Relative cohomology of groups

Comment. Math. Helvetici 59 (1984) 149-164 0010-2571/84/010149-16501.50 + 0.20/0 (~ 1984 Birkhauser Verlag, Basel

Relative cohomology of groups

NATHAN HABEGGER, VAUGHAN JONES,* OSCAR PINO ORTIZ and JOHN RATCLIFFE

w Introduction

If G is a group and A is a G-module, the cohomology groups H"(G, A ) can

be interpreted either as the cohomology of the cochain complex C*(G, A) given by the bar resolution (see [M]) or as equivalence classes of n-fold extensions. This

situation has reached perfection and is summarized in an historical note by

MacLane in [Ho]. If N is a normal subgroup of G and B is a submodule of A on

which N acts trivially (and hence B is a Q-module with Q = G/N), one may define the relative cohomology groups H" (Q, G; B, A) as the cohomology of the quotient complex C*(G, A)/C*(Q, B). In this paper we present another cochain

complex L*(G, N; B, A ) whose cohomology is the relative cohomology but which is more explicit than the quotient complex and which grew out of certain

"crossed" extensions which give an interpretation of the group H2(Q, G; B, A),

analogous to the n-fold extensions mentioned above. A crossed extension is an exact sequence 1 --~ A --~ C A N--~ 1 together with

an action (g, c) --~ gc of G on C such that O(c)d = cdc -I for c, d c C and O(gc) = gO(c)g -1 for g~ G, c ~ C. It follows that A is in the centre of C and that the

action of G induces the structure of a G-module on A. This idea goes back to Whitehead ([W]) and was studied extensively in [HI, [L], [O] and [R]. One may

define a sum of two crossed extensions and equivalence classes of such form a group. It can be shown that this group is isomorphic to the second relative

cohomology group. One way to establish this isomorphism is to give a cocycle

description of crossed extensions. More precisely, if s : N ~ C is a section for O,

define c to :NxN-- -~A and c q : N x G - - ~ A by

Cto(n, m) = s(n)s(m)s(nm) 1

ctl(n, g) = s(n)(gs(g-Xng)) -1

* Supported in part by NSF Grant #MCS 79-03041

149

150 N A T H A N H A B E G G E R E T AL.

One can easily show (see [R]) the following cocycle relations

(1) ao(m, s) - ao(nm, s) + ao(n, ms) - ao(n, m) = 0 (2) ga0(g-lng, g - l ing) -o t0(n , m ) = al(m, g ) - a l ( n m , g )+ al(n, g) (3) ga l (g - lng , h)-al(n, gh)+ al(n, g ) = 0 (4) al(n, m) = ao(n, m)-ao(m, m-into)

The dependence of ao and a l on the section s gives natural coboundary relations.

Crossed extensions arise naturally in algebraic topology and in the study of group actions on algebras. In [J] it was shown that the above relations can be interpreted as forming the beginning of a cochain complex derived f rom the double complex C*(G, C*(N, A)). In [O], this complex L*(G, N; A) is defined in all dimensions and even in the case where A is an arbitrary G-module . The pair (ao, oq) is then a 2-cocycle. Thus H2(L*(G, N; A)) classifies crossed extensions when A is a Q-module . It was conjectured in [O] that the cohomology of L*(G, N; A) is the same as the relative cohomology. The conjecture was proved for H 3. Also a "crossed module" formalism for arbitrary G-modules was worked out in [O] and [CW]. Another version appears in [HI.

In w of this paper we will show that L*(G, N; B, A) has the cohomology of the quotient C*(G, A)/C*(Q, B), thus establishing the conjecture of [O]. As an application, we immediately obtain the 8- term exact sequence of [H], JR], ILl.

In w a spectral sequence converging to H*(Q, G;B, A) is given. For B = {a I ga = a for all g e N} this has a small calculational advantage over the spectral sequence of Hochschi ld-Serre [H-S] in that the bot tom line has disappeared.

In w we give an example where results are easily obtained using the cocycle relations of L* which are not obvious using C*(G, A)/C*(Q, B).

At this stage there is no "n- fo ld crossed extension" theory for the relative cohomology groups although some cases are t reated in [O], and the third cohomology relations for the complex L* do arise naturally in the study of Connes ' invariant x(M) for von Neumann algebras (see [C]).

w Shuffles and partial shuffles

Let cr be a permutat ion of {1, 2 . . . . . n}. For a group G we define a map

1~ : G " ~ G " by l~(gl . . . . . gn) = (b~lalbl . . . . . b~la, b,) where a,,ci ~ = gi, bo~i~ = gJ, gJ~" " " gh where i < j ~ < . . . <Jz are all the ik'S satisfying t r ( i )>tr( jk) . In words, we move the variable gi into the or(i)th position, and then conjugate g~ by the product of those gk, k > i, in their order, which are now in a position inferior to gi. (Note that b~ = 1).

