Journal of Pure and Applied Algebra 58 (1989) 305-320 305

North-Holland

ON THE HOMOLOGY OF THE HIGHER RELATION MODULES

Rainer ZERCK
Akademie der Wissenschaften der DDR, Karl- WeierstraJ-Institut fur Mathematik, PF 1304,
Mohrenstr. 39, DDR-1086 Berlin, German Democratic Republic

Communicated by K.W. Gruenberg

Received 25 September 1987
Revised 8 February 1988

Let 1 --t R + F+ G + 1 be a free presentation of the group G. In the present paper we study
the homology groups of G with coefficients in the higher relation modules R,= yCR/y,+ ,R
(c?2), where yCR denotes the cth term of the lower central series of R. It is shown that
H,,(G, R,.) for n 2 1 as well as the torsion subgroup of H,,(G, R,) are annihilated by multiplication
with c*, where c* = 4 for c = 2 and c* = c for cz 3. For finite G a similar result holds for the Tate
cohomology groups: c* . fi”(G, R,.) = 0, n E Z. Moreover, the rank of the free abelian direct sum-
mand of H,(G, R,.) will be determined.

1. Introduction

For an arbitrary non-trivial group G and a free presentation

l-R-FAG-1 (1)
let ycR denote the cth term of the lower central series of R: y,R = R, y,R = R’=
[R,R], yc+l=[ycR,R], ~21. The factors R,:=y,R/y,+,R, ~21, may be regarded
as left G-modules by setting

wR-(uyc+,R):=wuw-‘yc+,R, WEF, u~y~R. (2)

R, is called the cth relation module of G associated with the free presentation (1).
R, = R/R’ is the ordinary relation module.
From the group-theoretical point of view the O-dimensional (co)homology groups
of G with coefficients in R, are of special interest.
Case of (HO). The homology group H,(G, R,) is isomorphic to y,R/[y,R, F], the
kernel of the free central extension

l+y,R/[y,R,F]+F/[y,R,F]-+F/‘y,R+l.

It has been studied by a number of authors. A detailed survey of the corresponding

results may be found in [ 191. The long-prevalent opinion that y,R/[ ycR, F] for
cz2 is torsion-free was shaken by C.K. Gupta [7] in 1973. She discovered elements

0022-4049/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

306 R. Zerck

of order 2 in the free centre-by-metabelian groups F/[F”, F] of rank greater than 3.

By a result of Shmel’kin [17] the groups F/y,R do not contain elements of finite
order for ~12. Therefore the torsion elements mentioned form a subgroup of
F”/[F”, F] = y#“/[ y2F’, F]. Passing on to the general situation Stohr [19] has
shown that y,R/[ ycR, F] decomposes into the direct sum of a free abelian group A,
and the torsion subgroup, which is of exponent dividing c*, where c* equals c for
~13 and c*=4 in case c=2:

c* . tors(HO(G, R,.)) = 0. (3)

In [18] the same author characterized this torsion subgroup as the kernel of a certain
matrix representation for F/[y,R,F], which was given by C.K. Gupta and N.D.
Gupta [8]. We also mention that by using this representation y,R/[y,R, F] is
shown [8] to be the centre of F/[ ycR, F].
Case of (HO). There is a similar assertion concerning the group of fixed points
H’(G, R,) of the cth relation module: H’(G,R,) is the centre of F/y,+,R. This
centre is non-trivial if and only if G is finite. In case c= 1, this is a classical result
of Auslander and Lyndon [ 11. The case c> 2 is treated by C.K. and N.D. Gupta [8].
The purpose of this paper is to study the homology groups of G with coefficients
in the higher relation modules (~12). In this way some of the cited results can be
integrated into a more general context. So we get together with a new proof of (3)
our main

Theorem 4.1.

c*.H,,,(G,R,)=O for ~22, mzzl.

Concerning the structure of the cohomology groups with coefficients in R, we

again have to distinguish between finite and infinite G. In general, for infinite
groups the quoted cohomology groups are not annihilated by c*. In Section 7 we
show that for all dimensions m L 1 there is a group G with a free presentation (1)
such that H”(G, R,) contains elements of infinite order for all c> 2.
Sections 5 and 6 of the present paper are devoted to finite G and its Tate cohomo-
logy groups i?“(G, R,), m E Z, CI 2. A slight modification of our methods leading
to Theorem 4.1 allows us to show ((s; t) denotes the greatest common divisor of the
positive integers s and t)

Theorem 5.3. Let G be a finite group, ICI the order of G. Then

(IGl;c*).A”‘(G,R,)=O for ~22, me??.

