ON Kst{Z/p2Z) AND RELATED HOMOLOGY GROUPS … · The prime to p homology has been calculated by...

TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 270, Number 1, March 1982

ON Kst{Z/p2Z) AND RELATED HOMOLOGY GROUPSBY

LEONARD EVENS AND ERIC M. FRIEDLANDER1

Abstract. It is shown that, for/? » 5,

R = Z/p2Z,K3(R) = Z/p2Z + Z/(p2 ~ l)Z

and K4(R) = 0. Similar calculations are made for R the ring of dual numbers over

Z/pZ. The calculation reduces to finding homology groups of Sl(/Î). A key tool is

the spectral sequence of the group extension of Sl(«, p1) over Sl(w, p). The terms of

this spectral sequence depend in turn on the homology of Gl(n, p) with coefficients

various multilinear modules. Calculation of the differentials uses the Charlap-Vasquez

description of d2.

Introduction. In this paper we shall show that, for/? > 5,

K3(Z/p2Z) = Z/p2Z © Z/ (p2 - l)Z

and

K4{Z/p2Z)=0.

These groups can be determined from the integral homology groups of SL(Z//?2 Z).

The prime to p homology has been calculated by Quillen. In this paper, we shall

calculate the /7-primary component of Hr(SL(Z/p2Z), Z) for r = 1,2,3,4.

Our method is to consider SL(«, p2) as an extension of SL(«, p) and to use the

Lyndon-Hochschild-Serre spectral sequence. To determine the E2 term of the

spectral sequence requires rather extensive calculations of homology for GL(w, p)

with coefficients in various modules associated with its multilinear algebra. (For

example, we show for/? > 5, n 3= 2, that i/2(GL(n, p), Vn A Vn) = Z/pZ where Vn is

the module of « X « matrices of trace 0.) To calculate the differential d2 in the

spectral sequence, we restrict to the case n — 2 and use the theory of Charlap-

Vasquez. That theory splits d2 into two terms, one determined by the group

extension and the other by the corresponding split extension, d2 for this split

extension is calculated by cocycle calculations in an explicit double complex.

With minor modifications, our calculations also apply to the split extension

SL(«, Z/pZ[e]) -* SL(«, Z/pZ) where Z/pZ[e] is the ring of dual numbers. We thus

obtain the additional results

K3(Z/pZ[e]) = Z/pZ® Z/pZ © Z/(p2 - l)Z,

K4(Z/pZ[e]) = 0 (p>5).

Received by the editors February 29, 1980 and, in revised form, July 8, 1980.

1980 Mathematics Subject Classification. Primary 18F25; Secondary 20J10, 20J06.

Key words and phrases, /^-theory, homology of groups, Lyndon-Hochschild-Serre spectral sequence.

Charlap-Vasquez theory, linear groups, dual numbers, Z/p Z.

' Partially supported by the National Science Foundation.

© 1982 American Mathematical Society

0002-9947/81/0000-1021/$ 12.50

1License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

2 LEONARD EVENS AND E. M. FRIEDLANDER

We gratefully thank Stewart Priddy and Brian Parshall for many valuable con-

versations. We also thank Victor Snaith for suggesting in his study of the analogous

questions for p = 2 that iT-groups might be determined by direct computation.

Above all, we wish to thank the referee for his careful reading of the paper. In

particular, he told us how to remove unnecessary (and annoying) conditions from

the crucial Propositions 7.6, 8.1, and 8.4.

Table of notations, p: a prime > 5, unless otherwise noted.

F = Z/pZ.F[e] = F[X]/(X2) = the ring of dual numbers over F.

GL(«, p) = the group of n X n nonsingular matrices over Z/pZ. GL(« — 1, p) is

imbedded in GL(«, /?) in the upper left-hand square. Similarly, GL(n, p2), SL(«, p),

SL(«, p2) are defined as usual. SL(Z//?Z) and SL(Z//?2Z) are the infinite limits.

SL: a related group defined in §2.

Mn: the module of « X « matrices over F. GL(«, p) acts by conjugation.

Vn: the submodule of Mn of matrices of trace 0.

A * A : the rc-fold wedge product over F of the vector space A.

§kA : the A;-fold symmetric product over F of A.

A ° A: the 2-fold symmetric product, a ° b denotes a typical generator.

Â = HomF(^,F).

Bn: the subgroup of GL(n, p) consisting of all upper triangular matrices with

nonzero diagonal terms.

Un: the subgroup of Bn consisting of all upper triangular matrices with l's on the

diagonal.

Hn: the torus subgroup of Bn consisting of all diagonal matrices with nonzero

entries.

Wn_x: the n — 1 dimensional vector space Fn_1. Wn_] is imbedded in Bn as the

matrices with l's on the diagonal and all other entries 0 except those in the last

column. Then Wn_x is a normal subgroup of U„ and conjugation yields the usual

action. Elements u of W„_] are considered column vectors, elements/of Wn^x are

considered row vectors, fu is a scalar, and uf is a matrix.

C„: the normal subgroup of Bn consisting of all matrices with l's in the first n — 1

diagonal positions, a nonzero entry in the last diagonal position, and zeroes

elsewhere except in the last column. C„ is the semidirect F* X Wn_, where F* acts

on Wn_] by multiplication by the inverse.

e¡j\ the matrix with 1 in the i, /-position and zeroes elsewhere. The diagonal

matrix (i,, i2,...) acts on e{j as t¡tjx.

"i — eii ~~ ei+l,i+l-

*ij = I + eij-

[G, A]: for a G-module A (G a group), the submodule generated by all g(a) — a.

X ®z[K] A: X and A are supposed left K-modules and vx ® va = x ® a. If V is

written additively, vx ® a = x ® ( — v)a.

[«,, u2,... ,vr]: a typical basis element for the standard Z[F]-free resolution X of

Z. For r = 0, [•] denotes the single basis element. d[u] = u[-] — [■] (see §11). On

occasion, the image in X ® Z[ v] F is denoted by the same symbol.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Kíf{Z/p2Z) AND RELATED HOMOLOGY GROUPS 3

x n f: the Pontryagin product H\{V,¥) <8> H¿V,F) -> Hjy,¥) induced by the

multiplication homomorphism VX V -» F for F an abelian group. The cup product

in H*( V, F) is denoted simply by multiplication.

Ô: H'(V,Z/pZ) -» H'+\V,Z/pZ) is the Bockstein morphism, i.e. the connecting

homomorphism arising from the exact sequence 0 -» Z/pZ -* Z/p2Z -* Z/pZ -> 0.

p(u): the element of H2(V,F) represented by 2*=o [u> 7°1- P: ^^ ^(^F) is a

monomorphism.

//,.(, ; p) the/?-primary component of the relevant homology group.

Chapter I. The basic argument

1. Reduction to/?-primary homology. From Quillen [Q], we know

/ , x s fO, r= 1,2,4,Hr(SUZ/PZ),Z)=^/(p2_l% r = 3

Since for each n, Ker{SL(«, p2) -» SL(«, /?)} is a /»-group, it follows that

Hr{SL(Z/p2Z),Z; l)=Hr(SL(Z/pZ),Z; I), I*p.

On the other hand, we shall show as a major result of this paper

Theorem 3.4. For p s= 5,

Hence, we have

Corollary 1.1.

Hr{SL{Z/p2Z),Z) = 0, r= 1,2,4,

#3(SL(Z//>2Z),Z) = Z//?2Z ®Z/(p2- l)Z.

From this we obtain our main result.

Theorem 1.2. Let p ^ 5. Then

Ki(Z/p2Z) = Z/pZ © Z/ (/> - 1)Z, K2(Z/p2Z) = 0,

K3(Z/p2Z) = Z//?2Z © Z/ (/?2 - 1)Z, K4(Z/p2Z) = 0.

./Vote. #2 has been calculated by Milnor [M], K\ by Bass [B].

Proof. We recall that the AT-groups of Z/p2 are defined to be the homotopy

groups of (BGL Z/p2)+ , where ( )+ is the Quillen plus construction with respect to

the (perfect) commutator subgroup of the fundamental group (see [W]). By Corollary

1.1, //,(BSL Z/p2) = 0 so that SLZ//?2 is the commutator subgroup of GLZ/p2.

This implies that (BSLZ//?2)+ is the universal covering of (BGLZ//?2)+ [W].

Consequently,

Kx{Z/p2) =*,((BGLZ/p2)+) = (GLZ//?2)/(SLZ//,2) - Z//? - 1 © Z//?,

K,(Z/p2 ) = 77, ((BSL Z/p2 )+ ) for i > 1.

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Because BSLZ//72 -> (BSLZ//>2)+ is a homology equivalence, Corollary 1.1 and the

Hurewicz theorem imply that

772((BSLZ//?2)+ ) ä H2(BSLZ/p2) = 0,

7T3((BSLZ//?2)+ ) =* //3(BSLZ//?2) = Z//?2 - 1 © Z//?2 = n.

Moreover, because (BSLZ//?2)+ -» (BSLZ//?)+ is an /-equivalence of simply con-

nected spaces for primes l^p, 7r4((BSLZ//?2)+ ; /) -=* 7r4((BSLZ//?)+ ; /), and the

latter is zero by [Q]. Hence, tt4((BSLZ//?2)+ ) = t74((BSLZ//?2)+ ; /?).

Let (BSLZ//?2)+ -* Àïn,3) be the natural 3-equivalence obtained by attaching

cells to (BSLZ//?2)+ to kill the homotopy groups above dimension 3 and let F

denote the homotopy fibre of this map. The Hurewicz theorem and the long exact

homotopy sequence imply that F is 3-connected and 7r4((BSLZ//>2)+) — 7r4(F) —

H4(F). For p>3, H¡(K(U,3),Z/p) equals Z/p for ¿ = 3,4 and equals 0 for

/ = 5,6 [C]. Consequently, the Hurewicz theorem and the universal coefficient

theorem imply that

H3(K(U,3)) - ^(K(U,3)) = Z/(p2 - 1) © Z/p2,

Hi(K(U,3),Z;p)=0, 3<i<7,3<p.

Using the Serre spectral sequence

E2S = Hr{K{Z/p2,3), HS(F)) -//r+J((BSLZ//)+ )

we conclude that H4(F,Z; p) ~ //4((BSLZ//?2)+ , Z; p) s H4(SLZ/p2, Z; p).

Consequently, Corollary 1.1 implies that

7T4((BSLZ//?2)+) =0. D

Similar arguments will allow us to conclude for the ring of dual numbers

Z/pZ[e | e2 = 0] the following. (Use Theorem 4.1.)

Theorem 1.3. Let p > 5. Then

7/r(SL(F[e]),Z) =0, r= 1,2,4,

//3(SL(F[e]),Z) = Z/pZ © Z/pZ ®Z/{p2- l)Z.

Arguing as in the proof of Theorem 1.2, we have

Corollary 1.4. Forp > 5

K,(F[e]) = Z/pZ ®Z/(p- 1)Z, K2(F[e]) = 0,

K3(F[e]) =Z/pZ®Z/pZ®Z/(p2 - l)Z, if4(F[e]) =0.

K2 was calculated by van der Kallen [V].

2. Reduction to SL. Define

SL(n,/?2) = KerJGL(«,/?2)det<-^ U(Z/p2Z) .

These groups form a direct limit system, and we define

SL(Z//?2Z) = lim SL(«,/?2).License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Kit(Z/p2Z) AND RELATED HOMOLOGY GROUPS 5

Clearly, SL(«, p2) = Ker{SL(«, p2) " U(Z/p2Z) -» U(Z/pZ)}. Moreover, for

(n,p-l)=\,

SL (n, /?2) = SL(«, /?2) X Z/ (/? - 1)Z,

so //r(SL(«, /?2), Z; /?) ^ //r(SL(«, /?2), Z; p), and by direct limit

Hr(SL(Z/p2Z),Z;p)^Hr(SL(Z/p2Z),Z;p).

In what follows, we calculate the right-hand groups for r = 1,2,3,4.

3. Calculation of Hr(SL(n, p2), Z\p),r = 1,2,3,4, p > 5; the basic argument. The

homomorphism it: Z/p2Z -* Z/pZ induces an epimorphism 77: SL(«, p2) -»

GL(«, p). Its kernel consists of all n X n matrices of the form I + pA where

A <E Mn{Z/p2Z) and det(7 + pA)p~l = 1. However, det(I + pA)p~x = 1 +

(p — l)Tr(pA) = 1 if and only if Tr(pA) = 0. Since / + pAx — I + pA2 if and only

if Ax = A2moà p, I + pA »-» ir(A) identifies the kernel with Vn = s\(n, p) = the

additive group of n X n matrices over Z/pZ with trace 0. GL(«, /?) acts on the

kernel by conjugation x: T(-» xTx~l.

To calculate the desired homology, we use the spectral sequence of the group

extension described above, or rather its /»-primary component

Hs(GL(n,p),Hl(V„,Z);P)=>Hs+t('SL(n,p2),Z;p) (s + t>0).

Note that since all the relevant groups are finite, the spectral sequence does

decompose into independent primary components. Also, for t > 0, we may omit the

restriction to/?-primary component since Ht(Vn, Z) is a/?-group.

We list here the relevant homology groups of V„. The decompositions are as

GL(«, /?)-modules and hold for /? > 2 (but not p = 2). Also, they are natural with

respect to n. (These facts are easily established using the known structure of

H^(Vn, Z/pZ) for Vn an elementary abelian/?-group.)

H0(V„,Z) = Z, Hl(V„,Z) = Vn, H2(V„,Z)= A2Vn.