Relative cohomology of groups 151

E X A M P L E . (1) if cr is the pe rmu ta t i on (123) then

/~,(gl, g2, g3)= (g3, g~lglg3, g319293) -

(2) if cr is (132) then

(gl, g2, g3)= (g2, g3, g31g21glgzg3)

(3) if cr is (12) then

l,, o/ ,(g, h ) = l,,(h, h - l g h ) = ( h - ' gh , h - l g - ~ h g h )

E x a m p l e (3) above shows that in general l, o l,, :/: l,o,,. H o w e v e r

P R O P O S I T I O N 0. Let o- be a perumtation of {1 . . . . . p + q } fixing the sets {1 . . . . . p}{p + 1 . . . . . p + q} and let r be a permutation satisfying r(i) < I"(i + 1) for iCp . Then l, o l , ,= l~ .

Proof. Let l,~(gl . . . . . gw g,+l . . . . . gp+q) = (x~ . . . . . x~, xp+l . . . . . xp+q) x~ = bf~cqbi. For p + l - - - i - - ~ p + q we find in bo th l, ol~(gl . . . . . gp+q) and /.o,(gt . . . . . gp+q) the var iable x~ in the 1-(i)th place. H o w e v e r for 1-----i<--p in l, ol ,(gi) we find xi in the ~-(i)th place con juga ted by x~+~ .- �9 xp+r where r is the m a x i m u m j satisfying ~-(p+])<~-( i ) , whereas in /,~(g~) we find x~ in the 1-(i)th place con juga ted by the p roduc t of the %+i =g~-~,+~l, l<--i<--r, in their original order . F r o m the definit ion of the bi, we see, by induction, that these p roduc t s are identical.

D E F I N I T I O N . A p e r m u t a t i o n r is a par t ia l shuffle of type (a, b, c, d), a + b +

c + d = n if (1) (~ ( i )= i for l < - i < - a o r a + b + c + l < - i ~ n (2) r o r a + b + l < - i < - a + b + c - 1

(In words , the sets {1 . . . . . a}{a + b + c + 1 . . . . . n} are left fixed, while the sets {a + 1 . . . . . a + b}{a + b + 1 . . . . . a + b + c} are shuffled).

A p e r m u t a t i o n is a 3-shuffle of type (a, b, c), a + b + c = n, if o-(i) <o- ( i + 1) for i r i ~ a + b .

P R O P O S I T I O N 1. (A) I f ~ is of type (0, a, b, c) and -r of type (0, a + b, c, O) then .root is of type (a, b, c). A n y "q of type (a, b, c) has a unique decomposition as above. Furthermore l.~,, = l.o l,,.

152 NATHAN HABEGGER ET AL.

(B) I f o- is of type (a, b, c, O) and T of type (0, a, b + c, O) then ToO" is of type (a, b, c). A n y rl of type (a, b, c) has a unique decomposition as above. Furthermore

Proof. The formula ltd. = l. o l~ follows f rom Proposition O. The rest is obvious.

Q.E.D.

Let G be a group, N a nomal subgroup. If we let X, Y, Z denote either G or N then l~(X a x Y b + c x Z a ) C X a x Y b + c x Z a for o- of type ( a , b , c , d ) . Thus if a : X a x Y b + c x Z a - ~ A is any map, A a 2[-module, we may define S(a, b, c, d )a : X ~ x yb+c X Z a -* A by

S(a, b, c, d)a = ~'. (-1)1'%t o/~ type

(a,b,c,d)

( -1 ) t'l = signature o-.

P R O P O S I T I O N 2. Let a : G" --> A. We have the formula

IS(a, b, c, 0)S(0, a, b + c, O)a = S(O, a, b, c)S(O, a + b, c, 0)a J

Proof. By Proposition 1 both sides are equal to

Y. jx type (a,b,c)

w R e l a t i v e c o h o m o l o g y o f g r o u p s

Let G be a group, N a normal subgroup. We write Q for the quotient GIN. If A is any G-modu le we denote by A N the submodule of elements fixed by N. A N

is a Q-module . Let B be a Q-submodule of A N. We denote by C*(G; A) the complex of normalized cochains a : G" ~ A with

coboundary

da(g0 . . . . . g,) = goa(gl . . . . . g,)

+ ~ ( -1) ia(g0 . . . . . gi-lgi . . . . . g,) A =1

+ ( - - 1 ) n + l o l ( g o . . . . . g n - 1 )

The projection G --~ Q induces an inclusion of complexes C*(Q; B ) c C*(G; A). Let C*(Q, G; B, A ) be the quotient complex.