The case m = 0 was first investigated by Stohr [20]. In non-homological terms his
result reads as follows: the group of fixed points of R, is exhausted up to exponent
c* by its trivial part N,R,={N,a= CgEGgala~R,}, the norm subgroup, i.e.
c*RGcc - N,R,c RGc.
 Homology of higher relation modules 307

It is shown by Baumslag, &rebel and Thomson [2] that in the case of infinite G
the free abelian direct summand A, of y,R/[ y,R,F] is of infinite rank. This result
may be completed by the corresponding one for finite G with finite presentation (l),
which we get using Theorem 5.3. We prove that rankA,=rank R,?. The rank of
RF was computed by N.D. Gupta, Laffey and Thomson [9]. We will state

Theorem 5.4. Let G be finite, IGl the order of G.

(i) The torsion part of y,R/[y,R, F] is isomorphic to l?‘(G, R,,) and of expo-
nent dividing (c*; ICI).
(ii) The rank of the free abelian part of y,R/[ yCR, F] equals

r,= & &Ad). n,(G). (mc’d- I),

c
where p is the Miibius function, m= 1 +(j- l)/GI =rank R and n,(G) is the
number of elements g in G with gd= 1.

Thus, in the case of a finite group the torsion part of y,R/[ yCR, F] may be inter-
preted as a certain cohomology group and the result (3) may be strengthened.
For a given group G, the G-module structure of R, depends on the choice of the
free presentation (1). Nevertheless, we get some independence results concerning the
homology groups. We will restrict ourselves to finite groups G and free groups F
of finite rank while discussing independence questions. So fix a further representa-
tion of this type with an arbitrary free group P and a free subgroup f?

l-l?-PZG-1. (1)

Write 8, for the corresponding relation modules. By a result of Williams [21] two
non-minimal relation modules of a finite group are isomorphic, if they have the
same rank (i.e. rank F= rank P> d(G) = the minimal number of generators of G).
In case rank F= rank P= d(G) examples for R, + l?, were constructed by Dyer and
Sieradski [4] thereby answering a question of Gruenberg (see [6, Lecture 51) in the
negative. So R,=& cannot be expected.
Nevertheless we shall show

Theorem 6.2. If rankF=rankF, then l?“‘(G,R,)=tim(G,Rc), me& ~21.

The exceptional role of c = 2 (c* = 4 for c = 2, whereas c* = c for CL 3) is confirmed

by Kuz’min [l 11, who constructed a free presentation (1) such that y2R/[ y2R, F]
contains elements of order four. The 4-part of fi”(G, R2) does not at all depend on
the presentation, i.e. even for rank Ffrankp we get

Theorem 6.4. For finite G and finite presentations (1) and (i):

2.Am(G,R2)‘2.Am(G,a,), meZ.
308 R. Zerck

In his recent paper [12] Kuz’min gives a detailed description of the torsion part
of Y#/] Y& Fl = ffo(G, Rd.
Our main tool is the cap product reduction theorem of Eilenberg and MacLane
(see [lo]) in connection with the canonical action of the symmetric group on c letters
on the cth homogeneous component of the universal enveloping unitary ring of the
free Lie ring @_, R,. The basic homologic and Lie-theoretic preliminaries are
collected in Section 2. The &-action on the homology level is studied in Section 3.
Section 4 is devoted to the proof of the Main Theorem. In Sections 5 and 6 we deal
with finite G: at first we adapt the Main Theorem to the Tate cohomology theory
and-derive the results mentioned on the torsion part and on the free abelian part
of y,R/[ yCR, F]. In Section 6 we discuss whether I?“(G, R,) depends on the
presentation (1). Finally, in Section 7 we give an example for c*. W’(G, R,) ~0,
mrl, Cll.

2. Preliminaries

2. I. General

Our notation is standard. All modules under consideration are left. For an abelian
group A the torsion subgroup is written tors A. The symbol SC is adopted for the
full symmetric group on c letters. sgn denotes the sign homomorphism SC+
{ 1, - l} I C*. Further notations have just been explained in Section 1.
A group G acts on the tensor product of the G-modules B,, . . . . B, (c> 1) dia-
gonally:
g(b,O...O6,):=gb,O...Ogb,, gEG, b;eB;.

