The Cohomology of the Groups O~(q) for q Odd

237

Example 5.5 Let ,B be a non-square in lFq and suppose q == 1 mod (4). Then the two subgroups (Z/2)2 C ot(q) described above are

For I I the eigenvector splitting is (el - e2, el + e2) which corresponds to the splitting of the quadratic form as (-2)~(2). For I the eigenvector splitting is (,Bel + e2, ,Bel - el) which gives the form (2,B)~( -2,B). Corollary 5.6 The normalizer of the maximal elementary 2-subgroup of O~m(q) or 02m+l(q) with. splitting space (1)8 ~(,8)2m+c-8 is

(Z/2 2 Ss) x (Z/2 2 S2m+c-s) so the Weyl group is S8

X

S2m+c-8'

Corollary 5.7 The restriction map taking H* (O~m (q); IF 2) to

H*(O~(q) x

(ot(q)):=>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<-,?

is onto those elements which evaluate equally on monomials of the form

and eil

® ... 0

ei 21 _2

®

L

W2p-l ® W2q'

p+q=k This notation is meant to indicate that the restriction of the element to each of the conjugacy classes of elementary 2-groups has the indicated form. These elements satisfy the constraint conditions and consequently are in the image of H*(Ois(q); 1F2) or H*(02s+1(q); IF2)' Indeed one has the following theorem.

(1 + x)(l + x 2 ) •.. (1 + xm - 1 ) (1-x)(1-x 2).··(1-xm ) . Further details, including the relations between the generators are explained in [FP]; pp. 264-277.

X

----------~--~~----~+----~~--~ 2

5.12

W2p 0 W2q-1,

Theorem 5.13 H*(02n+l(q); lF 2 ) is the subring of 6.8 generated by the elements Xk, Xk. Similar generation results hold in the remaining cases. In all cases the Poincare series of H*(0in(q);1F2) is

x

1

L

p+q=k

Now we describe H*(03(q); 1F2)' There are two conjugacy classes of (71/2)3 in o (3), the first associated to the form (1) 1- (1) 1- (1) and the second to the form (,8)1-(,8)1-(1). The first contributes a polynomial algebra H*((71/2)3;1F 2)S3 = IF 2 [0"1, 0"2, 0"3] and the second contributes IF 2[0"1, 0"2]0"1 0IF 2(0"1]. This can be seen since the terms in homology are all of the form fdj ® es with 0 :::; i < j. Consequently its Poincare series is

(1 - x)(l - x 2)(1 - x 3) (1 + x)(l + x 2)

0 W2(k-s),

ei 21 _ 1

®

ei21_1

® ....

Examples of such elements are easily constructed. Note that

H*((Z/2)m)Sr XS rn-r = lF 2 [W1, .. ·, wr ] ® lF 2 [W1,.··, w m- r ] where Wi is the ith symmetric monomial in the first r coordinates and Wj is the /h symmetric monomial in the last m - T coordinates. Then set

6. The Groups H*(SP2n(Q);IF2) The symplectic groups differ from the orthogonal groups we have been considering in one important respect. It turns out that H*(SP2n(q); 1F2 ) is not detected on elementary 2-subgroups. But there is an important similarity. Using the inclusion Hn c Hn 1-H we obtain inclusions SP2n(q) C SP2n+2 (q), and passing to limits as in the orthogonal case we obtain a group Sp(q) = lim __ (Sp2n(q)) together with a bicommutative, biassociative, Hopf algebra structure on H* (Sp(q); IF 2 ). The Hopf algebra structure here forces the structure of the groups H*(Sp(q); 1F 2), and by restriction, the groups H* (SP2n (q); IF 2) as well. We have already seen that Sp2(q) = SL 2 (q) and we have for all odd q the result H*(SL2(q); 1F2 ) = E(e3) ® lF 2 [b 4 ], [Q3]. SP2n(q) contains the subgroup Sp2(q) 2 Sn = SL 2 (q} 2 Sn and this subgroup always contains a Sylow 2subgroup for q odd. It follows that the detecting groups for H*(SP2'n(q); IF 2 ) have the form SL 2(qr x (71/2)S with r+s :::; n. Moreover, following the lines of argument in §5 we have that each of these groups is conjugate to a subgroup of SL2(q)n C SL 2(q) 2 Sn- In particular

Lemma 6.1 The two inclusions

'-""'''' ...... p''''-'..1.

,..L ........... . 1. ....................

and SP2n(1)

X

_ .........

'--.1._

Z/2~SP2n(q)2

We now turn to the proof of the main theorem C

SP2n(q) (Z/2 C SP4n(q)

are conjugate in SP4n(q). Here f.L(B, T€) = (B, B7€) where element in the center of SP2n (q).

7

is the non-trivial

Proof. The ±1 eigenspaces of T give a splitting of lF4n into lF 2n ~lF2n and the action of Ll(SP2n(q)) is diagonal on these two subspaces. On the other hand T acts as +1 on the first summand and -1 on the second. In view of the fact that SP2n(q) lZ/2 is exactly the stabilizer of such an orthogonal splitting, the result follows. 0 Lemma 6.2 The restriction map

with dim(ei)

=

3, dim(bi ) = 4, 1

~

i ~ n is injective in cohomology.

Dually, this gives

Theorem 6.5

H*(Sp(q)i lF 2 )

=

lF 2 [b 4,.·., b4n ,·· .]0 E(e3,"" e4n-l,"') ,

and H*( SP2 m(q);lF 2) injects into H*(Sp(q);lF 2) as the suspace spanned by all monomials b~l ... bile;l ... e~~_l with 2: i j + 2: Ej ~ m. Proof. The main step in the proof of (6.5) is the following preliminary result. Lemma 6.6 a. There is a unique conjugacy class of maximal elementary subgroups (Z/2)n C SP2n(q) for each n and any elementary 2-group (Zj2)S in SP2n (q) is contained in a subgroup (Zj2)n. b. (Z/2)n x (Z/2)n C SP2n(q) (71/2 C SP4n(q) is a closed syste~ for n > 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
II


W ----54

where 54 C GL3(2) is the parabolic subgroup of all matrices in G L 3(2) of the

form

(~ ~) H2

with A E GL2(2). We write Hz for the group

(Z/4)2 XT Z/2 {a,b,y!a 4 =b 4 =y2=1, ab=ba, ay=ya-l, by=by-l}

and H for the extension H2

XT

71,/2

= Syl2(G 2 (5)).

Remark. It is also shown in [FM) that when we include G(q) C G(q2) then both the image of IC(q) and the image of IIc(q) are conjugate to IC(q2). Remark. W also occurs as the normalizer of a (71,/2)3 in the Mathieu group M 1 2. It is discussed in Chapter VIII.

248

7. The Exceptional Chevalley Groups

Chapter VII. Finite Groups of Lie Type

The data above gives a complete determination of the poset space for G in [FM]. The quotient of the poset space by the action of G is given below with the isotropy groups of the faces indicated.

249

Theorem 7.3 1. H*(Sy12(G)) is detected by restriction to its maximal 2-elementary sub-

groups (7L/2)3. . The Poincare series for 8yI2(G) is independent of G and given as (1!x)3' 3. H*(SyI2(G)) ~ IF2[Cl,T4,T2](1,h,,2,1-"3)EBIF2[cl,T4,el](e!,elI2) with the subscripts denoting the dimensions of the generators. The keymultiplicative relation is eT2 = O.

2.

Since E contains representatives of both conjugacy classes of (Z/2)3 res

in G it follows that H* (G) '----+ H* (E) is injective. Moreover, applying the Cardenas-Kuhn theorem to the closed system I
IF 2 [d 4, d6 , d7 ]. Here the labels in smaller type are the isotropy groups of the edges. Also, Nl = (7L/(q - 1) x 7L/(q - 1)) XT (7L/2) since q == 1 mod (4). There are two useful cancellation rules that can be used to simplify the diagram without changing the homology type of the associated classifying space. If a "free" edge, i.e. an edge incident on only one face, and the corresponding face have the same isotropy group then we can remove both edge and face. Likewise, if a free vertex, i.e. a vertex incident on only one edge, and the corresponding edge have the same isotropy group we can again remove both the edge and the vertex. By iterating these reductions the homotopy type of the resulting classifying space corresponds to the diagram

Thus we have the following theorem which reduces the determination of H* (G; IF 2) to a much simpler problem. Theorem 7.2 The intersection of E and 8L2(q) * 8L 2 (qV) isomorphic to W = N G (11). AIoreover the homomorphism

XT

(n) in G is

is surjective and induces isomorphisms in cohomology with (7L/2) coefficients. Here A *B C is the free product of A and C, amalgamated over the common subgroup B. These groups are studied as follows. First 8y12( G) S;;; D2 n ( Z/2 as an index(2) subgroup, and from the Gysin sequence for this pair together with the fact that H*(D2n ( Z/2) is detected on elementary 2-groups it follows that H* (8y12 (G)) is also detected on elementary 2-groups and its cohomology can be explicitly determined. In summary one has the results of Fong and Milgram, [FM] , (M2],

But the system I
(1 + x 3 )(1 + x 5 )(1 + x 6 ) (1 - x4)(1 - x 6 )(1 - x 7 )

•

250

Chapter V11. Finite Groups or LIe .type

2. H* (G) is free (and finitely generated) over a subpolynomial ring

C,hapter VIII. COhOlllOlogy of Sporadic

Simple Groups

IF 2 [d 4 , d6 , d7 ] and if q' = qW then G(q) C G(q') and the resulting restriction map takes the subpolynomial algebra for G(q') isomorphically to the subpolynomial algebra for G(q). 3. As a ring H*(G) is generated by d4, d6 = Sq2(d 4), d7 = Sq1(d 6 ), a generator f3 in dimension 3, and f = Sq2(f3) in dimension 5. The relations are f34 = d7 f + d6 f3 2, f2 = d4f32 + d7f3, both of which are implied by the formula Sq4(f) = d4 f + d 6 f3.