D E F I N I T I O N . The group H*(Q, G; B, A ) = H*(C*(Q, G; B, A)) is called the relative cohomology group. T he associated long exact sequence

�9 . . - -~H*(Q,B)- -*H*(G,A)- -~H*(Q, G;B,A)---~ . . .

is called the fundamenta l long exact sequence.

Remark. W e justify our use of the term "relat ive".

If A is a trivial G module , and B = A , then H " ( Q , G ; B , A ) = H"+1(Mo, BG; A) where Mp is the mapping cylinder of the fibration p :BG--> BQ.

Proof. If C.(X, A ) (C*(X ,A ) ) denotes singular (co)chains then C*(G, A)~- C*(BG, A), C*(Q, A) ~- C*(BQ, A ) so

C * ( G , A ) H o m z (C.(BG), A ) = H o m z (ker p . , A)

C*(Q, A ) - H o m z (C.(BQ), A)

where 1 --> ker p . --~ C.(BG) p* > C . ( B Q ) --~ 1 is exact. Finally, the suspension of ker p . is i somorphic to coker i . , where

1 ~ C. (BG) i, > C.(Mp) --~ coker i . ~ 1 is exact.

w A spectral sequence for relative cohomology

W e review briefly parts of the fundamenta l paper of Hochschi ld and Serre [1]:

Regard C*(N,A) as a G - m o d u l e by (g/3)(n~ . . . . . n~ )=g /3 (g - an lg . . . . . g-an~g). Then C*(G, C*(N, A)) is a double complex with coboundar ies

dNa(gl . . . . . gr)(no . . . . . ns) = noa(gl . . . . . g~)(nl . . . . . ns)

+ t ( - l y a ( g l . . . . . gr)(no . . . . . ni lnl . . . . . n~) i = l

+ ( - 1 ) s + l a ( g , . . . . . g,)(no . . . . . n s - 0

and

d~a(go . . . . . g,)(nl . . . . . ns) = goa(g l . . . . . g , ) (golnlgo . . . . . go~n~go)

+ t (-1)*a(go, - - . , g~-lg~ . . . . . g,)(n~ . . . . . n~) i=1

+ ( - l y + l a ( g o . . . . . g r - 1 ) ( n l . . . . . /'Is)

154 N A T H A N H A ] B E G G E R E T AL.

For reasons of conven ience we will wri te a(nl . . . . . n~, gl . . . . . g,) in place of

or(g1 . . . . . g~)(nl . . . . . ns). Let T*( G, N; A ) =~p=q=. C"( G, Cq(N, A )) be the total complex associated

to the double complex with total differential do + (--1)i+ldN on C*(G, C~(N, A)). W e define a m a p l : C*(G, A) ~ T*(G, N; A) by

l(a)=(ao . . . . . a,), [ ap=S(O,q,p,O)alN.• p + q = n .

Note in par t icular tha t an = a and a o = a l N o .

P R O P O S I T I O N 3. l : C*(G, A) ~ T*(G, N; A) is a map of complexes.

Proof. This is the "genera l ident i ty" of [H-S] . Q . E . D .

Def ine a fi l tration R o of C*(G, A) where a ~ R o if a ( h l . . . . . hq, gl . . . . . gp) depends only on the hl and the classes m o d N of the gv Also Rp fq C"(G, A ) = 0 if p > n .

Def ine a filtration Sp of T*( G, N; A) by S, = ~i~p Ci ( G, C*(N, A ) ). If a ~ R o and l (a ) = ( a 0 . . . . . or,), then % = 0 , j < p , so l (a)eS o. Thus l is a m a p of fi l tered complexes .

T H E O R E M 4 (Hochschi ld-Ser re) .

l* : E~o,,(R) ~ E~.,(S) = C~ H"(N, A ))

is an injection and image l * = CP(Q, Hq(N, A)).

C O R O L L A R Y 5. E~.q(R) = H~ Hq(N, A))

W e now study the quot ien t complex C*(Q , G ; B, A) . Def ine a fi l tration of C*(Q, G ; B , A ) by /~j = i m a g e of R i.

T H E O R E M 6 (Spectral s equence for re lat ive cohomology) .

HP(Q, Hq(N, A)) q > 0

E~'q(R)= H p Q, q=O

=> H*(Q, G; B, A)


Remark. If B = A N, the E2 te rm of this spectral sequence is identical to the Hochschild-Serre E2 term except that the bot tom line has disappeared.