The cth tensor power of a G-module B is given by T,B :=Z, T,B := B@ T,_ ,B,
CL 1. T,B is a left &.-module via

a(b10...06,):=b,~l~,,0...06, I(~), a~&, ~;EB.

The G- and the &-action on T,B centralize one another.

2.2. Group (co)homology

Our terminology is that of [3]. In particular, we take over the sign conventions.
For a group G and a G-module homomorphism ly :A + B denote the induced
homological map H,(G,A) -+ H,(G, B) by H,(w). The augmentation ideal of G is
written Io. The sequence

aw
O-I G -ZG-Z-O (4)

splits over L.
One obviously gets
 Homology of higher relation modules 309

As is well known, the Tate cohomology groups Am(G,B) are defined for finite
groups by setting

whereas &‘(G, B) and A”(G, B) are the kernel and the cokernel of the norm map

No : H,(G, B) --t H”(G, B) : b + IGB ++ c gb,

gee
respectively, i.e. the sequence

O-I?-l(G,B)+Ho(G,B)-tHo(G,B)+~o(G,B)-tO
is exact.

2.3. The cap product reduction theorem

For the material of this subsection we refer to [lo] and [ 141. The tensor product
P:=~G~FZFisaleftG-modulebysettingg(h~m):=gh~m;g,h~G,m~I~.It
is well known that P is G-free on { 1 @(u - 1) /YE Y}, where Y is a set of free
generators of F. P is contained in the exact and z-split G-module sequence

0-R+P%o-0, (5)
called the relation sequence, where

~(u~~R)=I@(u-~), UER, o(l@(y- l))=rc(y)- 1, JVE Y.

For an arbitrary chosen G-module D we get by dimension shifting

H,+,(G,D)=tH,+,(G,Z,OD)

_ H,(G,R, 00, WIT 1,

+ i ker(Ho(G,R,@D)-+Ho(G,P@D)), m=O.
The isomorphism

o:H,+,(G,D)s H,?2(GyRi@D)3 mr 1,
i ker(Ho(G,R1@D)-+HO(G,P@D)), m=O

may be alternatively obtained by the cap product with the characteristic class
XEH’(G,R~) corresponding in the sense of Schreier’s theory to the group exten-
sion 1 -tR/R’+ F/R’-+ G+ 1, viz., (Y is induced by

xn-:H,,,+,(G,D)+H,(G,R,@D), mr0.
310 R. Zerck

The c-fold cap multiplication with x yields the isomorphisms

H M+ZAG, D)

_ HAG, T,R, 0 D), m2 1,

(6)
+ ker(H,(G, TCR,OD)-+HO(G,PO T,_,R,@D)), m=o.

Lemma 2.1. The kernel in (6) coincides with the kernel of H,(G, T,R, @D)+
H,(G, KPOD).

Proof. The Z-decomposition P=R,@IG shows that there is an exact L-split G-

module sequence 0 + T,-, R, + T,-, P + U-+ 0 with U free abelian. This implies
that after tensoring with P and D we get the exact G-module sequence

O-POT,_,R,OD-*T,POD-tPOUODjO.
The G-module P@ U@D is induced and therefore homologically trivial. Thus
H,(G, P@ T,_ ,R, @D) + H,(G, T,P@ D) is injective. Consequently,

H,(G, T,R, OD), mll, (7)

H m+zc(G,D)s
i ker(H,(G, T,R,@D)+H,(G, T,P@D)), m=O.
0
2.4. Lie-theoretic preliminaries

For proofs and detailed discussions of the following we refer to [ 151 and [ 161.
Fix a free presentation (1). The normal subgroup R is itself a free group by
Schreier’s theorem. Let Xbe a set of free generators for R. The graded abelian group

9(R) := @ yCR/y,+, R = @ R,
L-21 C?l

is associated with the lower central series of R. S?(R) is a Lie ring with Lie multi-
plication given by

[w,, ,R UY~+IN := [u, ~1~c+d+lR

= ylupl UE ycR, vEydR. (8)
uvYc+d+ ,R,

The G-module structure (2) is compatible with the Lie multiplication (8). 9(R)
turns out to be the free Lie ring on X (more precisely: on {xR’ IxEX}). For this
we write 9(R) = L(X) and pass on to the additive notation for the abelian group
structure of R,. The tensor ring

g(R) := @ T,R,,
CZO

which is the free unitary associative ring on X, is the enveloping unitary associative
ring of S??(R)with injective G-homomorphism

v : g(R) + g(R).
 Homology of higher relation modules 311

For an explicit description of v in degree c consider the element Q2, of the integral
group ring ZS,. (~22):