(H2 is spanned by all Pontryagin products of elements of Hx(Vn, Z).)

H3(VH,Z)= AV„©S2K„,

H4(Vn,Z)= AV„©Ker{K„® A2F„-A3F„}.

Proposition 3.0. We have the following basic table of results for n > 2,p s= 5.

(a) Hs(GL(n, /?), Z; p) = Hs(GL(n, p), Z/pZ) = 0, 1 < s < 2p - 4.(b)

Í0, 5 = 0,1,3,Z/pZ, 5 = 2 (stably, i.e. H(GL(2, p), V2)

-»//2(GL(n, p), V„) is an isomorphism).

Hs(GL(n,p),Vn)

(c)

Hs(GL(n,p),/\2Vn)={Q¿/Z/pZ, s = 2.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

(d)

(e)

LEONARD EVENS AND E. M. FRIEDLANDER

Hs(GL(n, p), %2Vn) = ( Z^Z' s = ° (stab'yî>

10, 5=1.

Hs{GL(n,p),A>Vn)= {Z^Z' J = * <*"**).

(f) //0(GL(«, /?), F„ 0 A 2 K„) = Z//?Z (ííró/y).

(g)tf0(GL(«,/?),A4Kn) = 0.

The/?/-oo/of this proposition takes up the major part of this paper. See Chapter II,

Propositions 5.1, 5.2, 7.1, 7.3, 7.5, 7.6, and 8.1-8.4.

Using this basic table, we may conclude that the E2 term of the spectral sequence

looks like:

Z/p « -z/B

0

O

*&

**

0

In more detail, we have

Proposition 3.1. (p > 5, n s* 2.)

(i) Hs(GL(n, p), H0(V„, Z); p) = 0,5 = 1, 2, 3,4.(u)

(0,^(GLK/?),//,^,^)5 = 0,1,3,

[Z/pZ, s = 2(stably).

(iii)

(iv)

^(gl(«,/?),//2(f„,z))={^Zj * = °;J'

Hs(GL(n, p), H3(Vn,Z)) =[Z/P® Z/P> s = ° (stab'y)(.0, 5=1.

(v)tt0(GL(«,/?),//4(F„,Z)) = 0.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Kt(Z/p2Z) AND RELATED HOMOLOGY GROUPS 7

Proof, (i) is a restatement of (a) in the basic table, (ii) is (b) and (iii) is (c). (iv)

follows from (d) and (e). To prove (v), consider the exact sequence 0 -» Ker -» V„ 0

A2 V„ -» A3 Vn -» 0. Since H,(GL(n, p), A3 F„) = 0 by (e), we obtain

H0(GL(n, p), Ker) = 0 from:

0 - HQ(GL, Ker) -+ #0(GL, K„ 0 A 2 K„) - H0(CL, A 3 F„) - 0

II IIZ/pZ Z/pZ

Since H0(GL, A 4 F) = 0 by (g), the desired result follows. D

To complete the calculation, we must investigate the differential:

^2,2: £2,2(Z) ~* £02,3(Z)

II II

Z/pZ Z/p © Z/p

We first consider the case n — 2. From the basic table (c), for n = 2, E22(Z) =

Z/p. Moreover, we have

Proposition 12.4(b). For « = 2, p > 3, and coefficient module Z, d\2 is a

monomorphism.

The proof of this fact is based on the Charlap-Vasquez theory and is discussed at

length in Chapter III.

To deal with n > 2, consider the commutative diagram:

H0(GL(n,p),H3(Vn,Z))

t«

H0(GL(2, p), H3(V2,Z))

di.2

"2,2

mono

H2(GL(n,p),H2(V„,Z)) = Z/pZ

Î

H2(GL(n,2), H2(V2,Z)) = Z/pZ

We conclude that i/| 2 is a monomorphism for n > 2.

To continue with the basic argument, we can now conclude that the E3 term of

the spectral sequence looks like:

O

O

O ^p

It follows that for total homology //, =0, H2 = 0, 773 = Z/p2 or Z/p © Z/p,

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■s LEONARD EVENS AND E. M. FRIEDLANDER

To determine precisely what happens for H3, we must repeat the above argument

for coefficient group Z/pZ. We have

Hs(GL(n, p), H,(Vn,Z/pZ)) -» Hs+,(SL(n, p2),Z/Pz).

Also, forp > 2, we have decompositions of GL-modules

H0(V„,Z/pZ) = Z/pZ,

Hx(Vn,Z/pZ)=Vn,

H2{Vn,Z/pZ)=Vn®A2V„,

H3(V„,Z/pZ)= A3K„©K„® F„= A3F„© A2F„©S2F„.

Vn is imbedded in H2(Vn, Z/pZ) as all p(u), o G V„, where p(v) is represented by

the chain SfJo' [v, iv].

A 2 Vn is imbedded in H2(Vn, Z/pZ) as Pontryagin products u n v, u, v G K„, and

similarly for A 3 V„ in i/3(F„, Z/pZ).

Vn 0 F„ is imbedded in #3 as all m n p(u).

Hence, from the basic table, we have

Proposition 3.2. For p > 5, n > 2,

//J(GL(«,p),//0(K„,Z/pZ)) = 0,

(0,

5= 1,2,3,4;

5 = 0,1,3,5 = 2;//5(GL(«, p), Hx(Vn,Z/pZ)) = | ¿

//J(GL(«,p),//2(F„,Z/p))={^'

7/0(GL(«, p), tf3(F„,Z/p)) = Z/p ® Z/p (stably).

5 = 0,1,

1 Z/p ©Z/p, 5 = 2;

Again, we will show in Chapter HI

Proposition 12.3(a). d\2: £222(Z/pZ) -> £23(Z/pZ) ¿s an isomorphism for n = 2.

It follows as before that ¿f2 is an isomorphism for Z/pZ for n > 2. Hence,

E\Z/pZ) looks like:

0

0

0

O

0

O

O

*/p

o

o

o o

It follows that H3(SL(n, p2),Z/pZ) = Z/pZ. Hence, by the universal coefficient

theorem we may conclude H3(SL(n, p2),Z; p) = Z/p2. DLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

KJf(Z/p2Z) AND RELATED HOMOLOGY GROUPS 9

To summarize, we may now state one of our basic results.

Theorem 3.3. For p >5,n>2,we have stably

Hx(SL(n,p2),Z;p)=0,

H2(SL(n,p2),Z;p) = 0,

H3(SL(n,p2),Z;p) = Z/p2Z,

H4(SL(n,p2),Z;p)=0.

Proof. The only thing to check is stability for degree 3. However, we know

comparison with n — 2 yields isomorphisms for E2t(Z) in all relevant positions.

Thus, we conclude that comparison with « = 2 yields an isomorphism also for the

relevant total homology. D

We have now proved

Theorem 3.4. Forp > 5,

Hr(SL(Z/p2),Z;p)\Z/p2Z,

1,2,4,

3.

4. Calculations for the split extension. For the ring of dual numbers F[e], we may

argue very much as above. Define

_ I det()''-1

SL(n,F[e]) = Ker GL(n,F[e]) -* i/(F[e])

and

SL(F[e]) = lim SL(«,F[e]).

The latter group is related to /i-theory as before. Its prime to p homology is given by

Quillen's results. To calculate Hr(SL(n, F[e]), Z; p), we argue as above except that in

this case the extension 0 -» Vn — SL(n, F[e]) -* GL(n, p) -» 1 splits. The E2-terms of

the spectral sequences are as before.

O

O

0

o

%

^p

0 o

O

o

o

ype^p

2% O

o

E¿Ca) E(5/p2)

To calculate the relevant d2,s, we rely on the following result from Chapter III.

Propositions 12.4(a), 12.3(b). For n - 2, p > 3, Im d\2 is 1 dimensional both for

ZandZ/pZ.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


To determine d\ 2 for « > 2, consider the commutative diagrams

H0(GL(n,p),H3(Vn,C)) '" H2(GL(n, p), H2(V„, C))

H0(GL(2,p),H3(V2,C)) "" H2(GL(2,p),H2(V2,C))

for C = Z or Z/p Z. Note that as a result of our previous analysis, we know both

vertical arrows are isomorphisms. Hence, dim Im d22= 1 for arbitrary n > 2.

Thus the relevant E3 terms look (stably) like:

O

*/p

0

0

o

o

o

o

o

Z/c

o

o

o o

o

o

o

o

o

2/e

o

o

o o

E3C2:) E3U/p20

Hence, we have proved

Theorem 4.1. For n s» 2,p 3s 5, we have stably

Hl(SL(n,F[e]),Z;p)=0,

H2(SL(n,F[e]),Z;p) = 0,

H3(SL(n,F[e]),Z;p) = Z/p © Z/p,

#4(SL(n,F[e]),Z;p)=0.

Chapter II. Homology calculations to fill in the basic table

5. Initial calculations.

Proposition 5.1. Forp > 2, n > 1, 1 < r < 2p — 4,

//r(GL(«,p),Z;p) = 0, Hr(GL(n,p),Z/pZ)=0.

Proof. Using the universal coefficient theorem and Bn D U„ (ap-Sylow subgroup),

we need only prove Hr(Bn, Z/pZ) = 0, l<r<2p-4.

To establish this, note first that since Bn = HnÎX Un, we have

Hr(B„,Z/pZ) = Hr(Un,Z/PZ)Hii.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

K.JZ/p2Z) AND RELATED HOMOLOGY GROUPS 11

We look for the irreducible constituents (all one dimensional) of Hr(Un,Z/p) as a

Z/p(//„)-module. We shall see that, for the relevant r, these are all nontrivial-from

which Hr(Un, Z/pZ)H = 0 follows. U = Un may be built up as successive extensions

1U

Lk+\ I/fc+l

u1, k = 2,3,...,« — 1,

where Tk is generated by 1 + e¡¡,j — i > k.

1 0

0

1

The £2-term of the spectral sequence of this central extension is H^(U/Tk, Z/pZ) 0

HjiTk/Yk+,, Z/p) so we need only look at the irreducible //„-constituents of this, or

by induction the irreducible constituents of Hi¡(U/T2) ® HJt.(T1/T3)

0 • • • 0//„;(rn_I). Recalling that this is a graded tensor product, we may identify

this as the homology of the elementary abelian group

©yri+l*:

e Z/pZe,j-i>\

As usual,

(upper triangular matrices with O's on the diagonal).

H*(Ln,Z/pZ) = /\L„ 0 9(pLn) *- Divided power algebra

î Î

degree 1 degree 2

Given the action, h ■ etj — <,*r'e,/ where' U 0

h =

0

it is straightforward to see that the first trivial irreducible constituent in a positive

degree occurs in degree 2(p — 2) + 1 = 2p — 3. (This is easier to see in the dual

cohomology module which is an exterior algebra tensored with a symmetric algebra.)

D

In what follows, we shall make frequent use of the spectral sequence of a filtered

module. The general theory is as follows.

Let G be a group and A a (7-module. Suppose in addition A is filtered by

submodules A = A0 D At D A2 D ■ ■ ■ D A,D Al+i = 0. We obtain from this a

filtration of the standard chain complex

Cr(G,A) = F0Cr D F_ ,Cr 3 F 2Cr D ---D F_,Cr D F_,_ ,Cr = 0License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


with FsCr = Cr(G, A_s) viewed naturally as a submodule of Cr(G, A). In general, a

filtered complex yields a spectral sequence. (See [C-E, Chapter XV] for an extensive

discussion.) The £,-term is given by

i Í F*c* \ (IaEs.t - Hs+t\ "i ~F~ =HS+A C*\ G,

= Hs+l\G,

A.— s

A-s+\

Note that E¿, = 0 outside the set defined by s + t s» 0, — / < s < 0. To calculate

d*t(ck) for ck G £*„ choose c G FsCs+l = Cs+l(G, A_s) representing ck such that

de G Fs_xCs+r (That is possible because dxcx = ¿Pc2 = • • • = dk~xck~x = 0 where

c represents c' in E's,.) Then dk(ck) is represented by Jc.

Proposition 5.2. Forp » 5, « > 2,

ff0(GL(«, p), F„) = 0, ff0(GL(«, p), Mj = Z/pZ.

//,(GL(n, p), Fj = Hx(GL(n, p), M„) = 0.

//2(GL(«, p), Fj = ff2(GL(n, p), M„) = Z/pZ.

Moreover, this isomorphism is stable; it commutes with the induced map from n = 2.

H3(GL(n, p),V„) = H3(GL(n, p), M„) = 0. Similar formulas hold for Bn.

TrProof. The sequence 0 -» FM -> Mn ->F — Z/pZ -> 0 together with Proposition

5.1 allows us to relate the homology of Vn to that of Mn or vice versa.

For V„, we first calculate Hr(Bn,Vn) (r = 0,1,2,3). We use B„ = B„_lX C„,

Cn = F* X W„_l. Recall [W, V] is the submodule of V spanned by all u(s) — s,

u G W, s G F. Consider the tower

A0

K~t

A<»-,

rç,-l

where IF„_, acts trivially on the factors, F* acts trivially on M„_,, by multiplication

on JF„_,, and by reciprocal multiplication on W„_]. Note H0(Cn, V„) =

HQ{Wn_x,Vn)¥. = {Wn_,)¥. = 0.

The given filtration yields a spectral sequence

El, = Hs+t(C„, A-s/A_s+l) ~ HS+I(C„,V„).