Remark. The Poincare series in 7.5.1 is exactly the same as the Poincare series of the sporadic group J 1 studied in II.6.9 and IIL1.9. We will try to explain the relation between J 1 and G 2 (q) in Chapter IX where we use the plus construction. The group E studied above will playa crucial role in this analysis as the next remark will help to explain. Remark 7.6 The group E also appears as a maximal finite subgroup of the topological Chevellay group G 2 of dimension 14 given as the automorphism group of the Cayley plane. G 2 is a subgroup of the orthogonal group 0(7). This embedding of E in was given by H.S.M. Coxeter, [Cox], and a particular representation is given in [CW], pg. 448, as the span of the permutation matrices (1,2,3,4,5,6,7), (1,2,4)(2,6,5), and the matrix 15(1,2,4,7)(1,2)(3,6) where b(a,b,c,d) is the diagonal matrix in 0(7). In fact, under this embedding I turns out to be a maximal torus in G 2 and E is its normalizer. Thus, by a general theorem of Borel, the inclusion induces an isomorphism H* (BG 2 i F2)---tH* (I; F2)GL 3(2) . An independent verification of this isomorphism is given in [M2] by using the explicit embedding to evaluate the composition

H* (B O (7); F2 )---tH* (BG 2 ; F2 )---tH* (I; F2)GL 3(2) on the Stiefel-Whitney classes (which generate H*(Bo(7);F2)' [MS]).

Remark. It is not hard to see that H*(G 2(F p oo); F 2 ) = lim (H*(G 2(pn); F 2 ) +---

and that 7.4, 7.5 show that this limit is exactly F 2 [d 4 , d6 , d7 ] which is, by 7.6 also equal to H*(G 2 ;JF 2 ). This result, due originally to Grothendieck is an aspect of his theory of Etale cohomology. A discussion of this remarkable theory lies outside the scope of this book however. We refer the reader to the work of Kleinerman [Kl] for more on this topic.

O. Introduction In this chapter we will describe recent progress towards understanding the cohomology of the sporadic simple groups. Briefly we recall that from the classification of finite simple groups, [Gor], it was shown that there exist 2'6 simple groups not belonging to infinite families (i.e. not of alternating or Lie type) and we study six of these groups here: four of the five Mathieu groups, the first Janko group, J 1 , and the O'Nan group 0' N. Here are some of the reasons, (aside from pure curiosity), for understanding the cohomology of these groups. As we will see in Chapter IX, we can add two and three dimensional cells to the classifying space of a perfect group to obtain a simply connected topological space, Ei], with the same homology as BG , and the homotopy groups of these spaces are of basic importance in homotopy theory. The symmetric groups build up Q(SO), a process discussed in the introduction to Chapter VI and proved in IX.3. The general linear groups over a finite field similarly build the classifying spaces Eo and Eu as well as certain fibers of "Adams operations", pk - I, known as Im(J)-spaces (IX.3 again). Indeed, in IX.3.2 we point out that these lead to a product splitting Q(SO) = Im(J) x Coker(J) with the Im(J) space completely understood. It is natural to expect that the sporadic groups should playa role in the structure of the C oker( J) space, though we are only beginning to understand some of the smaller sporadic groups in this framework. The group Mll which has 2-rank two has been studied by F. Cohen and the three 2-rank three sporadic groups M 12 , hand 0' N are currently being analyzed from this point of view. We only have partial information about 0' N, but both hand M12 are closely tied to the exceptional Lie group G 2 , facts which are still slightly mysterious, but which we will explain in Chapter IX. Among the rank four sporadic groups we have determined the cohomology of M22 and M 23 and the cohomology rings are discussed in §5. Neither one is Cohen-Macaulay so the rings are quite complex, but M 23 turns out to be the first example of a finite group for which Hi(G; Z) = 0 for i ~ 4. Indeed, it had been conjectured by C.Giffen [Gi] that the only finite group for which this is possible is the trivial group. Also, the Coker(J) space is 5-connected with H6(Coker(J); Z) = Z/2, and the natural inclusions M 23 C 5 23 C 5 00 induce an isomorphism of

252

2. The Cohomology of h

Chapter VIII. Cohomology of Sporadic Simple Groups

H 6(M23 ; Z) with the Z/2 which generates H6(Coker(J); Z) in this dimension. Again, we expand on these remarks in §5 and Chapter IX. Likewise, from the point of view of modular representations, the sporadic groups are key examples and it is apparent that cohomological invariants play a fundamental part in recent and ongoing research in this field. As was mentioned in our introduction to this book the ideas here are discussed in the books of Benson and Evens. What we shall concentrate on is furnishing detailed calculations of the cohomology of a few of these groups. We will apply all the different techniques described in the previous chapters, hoping perhaps that this will serve as an additional justification and description of our methods. In particular, our calculation of H*(Mll; W2 ) in §1 completely replaces the much more complex original calculation by Benson and Carlson. Also, our calculation of H*(M12' W2 ), where M12 is the Mathieu group of order 95,040, replaces the less natural and longer original calculation in [AMM2]. We use W2 coefficients throughout, so they are often suppressed.

and so the Poincare series for Mu is (as first computed in [We], compare [BC3])

(1 - t 3 )(1 - t 4 ) The group GL2(W3) contains a Sylow 2-subgroup of Mu and has its mod(2) ~ohomology detected on its elementa~y 2-subgroups. Consequently the same IS true for Mu. More restrictively it follows from VII.4.4 that

H*(Ml1; lF 2 ) ~ W2 [Xl, X2]GL 2 (F 2 ) = W2[d 2, d3] , ~the Dickson algebra described in II1.2.3) and since there is only one element m each 20f the dimensions 3, 4, and 5 in this ring we see that H*(Mu) ~ W2[d 3, d 2 ](1, d2 d3)' Hence we have

Theorem 1.2

H*(Mu) ~·lF2[V3, V4, vsl/v~v4 + vg = 0 with Poincare series p(t)

1. The Cohomology of

253

= U- t\tS-t4 )'

MIl

2 Let G = M ll , the first Mathieu group having order 7920 = 24 3 5 11. Mll has 2-rank two with one conjugacy class of groups (Z/2)2 and one conjugacy class of involutions. From the Atlas, [Co], N(2A) = 2.174 = GL2(W3), (that is to say, the ~ormalizer of an involution is the non-split extension of the form

Remark 1.3 From the action of the Steenrod Algebra on the Dickson algebra, we have the following table giving the action of the Steenrod algebra on H*(Mll)' gen.\Sq

Sql

Sq2

Sq3

V3

0

Vs

v 32

V4

0

V~

0

Vs

V5

0

0

Z/2 <J N(2A)-tS4 isomorphic to GL 3(W3)), and we can also check that N(('1.,/2)2) = 174. Thus the quotient IA 2 (NI ll l/ Mll has the form

174 •

•

GL 2 (W 3 )

Sq4

v 42 V~

+ V4VS

D8

and we can apply V.3.3 to obtain the formula

2. The Cohomology of J I We already know from IV.2.7 that H* (D8; lF 2) which has Poincare series 2 1 PDs(t) = (1 _ x)(l - x 2 ) - 1 - x 2

0) 1

(1-x)2'

. . ' . ,. . 1±t±t2 ±t 3 Simllarly, usmg VI.1.13, the Pomcare senes for 174 lS (1-t2)(1-t3)' From VII.4.5 and 111.4.2 the Poincare series for H*(GL2(lF3)) is

(1 + t) (1 + t 3 ) (1 - t 2 )(1 - t 4 )

1 +t

+ t2 + t 3 + t 4 + t 5

(1 - t 3 )(1 - t 4 )

J 1 is the first Janko group, of order 175,560. It has a 2-Sylow subgroup ~somorphic .to ('1.,/2)3. Using this, we have already calculated its cohomology

m II.6.9 usmg the invariant ring determined in III.1.9. For further remarks on H*(J 1 ) using a poset decomposition see V.2.11 and V.3. For convenience, we recall the result from 11.6.9, Theorem 2.1

,2 +

H*(J 1 ) ~ lF 2 [X3, Y4, z71bsl f.16)/ f.12

Yf.1

+ xz =

0

+ x4 + x 2J.L + y3 + ,Z =

O.