Q,=(l-i(c))(f-Uc-l))...(l-C(2)),

where [(n)=(l . . . n) denotes the cyclic permutation 1 H 2 ++ 3 c ... ct n H 1. For the

cth component V~ of v we get

v,]q, ... f a,.] = n&l, 0 ... @a,),

where [al, . . . , a,] = [[a,, . . . , a,_ 1], a,] is a left normed commutator, aieR,. Note
that R,. is as an abelian group generated by left normed commutators.
In the opposite direction there is a surjective G-homomorphism

Q : 9(R) --f g(R),

which is given in degree c by

e,(alO...Oa,)=[a,,...,a,.l, a;eR1. (9)

The results in the following sections are essentially based on the fact that the restric-
tion of QV to the cth component of 9(R) is the c-fold identity map (Wever’s for-
mula, see [15, Chapter 5]), i.e. for any PER,:

&V,(b) = cb. (10)

3. SC-action

As explained in Subsection 2.1, T,H”(G,B) carries a left S,-module structure.

On H,(G, T,BOD) an &-action may be defined via H,(a@idD), DES,, B and D
being G-modules. Define

K: T,H”(G,B)OH,,,+., (G,D)-+H,(G, T,BOD),

~,0...0P,0s-~~n(...n(~,ns)...)
to be the c-fold cap product, where pie H”(G, B), 6 E H,+,,(G, D).

Proposition 3.1. For all DES, the diagram

T,H”(G, B) 0 H,,, +,,,(G D) A HAG, T,BOD)

T,H”(G
I(sgn u)“(o@

B) 0 H,,, + .,(G
id)

0 A H,,,(G WOO
I
0

is commutative.
312 R. Zerck

Proof. The cap product is adjoint

to the cup product in the sense that for pi E
H”l(G, B,), /3zE Hn2(G, B2), and 6 E H m +n, +.,(G, D) the equation PI n (P2 n 6) =
(fl,Ufl,)nd holds. Hence K(P~O...OP~O~)=(P~U...UP~)~~. Now the asser-
tion follows from the associativity and the semi-commutativity of the cup product.
0

The cap product reduction theorem in its iterated version (7) seems to be tailor-
made for proving

Proposition 3.2. The &-actions on ker(H,(G, T,R,) + H,,(G, T,P)) and H,(G, T,R,),
rnz 1, are trivial for ~22.

Proof. We have seen that the isomorphism (7) is realized by the c-fold cap multi-
plication with x:

KOIO...OXO-_):H,+~,(G,~)

HAG, T,R,), m>l,

(11)
? t ker(He(G, T,Ri) + He(G, V)), m =O.
By Proposition 3.1 we obtain that the SC-action on the right side of (11) solely per-
mutes the x’s on the left side without changing the sign because of n = 2. 0

4. The Main Theorem

The purpose of this section is to prove

Theorem 4.1. H,,,(G, R,), ml 1, and the torsion part of H,,(G, R,) are of exponent
dividing c*, c 12.

Proof. Note first that H,(v,): H,(G,R,)+ H&G, T,R,) maps the torsion part of
H,(G, R,) into ker(H,(G, T,R,) + H,(G, T,P)). Indeed, since T,Pis a free G-module,
H,(G, T,P) is torsion-free.
Now choose the element A, E ZS,
A .= (1)+(12)9 c=2,
” I (123)+(132)-(12), cr3
and consider the diagrams

torsWo(G,R,))

Ho(vd (12)

ker(Ho(G,
I
T,R,) + H,(G, T,P)) kerWo(G
I
T,R,) + H,(G, T,P))
 Homology of higher relation modules 313

and
HAG, R,) HAG, R,)

bC1;) :_:r. 1H&c) (13)

H,(G,T,R,) m ‘ ’ ff,n(G, T,R,)

By Proposition 3.2, H,(A,) =aug I,. id =c*/c. id. Together with (10) this fact
shows the commutativity of the diagrams (12) and (13):

HA@,). H,bL)~ fC,,(v,) = ff,Ae,) . bug A,). fMv,)

= (aug A,). H,(Q~v,) = (aug ,I,). H,(c* id)

=(aug&).c=c*.