Proof. Considering B c A ~ C~ A ) we have an inclusion C*(G, B) T * ( G , N ; A ) of complexes since dN(C*(G,B))=O. Since l ( C " ( O , B ) ) c Cn(G, B), we get

l: C"(Q, G; B, A ) --~ T N ( G , N ; A )

c"(G, B)

If we let ~ = image of Sj in T"(G, N; A) IC"(G, B) then r is a map of filtered complexes. Theorem 6 is then a corollary of the following analog of Theorem 4.

P R O P O S I T I O N 7. r . :E~. ,( /~)--~ E10,q(S) is an inclusion with image

{ C"(Q, Hq(N, A) ) q > 0

Proof. Cn(O, B) c Rn c Rp, if p ----- n, so E~ -~ E~ if q > 0 and

EO,o(~) E~ o(R) c~(o, B)"

Since d (C"(O, B)) ~ C"+X(Q, B) ~ R,+I , it follows that

0---~ C"(Q, B)---~ 0 - - ~ . . . is a subcomplex of

0 ~ U~ ~ E~ ---~. �9 �9 with quotient

0--, E~ --, E~ - - , . �9 �9

From the associated long exact sequence, we get

0 ~ C"(Q, B) ---. E~,o(R) --~ E~,o(/~) ~ 0

is exact and E~.,(R) = E~.q(R), q > O.

Similarly 0 ~ C"(G, B) ~ EX,,o(S) ~ E~.o(g) ~ 0 is exact and E~o,q(S) ~-


E~,q(g), q > 0. A look at the commuta t ive diagrams

E~,q(R) ~ CP(Q, Hq(N, A)) ~ E~,q(S)

E~,q(/~) , E~.q(g)

and

0 ~ C"(O, B) --~ E~,.o(R) ~ E~,o(-R) ~ 0 LI

Cn(O, A N)

N C"(G, A N)

0 ~ Cn(G, B) --~ E~.o(S) ~ E..o('S) ~ 0

establishes the proposi t ion.

q > 0

w The complex L*(G, N; A, B)

W e write an e lement of T"(G, N; A)/Cn(G, B) as (s0 . . . . . a.-1, at.) where

Cn(G, A) at,, c C"(G, A/B).

C " ( G , B )

D E F I N I T I O N

T"(G, N; A) L " ( G , N ; A , B ) = (ao . . . . . ~,, 1,&,,)~ C"(G,B)

such that ak --= S(0, n - k, k, 0)at.[N--k• m o d B

and S(O, n - m , m - k , k)ak = S ( n - m , m - k , k, O)a,.IN- k• ~}

Remark. If A = A N = B we m a k e abbreviat ions such as Ln(G, N; A, A) to

L"(G, N; A). Note that in this case

C"(G, A) &n e C . ( G , A ) =0


SO we abbrev ia t e (a0 . . . . . a . 1, ~ . ) to (s0 . . . . . a . _ 0 . T h e reader rhay check that if A = A N = B, then (s0, a l ) is in L2(G, N ; A) if and only if the relat ion (4) of the in t roduct ion is satisfied and a cocycle iff (1) (2) and (3) are satisfied.

It is a direct consequence of Propos i t ion 2 that the m a p

l: C"(G, A) ---> T"(G, N; A) C"(O, B) C"(G, B)

of w (given by l-(&) = (So,. � 9 a , - 1 , &,), ak = S(0, n - k, k, 0)a[N- k• has image conta ined in L"(G, N; A, B). In fact

P R O P O S I T I O N 8.

I m a g e l = L"(G, N; A, B)

C O R O L L A R Y 9. L * ( G , N; A, B) is a subcomplex of T*(G, N; A)/C*(G, B).

D E F I N I T I O N . A *(G, N; A, B) = H*(L *(G, N; A, B))

T H E O R E M 10. l* : H * ( Q , G ; B, A) --> A*(G, N; A, B) is an isomorphism.

Proof of Proposition 8 and Theorem 10. Consider the following d iag ram of complexes

C*(G , B) C*(G, A) C*(G, A) 0---> > > ->0

C*(O, B) C*(Q, B) C*(O, B)

C*(G, A) 0 --~ L*(G, N; B, B) --. L*(G, N; A, B) -~ ---> 0

C*(G, B)

where zr(a0 . . . . . &.) = ~.. T h e top row obviously is exact and the first square commuta t ive . Commuta t i v i t y of the second square follows f r o m the identi ty S(0, 0, n, 0 ) a = a, so that if l(a) = (a0 . . . . . a , ) , then a , = a . H e n c e w is sur ject ive since w o 1-2 is.