254

3. The CoholI1ology of .11112

Chapter VIII. Cohomology of Sporadic Simple Groups

with Poincare series

t = (1 +t )(1 +t ) p() (1- t 3 )(1- t 4 )(1 - t 7 ) 5

6

_

(1 +x )(1 +x )(1 +X

-

(1 - x 4 )(1 - x 6 )(1 - x 7 )

3

5

6

(1,2,3)( 4,5,6)(7,6,9)

a

(2,4,3,7) (5,6,9,8)

b

(2,5,3,9)(4,8,7,6)

x

(1,10)(4,5)(6,8)(7,9)

Y

(1,11)(4,6)(5,9)(7,8)

) .

The action of A(2) can be computed directly from the explicit form of the generators given in II.l.9, particularly x given in (II.l.I0). We make the following remarks anticipating our discussion of the relation between J 1 and the exceptional group G 2 which we will give in Chapter IX. First H* (J1 ; IF 2) is Cohen-Macaulay over the Dickson algebra lF2[d 4 , d6 , d7 ] and can be rewritten in the form

In particular the quotient algebra by the ideal generated by the Dickson elements is H*(J1 ;lF 2 )/(d4,d6 ,d7 ) ~ lF 2[x,Sq2(x)l/(X 4

u

= (Sq2(X))2 = 0,x 2 = Sq1(Sq2(X)))

and this is exactly H* (G 2 ; IF 2), a "coincidence" which we will try to explain in Chapter IX. (We are indebted to F. Cohen for this description of H*(J1 ; lF2) and the geometric explanation which we give in Chapter IX.)

255

z

(1,12)( 4,7)(5,6)(8,9). It is 5-fold transitive and has order 26 33 5 11 = 95,040. Note that (a,b) ~ Qs while (x,y,z) ~ 54. When we conjugate a,b by x,y,z we obtain xax = b, xbx = a, yay = ba, yby = b- 1 , zaz = a-I, and zbz = ba, so Qs is normal in the subgroup W = (a,b,x,y,z) which consequently has order 26 3 = 192 and contains a 2-Sylow subgroup of M 12. In particular

where

d

=

xyz = (1,10,11,12)(4,8,7,6)

k = xyxz = (1,12)(5,9)(6,8)(10,11). Note that bd = db so (b, d) = (Z/ 4)2, and k()k = ()-1 for each () E (b, d). 2 Likewise, ka = ak, ada- 1 = b- 1 d, so setting s = ad2 we have 8 = 1, (8, k) = (Z/2)2 and we can write H as a split extension (Z/4)2 XT (71/2)2 1 where the twisting by 8 is given by sds = b- 1 d, sbs = b- . There are seven conjugacy classes of elements of order two in H, the central element a 2 = (2,3)(4,7)(5,9)(6,8), one having two elements and representative d2 = (1,11)(4,7)(6,8)(10,12), four with four elements in each conjugacy class having representatives

3. The Cohomology of

M12

In this section we study the Mathieu group M 12 . It is considerably more complex than Mll and J 1 and we begin with a detailed analysis of the structure of its elementary 2-groups. We explicitly construct subgroups of M12 with respect to which each conjugacy class of maximal elementary 2-groups is weakly closed. Then we prove that H*(Syl2(M 12 ); lF 2) is detected by restriction to its elementary 2-subgroups and from that the determination of its cohomology is fairly direct. The discussion here is a modification, based on ideas in [FMJ, [M2], of the work of [AMM2] where H*(M12; lF 2) was first determined.

The Structure of the Mathieu Group M12 The Mathieu group M12 is the subgroup of S12 generated by the following 6 elements as given for example in [Ha, pp. 79-80]:

k

=

(1,10)(4,7)(5,9)(11,12), (1,12)(2,6)(3,8)(4,5)(7,9)(10,11), (1,10)(2,5)(3,9)(4,6)(7,8)(11,12), (1,11)(2,6)(3,8)(4,9)(5,7)(10,12),

and finally one class with eight elements and representative z. The centralizers of the elements in the last 5 classes are given as follows, four copies of Ds x Z/2 for the four classes with four elements and (Z/2)3 for the class with 8 elements. In 5 12 the involutions contained in M12 lie in two conjugacy classes, the class {4} consisting of elements which are products of 4 disjoint involutions and {6} consisting of those elements which are products of six disjoint involutions, so there are at least two distinct conjugacy classes of involutions in M 12 . In fact, it is well known, [Co], that there are exactly two. From the structure of the centralizers in H it is easy to see that every elementary 2-subgroup of H is contained in a (Z/2)3 and that there are exactly 8 distinct (Z/2)3'8, three of the form 47 (by which we mean that each

256

3. The Cohomology of

Chapter VIII. Cohomology of Sporadic Simple Groups

non-identity element is in the class {4}), three of the form 43 6 4 , and finally two of the form 4 1 6 6 . These last two are conjugate in H and consequently weakly closed in H C Nh2. In the other two classes two of the groups are conjugate and the third is normal in H.

Remark 3.1 These (7I.,/2)3'S are given as follows. The normal subgroup of type 47 is VI = (b 2 , d2 , k) and the two other subgroups of this type are V2 = (b 2 , d 2 , bd- 1k), (b 2 , d 2 , d- 1k). The normal subgroup of type 4 3 6 4 is V3 = (b 2 , d2 , bk), and the two non-normal subgroups of this type are V4 = (b 2 , k, s), (b 2 , d 2 k, bs), while the two subgroups of type 41 66 are Vs = (b 2 , bd 2 k, s) and (b 2 ,bk,ks).

Set H21 = (a, b, k, dkd- 1 ) = Qs XT K where K c 54 is the Klein group. Since k and dkd- 1 act on Qs by conjugation with a, ab respectively, it follows that (ak, abdkd- 1 ) ~ Qs commutes with (a, b) and H21 is the central product, Qs * Qs ~ Ds * D s , an extra special 2-group. H21 is also the subgroup of H which is spanned by the three groups of the form 4 3 6 4 , and the element of order three, xy

(3.2)

= (1,10,11)(4,9,8)(5,6,7)

which normalizes H21 permutes the three subgroups cyclically. Thus these groups form a single conjugacy class in M12 and are weakly closed in W C M 12 . H21 also contains the normal 47 subgroup, and xy normalizes this group as well. Hence, W can be rewritten (71.,/2)3 xT 54. Finally, H21 contains both of the 4 1 6 6 subgroups, and each is normal in (H21, xV).

M12

257

and VI is weakly closed in W' C M 12 . Summarizing the discussion above we have

Theorem 3.4 a. There are precisely three conjugacy classes of groups (Z/2)3 contained in M 12 . Under the inclusion M12 C 512 they remain non-conjugate. The first, I, has the form 4 7 , the second I I, has the form 4 3 64, and the third, III, has the form 4 1 6 6 • b. There are subgroups Wand W' of M12 described above so that W n W' = H and I C W', II c Wand III C H are all weakly closed in M 12 . c. The Weyl group of I in M12 and the Weyl group of II in M12 are copies of 54' d. The Weyl group of III in M12 is A4, the Weyl group of III in W. (There isn't much to add to the discussion above to complete the proof. In the Atlas, [Co], it is noted that the centralizer of the involution of type {4} is W, and the centralizer of an involution of type {6} is Z/2 x 55 and from this the result is direct.)

For later reference we give the explicit actions of xy and T on the eight (7I.,/2)3,s in H now. The notation is that the element in the nth position in the image group is the image of the nth element in the domain group. First the normalizing actions.

The action of d2 xy on a representative group V5 is

(3.6) The subgroup, H22 C H which is spanned by the three subgroups of the form 47 also haB order 32. It is (b, d, k) so H22 ~ (71.,/4)2 xT 71.,/2 with the element of order two acting to invert the elements in (7L / 4) 2 . Consequently there are exactly four subgroups of the form (7L/2)3 C H 22 , (b 2 , d2 , k), (b 2 , d2 , bk), (b 2 , d 2 , dk) and (b 2 , d 2 , bdk). The first, third, and fourth each have the form 4 7 and the second has the from 4 3 6 4 . Also, the element (3.3)

T

= (1,4,2)(3,11,7)(5,10,6)(8,9,12)

H22 xT 53

(3.7) and the acton of T on the three subgroups of type 4 7 ,

E M12

normalizes H22 and cyclically permutes the three 47 subgroups while normalizing the 4 36 4 . Finally, the element d 2 a = b2 S E H satisfies d 2 aTd 2 a = T- 1 , so

vV' = (H, T) =

Next we give the action of xy on the three subgroups of type 4 36 4

Using the results above, and, for example, the detailed subgroup results in [AMM2] or [BR] we have the following picture of the poset space of 2elementary abelian subgroups for M 12:

Ghapter V111. GohOmolOgy or "poramc ;:)lmple Groups

with generators x(x) = 1, x(y) = D, fj(x) = D, y(y) = 1. In H* (Ds x Ds; lF2) write Xl = x01, X2 = 10x, YI = y01,Y2 = 10y, WI = w01 and W2 = 10w. Then in the Gysin sequence for the inclusion H22 C Ds X Ds we have that the map UX:H*(Ds x D s ;IF 2)-+H*+I(Ds x DSi lF 2) is just U(XI +XI +YI +Y2)' We have

W

and this can be rewritten as lF 2[x, Xl

+ YI, Xl, X2, WI, W2)/(X X2 = X~ + X2(XI + YI), XI(XI + YI) = xD·

In particular, multiplication by X is injective, and from this the result is direct 0 where CI is the image of Xl + YI.