The F'-term is indicated below. (Subscripts n, n — 1 are deleted for convenience.)License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

K*(Z/p2 Z) AND RELATED HOMOLOGY GROUPS 13

H3(C,W) = 0 (A W®W for p = 5)

H2(C,W) - 0

H^C.W) = 0

HQ(C,W) = 0

H3(C,M) = 0

H2(C,M) = 0

Hj/C.M) = 0

H3(C,W) = 0

H2(C,W) = M

HQ(C,M) = M <-^t±- H^C.W) = M

HQ(C,W) - 0

s + t = 3

8 + t = 2

8 + t = 1

s + t = 0

(For example, H2(C, W) = H2(W, W)F, = {A2W®W®W® W)¥. = M.) Since

H0(C, V) = 0, í/¿ , must be onto and hence it is an isomorphism. Hence, £2 looks

like:

(/3w®W, p = 5)

s+t=0

Thus, we have

Lemma 5.3.

n3(c„, V„) = 0 (p > 5), H2(C„, V„) a MB_„

//,(Q,FJ = 0, H0(Cn,Vn)=0.

For p = 5, H3(C„, V„) is a quotient of A 3 W„_, 0 W„_,.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


We next consider the spectral sequence associated with the decomposition Bn =

Bn_x X C„. From H0(C„, Vn) = 0, we conclude H0(Bn, FJ = 0 for all n > 1. Re-

placing n by n - 1, we see H0(Bn_x, Mn_x) = Z/pZ. HX(BX, M,) = tf,(F*,F) = 0,

so we may assume inductively that Hx{Bn_x, Mn__{) — 0. It follows easily that

Hr(Bn, F„) = 0, r = 1,3, and H2(B„, Vn) = Z/pZ. See the E2 diagram below. (For

p = 5, it suffices to note that H0(Hn_

subgroup of diagonal matrices.)

Ol

*/p.

O

A3W„_X®W„_X) 0 for Hn_x the

To complete the proof, we need only show the existence of a stable nonzero class

in H2(GL(n, p), Mn). (For, we know by the above that H2(GL(«, p), Mn) =

H2(GL(n, p), F„) < Z/pZ.) To see this, we show that

H2(GL(2, p),M2)^ H2(GL(n, p),Mn)

is a monomorphism with the left-hand side ^ 0. Dually, we can show instead

H2(GL(n, p),Mn)^ H2(GL(2, p), M2) ^ 0 is an epimorphism. However, A® B

-» Tt(AB) provides a GL(/i, p)-pairing of Af„ 0 Mn -» F which gives an isomor-

phism Mn^ Mn. Under this isomorphism, the dual of the inclusion i: M2^> Mn is

the map /': Mn -* M2 which given A G Mn picks out the 2 X 2 block of A in the

upper left corner. Note y ° / = id.

From the comparison,

1 -» M2 -» GL(2, p2) 1: t2 G//2(GL2,M2)

1

1/ I

GL(«,p2)

GL(2,p)

J,â:

GL(n,p) - l:tnGH2(GLn,Mn)

we conclude /*(t2) = k*(in) G //2(GL(2, p), Mn). However, the map under con-

sideration is7„ o A:*: H2(GL(n, p), M„) -» H2(GU2, p), M2), so

j*(k*(ln)) =J*{i*(li)) = '2-

To complete the proof, we show that i2 is nontrivial. If i2 were trivial, we could

find an element z in GL(2, p2) in the coset of [J, {] such that zp = 1. But z can be

written z = I + E,,i2 + pA where

EX2 — A G Mn(Z/pZ).

Hence,

(1 + En +pA)p = 1 + p(F,2 + pA) + ( £ )(£I2 + p^l)2 + ■ • • + (EX2 + pA)p.

It is easy to see that (pj){Ex2 + pA)J = 0, 2 </' <p. Also, (F12 + pA)p = 0 for

p 3* 5 since every term has at least one factor F,22 or two factors pA. Hence,

z"= 1 +pEX2¥= 1.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

KjZ/p2Z) AND RELATED HOMOLOGY GROUPS 15

6. Calculation of //*(C„, V„ A F„). In what follows, we shall calculate

Hr(B„,V„oVn) for r = 0,l, Hr(B„,V„AV„) for r = 0,1,2, H0(B„, V„ 0 A 2F„),

Hr{B„,A3Vn) for r = 0,l, and //0(5„, A 4F„). Our basic method is to couple

the spectral sequence of a filtered module with appropriate coinvariant (dually

invariant) calculations. We proceed generally in the indicated order except

where necessary to pursue inductive arguments. In particular, we calculate

Hr(C„, V„ A F„) (to be used primarily for Hr(Bn, Vn A F„)) in this section.

Proposition 6.1. (0) Forn>2,p> 5, H0(C„, Vn A F„) = 0.

(i) For n > 3,p > 5, there is an exact sequence

0^Hx(Cn,Vnf\Vn)Ker{Mn_x®Mn_x^Mn_x}

dx(A2Wn_x®A2Wn_x)Mn_xAMn_x->0

which splits if n z -1 mod p.

(ii) For n 3s 3, p > 5 or n > 4, p = 5,

#2(C„,F„AF„)SMn_x®Mn_x

dx(®2Wn_x® A2W„_X)'

For n — 3, p = 5, /Aere is a« exaci sequence 0 -> F -» H2(C3, V3 A F3)

M2 0 Af2/Im d ' -> 0, wfcere £ « a quotient of A2W2® A2W2 = Z/5Z.

(A fuller explanation of the notation appears below.)

Proof. (0) for n = 2 follows since A A B \-> [A, B] defines an isomorphism

F2 A V2 a V2.

We consider the following filtration of Vn A Fn.

F A Fn n

V„ A Y

»;-i A ^,-

^3 = *n A A,

í<4 = J;a»;_i

i

wn_x 0 if„_

Io

^„-.am„-,

^-,®M,_,

r«-i rn-1

Af„_, 0 IF..



This induces the spectral sequence

El,, = Hs+\Cn,^-\ - Hs+[(C„,Vn A V„).

We exhibit the F'-term for p > 5. (Subscripts and Cn are omitted for brevity.) (See

diagram.) The necessary calculations and the case p = 5 are outlined in the appen-

dix. The differentials are given as follows: we have the complementary inclusion

onto a direct summand ( p > 2)

i: A2W ^W®W, i(uAv) = u®v-v®u.

Use V for the corresponding morphism for W. Then í/¿ 2 is the composition

« - tvíst

A2W® A2W -^ W® W®W®W -* W®W®W®W=M®M.

dl 3 is the composition

, „ -(twist)®/' „ twist

®2W®A2W -> ®2W® ®2W ^ M®M.

Note that dxQ 2 and dx0 3 are monomorphisms onto direct summands.

dl, 2: M 0 M -> M is given by

</iip2(s®r) = (r + Tr(r))s,

which is easily seen to be an epimorphism. It follows that F2 looks like

whereF2, 3 = Coker(¿¿ 3: ®2W® A2W ^ IF0M)and

2 Ker(rfl1|2:Af 0M->Af)

~''2 \m(dx02: A2W® A2W ^ M ® M) '

Then d2_X2 is the negative of the morphism obtained by restricting the standard

projection 77: M 0 M -» M A Af to the kernel indicated above. (One sees easily that

■n: M ® M ^ M A M kills Im dx02.)

Lemma 6.2. d2x 2 is an epimorphism; it is a split epimorphism if n z -1 mod p.


K*(Z/P2Z) AND RELATED HOMOLOGY GROUPS17

<3

9

CM -

<

3

®

II

<3

oII

<

o- Il


18 LEONARD EVENS AND E. M. FRIED LANDER

Proof. It suffices to show Ker(A/0 M -» M) -» M A M is an epimorphism.

etj A ekt (' ^ k OTJ ̂ ') is the image of the following element in the kernel:

eu 0 ekl if /' t^ / and £?*=/,

— etj 0 e,y if /' = /, but /' ¥=j and k ¥^j,

l(e„ ® e„ - ew 0 e„) if i=j = I (then * * i =j),

eu ® */< - ify ® *,> if í = /, fc = y (then i^k=j),

-ekk 0 ey if Ä = / but; ^k,i¥^j,

-lie, 0 e„ - e„ 0 <?,,) if * = / =j (then / * k =j),

■kk (<?,, 0 e„) if A: = / and i = j (then y = i¥=k).

For the splitting, define the GL(« - 1, p)-map \p: M A M ^ M ® M by

t(S AT) = S®T- T®S + I®[S,T] + I® ((TrS)T- (TtT)S).

Note that dx_x 2(<KS A 7")) = TS + (Tr T)S - ST - (TrS)F + [S, T] + 0 • / +

((TrS)F - (Tr T)S)I + 0 • / = 0. Read modulo Im dx02, </< splits 77 since

ir($(S AT)) =2S AT+ I A[S,T]+ I A (Tr ST - Tr TS).

[To see 771// is an isomorphism, examine its effect on the sequence 0 -» F ->XA/ A M

^VAV^0(V= M/FI) where A(S) = / A 5 (5 representing 5). On X( F) = / A

F, 77^(/ A T) = 21 A T + I A (n - \)T = (n + 1)1 A T. 77^ clearly induces multi-

plication by 2 modulo I A V. Hence, for p > 2, -rnp is an isomorphism «n+lïO

mod p.]

Looking at E \ we are done.

Appendix. We indicate briefly how some of the terms in E ' are calculated.

H0(c„, A 2 W) = H0(W, A 2 W)F, = ( A 2 w\. = 0,

//,(c„, a2if) = (w® a2if)f. = 0,

H2(Cn,A2w) = [(A2 W® W) ® A2IFJF, = A2W® A 2 IF,

H3(Cn,A2w) =[(A3IF© ®2IF) 0 A2IF]f, = ®2W® A2W,

Hx(Cn,W® M) = (W® W® M)F. =M0 M,

H2(Cn,W® M) = [(A2IF© W) 0 W® m\f, = W®W® M = M® M.

The other calculations are similar. For p = 5 we have exceptionally

H2(Cn,W A W) =[(A2IF© W) 0 IF A w\f, = A2IF0 A 2 IF

and H3(C„, M®W)= A 3 W ® M ® W (r4 = 1 in F5*). Note for n = 3 (n - 1 =

2), the latter term is zero, but H2(C3, W2AW2)=A2W2®A 2W2 s Z/5Z.

To calculate the differentials we follow the procedure indicated in §5. See the

table of notations for terminology.


Kif(Z/p2Z) AND RELATED HOMOLOGY GROUPS 19

^ô,2: A2W <Z H2{W,Z/pZ) is spanned by products u D v, so we need only

calculate í/¿ 2 on elements of the form u n u 0 / A g, f, g G IF. Choose

A0

/

0

0

to represent / G W and similarly for g. We use the fact that d is a derivation relative

to the Pontryagin product on the chain level and related facts outlined in §10.3. We

have

d([u] n [v] 0/Ag) = (d[u] n v - [u] n </[«]) 0/Ag

= [(«[•]-[]) n[.l-[.]n (•[•]-[•])] ®/Af

= [o]0[(-«)(/Ag)-/Ag]

-t«]0[(-o)(/Ag)-/A|].

However, calculation shows

(~u)f =uf

f-(/«)«

/«(note/« = Tr(«/))

and similarly for g. It is sufficient to calculate ( — u)(f A g) — /A g modulo A2.

That means we may ignore terms of the form

0 0A

0

L0and A

We are left with(-w)(/A g) -/A g =/ A (~ug) + (~uf) A g where

0 TrF

Hence,

¿([u]n[t>]®/Af)=[t>l0[fA(«/)-/A(«g)]

-[u]®[gA(Vf)-fA(^g~)].

In the identification M = W ® W, uf ss u ® f. Unraveling what the above expres-

sion represents in HX(C„, IF0A/)= IF0IF0M= IF 0 IF 0 IF 0 IF, we have

dr\,2(u A v ® f A g) = (v ® g) ® (u ® f) - (v ® f) ® (u ® g)

- (u® g)®(v®f) + (u®f)®(v® g)

which after twisting yields

(ü0«-u0u)0(g0/-/0g) = (M0u-ü0w)0(/0g-g0/)

as claimed.

dxQ3: ®2W CL H3{W,Z/pZ) is spanned by elements of the form u n p(v) where

p(v) G H2(W, Z/pZ) is represented by p[v] = Sf=~0' [t>, iv\. We have

d([u] n p[v] 0/Ag) = («[.] - [•]) n p[v] 0/Ag- [«] n dP[v] 0/Ag.



However,

dp[v] =d% [v,iv] = 2 (v[iv] -[(/+ l)o] + [v])

= v{l[iv])-l[iv].

Hence,

i/([«]np[ü]0/Ag)=p[t)]0[(-M)(/Ag)-/Ag]

-2["]n[iü]0[(-ü)(/Ag)-/Ag].

Calculating modulo A2 as above, we have

d([u] n p[v] 0/Ag) = p[v] ®[gA(iTf)-fA (ûg~)]

- 2 [«] n [iv] ®[g A (Vf) -/A (ig")].

Given //2(C„,IF0 M) = [(p(W) ® A 2 W) ® W 0 M]F. sp(fF)® ^®MsW

0 IF0MsM0Af, and noting that the second group of terms is in A 2 W ® W

0 M, we have

<3(W n p(u) 0/A g) = p(u) 0 (g 0 M ®/-/0 M 0 g).

Identifying p(V) = V, we have

d¿3(t/0t30/Ag) = ü0g0ií0/-t)0/®«0g

= -(v ® u) ® (f® g - g®f) (after twisting).

dLl2mddlX2:Hx(C„,W®M)= W ® W 0 M s M 0 M. Represent u 0 / 0 F

by[w]0/A f. Then

</([«] ®/Af) = («[.]-[.])®/Af

= []®[(-«)(/Af)-/Af].