Our next goal is to prove that

0, : CR, -+ R,

is the zero map. Put a, @ .e. @CI,E T,R,. Then

ec((l)+(12))(a,O...Oa,)=e,(a,OazO...Oa,+azOa*O...Oa,)

=[a,,a,,...,a,l+[a,,a,,...,a,l

=[[a,,a,l+[a,,a,l,a,,...,a,l
=o
because of the anticommutativity law in 9(R). We have e2A2=0 as claimed.
Moreover, for cz 3 we get

= [ha2,4 + ~J~,~~,Q~I
+ [a3,al,a21,a4,. . . ,a,1 =O
by the Jacobi identity in 61?(R).
For cr3 we have &=((1)+(123)+(132))-((1)+(12)), consequently &A,=O.
From this, H,(Q~). H,(A,) = H,@,A,) = 0 follows. Returning to the commutative
diagrams (12) and (13) we see that c* annihilates tors(HO(G,R,)) and H,(G,R,),
m L 1, respectively. Now the proof is complete. 0

The ‘&-trick is due to Kuz’min [l l] in case c= 2 and to Stohr [19] for arbitrary
CI 2. They were concerned with the group-theoretical relevant part of Theorem 4.1,
which we will state as
314 R. Zerck

Corollary 4.2. The torsion subgroup of y,R/[y,R,F] is of exponent dividing c*,

cr2. 0

5. The case of a finite group (fixed presentation)

Throughout Sections 5 and 6, G will be a finite group with finite presentation (1).
Propositions 3.1 and 3.2 as well as Theorem 4.1 may be reformulated in terms of
Tate cohomology groups. In this context the cap product must be replaced by the
cup product of Tate cohomology theory. The cup product reduction theorem reads
as follows: cup multiplication with the characteristic class x E fi2(G, R,) yields the
isomorphisms

A~(G,D)zP+~(G,R,~D)
in all dimensions m E Z. c-fold cup multiplication implies the isomorphisms

A”(G,D)~P+~C(G, T,R,@D).
Repeating the arguments for showing Proposition 3.1, 3.2, and Theorem 4.1 we get

Proposition 5.1. For all CJES, the diagram

TCI?“(G,B)@I?m(G,D) ---fi”+C’“(G,
K T,B@D)

(sgn o)“(o 0 id) c7

! I
TC~“(G,B)@~“(G,D)~~m+cn(G, T,B@D)
where K now denotes the c-fold cup product, is commutative. 0

Proposition 5.2. The SC-action on l?(G, T,R,), m E Z, CT 2, is trivial. 0

Similarly to Theorem 4.1 we get c *.l?“(G,R,)=O for all ~12. Moreover,

because of ICI. @‘(G, R,) = 0 we can state

Theorem 5.3. For al/ m E Z, cz 2: (c*; /Cl). fi”(G, R,) = 0. 0

In connection with the norm map

the groups F?‘(G, R,) and fi’(G, R,) are of special interest. Let j= rank F be
finite. Then R, as well as RF and NGRc are finitely generated free abelian groups.
Because of c* . fi’(G, R,) = 0 we get c * . RF c No R, c RF and, consequently,
r, = rank RcG= rank No R,. (14)
 Homology of higher relation modules 315

This rank was computed by N.D. Gupta, Laffey, and Thomson [9]:

rc=+q$
Pu(o%(G)~(mC’~-l) c
(15)

where m=l+(j-l)lGl= rank R, n,(G) is the number of elements in G with

gd = 1, and p is the Mobius function. Clearly, the sequence

0 -+ I?-‘(G, R,,) + H,(G, R,) + NGRc -+ 0

splits, i.e. there is an isomorphism

H,(G,R,.)=AP1(G,R,)@N,R,.
Thus we have proven the following:

Theorem 5.4. Let G be finite, ICI the order of G.

(i) The torsion part of y,R/[ ycR, F] is isomorphic to I?‘(G, R,.) and of expo-
nent dividing (c*; ICI).
(ii) The rank of the free abelian part of H,(G, R,) equals rC. 0

In closing this section we give an example which illustrates some aspects in deter-
mining yc R/[ yCR, F] explicitly by the methods considered.