There are 9 vertices,'17 edges and 9 triangles in this orbit complex. We have only shown the isotropy information for the vertices and a few edges. The full details are given in [AMM2, p. 1D6]. In Webb's formula, V.3.3, most of the groups cancel out and we are left with only, (3.1D)

H*(MI2) EB H*(H)

~

H*(W) EB H*(W').

Additionally, a close analysis of the structure of the finite Chevellay groups G2(q) sho:,s that for q == 3,5 mod (8), Syl2(G2(q)) ~ SyI2(MI2), and the configuratIOn W UH W' is also contained in G2(q). See, e.g. [M2]. This completes our discussion of the subgroup structure of M 12 . We now turn to the cohomology ring. The Cohomology of

M12

Write DB = {x, y I x2 = y2 = (xy)4 = I}. Then H22 c Ds X Ds with embedding given by b f---* ((xy)-l,xy), d f---* (l,xy), and y f---* (x,x),' and this extends to a map 1r: H -+Ds 271/2 by 1r(s) = E, the element which exchanges the two copies of Ds. Theorem 3.11 H*(H22; IF 2 ) is detected by restriction to its four maximal

elementary (71/2)3 subgroups. In particular H*(H22; IF 2) = IF 2 [WI, W2, c](l, Xl, X2, XIX2) with relations xi = Xl C, X§ = X2C where the w's are two dimensional and the remaining generators are one dimensional. Proof. Recall from IV.2.7 or IV.1.1D that H*(Ds; IF2) = IF 2[x, fj, W2]/(XY = D), where HI(Ds; F 2 ) = Hom(D s , IFt)

In IV.7.3 a special class, T(x) E H 2i (G (71/2; IF2) is construct~d for each X E Hi(G; IF2)' Moreover, these classes, their cup products with e~ where e E H I(GI71/2; lF 2) is dual to E, and classes of the form tr(y), y E H*(G x G; lF 2) generate H* (G 2 7L/2; IF 2)' Also, in the Gysin sequence for the inclusion 7r: H

'---t

Ds ( 7L /2

the map H* (Ds27L/2; IF 2) ~ H*+l (Ds27L/2; lF 2) is multiplication by tr(xi +YI) which restricts to H*(Ds x Ds; IF 2) as Xl + X2 + Yl + Y2· We can now state Corollary 3.12 a. H*(Hi lF2) is detected by restriction to the cohomology of its 5 conjugacy

classes of maximal elementary two groups. b. Up to extensions, H* (H i IF 2) = IF 2[c, 7f*tr( WI) ,7f* T( w)] (1, 7f*tr(xI), 7f* T(x), 7f*tr( Xl W2)) EB e I IF 2 (el, c, 7f* T(w))(I,7f* T(x)) where el is one dimensional and dual to E. Proof. We begin by consdering the Gysin sequence for the inclusion (*). The kernel of Ux is (e) = eIF 2(T(x), T(Y), T(w), e)/(T(x)T(fj)), but note that the square of any element in (e) is again a non-zero element of (e). Now, the transfer, tr: H*(H; lF2)-+H*(Ds 2 7L/2; lF2)' while it does not commute with cup products, does commute with squares (since they are stable cohomology operations), so there are no possible nilpotents in the cokernel of the restriction map H* (Ds 2 7L/2; IF 2 ) -+H* (H; IF2)' Next we need H*(Ds 2 7L/2)/(UX)· We have an exact sequence D-----+(e)-----+H*(Dsl7L/2;lF2)-----+H*(Ds x Ds;IF 2 Yl/2-----+D.

260

3. The Cohomology of

Chapter VIII. Cohomology of Sporadic Simple Groups

When we cup this sequence with X we obtain an exact sequence

O-r(e)-rH* (Ds l71/2; F2)/(X) -rH*(Ds x DS)71/2/(XH*(Ds

X

DS)71/2)-rO,

and since the right hand quotient is easily seen to have no nilpotent elements

(3.14)

the first statement follows. 1r We now use the spectral sequence of the extension H22
But each of the generators above, except c is in the image of 1r* and hence is an infinite cycle, while c is also an infinite cycle, since HI (H ; F 2 ) = H om(H, lFt) = (71.,/2)3, and the spectral sequence collapses. This proves the

From 3.12 it follows that the restriction images of all the generators to the five conjugacy classes of (71/2)3's are explicit. They can be determined as follows. From IV.7.1 and IV.7.2 we obtain the restrictions of r(e) to the basic subgroups K x K, S x K, K x S, S x S, (J1(K),E) and (J1(S),E), and the image of tr is zero when restricted to these last two groups. But also, under 1r we obtain the following inclusions where A = (xy)2:

(3.13)

VI = (b 2 ,d2 ,k) V2 = (b 2, d2, bd- I k) V3 = (b 2 ,d2 ,bk) V4 = (b 2 , k, s) V5 = (b 2 , bd 2k, s)

(3.15)

0

.

f-+

f-+

.

(AI + A 2, A 2, Xl + X2) c K x K (AI + A 2, A 2, YI + X2; c S x K (AI + A 2, A 2, YI + Y2 + A 2) c S x S (AI + A 2, Xl + X2, E; = (J1(K), E) (AI +A 2,YI +Y2,E) = (J1(S),E).

2

For definiteness, assume that W is given so that res* (w) = a + ah in both H*(K) and H*(S) where a is dual to A and h is dual to X in K, Y in S. 2 In the each of the groupS Vi. write ,\ as the dual to b , 7 is 2dual to the second generator, and h is dual to the third. Also, write n = 7 + 7h, v = ,\4 + ,\2(72 + h2 + 7h) + '\(7 2h + 7h 2). Then we have the following table for the restrictions to the five (Z/2)3,s in 3.13.

gen.\group

VI

V2

V3

V4

V5

h h

h

7

T

0

0

0

C

h

1r*tr(XI)

0

e 1r*r(x)

0

0

0

h2

0

0

h 7 2 +7h

0

1r*tr(WI)

n

n

n

0

0

1r*tr(XI W2)

7 2h + Th 2

,\2h + '\h 2

0

0

0

1r*r(w)

V

V

V

V

V

h

From ~his the detailed structure of H* (H; IF 2 ) can be easily obtained. However, we now have a simple criterion for determining the cohomology of M l2 , W, a~d ~' directly from the table above by using the actions of T and (xy) detaIled,Ill 3.5-3.8, which lead to the following maps in cohomology:

E2 = F2[C,WI +W2,WIW2](1,XI +X2,XIX2,XIW2+ X2WI) EEl eF 2[e, c, WIW2](1, XIX2)'

second statement.

261

Nf12

Map

On Elements

(xy)*: H*(Vt)-tH*(Vt) T*: H* (V2)-tH* (VI) T*: H*(V3)-tH*(V3) (xy)*: H*(V4)-tH*(V3) v'5)-tH* (TT5) v' (d2xy)*: H* (TT

h f-+ h + 7, T f-+ h,'\ f-+ ,\ h f-+ h, T f-+ A + T, A f-+ T h f--+ h, T f--+ h + T + A, A f--+ T h f--+ h, T f--+ T + h, A f--+ A + h h f--+T,Tf--+h+T,AHA+T

Now the stability conditions for elements to be in H*(W) H*(W') and H*(MI2) are easily written down. ' Theorem 3.16 a. E H*(H; F 2 ) is contained in the 'image ofres*: H*(W' IF )-tH*(H' IF )

c:

if and only if:es*(a) E H*(Vl)71/3,and also in H*(VS)Z/3,2 while the 'm~p above from H (V4 ) to H*(V3) stabilizes res*(a). b. a E res*(H*(W'; lF 2 )) if and only if res*(a) E H*(V3)z/3 and the map from H* (V2) to H* (VI) stabilizes a. c. a E. res* (l!* (M12 )) if and only if the conditions in both (a.) and (b.) are satzsjied, z. e., if and only if a E H* (W) n H* (W') c H* (H).

Example 3.17 1r* r(w)

+ c4 + e 4 + (ce)2 + 1r*tr(wI)2

restricts to the Dickson element d4 at each of the Vi. From this and the fact that d = S 2(d ) d = Sql(d 6 ) it follows that H*(M12; lF 2 ) contains a copy of IF 6[d d qd 1 4r ' f 7 t't t h * 2 4, 6, 7' n ac I urn~ out t at H (MI2 ; IF 2 ) is actually Cohen-Macaulay, that is to say, free and fimtely generated, over this subalgebra.