However,

(-u)f = f-(uf)-(fu)ü where ¿7=- .

Aho(-u)f= f+rû where Tu = (T + Tr F) u. Thus, modulo ^4 = X A W,

(*) d([u] 0/A f) =[■] ®[/A rä-Âf].

Hence, under the identification w ® / - «/ G M, W®W^W®WsíM, we have

di] 2(«/0 F) = (T'u)f = T'(uf). Since the matrices «/span M, we may write

rfi, 2(S® F) = (F+Tr(F))S

as claimed.

Examining formula (*) a bit more carefully, we see that dl, 2 is represented by the

restriction to Ker i/,12 of the morphism defined by uf ® Th-> -uf A T or S 0 T i-» -S

A F as claimed.

It remains to deal with the casep = 5. We show

dx_41: H3(Cn, M ® W)= A3W® M® W ^ H3(Cn,W A W)

= A2IF0 A2IF


Kjf(Z/p2Z) AND RELATED HOMOLOGY GROUPS 21

is an epimorphism. Then £25 6 = 0 and the argument proceeds as before. To

calculate dx_41, we calculate

d([u] n ([t>] n [w]) ®TAx), u,v,w,x<EW,T<E M.

Use (-u)T = T + T'u, ( — u)x = x and the fact that d is a derivation with d[u] =

"[ " ] — [ ' ]• The above expression is

[v] n [w] ®Yü a x - [u] n [w] ®Yv ax + [u] n [v] ®Yw ® x.

Hence,

dL4 7(u Av A w® T® x)

= v A w ® T'u A x - u A w ® T'v A x + u A v ® T'w A x

where T' = F+Tr(F).

To see this is an epimorphism, note that, for / ¥" i, j, k, e¡ A e^ A e¡ 0 ekl ® x i-»

e, A ej ® ek A x. For n = 4 (n — 1 = 3)

ex A e2A e3® ex3® xr^ ex A e2® ex A x.

Similarly, ex A e2 0 e2 A x, e2 A e3 0 e2 A x, e2 A e3 0 e3 A x, ex A e3 ® e, A x,

and e, A e3 ® e3 A x are in the image. Since x is arbitrary, we are done with the case

n 3= 4. For « = 3, there is an additional nonzero term A 2 W2 ® A 2 W2 s Z/5Z (at

the upper left corner of the diagram) in E2. It may persist in F°° leading to the exact

sequence given in the statement of the proposition. D

7. Homology of F„ ° Vn and F„ A V„.

Proposition 7.1. For n>2,

H0(Bn, Vn o Vn) = /F0(GL(«, p), F„ o F„) = Z/pZex2°e2x,

and

H0(B„, M„ o M„) = H0(GL(n, p), M„ ° M„) = Z/pZexxo exx + Z/pZe12°e21.

Proof. We start with M„ ° Mn. Write [x, a] = x(a) — a for x G ¿?n, a G Mn ° Mn.

Using the action of diagonal matrices, we see that etJ ° ekl = 0 mod[Än, Mn ° Mn]

unless í ¥=j, k =j, I = i or i —j,k— I. Hence, H0(Bn, Mn ° Mn) is spanned by the

cosets of e¡¡ ° e¡¡, i <j, and eu ° e^, i <j. Consideration of [xsi, e, ° ejs], shows

esj o ejs = e,: ° e„ for 5 < i <j. Similarly, eit ° eü = e¡j ° eJt for /' <j< t. Considera-

tion of [xsi, ejs ° ejj] shows ess ° e^ = e„ ° e}j for 5 < i <j, and similarly e„ ° ejj =

e„ ° eu for i <j < t. Finally, consideration of [x¡¡, e¡¡ ° e¡¡] shows e¡¡ ° e^ = eu ° en

— e¡j ° ejt for /' <j. It follows that e,, ° exx and ex2 ° e2x span. That these are

independent follows by considering the values assumed by the /^-invariant forms

defined by <t>(A ° B) = Tr(,4)Tr(5) and t(A ° B) = Tr(AB). In fact, since those

functionals are GL(«, p)-invariant, the result also follows for the larger group.

To obtain the result for V„°V„, consider the exact sequence 0 -> Vn ° Vn -* Mn ° Mn

->Mn -» 0 where X+ (A ° B) = (TrA)B + (TrB)A. Note X+ (ex2 ° e2X) = Q and use

Proposition 5.2. □



Corollary 7.2. In H0(Bn, Mn ° Mn),

exx°e22 = exx ° exx - ex2° e2x.

Proposition 7.3. For n>2,

HQ(Bn, Vn A V„) = 0, H0(Bn, Mn A M„) = 0,

H0(Bn,V„®V„) = Z/pZe]2®e2X,

H0(B„, M„ 0 M„) = Z/pZe„ 0 exx + Z/pZeX2 ® e2X .

Similarly for GL(«, p).

Proof. Use Proposition 6.1(0) to derive the first statement. For the second, use

the exact sequence

0^VnAV„-*M„AMn^V„^0,

where \~~ (A A B) = (Tr A)B — (TrB)A, and then use Proposition 5.2.

The third result follows from Proposition 7.1 and the decomposition Vn® Vn =

A 2 Vn © §2F„ ( p > 2). The last result is analogous. D

We shall need some congruences in H0(Bn, Mn® Mn) for later reference.

Proposition 7.4. For n>2,we have

e®f = f®e, exx®e22=exx®exx-eX2®e2X

modulo [Bn, Mn ® M„].

Proof. Use H0(Bn, A 2 Vn) = 0 and Corollary 7.2.

Proposition 7.5. For « 2* 2, Hx(Bn, V„ ° F„) = Hx(GL(n, p), V„ ° V„) = 0. Also,

Hx(Bn, Mn o Mn) = Hx(GL(n, p), Mn o Mn) = 0.

Proof. Note first that for each n the first statement implies the second. This\*

follows from Proposition 5.2 and the sequence 0 -» Vn ° Vn -> Mn ° Mn -> Mn -> 0.

We argue inductively. For n — 2, HX(B2, V2 ° V2) = HX(U2, V2 ° V2)H . Let x gener-

ate the cyclic group U2. As an //2-module

" Tx(V2oV2)Hx(U2,V2oV2)=Wx

where Tx = 1 + x + x2 + ■ ■ ■ + xp '. Direct calculation shows (F2 ° V2)Ul

Z/pZ(hx o hx + 4e2X o ex2) + Z/pZe,2 » ex2.

t =

acts trivially on the first factor and as (txt2 x)2 on the second. Since it acts as i, on

Wx, Wx ® (F2 o V2)Ü2/TX(V2 o F2) does not have a trivial irreducible /^-constituent.

Hence, HX(B2, V2 ° V2) — 0 as required. For n > 2, we consider the following

filtration.


Kif(Z/p2Z) AND RELATED HOMOLOGY GROUPS 23

A0 = S2F„

Ax = Vn o *„

S2^„-«—i-

^3 = a; o x„

A4 = Xno Wn_

A5=W„_xoW„_x

Wn-,®Xn

w„_x 0 if,_,

0

Wn_x®Mn_

S2Mn-l

M^,®^.,

I0

This filtration yields a spectral sequence

El, = Hl+t\Bnt-^-^ ^Hs+t(B„,A).

To calculate the F'-term, we proceed as follows: Each Hr(Bn, As/A_s+X) is calcu-

lated using the spectral sequence of the split extension Bn = Bn_x X C„ (C„ = F* X

IF _ ).

5 = 0. H0(Cn, %2Wn_x) = (S2IF„_,)F. = 0, so

H0(Bn,S2Wn_x) = tfoK-i, H0(Cn,S2Wn_x)) = 0.

//,(C„, S2^„_,) = (IF„_, ® S2^,)F. = 0, so

Hx(B^x,S2Wn_x) =HQl(Bn_x,Hx(Cn,%2Wn_x)) =0.

//2(C„,S2IFn_1)= A2^„_,®S2IF,_„so

ff2(5B,S2JFn_,) = H0(Bn_x, H2(Cn,%2Wn_x)) = H0(Bn_x, A2Wn_x 0 §2IF„_,).

By Proposition 7.3, H0(Bn_x, M„_, 0 M„_,) = Z/pZe,, 0 e,, +

Z/pZeX2® e2x. Also, A 2 IF 0 S2IF is a direct summand of (IF 0 IF) 0 (IF 0 IF)

= M ® M. Hence, H0(Bn_x, A 2 Wn_x 0 S2IF„_1) is generated by the projections of

exx 0 exx and e12 0 e2X. However,

exx® exx^ ex® ex® ex ® ex \-+ex Aex® e] ° ex =0,

12 ** c21 ex ® e2® e ® e' \-> ex A e2® e2 ° e2 „ „i

Going the other way yields

ex Ae2® e2°ex y^{(ex 0 e2 - e2® ex) ® {-(e2 ® ex + ex ® e2)

:\[ex2 0 e2X - e22 0 e„ + exx 0 e22 - e2x 0 ex2].



However, we know from Proposition 7.4 that eX2 ® e2X = e2X 0 eX2, exx ® e22 =

e22 ® exl, soe, A e2 0 e2 ° <>' h> 0. Hence, H2(Bn, §>2Wn_x) = 0.

5 = -1. H0(C„, Wn_x ® Mn_x) = 0, so H0(Bn, H>_, 0 Mn_x) = 0.

Hx(Cn, W„_x ® Mn_x) = 3/w_, ® M„_„ so //,(/?„, lFn_, 0 M„_.) =

//o(5„„„ M„_, 0 Mn_x) = Z/peu 0 eu© Z/pe12 0 e2X.

5 = -2. //0(C„, Jf„_^0 IF„_,) = M„_„ so H0(Bn, W„_x 0 IF„_,) =

H0(Bn_x,Mn_l) = Z/pZen.

Hx(Cn,Wn_x 0 IF„_,) = 0, so Hx(Bn,W„_x ® W„_x) = Hx(Bn_x, Mn_x) = 0.

s = -3. H0(C„, S2M„_,) = S2M„_„ so H0(Bn, S2M„_,) = //0(5„_„ S2M„_,) =

Z/pZexx 0en+ Z/pZeX2® e2X.

Hx(Cn, S2M„_,) = 0, so //,(/?„, S2M„_,) = //,(£„_„ S2M„-i) = ° (assumed in-

ductively).

s =s -4. /F.(C„, As/A_s+X) = 0, r = 0,1, in each case. Hence, //,(£„, As/A_s+X)

= 0,r = 0,l.

The Fj ,-term of the spectral sequence looks like

Since H0(Bn, Vn ° V„) = Z/pZeX2 ° e2X, it follows easily that Fi, 2 = 0, so

Hx(Bn, Vn o Vn) = 0. (In fact, since eX2 ° e2X is in position ( — 3,3), we can say further

that d]_x 2 is an epimorphism and dlx 2 a monomorphism.)

Proposition 7.6. For n > 2, /? > 5, w have

(a)

HX(B„, V„ A V„) = Hx(GL(n, p), Vn A Vn) = 0;

Hx(Bn, Mn A Mn) = Hx(GL(n, p), Mn A Mn) = 0;


Kit(Z/p2Z) AND RELATED HOMOLOGY GROUPS 25

(b)

H2(B„, V„AV„) = H2(GL(n, p),VnAV„) = Z/pZ.

Proof. First note that using Proposition 5.2 and the sequence 0 -» V„ A Vn -»

Mn A Mn - V„ - 0 we derive Hx(Bn, M„ A Mn) = 0 from Hx(Bn, Vn A Vn) = 0.

To prove (a) and (b) we argue inductively.

For n = 2, as in the proof of Proposition 6.1(0), HX(B2, V2 A V2) = HX(B2, F2) = 0,

and H2(B2, V2 A V2) = H2(B2, V2) = Z/pZ by Proposition 5.2.

For n > 2, use the spectral sequence of the group extension Bn = Bn_x X C„. To

prove (a), since Hx(Bn_,, H0(Cn, Vn A Vn)) = 0 by Proposition 6.1, we need only

show H0(B„_X, HX(C„, V„ A VJ) = 0. From Proposition 7.3,

//<,(/?„_,, M„_, A M„_,) = 0,

and we may assume inductively that Hx{Bn_x, Mn_x A Mn_x) = 0. Hence, by

Proposition 6.1,

H0(B„ x, HX(C„,V„ A FJ) S//0(5_,,Ker/Im)

which in turn is a direct summand of H0(Bn x,Ker). Since Hx(Bn_x, Mn_x) = 0,

0-//0(5n_„Ker) -. H0(Bn_x, Mn_x ® Mn_x) - H0{B„_X, M„_,)- 0

_ II _ II _Z/pZe]2®e2x + Z/pZexx 0 <?,, Z/pZexx

is exact. Since e12 0 e2, h-> e,, and en ® en h» 2eu,

^0(5n_„Ker) = Z/pZ(2en 0 e,2 - eu 0 en ).

Consider next the morphism d\ 3: A2IF0A2lF^®2IF®<g>2IF=M®M.

e, Ae20e' Ae2i^(e1 0e2 -e20e,) 0(e' 0e2 - e20e')

- en ® e22 - e12 ® <?21 - e2x 0 e12 + e22 ® <?,,

(Proposition 7.4) = 2(exx ® e22 — ex2 0 e21)

-2(*n ®e„ -eX2®e2X ~ eX2® e2x)

= 2(exx ®exx -2eX2®e2X).

Hence, H0(Bn_x, Ker/Im) = 0 as required.