Fu r the rmore , since ak--=S(0, n--k,k,O)6L,[N, k• m o d B , it follows that if ~ , = 0 then ak takes values in B. This p roves exactness of the b o t t o m line.

F r o m the d i ag ram it follows that 11 is sur ject ive if and only if 12 is. T h e five l e m m a appl ied to the long exact sequences in h o m o l o g y shows that 11 induces an i somorph i sm on h o m o l o g y if and only if 12 does.

158 N A T H A N H A B E C ~ E R E T AL.

So it suffices to prove 8 and 10 in case A = A N = B . Then

T" (G, N ; A ) - (]~ C~ C"(N, A ) )

C~(G, A ) p§ q : > 0

and

, T " ( G , N ; A ) L " ( G , N ; A ) = (ao . . . . . a . - l ) ~ C----~,-A-) such that for

m <-~ n -- 1, S (n - m, m - k, k, O)otmIN. ~• = S(O, n - m, m - k, k)O~k] k

W e begin with the following

L E M M A 11. Let a =(ao . . . . . a , -1) , b =(/30 . . . . . /3,-1) be elements of L " ( G , N; A ). Then a = b if and only if ao = /30 and ak (ml . . . . . m , - k , gl . . . . . gk) =/3k(mI . . . . . m~-k, gl . . . . . gk) whenever g l e N .

Proof o f I_emma 11. Since a k l N . = S ( O , n - k , k , O ) a o = S ( O , n - k , k , O ) / 3 o = /3k IN - we may assume some g~ ~ N. Suppose we have shown that ai =/3~ for i < k, and that ak =/3k whenever the first l for which g~ ~ N satisfies l ----- j < k. W e show

a k ( m l . . . . . m,,-k, hi . . . . . hi, gj+l . . . . , gk) = /3k(ml . . . . . m~-k, hi . . . . , hi, gi+l . . . . . gk) wheneve r hi . . . . . hj e N but gj+l r N.

For simplicity, write (m, h, g) for (ml . . . . . m,_k, hi . . . . . h~, gi§ . . . . . gk). Since

ak-i =/3k-j we have

S ( n - k , ] , k - i , O ) a k ( m , h, g ) = S(O, n - k , ] , k - ] )o tk_ i (m , h, g)

=S(O, n - k , ] , k - D / 3 k - j ( m , h, g )= S ( n - k , ], k - ] , O)/3k(m, h, g).

Fur thermore , for o- of type (n - k, j, k - j , 0), o'~: id, we have by our induct ion hypothesis that akol, , (m, h, g) =/3kOl,,(m, h, g) since in l,,(m, h, g) the value

gl+l g N will occur earlier. Thus

ak(m, h, g) = S(n - k, ], k - ] , O)ak(m , h, g ) - ~ ( - 1)J~lak o l,.(m, h, g) or t y p e ( n - k d , k - L 0 )

o - ~ i d

= S ( n - k , ] , k - j , O ) / 3 k ( m , h , g ) - ~, ( -1) t~l /3kol~(m,h,g) o" t y p e ( n - - k j , k - j , 0 )

o ' P i d

=/3k(m, h, g).

E n d of proof of Proposition 8. Le t (a0 . . . . . a . - 1 ) ~ L(G , N; A ) . W e will define m

/ 3 c C " ( G , A ) so that if l(/3)=(/30 . . . . . /3 . -0 , then /30=a0 and /3~(ml . . . . .

m._i, gl . . . . . g~)=ai (ml . . . . ~ m._i, gl . . . . . gl) wheneve r ml . . . . , m . - i ~ N and


g~ r N. By L e m m a 11, it will fo l low that /3~ = oq, all i, i.e. I(/3) = (So, . �9 �9 o~,-1).

Set /3 I N " = s0. T h e n /30 = so. Set /3(g~ . . . . . g , ) = 0 if glc~N. Suppose /3 is

def ined on N ~-~ x ( G - N) x G ~-1, for i > j > 0. Def ine

/3(ml . . . . . rn,_j, gl . . . . . gi)

= - }-" (-1)1~1/3 old(m1 . . . . . rn,_~, gl . . . . . gj) t r ty l~(O,n- - / , / ,O)

~r ~ id

+ a~(ml . . . . . m._j . gl . . . . . gi).

for rn~ e N and gl ~ N . (Note tha t /3ol,~(rn~ . . . . . m,_j , gl . . . . . gj) has a l r e a dy been

def ined) . I t fol lows tha t

/ 3 i ( r t l l . . . . . ran-i, g l . . . . . gi) = ~ (--1)1~1/3 ~ . . . . . ran-j, g l . . . . . gi) type(O,n-j,j,o)

= s i ( m l . . . . . rnn-i, gl . . . . . gj)"

Proof of Theorem 10 (A = A N = B) . I t suffices to show the fol lowing:

Le t K * be the ke rne l of the c o m p o s i t e

C*(G, A) C*(G, A ) r

--> L*(G, N; A). C*(Q, A)

Then the inclusion C*(Q, A ) c K* induces an i s o m o r p h i s m on homology .