Example 3.18 V5 is weakly closed in H C M12 and , from 3 .14, the'Image 0 f

res*: H* (H; 1F2)---+H* (Vs; 1F2 )

262

263

Chapter VIII. Cohomology of Sporadic Simple Groups

is lF 2 [h, r, V4] inH*(Vs). From 3.15 the action of (d 2xy)* fixes V4 and acts on lF 2[h,r] in the same way 'lL/3 acts in III.1.3. Consequently, H*(Vs)71/3 = 1F2[h2 h, ,2, h2, h,2, d4](1, h 3 h2, + ,3) is the image of restriction from H*(M12; 1F2)' Thus, besides the copy of the Dickson algebra there is one two dimensional generator a and there are two three dimensional generators, Sql(a) and l3 in H*(MI2; lF2)' They are constructed as follows: a = (rr*tr(xl)2 +71"* r(x) + e2 +c2 + ec which restricts to (0,0, h 2, h 2, h 2 +rh+ r2) and l3 = c3 + e3 + 7l'*tr(xl)3 + (c + e)7I"* r(x) + e2 c which restricts to (0,0, h 3, h 3, h 3 + h 2r + r 3).

+ +

+

+

Poincare series in 3.20. This was a critical step in [AMM2], but here, using the weak closure conditions, it only serves the role of assuring us that we have made no numerical errors.

Given a situation such as that of H, W, and WI, we can find a universal completion = W *H WI which makes the diagram below commute,

r

Example 3.19 The map T-*(xy)*T*: H*(V2)--+H*(V2) is given on elements 1-1- h+)", r 1-1- h+)..+r, ).. ~ h, so 7I"*tr(xlw2) is stable for T* and is also 'lL/3 invariant in both H*(V2)' H*(Vl)' Consequently, since it restricts to 0 in the remaining groups it is inthe image from H*(MI2)' This gives us a third independent generator m3 E H3(MI2), and a generator Sq2(m3) E HS(MI2 ). by h

J

W

The remaining details of the determination of H* (M12; IF 2) are direct and simplified considerably by the weak closure conditions of 3.4 as 3.18 shows. We leave them to the reader and content ourselves with quoting the result from [AMM2].

and such that any r l which satisfies this (generated by Wand WI) is a quotient of is called the amalgamated product of Wand W' over H. It is well known, (see [8e3]), that an amalgamated product as above will act on a tree with finite isotropy, and orbit space of the form

r. r

Theorem 3.20 H*(MI2;lF2) has the form lFda2,X3,Y3,Z3,d4"s,d6,d7l/R

where the d i are described above and R is the relation set a(x + y + z) = 0 x 3 = a 3x + ad4x xy = a 3 + x 2 + y2 x 2 y = a 3z + ad4z + yd6 + ad7 d7x = d4x 2 + a 2 x 2 d7y = a 2d6 + a 2y2 + d4x 2 + d4y2 d7z =,2 + a 2d6 + a 2x 2 + d4x 2 + d4z 2 Z4 = ,d7 + x4 + a 4d4 + z 2d6

+ xd6 XZ = a 3 + y2 yz = a 3 + x 2 = a 2y y, = ay2 X, = a 4 + az 2 d? =z3, + a 2d4d6 + aSd4+ zd4d7 + zd 6 b + az)

a,

+ d~(a3

+ xz + yz).

The Poincare series for H*(MI2; lF 2 ) is

1 + t 2 + 3t 3 + t 4 + 3t 5 + 4t 6 + 2t 7 + 4t 8 + 3t g + tID (1 - t 4)(1 - t 6 )(1 - t7)

r

W e _ _ _H_ _--4IG WI

In [Go], Goldschmidt analyzed the situation for actions on the cubic tree (the tree of valence 3) and obtained a classification of finite primitive amalgams of index (3, 3) (this refers to the indexes [W: HJ, [WI: H]). He shows that M12 is one of 15 such amalgams, necessarily a quotient of the universal one

r.

From this we deduce the existence of an extension (4.1)

l--+rl--+r--+M12--+1

where r l is a free group (it has cohomological dimension 1). Using the formula for Euler characteristics in [Brown], we have, on the one hand

+ 3tll + t 12 + t 14

and H*(MI2; lF2 ) is Cohen-Macaulay over lF 2[d4, d6 , d7]. (Note that all the generators have been constructed in 3.17-3.19 but x, y and z are linear combinations of S ql (a ), l, and m and not these generators themselves. ) As a test the reader should calculate the Poincare series for H*(W; lF2 ) and H* (W'; JF 2 ). Applying the result of Webb's formula, 3.10, then gives the

(amalgamated product), and also

x(r) Hence X(r l ) = 95, 040( - 1~2) group on 496 generators.

x(r')

IM12 1'

= -495 and it follows that r l ~ *i 96 'lL, the free

264

4. Discussion of H+ (1\!ft2; IF 2) .

Chapter VlIl. Cohomology of Sporadic Simple Groups

We can now state

265

The method they use is to construct a projective ZG-complex P* of dimenrkG

Theorem 4.2 The natural map T-tNh2 induces an isomorphism

sion

2:

(ni - 1) (where the {ni} are the dimensions of the generators of a

i=l

polynomial subalgebra over which the cohomology is free and finitely gener-

Proof. Consider the map

res~ EB res~': H*(W) EB H*(W')--+H*(H) . Its kernel is clearly im(resj¥') n im(resj¥") ~ H*(M12)' On the other hand, (3.10) gives that H*(vV)EBH*(vV') ~ H*(Nh2)EBH*(H). Hence resj¥' EBresj¥" is onto. On the other hand, from the structure of the orbit space of the tree described at the beginning of this section there is a classifying space for W *H W' of the form Bw U B /l B w ', and, applying the Mayer-Vietoris sequence we have a long exact sequence

ated), having the chain homotopy type of C*

III sni

rkG (

-1

)

.

This is done by

using cohomological varieties [BC1]. Then they consider the spectral sequence

Let Vi E Hni -1 (P*) be the cohomology generators; by construction they transgress to Pi E Hni (G) and we have E~O ~ H*((P*)G) ~ H*(G)/(Pi)

and so if q(t)

=

P.S. H*(P*G), then

j TI

rkG

PG(t) = q(t)

As it comes from a Mayer-Vietoris sequence the same map as before arises, hence the sequence splits and H*(vV) EB H*(W') ~ H*(T) EB H*(H).

Consequently, by rank considerations and the fact that the finite subgroups in T are mapped isomorphically into M12 under the projections the proof is complete. 0 Corollary 4.3 Hl(r'; IF 2 ) is an M 12 -acyclic IF 2(M12 )-module of rank 496

which is not projective. Proof. The proof follows from considering the spectral sequence over IF2 associated to (4.4) below and the observation that 64 does not divide 496. 0 This representation has radically different cohomological behavior at distinct primes dividing IM121. For ex?-mple, at P = 3 we have a sequence

and clearly the term on the left must be non-trivial. It appears, however, that this module restricted to Ml1 ~ M12 is the same one associated to the poset space for NI l1 . To complete our discussion on M 12 , we will explain the nature of its Poincare series. Recall from 4.1 that H*(M12; IF 2) is Cohen-Macaulay over the Dickson algebra IF 2 [d 4 , d6 , d7 ]. For any finite group G with Cohen-Macaulay cohomology Carlson and Benson, [BC2], have shown that the Poincare series must satisfy a functional equation which in our case is

(1 - t ni ) .

(p*)G is the algebraic orbit co chain complex, and hence will also satisfy Poincare Duality, from which pc(t) satisfies (*). The construction of a geometric complex X satisfying this is more delicate, and obstructions certainly exist in the general case. For M12 the existence of such a complex can be proved, [M2], by considering, besides the map T-tM12 of 4.2, also a map T-tG 2 (IF 3 oo) constructed in [M2] as a consequence of the remark following (3.10). Taking plus constructions as described in Chapter IX, we obtain a fibering B'j:. ----+B~2(F300) and the fiber is a (2-local) finite complex with the correct Poincare series. On the other hand the (2-local) homotopy equivalence B'j:. -tBt12 gives the desired map on the fiber complex. We do not know if this fiber is a manifold or not though it is a (2-1ocal) finite dimensional Poincare duality complex. On the other hand, there is a closely related complex which is a manifold. We now elaborate on this. Let Y be the graph associated to the Tits Building of L3(IF2) first described in Chapter V. We recall that one associates a vertex to each proper subgroup in (IF 2)3 and an edge to any proper flag. This is a trivalent graph with a transitive L3(IF2)-action, having as orbit space the edge

.'-------. B=Dg

PI = 174

174 = P2

The Tits Building has the equivariant homotopy type of A 2 (L 3 (IF 2 )), and we have (4.4)

266

Chapter VIII. Cohomology of Sporadic Simple Groups

5. The Cohomology of Other Sporadic Simple Groups

where P = HI (Y, Z(2») is an 8-dimensional projective module, the so-called

267

In [M2], PE(t) was determined to be

Steinberg representation.

r

The above also arises by considering the amalgamated product = 174 * 174 . The graph Y is a quotient of the cubic tree under a free normal

PE(t) =

l+t 2 +3t3+2t4+4t5+5t6+4t7 +5t 8 +4t 9 +2t 10 +3t 11 +t 12 +t 14 (l-t 4 )( I_t 6 ) (l-t 7)

.