We next consider (b). By Proposition 6.1, H0(Bn_x, H2(Vn A F„)) =

H0(Bn_x,(Mn_x ® Mn_x)/lmdx) which (as shown in the proof of 6.1) is a direct

summand of H0(Bn_,, M„_, 0 M„_,) = Z/pZe12 0 e21 + Z/pZexx 0 <?,,. (For «

= 3, p = 5, use the fact that //0(£2, A 2 H^2 0 A 2IF2) = 0.) Referring again to the

proof of 6.1, we recall that Im dx ss®2W„_x ® A2IF„_,. Note that eX] 0 exx =* (ex

®ex)®(ex ®ex)t^>ex ®ex®exAex=0, so HQ(B„_x,lm dx) < Z/pZ. On the

other hand, going the other way,

ex 0 e2 0 ex A e2 i-> ex ® e2 ® (ex 0 e2 - e2 0 ex)

-«ii ®e22-el20e21 ZO.



Hence, H0(Bn_ „ Im dx) = Z/pZ, and thus

H0(Bn_x,{Mn_x®Mn_x)/lmdx)=Z/pZ.

Continuing with (b), we shall show that Hx(Bn_x, Hx(Cn, V„ A V„)) = 0. We use

the short exact sequence

Ker(Mn_x®Mn_x-+Mn_x)0-^Hx(C„,VnAVn)

■Mn_xAMn_x^0

\md]

of Proposition 6.1(i). If « s -1 mod p, the sequence splits, so we are reduced to

showing Hx(Bn_,, Ker/Im) = 0. If n = -1 mod p, we obtain the same reduction by

means of the following argument (suggested by the referee). It suffices to show that

H2(Bn_x, Ker) -» H2(Bn_x, Mn_x A Mn_x) is an epimorphism. In the proof of 6.1,

we derived the splitting (for « s -1) by showing that 77^ is an isomorphism of

M„_, A M„_,. (Refer to Lemma 6.2 for notation.) That in turn was deduced by- A

examining the effect of 77 ̂ on the short exact sequence 0 — FB_, -> Mn_x 0

M„-1 * K-1 A K-1 ~* ° (where \(f) = /„_, A F). If m = -1,77^ is unfortunately

trivial on Im À. However, we still have \irip = 2x so Im(2id — 77t//) Ç Im X. Thus, it

suffices to show X^(H2(Bn_x,Vn_x)) Ç ■n¡l:(H2(Bn_x,Ker)). For this, consider a:

M„_2 -» M„_, 0 M„_, defined by

a(r) = /„_, 0 F- F0 *„_,„_, - (Tr(F)/ (« - l))/„_2 ® ln_2.

Note that Ima Ç Ker(M„_, ® M„_, - M„_x). Also, if we set t(T) = f, then

M„_2 - Ker(M„_x®M„_x^Mn__x)

11 177

Pi-i - Mn_xAMn_x

commutes. Since i„: H2(Bn_2, Mn_2) -» H2(Bn x, Fn_,) is an isomorphism by Prop-

ositions 5.1 and 5.2, it follows readily that Xif(H2{Bn_x, F„_,)) C Im77* as required.

To show that Hx{Bn_x, Ker/Im) = 0, we use the fact established in the proof of

Proposition 6.1 that Im is the direct summand of Ker; thus, we need only show that

//,(£„_!,Ker) = 0. Since #,(£„_„ Mn_x ® M„_x) = 0 by part (a) and Proposition

7.5, it suffices to show H2(Bn_x, Mn_x 0 Mn_x) ^ H2(Bn_„ Mn_,) is an epimor-

phism. This follows from Proposition 5.2 and the commutative diagram

Mn_,0Mn_, - M„„,

ÎP /

^-.

where we define p(T) — /„_, ® F.

Since H2(B„_X, H0(Cn, V„ A Fn)) = 0, we may now conclude

//2(GH>, p), F, A F„) =£ H2(Bn, V„ A V„) « Z/pZ.

To conclude that H2(G\(n, p), Vn A F„) = Z/pZ, we need only show it is nontrivial.


K.J/Z/p2Z) AND RELATED HOMOLOGY GROUPS 27

For this, consider

//2(Gl(2,p),F2AF2) - H2(G\(2,p),V2)

1 I

H2(G\(n,p),V„AVn) - H2(G\(n,p),V„).

The upper horizontal arrow is an isomorphism since F2 A F2 s F2, and the right-hand

vertical arrow is an isomorphism by Proposition 5.2. It follows that the lower

horizontal arrow is onto something nontrivial, and we are done. (This argument was

also suggested by the referee.)

8. Higher multilinear modules.

Proposition 8.1. For «3=2, H0(Bn, Mn A Mn ® Mn) = Z/p © Z/p. Also,

HQ(GL(n, p), F„ A V„ 0 Vn) = H0(B„, VnAVn® F„) = Z/pZ (stably).

Proof. First note that the exact sequence 0 -» A 2 Mn 0 Vn -» A 2 M„ ® Mn -»

A 2 Mn -> 0 yields

H0(Bn, A2Mn®V„)^ H0(B„, A 2 M„ ® M„).

In addition, 0 -» A 2 F„ 0 V„ - A 2 M„ ® Vn -> Vn 0 V„ - 0 yields

0 - #„(*„, A 2 V„ 0 F„) - //0(ß„, A 2 M„ 0 F„)

-+H0(Bn,Vm9VH) = Z/p->0

(similarly, for GL(«, p)).

For n = 2, A 2 F2 0 F2 s V2 0 F2, and #0(fl2, F2 0 F2) = Z/pZ. The exact se-

quence above gives the desired result for A2 M2 0 M2. On the other hand, for

«3=3, using those sequences, we need only show H0(Bn, A2 Mn ® Mn) =

Z/p ® Z/p (stably) (similarly for GL(n, p)).

Assume n 3= 3. By duality, we have

"oí*,,. MnAMn® M„) ^ H°(Bn, M„AM„®Mn)^ H°(B„, Mn A M„ ® M„).

(Recall that M„ is self-dual as noted in the proof of Proposition 5.2.) Since it is

technically easier to calculate invariants, we calculate the last group.

Let z G Mn A M„ ® Mn be invariant under Bn. Using the action of the torus, we

conclude that z must be of the form

1 «i,**/; A *;* ® **/+ 2 ßukeu A eß.0 ekki¥=j=£k¥=i (</'

i+k+j

+ 1 y,JkeIJêkk®ejl+ 2 *,jeU A eß ® ««i¥=j¥=k¥=i i¥*j

+ 2 e,,^ A *„ ® *,, + 2 f,ye„ A ejJ 0 e,,

i¥=k¥=j



We now make use of the following argument suggested by the referee. Any z of

the above form which is invariant under Bn (defined over the the prime field F) is also

invariant under Bn(F) (defined over the algebraic closure). (Note that MnAMn® Mn

is contained naturally in the corresponding object defined over the algebraic

closure.) Namely, such a z is a sum of terms each of which has weight 0 and hence is

invariant under the torus //„(F) over the algebraic closure. If such a z is also

^„-invariant, then it must also be 5„(F)-invariant because Bn(F) is generated by Bn

and //„(F). On the other hand, it is well known that the 5„(F)-invariants are identical

with the GL„(F)-invariants. (To see this, note that the orbit of a /?„(F)-invariant

element is the image of the projective variety G1„(F)//J„(F) in the affine variety

(Mn A Mn 0 Mn) ®FF and is thus a single point.)

In particular, any ß„-invariant element z of the above form is invariant under the

symmetric group 2„ (contained in G1„(F)). Using the transpositions (ij), we con-

clude that ßiJk = r¡¡jk = 0 for all /' ¥=j # k ¥= i. Using permutations sending i to i',j

to/', and k to k', for various triples of indices, we see that aik = a is independent of

i,j, and k. Similarly, we may write yijk = y, 6, = 8, eu = e, \i} = f, and 0,- ■ = 0 since

these coefficients are also independent of the relevant indices.

Consider the coefficient of e¡-A ejk 0 e¡¡ in xik(z) — z where /' =£ j ¥= k =£ i.

Because ßkjj = 0, this coefficient is aijk — 8ijk. Thus, a = 8.

Consider the coefficient of e¡¡ A e¡¡ 0 e¡¡ in x¡,(z) — z where i =£ i. It is ô/; + ô„ +ij j, ji ji\ / j ij ji

Eij ~ tip so tnat f = 26 + e.

Consider the coefficient of ei} A e k 0 e¡¡ in xjk(z) — z where i ¥=j =£ k ^ i. It is

aijk - Su + yiJk, so that y = f - a.

Consider the coefficient of eu A e(j ® eu in x¡Az) — z where i ¥=j. It is Ö,- ■ — e¡¡ —

6,7, so that £ = 0-8.

Solving these equations yields a = 8, f = 6 + 0, y = 0, and e = -6 + 0, so that

we may conclude that a ^„-invariant z (if such exists) must be of the form

« 1 e,jêjk®ek, + 0 2 eiJAekk®ejii^j*kj=i i^j¥=kî

+ 8 2 eu A eJt 0 e„ + (0 - 8) 2 et) A e„ 0 eß

+ (0 + 8)2eiJA ejJ ® eJt + 0 2 eu A ßjJ ® eu

= 8 2 eIJAejk®eki + 0 2 e,JAekk®ejl.i J, k i J, k

Thus, by duality, H0(Bn, A 2 M„ 0 M„) < Z/p © Z/p. To see that we have

equality (even if Bn is replaced by GL(«, p), and even stably), consider the (stable)

GL(«, p)-invariant forms defined by

<¡>(AAB®C) = Tr(^)Tr(BC) - Tr(5)Tr(^lC),

and

\¡s(A AB®C)= lr(ABC - BAC).


K.J(Z/p2Z) AND RELATED HOMOLOGY GROUPS 29

These are independent since <j>(ex2 A e2x 0 exx) = 0, 4>(e]X A e22 ® exx) = -1,

Hel2 A e2X 0 e„) = 1, ^(exx A e22 0 exx) = 0.

Corollary 8.2. For « 3= 2, H0(Bn, A3Mn) = Z/pZ. Also, H0(Bn, A3Vn) =

H0(GL(n, p), A 3 F„) = Z/pZ (stably).

Proof. Consider the exact sequence 0 -> A 3 Vn -» A 3 M„ -» A F„ -» 0, where

À(yl A B A C) = Tr(A)B AC - Tr(B)A AC + lr(C)A A B. (Note that the co-

domain of X is A2 M, but that in fact Im X = A F.)

H0(Bn,A3Vn)^H0(Bn,A3Mn),

(and similarly for GL(«, p)), so we need only consider A3^,. From Proposition

8.1, it suffices to show the epimorphism

Z/pZ = H0(Bn, A2Vn 0 V„) - H0(B„, A3F„)

is an isomorphism (similarly for GL(«, p)). However, this is clear since the right-hand

side is non trivial. Indeed, the GL(«, p)-invariant form defined by (¡>(A A B A C) =

Tr(ABC - ACB) is nontrivial on A 3 F„ (<¡>(hx A ex2 A e2X) = 2). D

Proposition 8.3. Forp 3= 5, n 3= 2,

^,(^,A3F„) = //,(GL(«,p),A3F,) = 0.

Proof. For« = 2, A3F2 = Z/pZeX2 A«, A e2x.

Since ex2 A «, A e2x is fixed by xX2 and H2, B2 acts trivially, and the result follows

by Proposition 5.1.

For « > 2, we proceed by induction. We form the tower shown as Table 1

(subscripts are omitted). The calculations in the table are done as previously.

One calculation requires further comment.

nx(Bn, A2W„_X 0 W„_x) = H0(B„^X, Hx(Cn, A2Wn_x® Wn_x))

= H0(Bn_,, ®2W® A2W).

However, ®2 IF 0 A 2 W is a direct summand of ®2W ® ®2W ^ M ® M. Thus

H0(Bn_x, ®2W® A 2 W) is generated by the images of e,, 0 exx ^ ex 0 e, 0 e' 0

<?' i-^ e, 0 ex 0 e' A e1 = 0 and ex2 0 e2X - ex ® e2 0 e2 ® ex h» e, 0 e2 0 e2 A e'.

On the other hand, going the other way, we get

e, 0 e2 0 e2 A ex \-+ \ex ® e2® (e2 0 el - ex ® e2)

= 2-[eX2®e2x -exx ® e22]

= \[eX2®e2x -exx ® exx +e120e21]

= 2-[2eX2®e2X-exx®exx]mO.

Hence, HX(B„, A2W„^x®Wn_x) = Z/pZ.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Table 1

Level

0

Factor H (C , -)r n w -)

a2v Ax

2 A¿W @ X {

V A A'X

W © A¿X

A3X

A2X Aw

x A aw

10

3»yiw

AW© M

W© AM

W © (M ® W)

AM® W

M© AW

H =H =0, H ■■A2« © A2« © M

H.=0, H = ©2W © A2W

H =0,H - M® A M

H =M ® M, H,"0

II >H =0 (For P"I' 3H0 Hl U HX=W © AJW)

W°H_=H„(B ,,A2W®A2W®M)

¿ U n-1

H=H (B lf®2W® A W) = E/pZZ

H1=H0(Bn_1,M® A M) = Z/p © Z/p

H-=H„(B ,,M®M) = ZZ/p© ÍZ/p

H ,-H_(B n,M© M) = 0l l n-l

VH0(Bn-l'AM)=2/f2612*lAE21

H,=H,(B -,A3M) = 01 1 n-l

H =H =0 (including p=5)

We examine the spectral sequence

El, = Hs+l(Bn, A_s/A_s+x)^Hs+!(Bn, A)

as previously. If we exhibit only the range containing nonzero terms, the F'-term

looks like the following

O

ï/pZ«n.*hiA€at

•Bip ET1. ,= K<BM., ,A* «/„.^^¿»Oté-^-r V


Kie(Z/p2Z) AND RELATED HOMOLOGY GROUPS 31

To complete the proof it suffices to show dx_ x 3 is an epimorphism. (Then, dL3 4 is an

isomorphism since HQ(Bn, A 3 F„) = Z/peX2 A A, A e2X.)

