C AHIERS DE
TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE
CATÉGORIQUES
G. J. E LLIS
C. RODRIGUEZ -F ERNANDEZ
An exterior product for the homology of groups
with integral coefficients modulo p
Cahiers de topologie et géométrie différentielle catégoriques, tome
30, no 4 (1989), p. 339-343
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VOL. XXX- 4 (1989)
CAHIERS DE TOPOLOGIE
ET
G.8OMrTRIE DIFF.8R.ENTIELLE
CATÉGORIQUES
AN EXTERIOR PRODUCT FOR THE
HOMOLOGY OF GROUPS
WITH INTEGRAL COEFFICIENTS MODULO p
by G. J. ELLIS 1 and
C. RODRIGUEZ-FERNANDEZ
R2SUMfi.
Les auteurs g6n6ralisent la suite exacte a 8 terde groupes d’homologie entiere obtenue par Brown et
Loday en une suite exacte de groupes d’homologie a coefficients dans ZP, ou p est un entier non-n6gatif.
mes
In this article
we generalize Brown and Loday’s eight term
integral group homology [2] to an exact sequence in group homology with coefficients in Zp, where p is
any nonnegative integer.
exact sequence in
Let G be
Zp=Z/pZ
THEOREM
as
a
a group with a normal subgroup N, and consider
trivial G-module. We prove
1. There is
a
natural exact sequence
Here N**pG denotes the subgroup of N generated by the
elements [n,g] and nf’, for g E G, n E N. (When x,y are elements
of a group,
we
write
[x,y]
=
xy x-1y-1 and ’ w- = xi- x- 1.)
The group NAPG is a new construction. It is generated by
the symbols nAg and Inl for n E N, g E G, subject to the relations
1 This author would like to thank the
University of
its hospitality during the preparation
Compostela for
ticle.
339
Santiago
of this
de
ar-
for g, h E G, m, n E N. Note that if N = G,
redundant.
Clearly NAPG
It is routine to check that 6 is
and that its image is NUpG .
As
an
COROLLARY
Also, for
a
is defined
by
well-defined
homomorphism,
immediate consequence of Theorem 1
2. There is
aw
are
is functorial in N and G.
homomorphism 1: NAPG-G
The
then (2) and (7)
an
we
have
isomorphism
presentation R &#x3E;- F - G of G, there
is
an
isomor-
phism
In order to prove Theorem 1
LEMMA 3.
FAPF rr
If F is
a
we
need the
free group, then 6 induces
following
an
isomorphism
F#PF.
from 121 that NAG is the group generated by the
n A g for g E G, n E N, subject to relations (1), (2), (3).
There is thus a homomorphism i: N A G- N APG, n A g + nAg. By
(4) the image of i is normal in NAPG. On taking G = N = F we
thus have a commutative diagram
PROOF. Recall
symbols
where 8" is induced by d, and where 1’ is the isomorphism proved in [21 (see [3] for an algebraic proof of this isomorphism).
340
The homomorphism 8" is clearly surjective, and hence has a
splitting since pFab is free abelian and FAPF/lm(L) is abelian by
(6). This splitting is surjective because of (5). Therefore 1" is an
isomorphism. Since the rows of the diagram are both short
exact, it follows that d: FAPF-F#pF is an isomorphism. ·
In [I] the
following
natural exact sequence
is obtained. Thus to prove Theorem 1 it suffices to exhibit
isomorphism rr:NAPG-L0VP1(a)
an
such that
commutes.
For any surjection s: F-G with F a free group let S be
the kernel of the composite homomorphism
Let i : S’-S be
an
isomorphism.
Then it is shown in ([1],
Let T be the kernel of
Propositions
6.4 and 8.1)&#x3E; that
As in [1] let (1: G-F be an), set theoretic section of E:
F-G. Then (1 induces a section N-Srr S’; under this section we
denote the image of n E N by u ( n)’ E S’ . Let
With this notation,
LEMMA 4.
There is
we
a
have
homomorphism h:NApG-L0V1P(a) defined
by
341
PROOF. We need to show that h preserves the relations
By
[1]
h preserves (1)-(3). Relation (4) is
clearly
(1)-(7).
preserved be-
cause
from which
we
Relation (5) is
see
that
preserved
because
preserved
because
and
which
implies
Since
we
have
Relation (6) is
Clearly
relation (7) is
Consider the
By Lemma 3
defined by
we
we
have
preserved. o
homomorphism
have
a
homomorphism
We therefore have a set theoretic map
defined as follows (cf. [1], 8.12):
LEMMA
5. The composite function
342
cp:
S"#PS* =S’APS’-NAPG
g: S’S*F#p(S’*F)-NAPO
is
a
homomorphism,
and induces
a
homomorphism
The first homomorphism is clear, and certainly D is in
the kernel of the first homomorphism. By Lemmas 4 and 5 we
have a commutative diagram
PROOF.
where h’ is the restriction of hc, and is an isomorphism by Section 8 of 111. Clearly LoVO(a) lies in the kernel of 4J, and so 4J
induces a splitting of h . But 4J is surjective and hence h is an
isomorphism. It follows that h is an isomorphism. It is readily
seen that the above diagram (*) commutes. So Theorem 1 is proved.
REFERENCES.
1. J. BARJA
eight-term
&#x26;
C.
RODRIGUEZ,
exact sequences,
Homology
Top.
Cahiers
groups Hqn (-) and
et Géom. Diff. Cat.
XXXI (1990).
2.
Van Kampen Theorems for diagrams
26 (3) (1987), 311-335.
3. G. J. ELLIS, Non-abelian exterior products of groups and exact
sequences in the homology of groups, Glasgow Math. J. 29
(1987), 13-19.
R. BROWN &#x26; J.-L. LODAY,
of spaces,
Topology,
Mathematics
de Algebra
Universidad de
Department
University College
Departamento
GALWAY
SANTIAGO
IRELAND
DE COMPOSTELA
ESPAGNE
343