Le t B~ b e the f i l t ra t ion of C*(G,A) w h e r e a c B ~ M C " ( G , A ) if a is

unchanged w h e n e v e r any of the last i va lues is mul t ip l i ed by an e l e m e n t

of N, B i N C " ( G , A ) = C " ( Q , A ) if i>--n. F o r i<p<--n, le t B~,p=

{ s ~ B i l a ( g l . . . . . g , ) = 0 if gp-i . . . . . g , - i a re all in N}, B i , i = B i , and

B~,p M C" ( G, A ) = B~ + ~ O C" ( G, A ) if p > n. Set K~ = K n B~, K~,p = K M B~,p. W e will show the fo l lowing

(a) K, = K,.,+x

H,,( K,.,, '~ (b) \ K i , . + l ] = 0 i < n

H"( K"~ ) (C) \ K i . p + I / = 0 i < p < n.

It fol lows tha t

H"( K, ~= H.(K,.,+I~ \Ki+l] \Ki,n+2] = 0


a n d h e n c e t ha t

\ C * ( Q , A ) ] - H \ K , + I / = O.

Proof of (a). L e t a b e of t y p e (O, n - i, i, O). If a~:id a n d a ~ B i , t h e n

a o l~[N- ' • = 0. H e n c e the f o r m u l a , va l id for at �9 Bi :ati = atlN- '• In p a r t i c u l a r if

at~Ki, 0=at~ =at[N',- ,• i.e. at E K i , i + 1.

Proof of (b). F o r at ~ B~,, 0 C"(G, A ) , d a ~ B~,,+I we h a v e for m r N

0 ---- d x l ( g I . . . . . g n - i - 1 , g, m , h I . . . . . hi)

= ( - 1 ) " - i ( a t ( g l . . . . . g , - i - 1 , gin, h i . . . . . h i ) - a t ( g l . . . . . g , - i 1, g, h i . . . . . hi-))

i .e. at ~ Bi+l N C" (G, A ) = Bi,,~§ (3 C"(G, A) , It fo l lows tha t if a c

Ki,n N C"(G, A ) , dot ~ Ki,,,§ t h e n at ~ Ki , ,§

Proof of (c). C h o o s e a sec t ion * : Q --> G of the q u o t i e n t m a p zr : G --> Q such

tha t 1 " = 1.

W e first s h o w (c) in t h e specia l case i = 0.

S u p p o s e a E K o , p f"l C " ( G , A ) , d a E Ko,p+ 1. D e f i n e for

O', r a p + 2 . . . . . m n ~ N,/3(gl . . . . . g p - 1 , q * o r , r a p + 2 . . . . . ran)

= at(g1, �9 � 9 g~-l , q*, ~r, m~§ . . . . . m , ) .

If fi is a n y e x t e n s i o n o f / 3 to all G "-~, t h e n fi ~ B0,p. M o r e o v e r

d f i (g l . . . . . gp-1, q*tr, mp+l . . . . . raM)

= ( -1)P/3(g l . . . . . gp-1, q* t rmp+l , rap+2 . . . . . m , ) + " �9 �9

+ ( - 1 ) P / 3 ( g ~ , . . . , gp_~, q*or, rap+ 1 . . . . . r an)

= (-1)Pat(g1 . . . . . gp-1, q*, o-rap§ rap§ . . . . . m~) + . �9 �9

+ ( - 1 ) ~ a t ( g l . . . . . gp-1, q*, or, mp§ . . . . . r a m - 0

= - d a ( g l . . . . . gp-1, q* , tr, mp+l . . . . . m , )

+ ( - t ) P a ( g l , �9 �9 - , gp-1, q*o' , r r t p+ l , . . . , ran)

= ( - 1 ) P a ( g l , - � 9 gp-1, q*cr, mp+l . . . . . m , )

s ince d a ~ Ko,p+l, i .e. at - ( - 1 ) p dfi ~ B0,p§


T h u s it suffices to s h o w tha t 18 can be e x t e n d e d to a /3 �9 K. N o w 18 is a l r e a d y

d e f i n e d on G p x N ~-p-~. E x t e n d 18 as fo l lows:

L e t 18(gl . . . . . g . _ 0 = 0 if g ~ r a n d s o m e g i C N , p < i < - - n - 1 . A s s u m e n o w

tha t 18 is d e f i n e d if s o m e g~ r N, i ~ k < n - 1. T h e n if gk+l r N b u t m l . . . . . rnk �9 N

(and if k < p s o m e g ~ N , p < i < - - n - 1 ) de f ine

18(ml . . . . . ink, gk+l . . . . . g . -~)

= -- ~ (--1)1~118 ~ l~(rnl . . . . . ink, gk+l . . . . . g . -~) o" t y p e ( O , k , n - - k - - 1 , 0 )

cr v~ id

t he r igh t s ide a l r e a d y h a v i n g b e e n de f ined . I t f o l l ows tha t

18.-k l(rnl . . . . . mk, gk+l . . . . . g . _ l ) = 0 fo r g k + l ~ N (and p r o v i d e d s o m e g i ~ N ,

p < i --< n - 1, if k < p). T h u s , by L e m m a 11, it wil l f o l l o w tha t 18 c K if w e can s h o w

18. k_~(m~ . . . . . rnk, gk+l . . . . . go, mo+l . . . . . m . 1) = 0

for

Bu t s ince 18 �9 Bo,o a n d a �9 Ko,o, a c a l c u l a t i o n s h o w s

g ~ / p + l , �9 �9 �9 , /7"/n 1 � 9

18.-k - l ( m l . . . . , ink, gk+x , . . �9 , g, 1, q*cr, r a p + l , . . . , m . 1)

= a . - k ( m l . . . . . ink, gk+l . . . . . go 1, q* , or, too+ 1 . . . . . m . l) = 0 .

N o w s u p p o s e a �9 K~,p, da �9 Ki,p+l, i > 0 . D e f i n e a ' (q l . . . . . qi ) (g l . . . . . g . - i ) =

a ( g l , g . i, q* , * ' . . . . . . . , q ~). T h e n ~ (ql . . . . . q~) �9 Bo,p_~ a n d a ca l cu l a t i on s h o w s

a ' (q l . . . . . q i ) k (m l . . . . . mn i-k, gl . . . . . gk)

= o , k + , ( m l . . . . . r n . _ , _ ~ , g , . . . . . gk. q * . . . . . q*) = 0

so a ' ( q l . . . . . qi) �9 K0,p-i. M o r e o v e r d a ' ( q l . . . . . qi) �9 Ko,p-i+l s ince

d a ' ( q l . . . . . q~)(gl . . . . . g._~, r n ) = d a ( g ~ . . . . . g._~, re, q* . . . . . q*) fo r m e N .

T h u s t h e r e is a 18'(q~ . . . . . qi) �9 K0.p-~ such tha t a ' ( q l . . . . . q~)-d18 ' (q l . . . . . q~) �9

K o , p - i + l .

D e f i n e 18 by 18(gx . . . . . g . ) = 18'(~r(g._,) . . . . . r r ( g . ) ) ( g l . . . . . g . - i - 1 ) . W e h a v e

/3 �9 B~,p (so 18i = 0 fo r ] < i). M o r e o v e r

18i+k(ml . . . . . m . - i - k - 1 , gl . . . . . gk, hi . . . . . hi)

= 18'(Tr(hl) . . . . . ~r(hi))k(rnl . . . . . r n . - i - k - 1 , gl . . . . . gk) = 0.


So /3 e K. Since

( o r - d/3)(gl . . . . . g , - i - a , m, hi . . . . . /1i)

= (or'- d/3')(zr(hl) . . . . . 7r(h~))(g 1 . . . . . g ._,_, , m)

it follows tha t ot - d/3 ~ K~.o+l.

f o r m E N ,

w A calculation using L *( G, N; A)

P R O P O S I T I O N 12. Let A be a divisible abelian group. Let N be a central subgroup of a perfect group G. Then H3(G/N; A)--~ H3(G, A) is injective and HE(G/N; A ) --~ HE(G, A ) is surjective (trivial coefficients).

Proof. I t is equiva len t to show H E ( G / N , G ; A ) = O . By T h e o r e m 10, it is equiva len t to show A E ( G , N ; A ) = O . So let a = ( o t o , otl) be a cocycle of L*(G, N; A) . W e must show a = db, where b = (/3) and db = (dN/3, d~/3). Now

0 = d~a l (n , g, h) = gcq (g - lng , h ) - a l (n , g h ) + a l (n , g)

=Otl(n,h)-otl(n, gh)+ax(n,g) (since A is a trivial G module and N is central

so ax(n,-):G--~ A is a h o m o m o r p h i s m . Since G is per fec t this shows O/1 ----0.