To show dx_X3 is an epimorphism, we argue as follows. The corresponding

morphism/or C„,

II II

A'^Â^.-^M,., 02W„„I ® A2Wn_x = W„-X®(A2Wn_x® ¡V„_x)

is given by

dx_x 3(u Av®fAg®S) = -[u®(S + TrS)v-v®(S + Tr S)u] ®fAg

(see below). Since we showed above that

H0(Bn_x,®2W„_x ®A2Wn_x) = Z/pZex®e2®e2Aex ,

it suffices to note that

dx_x3(ex Ae2®e2Aex® exx) = ~[ex® e2~ 2e2 0 ex] 0 e2 A ex,

dlX3(ex Ae2®e2 Aex 0 e22) = -[2e, 0 e2 - e20e,] ® e2 A ex.

(Subtracting twice the second from the first yields +3ex ® e2® e2 A ex as required.)

Appendix. Calculation of dx_x 3 for C„:

d([u] H [v] ® f A g A S) = (d[u] (1 [v] - [u] H d[v]) ® f A g A S

= [v] ®[(-u)(f A g A S) - f A g A S]

-[u]®[(-v)(fAgAs)-fAgAS].

However, ( — u)f = f— (uf) — (fu)ü, similarly for g, and

(~u)S = S+Hrü, 5" = 5 + Tr(S).

Hence, modulo

A3= VAXAX,(-u)(fAgAS) -/A g A S =f A f A 5^.

Hence, î/([m] n [v] ® f A g A S) = [v] ® (f A g AS^û) - [u] 0 (/A g A 5^).

Hence, with some twisting, we get

dx_X3(u A v ® f A g ® S) = (v ® S'u - u® S'v) ®fAg

as claimed. □

Proposition 8.4. For n>2, H0(Bn, A 4 Mn) = Z/pZ, //0(GL(«, p), A 4 F„) =

Proof. We calculate H°(Bn, A 4 Mn) instead and use duality. Let z G Mn A Mn

AMn A Mn be invariant under Bn. Using the action of the torus, we conclude that z

must be of the formLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


2 <*ijkeijAeßAekiAeik+ 2 ßijkieu ^ eJk A ekl A eni<j,k i<j,k.lk<l j=£k¥=l¥-j

k¥=jl=l

+ 2 yijkeijAeßAeikAekij<k

j¥=i¥=k

+ 2 W<7Aey*A<?*/A<?//+ 2/</,* i¥=j¥=k¥=i

i, j, k.ldistinct

*tjlfilj A ejk A **/ A *«

+ 2 îjkieijAejiAekkAell+ 2 Vukeu A <?,-/ A eu A ekk

k<li, j, k, /distinct

i, j, k distinct

+ 2 9uetJ A eß A eu AÉ]J+ 2 \jUeu A eß A ekk A e„.i<j /</<*</

For « = 2 only terms of "type 0 " occur. For « = 3 only terms of type à, y, e, r/,

and 0 occur. However, with minor modifications, the arguments we shall introduce

below are also valid in those cases.

As in the proof of Proposition 8.1, we use the referee's argument to conclude that

z is in fact Gl(«, p)-invariant and hence invariant under the symmetric group 2„.

Using the transposition (//), we conclude that aijk = ÇiJk = XiJk = 0 for i, j, k

distinct. Using the cycle (ijkl), we conclude that ßiJkl = 0 for i, j, k, I distinct.

Using permutations sending i to i',j to/, and k to k' for various sets of indices, we

conclude that yijk = y, 8jjk = 6, £ijk = e, t\ijk = r\, and 0¡jk = 0 are independent of

the indices.

Consider the coefficient of e¡j A eß A eik A ekk in xik(z) — z,j < k. It is —yjjk —

Vijk + BkJ„ so y = e - 7,.

Consider the coefficient of e¡¡ A eji A eki A eu in xki(z) — z, i <j. It is r¡¡¡¡ — 8¡Jkl,

so TJ = 6.

Consider the coefficient of e,j A ejt A ekj A eß in xki(z) — z, i </'. It is 0,- ■ — Tjy/A.

— £jkj, so 0 = r/ + e.

Consider the coefficient of e,y A eJk A eik A ekj in xik(z) — z,j < i and/' < k. It is

ckij lijkY,.-t,so-2y = 0.-ykji + euk

From these equations we conclude that e = tj = 8 and 0 = 2e (with appropriate

deletions for « = 2, 3). It follows that an invariant z, if such exists, must be of the

form given below. (Make the appropriate deletions and modifications for « = 2,3.)

2 eIJAejkAeklAe//+ 2 etJ A eß A eki A <?,¡<j,k i, j, k distinct

, j, k, /distinct

+ 2 eu A eJt A e„ A e„ + 2 ^ etJ A eß A e„ A e}i, j, Ar distinct i<J

= 6 2 e,j A ejk A eki Ae„+ 2 eu A eß A e„ A ekki<j.k i¥°j


Kj(Z/p2Z) AND RELATED HOMOLOGY GROUPS 33

Thus, by duality, H0(Bn, A4Mn) < Z/pZ. To complete the proof, consider the

exact sequence 0->A4Fn->A4M„->A F„^0 where

4

X(AX AA2 AA3 AA4) = 2 (-l)i+^r(Al)Ax A ■ ■ ■ AAÂ ■ ■ ■ AA4./=i

(Note as in the proof of Corollary 8.2 that A initially takes its values in A 3 M„, but

it is possible to show that Im X = A3 F„.) Since Hx(Bn, A3Vn) = 0, H0(Bn, A 3 V„)

= Z/pZ, it follows that H0(Bn, A 4 Mn) = Z/pZ and H0(Bn, A 4 F„) = 0. D

Chapter III. Calculations of d2 for n = 2

9. The relevant p-groups. We now turn to the needed calculations in the spectral

sequence for « = 2. We start by considering the spectral sequence of the group

extension l = F-»t/-»F-*l where (a) U is the p-Sylow subgroup of SL(2, p2)

consisting of all matrices over Z/p2Z of the form

1 + pa b

pc 1 + pda + d — be = 0 mod p,

(b) P = U2 consists of all matrices over Z/pZ of the form [l0*], and (c) V = V2 as

before.

Using multiplicative notation, we choose the following generators:

ForP,

yi -i

o i(Letj? = [¿-{]intt)

For F,

1 0

P 11 + pF2i where F2,

1 +p 0

0 1 -p

1 P~

1 + pHx where Hx = h,,

0 11 + pEx2 where Ex

Then,

(9.1) y~xzxy = zxz2z3x, y~xz2y = z2z32, y~xz3y = z3.

The calculations will be simpler if we change the basis for F to a:, = z,, x2 = Z2Z3-1,

x3 = z32. Then

(9.2)

(Additively, _y-1 has matrix

y lxxy = xxx2, y xx2y = x2x3, y xx3y = x3

1 0 01 1 00 1 1License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


on F.) Since the Charlap-Vasquez theory is for cohomology, we shall make our

spectral sequence calculations for cohomology with coefficients in F = Z/pZ and

dualize later. Thus we shall need the dual bases. Define a,(x ) = 6, . Then,

(9.3) yax = ax, ya2 = ax + a2, ya3 = a2 + a3.

10. The Charlap-Vasquez description of d2. Let l->F->t/-».P->lbea group

extension with F a finitely generated abelian group. Charlap and Vasquez [C-V] have

described

ds2':E¡-'(A)Ê¡+2-'-x(A)

for A any F(i/)-module on which F acts trivially. (F should be a PID and each

Ht(V,F) should be F-free.) We shall use their description in the case F is an

elementary abelian p-group, F = Z/pZ, and A = F.

In order to state the Charlap-Vasquez result, we need to introduce some terminol-

ogy involving certain standard identifications. We have as usual

H'(V,F) = H,(V,F).

(In particular, for a G H'(V,F), p G Ht(V, F), a(p) is calculated by evaluating a

cochain representing a on a chain representing p.) Moreover, H'~X(V, Ht(V,F)) =

H'-](V,F)® H,(V,F), so we have a pairing q: H'(V,F) 0 H'~\V, H,(V,F)) ->

H'-\V,F) defined by

(10.0) q(a®(ß®ix)) = a(n)ß.

q in turn induces a pairing

U,: HS(P, //'(FF)) ®H2(P, H'~X(V, H,(V,F))) - HS+2(P, H'~\V,F)),

that is, U q. E¡-'(F) 0 F22-'"'(#,( F F)) -> F2í+2'_1(F). Then Charlap and Vasquez

assert that for | G F^F),

(10.1) d2-'U) = (-l)Û?(F'-o#(X))

where x G H2(P,V) is the class of the extension, Q: V ̂ H'~\V, Ht(V,F)) is

derived from the Pontryagin product ¡i n v in H^(V, F), and F' G

H2(P, H' \V, Ht(V, F))) is a certain universal characteristic class.

Charlap and Vasquez (and later Sah [S]) have investigated V for F = Z and F

free abelian, but little is known about it in our case. We shall rely simply on the fact

that for the corresponding split extension (x = 0)

d?(t) = (-iyiuqv'.

Thus, we obtain the first term in (10.1) if we can calculate d2pX(i-) by some other

method. This is done in §11.

We concentrate now on calculating the second term (— l)i+'£ U qQj(x)- Let

p,,p2,...,p„ be a basis for H,_X(V,F), y1,Y2,...,Y„ the dual basis for H'~X(V,F),

and f„ £,,... ,!m a basis for Ht(V, F). For z G HX(V, F) = F, let p, n z = 2?=, a,^.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Kjf(Z/p2Z) AND RELATED HOMOLOGY GROUPS 35

Then one may translate the definition of Q in Charlap-Vasquez (called P there) into

the formula:

(10.2) Q(z) = 2a„y,®t;lGH'-](V,F)®Hl(V,F).

We shall be especially interested in the case V = V2 (as in §9), t = 1,2,3. In order

to calculate Q^ explicitly, we need bases and dual bases in the relevant dimensions.

We start with a description of the Pontryagin product in terms of standard chains.

(See the table of notations for definitions of these terms.)

Proposition 10.3. Let X be the standard chain complex over Z for the abelian group

V. The Pontryagin product is induced by a map (of complexes over Z) D : X 0 X -* X

which is consistent with the group homomorphism V X F -» F defined by multiplica-

tion. (X ® X is viewed as a FX V-complex by componentwise action.) That map

necessarily satisfies the rule d(x C\ y) = dx C\ y + (— l)degJcjc n dy. One may choose

this map of complexes so that in degrees 0, 1,2, and 3 it is given by

[]n[] = [-],

[u]n[.] = [-]n [«] = [«],

[u,v] n [•] = [•] n [u,v] = [u,v],

[u] n [v] = [u,v] - [v,u],

[u, v,w] n [•] =[•] n [u,V,w] = [u,V,w],

[u] n [u, w] = [«, D, w] — [t>, w, w] + [u,w,w],

[w, u] n[w] = [u, t>, w] — [w, w, u] + [w, w, u].

Proof. The Pontryagin product may be defined as the composite map Hjy, F) 0

H*(V,F) -> Z/*(FX FF) -> H ¿y, F) where the first component is the Künneth

isomorphism and the second is induced by the group homomorphism V X V -> V.

To compute it on the chain level, we note that X 0 X is a V X F-resolution of Z.

Hence, we need only define a map of complexes X ® X -* X (consistent with

augmentations and differentials) which is consistent with the group homomorphism

V X V -> V. (The map is unique up to chain homotopy.) Since the differential in

X 0 X is given by d(x ® y) = dx ® y + (— l)de%xx 0 dy, the map is necessarily a

derivation as stated. The main content of the proposition is that the chain map

defined above in low degrees has the right properties. One checks this by direct

calculations. For example, </,([■] n [u]) = dx([u]) = «[•] — [•], and

n ((d0 ® id + id 0 <*,)([.] 0 [«])) = n ([•] 0 dx[u])

= n ([■]•(«[■]-[•])) = Nn«[.]-[.]n[.]= «([-]n[.])-[-]n[.] = «[.]-[.].

The other calculations are similarly tedious and we omit them. Note that the map of

complexes has been defined on a Z[V X F]-basis for X ® X, and semilinearity with

respect to V X V -» V has been used extensively. D

Recall the notation p(x) G H2(V, F) for x G V, 8a G H2(V, F) for a G F



Proposition 10.4. Let V = V2, and choose a basis xx, x2, x3 for V = HX(V, F) and

dual basis ax, <x2, a3 for V = //'(FF). Then the following are bases and dual bases for

H,( V, F) and H'( V, F) respectively.

For t = 2, xm n xn, m < n, p(xm), m = 1,2,3, with dual bases aman, m<n, 8am,

m= 1,2,3.

For t = 3, x, D xm n xn, I < m < n, x, C\ p(xm), l, m = 1,2,3 with dual bases

a,aman, / < m < n, a,8am, l,m=\, 2,3.

Proof. Use the formulas in Proposition 10.3 to evaluate the cohomology classes

on the homology classes. For example, (a¡8am)(x, D p(xm)) is calculated as follows.

