An exterior product for the homology of groups with integral coefficients modulo <span class="mathjax-formula">$p$</span>

C AHIERS DE

TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES

G. J. E ^LLIS

C. R ^ODRIGUEZ -F ^ERNANDEZ

An exterior product for the homology of groups with integral coefficients modulo p

Cahiers de topologie et géométrie différentielle catégoriques, tome 30, n

^o

4 (1989), p. 339-343

<http://www.numdam.org/item?id=CTGDC_1989__30_4_339_0>

© Andrée C. Ehresmann et les auteurs, 1989, tous droits réservés.

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AN EXTERIOR PRODUCT FOR THE HOMOLOGY OF GROUPS

WITH INTEGRAL COEFFICIENTS MODULO p

by G. J. ELLIS 1

and C. RODRIGUEZ-FERNANDEZ

CAHIERS DE TOPOLOGIE ET G.8OMrTRIE DIFF.8R.ENTIELLE

CATÉGORIQUES

VOL. XXX- 4 (1989)

R2SUMfi.

Les auteurs

g6n6ralisent

^{la suite exacte a 8 ter-}

mes de groupes

d’homologie

^{entiere obtenue} par Brown et

Loday

^{en une suite exacte} de groupes

d’homologie

^{a coef-}

ficients dans

ZP,

^{ou p est un entier}

non-n6gatif.

In this article we

generalize

^{Brown and}

Loday’s eight

^term

exact sequence in

integral

group

homology

^{[2] to an exact se-}

quence in group

homology

^with coefficients in

Zp,

^{where p is}

any

nonnegative integer.

Let G be a group with ^a normal

subgroup ^N,

and consider

Zp=Z/pZ

^{as a trivial} G-module. We _prove

THEOREM 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 ele- ments

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 rela-}

tions

1 This author would like to thank the University ^of Santiago ^de Compostela ^{for its} hospitality during ^the preparation ^{of this ar-}

340

for

g, h ^{E G,}

^{m, n E N.} Note that if

N = G,

then (2) and (7) are

redundant.

Clearly

^{NAPG is} functorial in N and G.

The

homomorphism

^{1: NAPG-G is defined}

by

It is routine to check that 6 is a well-defined

homomorphism,

and that its

image

^is

NUpG .

As an immediate consequence of Theorem 1 we have COROLLARY 2. There is an

isomorphism

Also,

^for aw

presentation

^R &#x3E;- F - G of

G,

^{there is an isomor-}

phism

In order to prove Theorem 1 we need the

following

LEMMA 3. If F is a free group, then 6 induces ^an

isomorphism

FAPF rr

F#PF.

PROOF. Recall from 121 that NAG is the group

generated by

^the

symbols

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

where 8" is induced

by d,

and where 1’ is the

isomorphism

^pro-

ved in [21 (see [3] for an

algebraic proof

^{of this}

isomorphism).

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 an

isomorphism rr:NAPG-L0VP1(a)

^{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.

Let T be the kernel of

Then it is shown in

([1], 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, ^we have

LEMMA 4. There is a

homomorphism h:NApG-L0V1P(a)

^defined

by

342

PROOF. We need to show that h _preserves the relations (1)-(7).

By

^[1]

clearly

^h preserves (1)-(3). Relation (4) is

preserved

^be-

cause

from which we see that

Relation (5) is

preserved

^because

and

which

implies

Since

we have

Relation (6) is

preserved

^{because we have}

Clearly

^{relation (7) is}

preserved. o

Consider the

homomorphism

By

^{Lemma 3 we have a}

homomorphism

cp:

S"#PS*

=S’APS’-NAPG defined

by

We therefore have ^a set theoretic map

g: S’S*F#p(S’*F)-NAPO

defined as follows (cf.

[1],

^8.12):

LEMMA 5. The

composite

^function

is a

homomorphism,

and induces a

homomorphism

PROOF. 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

where h’ is the restriction of

hc,

and is an

isomorphism by

^Sec-

tion 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 pro- ved.

REFERENCES.

1. J. BARJA

&#x26; C. RODRIGUEZ,

Homology

groups

Hqn (-)

^and

eight-term

^exact sequences, Cahiers

Top.

^{et Géom. Diff. Cat.}

XXXI (1990).

2. R. BROWN &#x26; J.-L. LODAY, Van

Kampen

^{Theorems for}

diagrams

of _spaces,

Topology,

^{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.

Mathematics

Department University College

GALWAY IRELAND

Departamento

^de

Algebra

Universidad de SANTIAGO DE COMPOSTELA ESPAGNE