Ds

subgroup r' ~ r, with quotient L3(IF2)' As for M 12 , H*(r) ~ H*(L3(IF2))' We consider the non-split extension E:

The numerator represents the Poincare series of the manifold G 2 / E, and it clearly dominates our answer for PM12 (t), explaining the leading terms. As a corollary we obtain that the Poincare series for (H*(Q) ® St)L 3 (IF 2 ) is

(4.5)

t 4 + t 5 + t 6 + 2t 7 + t S + t 9 + t IO z(t) = (1 _ t4)(1 _ t6)(1 _ t7)

E is a group of order 1344 and it contains the subgroups W, W' which appear in M I2 ,. realized as (4.6) and Syl2(E)

1

--+

(Z/2)3

--+

W

1

--+

(Z/2)3

--+

W'

PI

--+ --+

--+

P2

--+

1

Algebraically, the denominator is explained by the action of the Dickson algebra

1

= Syl2(MI2), realized as 1 --+ (Z/2)3

Denote Q

= (Z/2)3,

G

--+

H

--+

Ds

--+

1.

= L 3 (IF 2 ) as before. Then

H*(E) ~ H*(HomE(F*,IF 2)) ~ H*(HomQ(F*,IF 2)G) ~ H*(G, HomQ(F*, IF 2 ))

where F* is a free resolution of Z over ZE. From this and Shapiro's formula, we deduce

5. The Cohomology of Other Sporadic Simple Groups The O'Nan Group Q'N The O'Nan group 0' N has order 460,815,505,920 = 29 34 5 73 11 19 31, and in [AM3] we determine the poset space IA 2 (0' N)I/O' N, obtaining the following picture:

H*(E) E9 H*(H) ~ H*(W) E9 H*(W')

E9 H*(G, HomQ(F*, IF 2 ) ® St) . Rearranging, we obtain Theorem 4.7

From this picture some easy cancellations give

The group E is the normalizer of a (Z/2)3 in the compact Lie group G 2 , which is a 14-dimensional manifold, with 5.1 ([Bo2]) Now E acts freely on G 2 and one can in fact show Theorem 4.8

H*(O'N)E9H*((Z/4)3 . .04 )

~ H*((Z/4)3 . GL 3 (IF 2 )) EEl H*(Z/4· SL 3(lF 4 )

XT

Z/2).

Our calculations show that the cohomology will be Cohen-Macaulay. Indeed, in this case the cohomology of SyI2(O' N) is already Cohen-Macaulay, but is not detected by restriction to elementary 2-groups. We refer to [AM3] for complete details.

268

The Mathieu Group

M22

The third Mathieu group 1''1£22 of order 443,520 of 5 22 generated by the three elements X

5. The Cohomology of Other Sporadic Simple Groups

Chapter VIII. Cohomology of Sporadic Simple Groups

= 27 32 5 7 11 is the subgroup

A

(1,21,6,7,19)(3,22,11,12,4)(5,18,20,15,8)(9,13,16, 10, 14)

k

(1,19,14,5,16,18,15,3)(2,12,20,9,13,7,17,10)(4, 22)(6, 21,11,8).

There is a sporadic geometry described in [RSY] for M22 which also satisfies the hypotheses necessary for Webb's Theorem to apply. The associated complex has the following form, where Vi = (71/2)i and: denotes semi-direct product:

We apply V.3.3 to this picture and, after some cancellation and arguments of the type described from (4.5) on in §4, we have

H*(lVh2) EB H*(Syl2(M22)) ~ H*(V4: 174) EB H*(V4: 17s)EB (H*(V3) Q9 Stl)GL 3 (1F'2) EB (H*(V4)

Q9

St 2 )A6

where the Steinberg modules are those associated to GL 3 (lF 2) and A6 respectively, thought of as alternating sums of the mod 2 homology of the poset spaces at p = 2. However, it turns out that this is not the best way of determining H*(M22) because determining the cohomology of the components on the right hand side of 5.2 is comparable in difficulty to determining H* (M22) directly. Instead we first determine H*(Syl2(}.;f22)) and use detection on elementary subgroups, as in our analysis of Nh2, to obtain enough information to understand the cohomology. There is one added complication here, though. It turns out that H* (S'yl2 (Nh2); IF 2) is not entirely detected by restriction elementary abelian 2-subgroups. A sylow subgroup of M22 is generated by the three elements k, m

5.3

H

B

H

D

/3 1 C

1

a Finally, Sy12(M22) is a split extension L XT (71/2), where the new generator e can be chosen so that e(a) = /3, e(/3) = a, and

e(A) e(B) e(C) e(D)

BCD, ABD, ACD, ABC.

Then the group F = (e, a/3, AD, BC). Another group which is basic in Sy12(M22) is the subgroup

V3: GL 3(lF 2)

5.2

is a split extension L = E XT (71/2)2. Let a, /3 generate the (71/2)2 then they satisfy a(A) = B, a(B) = A, a(C) = D, a(D) = C, while ,8(A) = C, ,8(C) = A, ,8(B) = D, ,8(D) = B.

(1,19, 7,9,12,11,15)(2,5,8,22,4,14,18)(6,17,21,20,10,16,13)

Y

269

(1,18)(2,12)(3,17)(5,20)(7,13)(9,15)(10,14)(16,19)

n = (1,16)(2,13)(6,21)(7,18)(8,11)(12,19)(14,20)(15,17).

The group structure of SyI2(M22) has been studied in detail by Janko, [Ja]. He shows that there are exactly two subgroups isomorphic to (71/2)4 contained in Sy I2(M22 ), E and F, with intersection V2 ~ ('£/2)2, and each is normal in Syl2(lVf22)' A specific description of Syl2(M22) can be given as follows. Let (A, B, C, D) = E, then Sy12(Nf22) has an index 2 normal subgroup, L, which

LL

=

(a, /3, ABCD) .

The normalizer of E in M22 is the semi-direct product EXT A6 and the normalizer of F is F XTSS. Finally, the remaining conjugacy class of extremal elementary 2-groups in M22 has LL as a representative, and the normalizer of LL in M22 is the split extension LL XT GL3(lF2)' All three groups E, F, and LL are weakly closed in Sy12(M22) C M 22 • Also, in [AMI], the ring H*(SyI2(M22)) is determined and the image of restriction to these three subgroups can be easily read off from the results there. Thus the Cardenas-Kuhn theorem determines the image of the restriction map from H* (M22 ) in the cohomology of each of these three groups. The image in H*(E) is H*(E)A6, the image in H*(F) is H*(F)S5, and the image in H*(LL) is lF 2 [d 4 , d6 , d7 ] n lF 2[(8 4 )2, D 2 , D3]. Here, if e, h, k are the one dimensional generators of H* (LL), then D2 = h2 + hk + k 2, D3 = h2k + hk 2 and 84 = e 4 + e2 d 2 + ed 3 . This intersection is not the full Dickson algebra but lF2[d~, d6 , d7 ](1, d4 d6 , d4 d7 ).

The algebra H* ((71/2)4; lF2)A6 has been discussed at the end of Chapter III, and is the ring lF 2 [W3, IS, dg, d 12 ](1, Ig, b1S , blS)

,g

,s,

,9

where W3 = SX;Xj, IS = Sq2(W3)' = Sq4 but the 55 invariant subring is more complex. It is determined in [AMI], [AM4J, and has the form

where Sq2(n6) = ns, Sq4(n6) = nlO, n12 n6 n S· Moreover, in the invariant subring

= n~, X12 = Sq4(ns) and X14 =

Sql(n6)

0,

Sql(ns)

iD3n6,

Sql(nlO) Sql (XI2) Sql(X14)

+ ,Sn6, iD3nlO + ,sns

where M is the image in H*(E) EB H*(F) EB H*(LL) described above does not split as rings even though it does split as JF 2-vector spaces. For example ,s(1),s(2) = a2 ds is one of the many cup products which are non-zero in H*(M22) even though they are zero in M.

iD3nS -

2

W3 n 6'

Consequently, Sql (iD3X12 + ,SnlO) = ,gn6' The exact structure of the elements XIS, X16, XIS and X24 are not known to us currently. In any case, the image of H*(M22; JF 2 ) in the direct sum

JF 2[a, b, c, dJA6 EB JF 2[a, b, c, d]S5 EB JF 2[a, b, cJGL 3 (2)