NOW 0 = o q ( n , m)=o t0 (n , m)--Oto(n, m-into)=Oto(n, m)--Oto(m, n) so Oto: N E --~ A is a symmet r i c cocycle. T h e cor responding extension 1 --~ A --~ C --~ N --~ 1 is thus abel ian and represen t s an e l emen t in Ex t ) (N, A ) = 0, since A is divisible (hence injective, c.f. [M2] p. 93). So this ex tens ion is split, i.e. 0 = [ a 0 ] e HE(N, A) , i.e., a0 = dN/3 for / 3 : N ~ A. N o w

SO

do/3(n, g) = g/3(g- lng) - / 3 ( n ) = /3(n) - / 3 ( n ) = 0

db = (dN/3, do~3) = (dN[3, O) = (Oto, O) = (ao, cq) = a.

The case N central, A trivial

Le t 1 --~ N- -~ G ~ Q ~ 1 be a central ex tens ion and A a trivial G module . T h e n equat ions 2 - 4 b e c o m e

(2) 0 = o q ( m , g ) - c t l ( m n , g )+o t l (n , g) i.e. oq is bi l inear

(3) 0 = o q ( n , h ) - a l ( n , g h ) + o q ( n , g)

(4) cq(n, m) = Oto(n, m)-cto(m, n)


It follows easily that

PROPOSITION 13. For a central extension with trivial coefficients we have an exact sequence

0 ~ A2(G, N; A) -~ H2(N, A) x Bilin(N x G, A) -~ Bilin(N x N, A)

(where i(ao, al) = ([a0], al) and

t([a0], cq) = oqlN• N - (ao - Tcto), Toto(n, m) = ao(m, n).)

COROLLARY 14. (1) Extl(N, A) • Bilin(N x Q; A) c A2(G, N; A) (2) If Bilin(N x G; A) = 0, A2(G, N; A) = Extl(N, A) (3) If H2(N, A) = O, A2(G, N; A) = Bilin(Nx Q; A)

COROLLARY 15. I[ G is perfect, A2(G, N; A ) = E x t ' ( N , A)

Remark. The results in this section may also be derived from known results from the homology theory of groups. [K] [EHS] [GS].

REFERENCES

It]

[cw]

[EHS]

[G-S]

[H-S]

[I-I]

[Ho] [J]

[K]

[L] [M] [O]

A. CONNES, Sur la classi[ication des facteurs de type II. C-R Acad. Sci. Paris, A-B, 281 (1975) 13-15. C. C. CHENG and Y1 C I W U , An eight-term exact sequence associated with a group extension. Mich. Math. J. 28 (1981) 323-340. ECKMANN-HILTON-STAMMSACH, On the homology theory of central group extensions II. The exact sequence in the general case. Comment. Math. Helvetici 47 (1972) 171-178. GUT-STAMMBACH, On exact sequences in homology associated with a group extension. J. Pure and Appl. Algebra 7 (1976) p. 15-34. HOCHSCHILD and J. P. SERRE, Cohomology of group extensions. Trans. A.M.S. 74 (1953) 110-134. J. HUEBSCHMANN, Group extensions, crossed pairs and an eight term exact sequence. J. Reine Ang. Math. 321 (1981) 150-172. D. F. HOLT, An interpretation of the cohomology groups. J. of Algebra 60 (1979) 307-320. V. F. R. JONES, An invariant for group actions in Algi~bres d'oI~rateurs. Springer Lecture Notes 728 (1978) 237-253. M. KERVAmE, Multiplicateurs de Schur et K-th~orie in Essays on topology and Related topics. M6moires d6di6s h G. de Rham. Springer-Verlag 1970 p. 212-225. J. L. LODAY, Cohomologie et groupe de Steinberg relatif. J. of Algebra 54 (1978). 178-202. S. MACLANE. Homology. Springer-Verlag 1975. O. PINO ORTIZ. Conjecture g~n&ale en cohomologie relative des groups. Th~se, Universit6 de Gen~ve, 1982.

164 N A T H A N H A B E G G E R ~ AL.

JR] [W]

J. RATCLIFFE. Crossed Extensions. Trans. A.M.S. 257 (1980) 73-89. J. H. C. WHITEHEAD. Combinatorial Homotopy II. Bull. A.M.S. 55 (1949) 453-496.

N. H., University of Geneva, Switzerland V. J., University of Pennsylvania, Philadelphia, USA O. 1:). O. University of Geneva, Switzerland J. R. University of Wisconsin at Madison, USA

Received March 11, 1983

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Relative cohomology of groups