Example. Let G be the dihedral group of order 8 given by the presentation

F=F(y,,y,), n(y,)=w, 7c(y2)=5, III*= 1, r2= 1, (c~r)~= 1.
Then S= { 1, y,, y:, yf, y,, y,y2, yiy2, yfy,} is a Schreier system of coset represen-
tatives of F/R. R is freely generated by all non-trivial elements of type S,V;(~(Syi)))’
with .sES, ie { 1,2} and q: F--+S a selecting function, i.e., R is free on x1 =yf,
-1 -1 -1 -1 -2
x,=Y,YlY, -’ Yl p3, x3 =Y,YzY,Y, 7 x,=Y:Y,Y,Yz Yl 9 x5 =Y:y2YlY2 Y1 , qi=y;, x7=

YlY22Yk 2 2
x8 =yly2yl -2, x9 =y;y;y;3. 2(R) is G-isomorphic to the additively written
free Lie ring on {x,, . . . , x9}. The G-action looks as follows:

i WX8 TX,

Xl x,+x,+x,+x,
-x, +xj -x2-x4-x5+x6-x7

x4 x,+x2-x6+x9

X5 x5+x,-x,

XI +x2 x4+x,-xg

Xl x6
X8 XY
x9 X8
x6 Xl

In order to simplify the calculations we define

x:.=x,,
I . iE{1,6,7,8,9}, x!, ‘ZX
. , -_x.,+4, iE{3,4,5},

x; :=x1 +x,-x,.
316 R. Zerck

cx;,***, x;} turns out to be a free generating set of S!?(R), too. G acts on these
generators by
I
OX; =X;(i) with O = (2345)(6789) E Ss, 1~ is 9,
9
7x; = -x; + c x;,
j=2
,
rXi =~fc;) with ? = (23)(45)(79) E Sg, 2 5 is 9.

g2(R) is free abelian with basis [xl, xj], 1~ i< j 5 9. IoXz(R) is generated by the
elements (w- l)[x;,xjl] and (7- l)[x,‘,xJ, 1 <i<js9. Then Hc(G,g!,(R)) is isomor-
phic to Z @ Z 0 Z, 0 Z, @ Zz 0 Z2, the direct summands being generated by [x&x;],
[&xd, i-$x;1 + L&x~l+ [xi,x~l, [xi,xil+ [&x;l, [&x;l, t&&l moddo L$Z2W,
respectively. Finally, the elements [x3,x9] [x7,x91P1, [xs,x,], [x~,x~][x~,x~][x~,x~]-~ .
[%x91-‘, [%x31, [x6,+1, [x1~~~1[~2,x~~[~~~x61[x~~~~1~1[~2,~-il-’ independently

generate the multiplicatively written group y2R/[ y2R,F]. They are of order 03, M,
4,4,2,2 modulo [ y*R,F]. The group I?‘(G,L?~(R)) is trivial because of _&(R)‘=
NGg2(R) = Z 0 Z with generating elements No [xi, x;] and No [xi, x;].

6. The case of a finite group (various presentations)

The group G is still assumed to be finite. The purpose of this section is to get some
information on how the Tate cohomology groups Am(G, R,) depend on the special
choice of the free presentation (1). Let

1-R-+~:G-l (0
be a further free presentation of G. The corresponding relation modules are denoted
by R”, . Assume both rank F= j and rank a=i to be finite.
We will make use of local arguments. For the basic facts we refer to [6]. Denote
by Z(o, the semi-local ring consisting of all rational numbers with denominator
prime to the group order. A G-module D is called a G-lattice, if the underlying
abelian group is free of finite rank. Clearly, the relation modules considered in this
section are G-lattices.
By setting g(d@q) :=gd@q, where g E G, LED, q E ZCG), the abelian group
D CG):=D@ZCG) is endowed with a G-module structure.

Proposition 6.1. For any G-lattice D and m E Z we have the isomorphism I?“(G, D) =
pm(G, DC,,).

Proof. The G-homomorphism

r:D-+DCG):dHd@l
is obviously injective. Denote by K the cokernel of this map, i.e. we have an exact
 Homology of higher relation modules 317

G-module sequence

O+D+D,,,-K+O. (16)

Take an arbitrary element d@a/b from DC,,, deD, a/b E L(,,. Then r(ad) =
b +(d@ a/b), i.e. K is a torsion group and the order of each element is prime to the
order of G. Hence, A”(G, K) = 0 for all m E Iz. The long cohomological sequence
associated with (16) gives the desired result. 0

Now we are in the position to prove the independence result.