:vill now be completely determined by specifying the "multiple image classes", I.e., those classes which must have non-trivial images in two or more of the summands. It turns out that they are generated by (iD3, iD3, 0), (0, n6, d6 ), (0, nlO, d4 d6 ) together with t~e polynomial ring JF 2 [ds , d 12 J, where d s I---? (d s , ds, d~), d12 I---? (d12 , d12 , d6 ). In fact the above completely describes the multiple image classes when we note that (iD~, 0, 0), (/s, 0, 0) and (0,0, d7 ) are also in the restriction image. It is important to notice also that the multiple image property changes the Sql operation on the elements which restrict respectively to ~O, n6'ld6) and (0, ~1O, d4d6), so in H*(M22) we have Sql(n6) = (0,0, d7) whIle Sq (nlO) = (0, w3nS + ,Sn6, d 4d7). . In summary, we can describe the non-nilpotent part of H* (M22 ) as the dIrect sum H*(V4)A6 EB H*(W4)S5 EB d7JF 2[d 4, d6 , d7] where the two copies of JF 2 [ds , d I2 ](1, iD3) in the first two rings are identified. The. key technical step in this determination is to show that (b IS , 0, 0) is in the Image of the restriction map from H*(M22)' Finally, the radical, i.e., the interesection of the kernels of the three restriction maps above, is determined in [AM4] and shown to have the form

JF2[ds , d I2 ](a2, a7, all, a14)

:v

here the m?d (4) Bockst~in ,64(a2) = iD3, while some higher Bockstein of a7 IS ~S' and a hIgher Bockstem of all is d12 . There are further higher Bocksteins whICh we do not completely understand at this time but aside from that [AM4) gives a complete determination of H*(M22)' '

Rema~k. The c?homology ring structure of H*(M22) is given by specifying extens~o~ data III the radical. It turns out that this extension data is highly non-tnvial so the exact sequence

Remark 5.4 The sporadic groups M 23 and MC L have the same Sylow 2subgroup as M22 and each contains M22 as a subgroup. Thus the analysis above for M22 should apply to determine H*(M23) and H*(MCL) as well and we discuss H*(M23; JF 2) next. At this time, however, the details for MC L have not yet been sorted out. The Mathieu Group M

23

For M 23 there is the double coset decomposition ]\1123 = M22 U M22gM22' Here 9 satisfies the condition, gM22 g- I n lvf22 = M2I = L3(4). Moreover there is an element 9 E M 23 -lv122 so g3 = 1 and 9 normalizes Syl2(L3(4)), [Co]. Consequently, we can assume the second double coset is M 22 gM22 and () E H*(M22) lies in H*(M23) if and only if res*(()) E H*(SYb(L3(4)))g. Thus, the determination of the stable elements becomes quite straightforward. Indeed, both E and F remain non-conjugate in M 23 but their Weyl groups become A7 and ('!l13 X As): '!l12 ~ GL 2 (JF 4 ) xT'!l12 respectively, while LL remains extremal. H*(E; JF2)A7 is Cohen-Macaulay over the Dickson algebra JF 2[ds , d I2 , d 14 , dIS], is given at the end of Chapter III, and has Poincare series 1 + XIS + x 20 + x2I + X24 + X25 + x27 + x 4S (1 - xS)(l - x 12 )(1 - x I4 )(1 _X I5 ) In particular, the multiple image classes (d s , ds, d~), (d I2 , d 12 , d~) remain, but the class (iD3, iD3, 0) is no longer invariant. However, since (0,0, d7) is still present in the restriction image, it follows that (0, n6, d6 ) must still be in the image, and this accounts for all the multiple image classes. In particular, it is shown in [M5] that the Poincare series for H*(F; JF 2)G2(1F 4 )X T Z j 2 is p(x)lq(x) where q(x) = (1- xlO)(l - x12)(1 - xIS)(l _ x 24 ) and p( x) is a polynomial of degree 57 with terms of the form 1 + x 6 + 2x s + x 9

+ xll + 2XI2 + x I3 + 3X I4 + ...

This describes the image under restriction. The kernel of the restriction map is JF 2[ds , d I2 ](a7, all), and this gives a complete description of H*(M23; JF 2). In particular note that Hi(M23; JF 2) = for i ::;; 5, and H6(M23 ; JF 2) = JF 2 with restriction image (0, n6, d6). Using the fact that H*(S23; JF 2 ) is entirely detected on its 2-elementaries, we can study the restriction map

°

272

Chapter Vlll. Cohomology of Sporadic Simple Groups

in dimension 6, and it is not hard to see that it is non-trivial.

Chapter IX. The Plus Construction and Applications

Remark 5.5 The remaining Sylow p-subgroups of M 23 and their normalizers are given as follows, Syl3(M23) = ('!l/3)2 with Weyl group equal to the 2Sylow subgroup of GL2(IF3). The 5-Sylow subgroup is '!l/5 with Weyl group '!l14. The 7-Sylow subgroup is '!l17 with Weyl group '!l13, the Weyl group for '!l/ll is '!l15 and finally the Weyl group for Z/23 is Z/ll. In each case it is direct to calculate the invariants, and only in the case of Z/7 is the invariant subring less than 6-connected,

O. Preliminaries

So putting this together we have

This is the first finite group known for which this happens. Indeed, it had long been conjectured that if G is a finite group with Hi(G; Z) = 0 for i = 1,2,3 then G = {I}.

Let G be a finite group. As we have seen, the classifying space BG has a very simple homotopy type as it isa K(G, 1). If G is perfect then HI (G; Z) = 0; suppose that we attach cells to BG to obtain a new, but simply-connected complex BG+ with the same homology as before. Or equivalently so that the homotopy fiber of BG-'tBG+ is acyclic, i.e. Hi(F; Z) = 0 for all i > O. The new complex will depend on G (as BG does) but the higher homotopy groups 7l'i(BG+) can be highly complicated invariants of G. In this chapter we will describe a construction as above, due to Quillen, and known as the plus construction. In general, given a group G with a perfect normal subgroup N we obtain a homotopy fibration X(N)-'tBG-'tBG+ with 7l'1 (X(N)) ~ N, X(N) acyclic. The main application of this is to afford a definition of the higher K-groups. For example, if G = GL(IF q ) and N = E, the subgroup generated by elementary matrices, then 7l'i(BG+) ~ Ki(IFq) i 2: 1 by definition.

1. Definitions We recall some notions from homotopy theory. Definition 1.1 a. A space X is acyclic provided fI* (X; Z) = O. b. A map j: Xl -'tX2 between path connected spaces is acyclic provided the homotopy fiber Ff of j is an acyclic space. We note the following: if j:X I -'tX2 is acyclic, then f*:7l'I(X I )-'t7l'I(X2 ) is an epimorphism with kernel a perfect normal subgroup. We now provide a criterion for the acyclicity of a map. We follow the exposition given by Hausmann and Husemoller in [HH].

Proposition 1.2 Let

f: X 1 -'tX2 be a map of connected spaces. Then f is

acyclic if and only if, for any coefficient system L on X 21 the induced map

!*: H*(X 1; j*(L))~H*(X2; L)

is an isomorphism, where j*(L) denotes the mduced local coefficient system on Xl.

2. Classification and Construction of Acyclic l\1aps We begin this section by proving a proposition which allows us to compare

Proof Recall that a local ~oefficient system L on X 2 is a module over 7f1 (X 2 ) and H*(X2; L)

= H*(C*(X2 ) o

for the homotopy fibration F

'l.7r 1 (X 2 )

i

~ Xl

L). Now consider the spectral sequence

f

---+ X 2 with

E;,q = Hp(X2; Hq(F; i* f* L))

=?

Hp+q(X I ; f* L) .

~ow

k*!* L is trivial on F, hence the spectral sequence only has one line if f IS acyclIc, and so the edge homomorphism Hp(XI; j* L)----+Hp(X2; L)

induced by f is an isomorphism. Conversely, assume f* induces a homology isomorphism with any coefficent syst~m L. In .particular consider the coefficients L = Z7fI(X2). In this case the Isomorphls~ can be interpreted geometrically as follows. Take the free 7fl(X2) bundle X 2---+X2 and pull it back to Xl using f, i.e.

acyclic maps. Proposition 2.1 Let h: X ---+XI' and 12: X ---+X2 be maps between CW complexes so that h is acyclic. Then there exists a map h: X I ---+X2 with hh ~ 12 if and only if ker( 7r 1(h)) ~ ker(7rI(h)) and h is unique up to homotopy. If 12 is acyclic then h is acyclic, and h is a homotopy equivalence if and only ifker(7rI(fd) = ker( 7r 1(h))·

Proof. Clearly, if h exists then 7r1(h) = 7r1(h) . 7r1(h) so ker(7r 1(h)) ~ ker( 7r1 (h))· For the converse assume that

h

is a cofibration and form the pushout

diagram

i

l

----+

X2

Xl Ux X2

Then 7r1«h):7r1(X2)---+7rI(Xl Ux X2) = 7r1(Xl)

----+

Hp(X1; f*Z7r1 (X 2)) ~ Hp(X2; Z7r1 (X 2))

and being filtration preserving, induces an isomorphism of the abutments

i.