Let a i denote a¡ viewed as a 1-cocycle on V, and let 8a, be the standard cocycle

representing 8at. Then x, n p(xm) is represented by

p-\ p-\

2 [x,\ n [xn, ixm] = 2 ([xl,xm,ixm][xm,x„ixm]+[xm,ixm,xl])i=0 i=0

and

(a,8am)(x,r\ p(xm))

= 2 (a,(xi)8am(xm,ixm) - a,(xm)8am(x„ixm) + al(xm)8am(ixm, x,)).

( = 0

If / ^ m, there is only one nonzero term, the first term with i = p — 1. If / = m, all

terms for ; = p — 1 are nonzero, but two cancel. In any event, the net result is 1.

The other cases are similar. D

Proposition 10.5. Let 1 -» V -> U -* P -» 1 be the extension in §9, and let p > 3.

Then choosing y as generator for P, we have H2(P, V) = Vp = Z/pZx3 and the class

of the extension x is given by \x3.

Proof. Generally, H2(P, V) = VP/TV where F = 1 + y + y2 + ■ ■ ■ +yp~l. In

our case, it is easy to see that TV = 0. To determine the class of the extension, let

1 -10 1

in SL(2, p2).

y

Then

'1 -/

.0 1

Consider now the commutative diagram:

1 - Gp(z3])=Z/pZ - Gp(y)=Z/p2Z

l<t> I

i - v -. U

Gp(y)=Z/pZIIP 1

The class of the upper extension is represented by z3~ ' ; hence so is that of the lower

extension.

Changing to additive notation, we may identify x with — z3 = jx3. DLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Kj/Z/p2Z) AND RELATED HOMOLOGY GROUPS 37

Proposition 10.6. Let P and V be as in §9, and let p > 3. Then

(Ï) H\V,F)P = Fax,

(ii)H2(V,F)p = Faxa2 © F6a„

(iii)

H3(V,F)P = Fa,a2a3 © Fa,ôa, © F(a,6a2 - a26a,)

©F(«,6a3 + a38ax — a28a2 + a28ax).

Proof, (i) is clear, (ii) follows easily since H2(V,F) = A 2 V + 8V (as an F(F)-

module), and A 2 F s V.

To prove (iii), note first that //3(F,F)= A3F© V8V (as an F(F)-module).

Moreover, A 3 F s F, and

V8V sK0F=A2FffiFoF.

Again, A 2 F = F and a basic invariant is a, A «2. Tracing back through the

identifications yields first ax® a2 — a2® a, and then ax8a2 — a28ax.

To calculate (F ° V)p, let y = 23=, o,a, ° a, + 21<7f»I7a( ° ay be invariant. Then

y(y) — bxax ° a, + b2a2 ° a2 + 2ft2a, ° a2 + o2at ° a,

+ o3a3 o a3 + 2Z?3a2 ° a3 + b3a2 ° a2

+ bX2ax ° a, + o12a, ° a2 + o23ö! ° a2 + ¿>23a2 ° a2

+ ô23a, ° a3 + 623a2 ° a3 + bi3a] ° a2 + bx3ax ° a3.

Hence, b2 + bx2 = b3 + b23 = 2b2 + b23 + bx3 = 2b3 = b23 = 0. Thus, b3 = b23 = 0,

bx2 = —b2, bX3 = —2b2, and

y = bxax ° a, + o2(«2 ° a2 — a, ° a2 — 2a, ° a3).

Tracing through the identifications yields a, » a, h 2cxx8ax and a2 ° a2 —

«i ° «2 " 2a, o a3 i-> 2a26a2 — a|6a2 — a28ax — 2a,6a3 — 2a36a,. Adding axda2

— cx28ax yields

2(a26a2 — a28ax — cxx8a3 — a38ax)

as required. D

Proposition 10.7. Let \ ^ V ^ U ^ P ^ \ be the extension in 10.5, and assume

p > 3. The second term A = (— l)i+l£ U 9(ö„(x)) m me Charlap-Vasquez decomposi-

tion (10.1) h given as follows (s = 0 in all cases):

(i)

£ = a,: A = 0 inH2(P,F) = F,

(Ü)

¿=a,a2: A = 0

| = 6«,: A = 01 <«"W) = ^



(iii)

è = «i«2a3:

£ = <xx8ax:

£ = ax8a2 — a28ax :

-2«ia2

A = 0

A = 0

£ = ax8a3 + a38ax — a28a2 + a28ax: A Ï8at

inH2(P,H2(V,F))=(A2v)P®(8V)P. D

Proof. Recall first that given a F-pairing q: A 0 B -* C, if a G /F(F, .4) is

represented by a G /I and /? G H'(P, B) is represented by ft e 5, then a U gß E

HS+'(P,C) is represented by q(a ® b) provided s or t is even. (See [C-E, Chapter

XII, §7].)

Abbreviate z = \x3, and use the bases and dual bases given in Proposition 10.4

(i)

1 Hz= ^3,soß(z) = \\ ®x3(\ G//°(FF)) (use (10.2)).

Hence (from (10.0)),

«, U ,0,(x) = M«i ® 0 ® *3)) = i«i(*s) = 0-

(ii) x, n z = jx, n x3, x2 n z = ^jc2 n x3, x3 n z = o, so

Q(z) = ¿(a, 0 x, n x3 + a2 0 x2 n x3).

Hence, a,a2 U, ß,(x) = i[^(«,a2 ® («, ® xx D x3)) + 9(a,o2 ® (a2 0 x2 n x3))]

= 0, and 6«, U,ß,(x) = 0.(iii) x, n x2 n z = ¿x, n x2 n x3, x, n x3 n z = x2 n x3 n z = 0,

p(x() n z = ^x3 n p(x,), i = 1,2,3, so

ß(0 = i a,a2 0 x, n x2 n x3 + 2 â, 0 x3 n p(x,)1=1

Hence,

«1«2«3 U ,Ô*(X) = 2?(«1«2«3) ® (ai«2 ® x, n x2 n x3)

= \axa2a3(xx n x2 D x3)ata2 = 2aia2>

«,6«, U,ß#(x) = 0, (a1fio2-a2fia,)U,ß,(x) = 0,

(a,ôa3 + a3ôa, - a28a2 + a28ax) U ?g*(x)

= i[(a3ôa,)(x3np(x1))Ôa1] =iÔa,. D

11. d2 for the split extension. Let Ü = P X F. Write F additively, and denote the

action of F on F by y(v). We shall calculate d|pl(£) by making use of explicit

resolutions. Let X = © Z[F][u,, v2,... ,vn] be the standard Z[F]-free resolution of

Z. Recall thatn-\

dv[vx,v2,...,vn] =vx[v2>...,vn] + 2 (-^)'[vi,...,vi + vi+x,...,vn]1 = 1

+ (-!)"[«,, v2,...,«„_,]License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Kj(Z/p2Z) AND RELATED HOMOLOGY GROUPS 39

and £„([•])= 1. Note also that P acts on X by y(v[vx, v2,. . . ,v„]) =

y(v)[y(vx), y(v2),...,y(v„)]. Also, dvy(x) = y(dvx), £(y(x)) = e(x), andj>(üx) =

y(u)yXx). Let Y be the usual minimal Z[F]-free resolution of Z. In particular,

7„ = Z[F]j„, dPy2„ = Ty2n_x, dPy2„_x = (y - l)y2n-2 (n > 0), and £Py0 = 1.

(7= 1 +y+y2+---+yp-].)

Make Y ® X into a Z[F X F]-resolution of Z as usual. In particular, define

y(ys 0 x() = y(Ä) ® y(xs), v(ys ® x,) = y, ® vxt.

Proposition 11.1. Y ® X is a Z[Ü]-free resolution of Z.

Proof. We have an isomorphism of Z[í7]-modules Z[Ü] ®VX,^ Z[P] ® X, s

Ys ® X, defined by vy ® x i-> y- ® u(yïx)). The left-hand side is Z[t/]-free since Xt is

Z[F]-free. D

It follows that H*(Ü, F) is the homology of the complex

C = UomZ[û](Y®X,F) - Homzm(y,HomZ[(/](*,F)).

C is in fact a double complex with differentials defined as follows: Since Ys = Z[P]ys,

we may identify Cs'= Hom/)(Z[F],HomK(A'„F)) = HomK(A-„F). Then d¡'=

(-\yd'vmâ

f_$y-$f, seven, s>0,

duf-\Tf, s odd.

As usual, dxdxx + dlxdx = 0.

The double complex yields a spectral sequence with E{-' — HS(P, H'(V,F)) =

H[X(H{(C)) and limit H*otaX(C) = //*((/, F).

Proposition 11.2. F«e spectral sequence is the Lyndon-Hochschild-Serre spectral

sequence of the split group extension l-»F-»f/-»P->l.

Proof. The Lyndon-Hochschild-Serre spectral sequence may be realized as the

spectral sequence of the double complex (as above) Hom^^Y-,Hom^ÎF,F))

where IF is any Z({/)-projective resolution of Z (see Evens [E]). Since Zis an acyclic

Z(i7)-complex over Z, there is a map of Z(i/)-complexes over Z, G: W -» X (and G

is unique up to Z(i?)-chain homotopy). Since such a G is also a Z(F)-map, and since

X and IF are projective Z(F)-resolutions of Z, G is an isomorphism up to

Z(F)-chain homotopy. G induces a morphism of double complexes

HomZ(/,)(y,HomZ(n(A',F)) -* Hom^^HomÎFF))

which by the above remark yields an isomorphism (the identity)

Hornby, H*(V,F)) - Wom^P)(Y, H*(V,F))

on the F,-level (and hence for Er, r > 1).

Arguing for IF exactly as for X, we see Y ® IF is also a Z(i/)-projective resolution

of Z and id ® G: Y ® W -* Y ® X induces the identity H*(Ü, F) -> H*(Ü, F) forcohomology of the two corresponding total complexes Hom^^y, HomZ(|/)(A', F))

and HornnP)(Y, Horn^V)(W, F)). DLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


To calculate d2-'a for a G H'(V,F)p, proceed as follows. Choose a cocycle

a G Homy(XnF) representing a. Since ya = a, we have (y — \)a = dvb where

b G HornV(X,_X, F). Then

d(a + ft) = (¿1+ </„)(<! +ft) = dva + (y- \)a + (-\)dvb+ Tb

= 0 + (y - \)a - dvb + Tb = Tb.

Hence,

(11.3) a + ft represents a E E°-' and t/(a + ft) = Tb represents d2a G E2-'~ '.

We are now ready to calculate </2• ' for / = 1,2,3. First, d2- ' =0 since the splitting

implies F2*'° = H*(P,F) is a direct summand of H*(Ü,F).

Proposition 11.4. Forp > 3,

d°22(axa2) = 0, d2°'2oa,=0.

Proof. Let a, be identical with a, but viewed as a 1-cocycle on F Then a,a2

represents a,^, and yax = a,, ja2 = ax + a2, >>a3 = a2 + a3, so >>(a,a2) =

a2 + axa2. Hence, (y — \)(axa2) = a2. Moreover, a2 = dvux where «,(«) = -{i2

for u = ixxx + i2x2 + i3x3 G F. Hence, c/2(a,a2) is represented by Tux. However,

by 9.2, y~l(v) = ixxx + (ix + i2)x2 + (i2 + i3)x3, so yux(v) = - {i\ and yux = ux.

Hence, Tux = pux = 0.

To show d28ax = 0, we use d2ax = 0 and the following result.

Lemma 11.5. The diagram below anticommutes.

H<(P,H'(V,F)) "S(^S) H2(P,H'+\V,F))

ids2' lds2',+ l

HS+2(P,H'~X(V,F)) "S+\P-S) HS+2(P,H'(V,F))

(By abuse of notation d28 + 8d2 = 0.)

Proof. We prove a somewhat more general result. Let A be a Z-free double

complex and put A = A/pA. (In particular, consider A = Hom^F, HomK(Xr, Z)).)

Let a G H¡x(Hf(A)) be represented by ä + ft where â G Is-', ft_G Jí+I''_', d^ = 0,

dlxä + dxb = 0. Then i/2a is represented byd(5 + ft) = i/nft. Let aG/F', ft G

î+i,r-i represent ä and ft" respectively. Then ¿jo = pg for g G As,,+ 1, and i/na +

i/jft — p« = 0 for « G /F+1''. Hence, dlxdxa = páng, and dxdxla — pdxh = 0; so

— dlxdxa — pdxh = 0, and p(dxlg + dxh) = 0. Since A is Z-free <ing + dxh = 0.

Clearly, g + h represents 8a G H[X(H[+\A)), and i/nA represents i/26a. However,

from dxxa + dxb — p« = 0, we also obtain dxxdxb = pdxxh or —dxdxlb = pdxxh.

Since ¿rift G /4î+2'f-' represents dxxb E As+2''~x (representing d2a), it follows that

— dxxh represents 6d2a as required. D


d°-3(ax8ax) = 0, i/20-3(a,ôa2 - a28ax) = 0.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Ä^(Z/p2Z) and related HOMOLOGY GROUPS 41

Proof. The first result follows since E2 is a differential ring and d2ax = d28ax = 0.

The second follows by Lemma 11.5 since 8(axa2) = (8ax)a2 — ax(8a2) and

d2(axa2) = 0. D

Proposition 11.7. For p > 3,

d°'3(axa2a3) = 0.

Proof.

(y - \)(axa2a3) = ax(ax + a2)(a2 + a3) - axa2a3

= a\a2 + axa3 + axa\.