Theorem 6.2. If j=j? then I?‘(G, R,)-I?m(G,Z?c) for WI E H, ~2 1.

Proof. The ordinary relation modules R, and I?, are locally isomorphic:

R&3+,-~,O&. (17)

Indeed, by Schanuel’s lemma we have the G-module isomorphisms

R,@ZGi=&@ZGj. (18)

After tensoring with IZ(o) we get

(RI 0 zcc,) 0 z,,,GJ= (R, 0 z(,,) 0 +)Gj.

Now because of j=Jthe semi-local cancellation theorem (see [6, Lecture 41) yields
(17).
The z(o)-Lie algebras g(R) @ Zcc) and 9(p)@z(,, are free and their first homo-
geneous components are G-isomorphic. Consequently, the same is true for the cth
components:

R,@ ZcG, = kc @ ZcG) as G-modules.

NOW the theorem is a consequence of Proposition 6.1. 0

Corollary 6.3. If j =jl then y,R/[ ycR, Fl = y,R/[ y,l?, PI.

Proof. This assertion immediately follows from Theorem 5.4 and 6.2. 0

Now we turn to the interesting case c = 2. We will show that for fixed m the 4-part
of fim(G, R2) is independent of the presentation (l), i.e. the rank of the 4-part of
A”(G, R2) is a group invariant.

Theorem 6.4. For finite j and j-and m E Z: 2. c?“(G, R2) = 2. I?(G, l?,).

Proof. We fix the notation L,(Z) for the cth homogeneous component of the free
Lie ring L(Z) on a non-empty set Z, cz 1. Then L(Z) = @,, 1 L,(Z).
Remind that 9(R) =L(X) and 9(R) =L(g). Consider the free Lie rings
318 R. Zerck

L(XUX&) and L(XUXj,) with

X$={x,JgEG, l<ili} and X&={2g,,,ig~G, l<i<j}.

Define G to act on X& and X& by h. Q = xhg,; and h . Zg,i = $,,;, where h E G,
X,;EXi,, $,;EXjc.
The first homogeneous components of the Lie rings Z,(XUX&) and ,C(XUXG)
are G-isomorphic by Schanuel’s lemma, see (18). For the second homogeneous com-
ponents we get the isomorphisms

R20(R10ZG/)OL,(X~)=L,(X)O(L,(X)OL,(X~))OL,(X~)

=L2(XUX~)=L*(XUX~)

Hence,
@‘(G, R,)@k??m(G,L2(X&)) =I?“(G,&) @A”(G, L,(Xjc)),

since Ri 0 ZGJ and k, @ ZGj as induced G-modules are homologically trivial.

In order to show Theorem 6.4, it now suffices to prove

Lemma 6.5. For j, cz 1, m E 2 c. H”(G, L,(XG)) = 0.

Proof. T,L,(Xi) is an induced G-module, hence homologically trivial. From the

commutativity of the diagram

I L
fi”(e,)
O=l)m(G, T,L,(X~))- r?"(G,L,(X%))

the assertion follows. 0

Corollary 6.6. The 4-part of y2R/[ y2R, F] is independent of the special choice of a
free presentation (1) of G. 0

7. On the cohomology of R,

Theorem 4.1 leads to the question as to whether the positive dimensional co-
homology groups of G with coefficients in R, are of exponent dividing c*. For
finite groups the answer is “yes”, see Theorem 5.3. But for infinite groups the struc-
ture of the cohomology groups Hm(G, R,) essentially differs from that of the
homology groups H,(G,R,). It seems to be difficult to say something about
H”(G,R,) in the general case.
 Homology of higher relation modules 319

The following example shows that H”‘(G,R,) may contain a non-trivial free
abelian subgroup.
Let G be the free abelian group of rank m: G = Zm. The cohomological dimen-
sion of G equals m. Moreover, G is an orientable Poincart duality group (see [3,
Chapter 8]), that means, for all G-modules A4 and i?O we have isomorphisms in-
duced by the cap product with 1 E Z- H,(G, 27):

H’(G,M) S H,_,(G,M).

In particular, for c> 1

Hm(G, 4) 1 H,(G, R,),

and the latter contains a free abelian direct summand of infinite rank [2].

Acknowledgment

It is my pleasure to thank R. Stohr for useful communications about the present

paper.

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