*

7r l(X)

7r1(X2) and if

ker( 7r1 (h)) ~ ker( 7rl (h))

f

Then 7f, 7ft are fibrations with fiber 7r1(X2). Comparing their respective spectral sequences, we see that f induces an isomorphism at the E2-level by hypothesis

-

H*(X3 ) ----+H*(X2) . To pr~ve 'acyclicity for f it suffices to prove it for j. From above we have that Ho(F) = 0. Now assume inductively that Hj(F) = 0, j < n. Consider the _spectral sequence E;,q = Hp(X2; Hq(F)) where F is the homotopy fiber of j. ~ook at EO,n; then, because the map from the total space to the base space IS ~ homology isomorphism, EO,n = 0 for r > n + 1. However it can only be hIt by n +l n +1 dn+l·. E n+l,O----+ E O,n , b_ut H n +1 (X 2 ) consists of permanent cocycles, so this is zero. We deduce that Hn(F) = and so, inductively, we have proved that F is.acyclic so f is an acyclIc map. ' D

°

----+

it follows that 7r1 «(P2) is an isomorphism. As (P2 is also acyclic (which may be verified using (1.2)), we see that (P2 must be a homotopy equivalence by Whitehead's theorem. Now assume X is a homotopy inverse for cP2 and let h = X . cP1; then h . h = XcP1 . h = X . cP2h ~ h· Clearly h is uniquely determined up to homotopy. If 12 is acyclic one can check that h must be D acyclic. The rest is clear. We are now ready to construct acyclic maps. Proposition 2.2 Let X be a path-c(mnected space and N a perfect normal subgroup of7r1(X), Then there exists an acyclic map j:X---+X+ with ker(f) = N. If X has the homotopy type of a CW -complex then so does X+ .

Proof. We divide the argument into two steps. (I) First assume 7r1 (X) = N is perfect. Let T1 be a wedge of circles indexed by a set of generators for Nand p: Tl---+X a map so that 7r1(P) is surjective. Now form the cofiber T X ---+X* of p, i.e. attach a 2-ce11 for each circle. Clearly 7r1(X*) = 1 and the homology exact sequence yields exactness for

276

3. Examples and Applications

Chapter IX. The Plus Construction and Applications

Now using the Hurewicz theorem we have that 71"2(X*) ~ H2(X*), Hence we can take a wedge T2 of 2-spheres and a map fL: T 2-+X* so that the composition H 2(T2)---tH2(X*)---tH l (T1 ) is an isomorphism. Next, let T2-+X*-+X+ be a cofibration; as before we have exact sequences q t2

~

4, q = 1

Jl.

0

I

11,·

0

---t

H 2(T2)

---t

I"· H2(X*)

l'

X

1

fo

---+

Xo

1

f

---+

xUxXo =x+

Once again, using (1.2), the fact that fo is an acyclic cofibration implies that X o) = 71"1(X)/71"1(X) ~ 71"1(X)/N as Xois SImply connected. Hence 71"1 (f) is an epimorphism with kernel N and the proof is complete. ' 0

! is also acyclic. Also, 71"1 (X Ux

The previous two results can be combined to prove the following classification theorem. . Theorem 2.3 Let X be a path-connected space with the homotopy type ola CW complex. The correspondence which assigns to an acyclic map f: X-+Y the subgroup ker71"1 (f) ~ 7r1 (X) induces a bijection

HI (T1 ) /~

(II) Nc:w let N ~ 71"1 (X) be a proper, perfect, normal subgroup, and denoted by g: X -+X the_ covering corresponding to N. Using (I) we construct an acyclic map fo: X -+Xo with X o simply connected. We now change it up to homotopy into a cofibration and for the pushout diagram

X

O---tH3(X*)---tH 3(X+) - - - t H 2(T2) ---+ H2(X*)---tH2(X+)---+O . Let f be the composition X -+X*-+X+; we claim it is a homology isomor:phism. This is clear in dimensions q ~ 4, or q = 1. In dimension 2 we have that fL*: H2(T2)-+H2(X*) is monic by the construction. It follows that H3(X*)-+H3(X+) is an isomorphism and so is H3(X)-+H3(X+), Now we have the following diagram with exact rows and columns:

277

r

---+

H2(X+)

---+0

H2(f)/

H 2(X)

I

equiv~lence classes Of} { acycl'lc maps on X

+-t

{set of normal, perfect} subgroups of 7r1 (X) .

The space X+ corresponding to a perfect normal subgroup N
0

which is (vertically) split by

H2(f)(X) =

0

t::

HI (Tl)-+H2(X*), Assume

==> r· s(x) = 0 ==> s(x) = fL*(W) ==> 0 = h*s(x) = thfL*(W) = v(w).

As v is an equivalence we conclude that s (x) = W = 0 and, as s is injective, x = 0 so H2 (f) is injective. Now let z E H2(X+), z = r(y), h*(y) =I O. Then h*(Y) = v(w) = th . fL*(W), and r(y - fL*(W)) = r(y) = z with th(X - fL*(W)) = 0, i.e. Y _ f-L* (w) E im( s). Hence H2 (f) is onto and we have shown that it is, in fact, an isomorphism. As X+ is simply connected every local coefficient system on it is trivial. Hence H * (f) is an isomorphism for for all coefficients and therefore f is acyclic with ker7rl (f) = 7rl (X) = N.

3. Examples and Applications The Infinite Symmetric Group

From work of Dyer and Lashof [DL] , there is an identification Q(SO)o ~ (nBC(SO))o' where C(SO) denotes the monoid iln>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<) ~ 1f'l(TL) *7\ C1f'

TX = T(ISXI")

-r

ISXI" ~ ISXI ~ X

Part (iii) follows from (i) and (ii) and the results in §l.

o

One can in fact prove that if K is a finite connected simplicial R emark . d' . J( (see complex, T K may be taken to be finite and of the same ImenSlon as [Mau]).

Chapter X. The Schur Subgroup of the Brauer Group

O. Introduction In this final chapter we apply the techniques of group cohomology to the representation theory of finite groups. Given G a finite group we know ,that IF( G) is semi-simple for any field of characteristic zero. Consequently, from the Wedderburn theorems there is a decomposition (0.1) where the Di run over central simple division algebras with center lKi a finite cyclotomic extension of IF. The question that we answer here is the determination of all the classes {Di} E B(lF) which arise in this way, that is to say, which division algebras occur in the simple components of the group ring of a finite group. When G is a finite group all the centers, Z(D i ), in the semi-simple expansion of IF(G) are cyclotomic extensions of IF, i.e., subfields of IF((m) , and m divides the order of G. Indeed, if lK = Q((/G/), then lK(G) is a direct sum of matrix algebras over lK. Conversely, if a sufficient number of roots of unity are not present, then JK( G) cannot split in this fashion, since it is already not going to be true for the cyclic subgroups of G. For this reason, when studying the division algebras which occur in (0.1), it is sufficient to assume that the field IF is cyclotomic. The division algebras which we discuss from here on are all assumed to be of finite dimension over their centers. The content of this chapter is unpublished work of Milgram in the mid 1970's. The question of identifying the possible division algebras which arise in (0.1) was first raised by Fields and discussed in his joint paper with 1. N. Herstein, [FH]. Later, M. Benard, [Br], Benard and M. Schacher, [BeS], and especially G. Janusz, [J], and T. Yamada, [YJ, did important work on the question. From our point of view the question becomes the determination of explicit maps of cohomology groups with twisted coefficients under restriction and change of coefficients. Hence we feel it is a suitable example with which to conclude this book.

............... .....-r ...........

~

-_.

--_ ..................... _--- - _ · · v

-

' ...

1. The Brauer Groups of Complete Local Fields

Valuations and Completions Definition 1.1 A non-archimedean valuation·· on a division algebra D is a map cp: JF ~lR+ where lR+ is the non-negative real numbers satisfying the following three conditions.

(1) cp(a) = 0 if and only if a = O. (2) cp(ab) = cp(a)cp(b). (3) cp(a + b) ~ Max(cp(a) , cp(b)).

The valuation ring Orp C D is the subset of D consisting of <111 those a E D with cp(a) ~ 1. That Orp is, in fact, a sub ring follows from conditions (2) and (3) in the definition, the first showing that it is closed u~der pro~ucts,. and the second showing that it is closed under sums. There IS a maXImal Ideal P C 0 defined as those a E D with cp(a) < 1. Thus, in the case of cpp, . 'm Q of t he form p a: w WI'th W as werp have rpthat Orp consists of those fractlOns above ,a> 0 and P rp consists of those fractions with a > O. In the case . where D is a field, IF, the quotient Orp/Prp = IFrp is a field, called the res~due class field oj the pair (IF,
valuation ring and Prp be the maximal ideal, then the quotient Orp/Prp is a division algebra.

A non-archimedean valuation trivially satisfies the triangle inequality. The valuation is discrete if the value group {cp( a) I a E JF, a i= O} is an infinite cyclic group. We also assume there is some a E JF with cp( a) i= to avoid trivial cases.

Proof. If a E Orp - Prp then
Example. The standard example is the p-adic valuation on the rationals. Let nlm = pO:w with a E Z and w = ;::, where both m' and n' are prime to p. Then cpp(nlm) = pO:. We will discuss examples of valuations on non-

Definition. Let