However, a\ = dvuk where uk(v) = - \i\ for v = ixxx + i2x2 + i3x3. Hence,

(y - l)(a,a2a3) = (dvux)a2 +(dvux)a3 + axdvu2

= dv(uxa2 + uxa3 — axu2).

Call the expression in parentheses c. Since yux = «,, yax = ax, we have Tc = uxTa2

+ t^F^ — axTu2. Because TV = 0, Fa2 = Fa3 = 0, and Tc = -axTu2. From (9.2),

we see that (yu2)(v) = - ¿(/, + i2)2. Hence,

(Tu2)(v) = -\P2 (Jh+i2f7 = 0

/>-! \ //>->

2 y2 h'. + 2 2 jlhh+pi = 0 forp > 3.y = 0 / \7' = 0 /

Thus Tc = 0 as required. □


i/2'3(a,6o3 + a38ax — a28a2 + a26a|) = 2a,a2.

Proof. Recall that 8ak is represented by 8ak defined by

'0, 0<ik+jk<p,8aAv, w) '1, F^'*+Á<2p,

where v = ixxx + i2x2 + i3x3, w =jxxx + j2x2 + j3x3, 0 < ik, jk<p.

We introduce some notation from computer programming. If k = pq + r, 0 =£ r <

p, write &/F = a and mod(fc) = r. Then, ôa^t», w) = (ik + jk)/p. (k/p and mod(k)

are integers, but we shall abuse notation by viewing them in Z/pZ = F. The context

will make clear which is meant.)

Lemma 11.9. For k = 2,3,

y8ak = 8ak_x + 8ak + dvfk

where fk(v) = ~(ik-\ + ik)/P- Also,y8ax = 8ax.



Proof. We do the case k = 2. From (9.2), y~\v) = ixxx + (ix + i2)x2 +

(¿2 + z'3)x3. Hence, (y8a2)(v, w) = 8a2(v', w') where

v' = /,x, + mod(/, + z'2)x2 + mod(/2 + /3)x3,

w' =jxxx + mod(jx +j2)x2 + mod(j2 +j3)x3.

Hence, (yôa2)(u, w) = (mod(/, + i2) + mod(/, + j2))/p. Hence,

(y8a2 — 8ax — 8a2)(v, w) = (*)

where

(*) = (mod(i, + i2) + mod(jx + j2))/p - (/, +jx)/p - (i2 +j2)/P-

On the other hand, dvf2(v, w) = (**) where

(**) = - (/] +j2)/P + (mod(ix +jx) + mod(i2 +j2))/p - (ix + i2)/p.

However,

'i +J\ =P(('\ +j\)/p) + mod(ix +/',),

H +J2=P((Í2 +Ji)/p) + mod(/2 +/2),

so,

i, +/, + i2 +j2 =p[(/', +jx)/p + (i2 +j2)/p]

+ mod(/, +/,) + mod(/2 +j2),

and

0\ +Jl)/P +(»2 +Jl)/P = ('l +Ji + »2 +Jl)/P

-[mod(i, +/',) + mod(/2 +/2)]/p-

Hence,

(*) =[mod(i, + i2) + mod(jx +j2)]/p +[mod(/, +/,) + mod((2 +/2)]/p

- Oi +j\ + h +j2)/p.

Similarly,

(»1 + >l)/P +(/'] +Jl)/P = ('l + »2 +/l +Í2)/F

-(mod(i, + i2) + mod(jx +j2))/p,

so

(**) = (mod(/, +/,) + mod(/2 +/2))/p

+ (mod(/, + i2) + mod(jx +j2))/p ~ (/, + i2 +jx +j2)/p.

Hence, (**) = (*) as required. D

To continue the proof of the proposition, calculate as follows using (11.3).

(y — l)(a]6fl3 + a38ax — a28a2 + a28ax)

= ax(8a2 + 8a3 + dvf3) + (a2 + a3)8ax — (ax + a2)(8ax +8a2 + dvf2)

+ (ax + a2)8ax — ax8a3 — a38ax + a28a2 — a28ax

= axdyf3 - (ax + a2)dvf2

= dv(-axf3 + (ax + a2)f2).License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

KÁ[Z/p2 Z) and related HOMOLOGY GROUPS 43

We need to calculate Tc where c = (ax + a2)f2 — axf3. We have

Tc = T(axf2) + T(a2f2) - T(axf3)

= axTf2+ T(a2f2) - axTf3.

However,

T(a2f2)=P2 (A2X//2) = P2 (kax+a2)(ykf2)k=0 k=0

= *. 2 fc//2 + «2(7/2).

Thus, Fc = (ax + a2)F/2 + fll(F'/2 - F/3) where F'/2 = 2£=<J fcyV2- We sha11 show

that F/2 = -a, and F'/2 — F/3 = 5 ax + a2. From this it follows that

Tc = - (a, + a2)a, + ax({ax + a2)

= a,o;2 — a2ax — {-a2 = axa2 — a2ax — \dvux.

Thus, Tc represents axa2 — a2ax = 2axa2 as required.

Thus, we need only establish

Lemma 11.10. F/2 = -ax, T'f2 - Tf3 = \ax + a2.

Proof. Define i2k\ k = 0, l,...,p, inductively by i20) = z2, ¿1 +/^ = FÍ* +

¿Oc+i) g ^ /<2*+1) </7. (/2* Ms the coefficient of x 2 in y "û for u = /,x, + z2x2 + z'3x3.)

Then adding yields

p-\ P-\ p

(t) p'i + 2 hm=p 2 ?*+ 2 *2<*)-fc = 0 ¿ = 0 k=\

From this we conclude i2p) = /(20) mod p so z'*/' = i2. Again, from (f) we get

U=Uk = «V However, (^*/2X») = /2(^) = -(*', + i?>)/P = -qk so (F/2)(u) =

-2^=0'?A = -J, = -ax(v), and F/2 = -a,.

Furthermore, multiplying ix + i2k) = pqk + z'2* + 1) by k and adding yields

p~l \ /»-1 p-\ p-\

2k\ix+ 2 ki(2k)=p2 kqk+ 2 ki2k+l\k=0 I k=0 A=0 k=0

But,

2 *#+» = 2 (*-1)#> = 2 *#>- 2 #>A: = 0 Ar=l fc=] /t=l

= ̂ fc^+F^-Y^/c=0 * = 0

(use i\p) = z'2u> — i2). Putting this in the above equation yields

PiPjl)h=pP2kqk+pi2- 2VA = 0 A:=0License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Hence,

(T'f2)(v)= 2 kf2(y-kv) = ~P2 kqkk=0 k=0

To calculate F/3, define i3k) inductively by z30) = i3, i2k) + i£*) = prk + i3k+]\

0 < i3k+]) < p. (i3k) is the coefficient of x3 in y~kv for v = z'|X, + z'2x2 + z'3x3.)

Adding yields

/> i />-i p-\ p

2#> + 2tk) = p2rk+ 2 4A)-/t = 0 A: = 0 A: = 0 A:=l

If /', = 0, ICZo i2k > = pi2, if »i * 0, zf, i2'>,... ,i£p- ') is a permutation of 0,1,2,... ,p

— 1, and 2£=¿ i2k) — p(p ~ l)/2. In either case, the sum is = 0 mod p. As above, it

follows that i3p) = if = i3, and

(3y3)(t0 = -2't = 4 sVA=l ^ A = l

Hence,

2 /l + '2 2'(r/2)(e) - (F/3)(ü) = -^V^'i + '2 = T'> + '2-

Thus F'/2 — F/3 = ja, + a2 as claimed. D

12. í/2 for GL(2, p). We may summarize the results of §§10 and 11 in a table.

Proposition 12.1. For p > 3, d2-'£ is given in Table 2 for t = 1,2,3.

Table 2

£ 5p/zr case Nonsplit case U

ax 0 0

axa2 0 0

8ax 0 0

axa2a3 0 —\axa2

ax8ax 0 0

a,6a2 — a28ax 0 0

ax8a3 + a38ax — a28a2 + a28ax 2axa2 2axa2 — 28ax

We now make the transition to GL(2, p) and then to homology. We need only

concern ourselves with t = 3. Using additive notation, we have z, = e2x, z2 — «,,

and z3 = ex2 and we denote the dual basis by e2\ «', and en. Referring to §9, we see

easily that e21 = a,, /z1 = «2, en = -a2 — 2a3. We know (by Proposition 3.2) that

H°(GL(2, p), H3(V,F)) (dual to H0(GL(2, p), H3(V, F))) is 2 dimensional, and it is

easy to see that it is the subspace of H3(V,F)p spanned by Xi = e21«'e12 =

-2a,a2a3 and x2 = e2X8en + en8e2X + 2h]8h] = -ax8a2 — 2a,6a3 — a28ax —

2a38ax+2a28a2 or

X2 = -2(a,6a3 + a38ax — a28a2 + a28ax) — (ax8a2 — a28ax).License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Kj/Z/p2Z) and related HOMOLOGY GROUPS 45

From Table 2, we have

Proposition 12.2. For p > 3, </2'3(£) is given by

£ Split case Nonsplit case

X, 0 axa2

X2 —4a,a2 —4a,a2 + 8ax

Note. d2(|) G H2(GL(2, p), H2(V,F)) but we may identify this group with

H2(P, H2(V,F)) = (A2V)p®Vp through restriction. For, by Proposition 5.2,

H2(GL(2, p), F2) = ff2(GL(2, p), V) = Z/pZ,

so res: //2(GL(2, p), V) -> H2(P, V) = Z/pZ is an isomorphism. Since F A F s F

also, the above contention follows.

It now follows thatd203: //°(GL(2, p), H3(V2,F)) -> H2(GL(2, p), H2(V2,F)) has

rank one in the split case and is an isomorphism in the nonsplit case. By duality, we

have

Proposition 12.3. For p > 3, we have that (a) dj2: H2(GL(2, p), H2(V2,Z/pZ))

->//0(GL(2, p), H3(V2, Z/pZ)) is an isomorphism for the nonsplit extension

1 - F2 -» SL(2, p2) -» GL(2, p) -> 1 and

(b) d22 has rank 1 for the corresponding split extension.

To deal with integral homology, we consider dually the coefficient group Q/Z.p

The exact sequence 0 -» Z/pZ -> Q/Z -» Q/Z -» 0 yields

0 - H2(V2,Q/Z) - H3(V2,Z/pZ) - H3(V2,Q/Z) - 0.

Since

/T(GL(2, p), H2(V2,Q/Z)) s #r(GL(2, p), //2(F2,Z)) = 0

for r = 0,1 (Proposition 5.2 and F2 A F2 s F2), we have

//°(GL(2, p), H3(V2,Z/pZ)) =F°(GL(2, p), //3(F2,Q/Z)).

Similarly, we have 0 -» Hl(V2, Q/Z) -» //2(F2, Z/pZ) -» Är2(F2, Q/Z) - 0, and

//3(GL(2, p), //'(F2,Q/Z)) « //3(GL(2, p), F2) = 0

(Proposition 5.2). Hence,

H2(GL(2, p), H2(V2,Z/pZ)) - H2(GL(2, p), H2(V2,Q/Z))

is an epimorphism. It now follows easily that for the nonsplit extension d2'3:

//°(GL(2, p), H3(V2,Q/Z)) -» //2(GL(2, p), H2(V2,Q/Z)) is an epimorphism. To

deal with the split extension, consider

0 -» Z/pZ -• Z/p2Z - Z/pZ - 0II II II

0 -» Z/pZ -* Q/Z ^ Q/Z - 0.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


It follows that 8ax e H2(V2,Z/pZ) is in the image of H\V2, Q/Z); hence, 8ax \-> 0

in H2(V2, Q/Z). Thus, under the epimorphism

H2(GL(2, p), H2(V2,Z/pZ)) - //2(GL(2, p), H2(V2,Q/Z)),

the image of axa2 (viewed as an element on the left through res-1) spans

H2(GL(2,p), H2(V2,Q/Z)). Since d2°'3X2 = -4a,a2, it follows that d^3 is an

epimorphism for Q/Z. By duality, we have

Proposition 12.4. Forp > 3, we have for both (a) split and (b) nonsplit extensions,

d2y. H2(GL(2, p), H2(V2,Z)) - //0(GL(2, p), H3(V2,Z))

is a monomorphism (so its image has dimension 1).

References

[B] H. Bass, Algebraic K-theory, Benjamin, New York, 1968.

[C] H. Cartan, Séminaire H. Cortan, 1954-55, Paris.

[C-E] H. Cartan and S. Eilenberg, Homologica! algebra, Princeton Math. Series, no. 19, Princeton Univ.

Press, Princeton, N. J., 1956.

[C-V] L. S. Charlap and A. T. Vasquez, The cohomology of group extensions, Trans. Amer. Math. Soc.

124(1966), 24-40.

[E] L. Evens, The spectral sequence of a finite group extension stops, Trans. Amer. Math. Soc. 212 (1975),

269-277.

[M] J. Milnor, Introduction to algebraic K-theory, Ann. of Math. Studies, no. 72, Princeton Univ. Press,

Princeton, N. J., 1971.

[Q] D. Quillen, On the cohomology and K-theory of the general linear groups over a finite field, Ann. of

Math. 96(1972), 552-586.

[S] C.-H. Sah, Cohomology of split group extensions, J. Algebra 29 (1974), 255-302.

[V] W. van der Kallen, Le K2 des nombres duax, C. R. Acad. Sei. Paris Sér. A-B 273 (1971), 1204-1207.

[W] J. B. Wagoner, Delooping classifying spaces in algebraic K-theory, Topology 11 (1972), 349-370.

Department of Mathematics, Northwestern University, Evanston, Illinois 60201


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