On certain homological invariant and its relation with Poincaré duality pairs
Let G be a group, S = {Sᵢ, i ∊ I} a non empty family of (not necessarily distinct) subgroups of infinite index in G and M a Z₂G-module. In [4] the authors defined a homological invariant E*(G, S,M), which is “dual” to the cohomological invariant E(G, S,M), defined in [1]. In this paper we present a...
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irk-123456789-1883572023-02-26T01:26:51Z On certain homological invariant and its relation with Poincaré duality pairs Andrade, M.G.C. Gazon, A.B. Lima A.F. Let G be a group, S = {Sᵢ, i ∊ I} a non empty family of (not necessarily distinct) subgroups of infinite index in G and M a Z₂G-module. In [4] the authors defined a homological invariant E*(G, S,M), which is “dual” to the cohomological invariant E(G, S,M), defined in [1]. In this paper we present a more general treatment of the invariant E*(G, S,M) obtaining results and properties, under a homological point of view, which are dual to those obtained by Andrade and Fanti with the invariant E(G, S,M). We analyze, through the invariant E*(G, S,M), properties about groups that satisfy certain finiteness conditions such as Poincaré duality for groups and pairs. 2018 Article On certain homological invariant and its relation with Poincaré duality pairs / M.G.C. Andrade, A.B. Gazon, A.F. Lima // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 2. — С. 177–187. — Бібліогр.: 7 назв. — англ. 1726-3255 2010 MSC: 20J05, 20J06, 57P10 http://dspace.nbuv.gov.ua/handle/123456789/188357 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let G be a group, S = {Sᵢ, i ∊ I} a non empty family of (not necessarily distinct) subgroups of infinite index in G and M a Z₂G-module. In [4] the authors defined a homological invariant E*(G, S,M), which is “dual” to the cohomological invariant E(G, S,M), defined in [1]. In this paper we present a more general treatment of the invariant E*(G, S,M) obtaining results and properties, under a homological point of view, which are dual to those obtained by Andrade and Fanti with the invariant E(G, S,M). We analyze, through the invariant E*(G, S,M), properties about groups that satisfy certain finiteness conditions such as Poincaré duality for groups and pairs. |
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Article |
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Andrade, M.G.C. Gazon, A.B. Lima A.F. |
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Andrade, M.G.C. Gazon, A.B. Lima A.F. On certain homological invariant and its relation with Poincaré duality pairs Algebra and Discrete Mathematics |
| author_facet |
Andrade, M.G.C. Gazon, A.B. Lima A.F. |
| author_sort |
Andrade, M.G.C. |
| title |
On certain homological invariant and its relation with Poincaré duality pairs |
| title_short |
On certain homological invariant and its relation with Poincaré duality pairs |
| title_full |
On certain homological invariant and its relation with Poincaré duality pairs |
| title_fullStr |
On certain homological invariant and its relation with Poincaré duality pairs |
| title_full_unstemmed |
On certain homological invariant and its relation with Poincaré duality pairs |
| title_sort |
on certain homological invariant and its relation with poincaré duality pairs |
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Інститут прикладної математики і механіки НАН України |
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2018 |
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http://dspace.nbuv.gov.ua/handle/123456789/188357 |
| citation_txt |
On certain homological invariant and its relation with Poincaré duality pairs / M.G.C. Andrade, A.B. Gazon, A.F. Lima // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 2. — С. 177–187. — Бібліогр.: 7 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT andrademgc oncertainhomologicalinvariantanditsrelationwithpoincaredualitypairs AT gazonab oncertainhomologicalinvariantanditsrelationwithpoincaredualitypairs AT limaaf oncertainhomologicalinvariantanditsrelationwithpoincaredualitypairs |
| first_indexed |
2025-07-16T10:22:55Z |
| last_indexed |
2025-07-16T10:22:55Z |
| _version_ |
1837798654761500672 |
| fulltext |
“adm-n2” — 2018/7/24 — 22:26 — page 177 — #15
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 25 (2018). Number 2, pp. 177–187
c© Journal “Algebra and Discrete Mathematics”
On certain homological invariant and its relation
with Poincaré duality pairs∗
Maria Gorete Carreira Andrade, Amanda Buosi Gazon,
and Amanda Ferreira de Lima
Communicated by V. Lyubashenko
Abstract. Let G be a group, S = {Si, i ∈ I} a non empty
family of (not necessarily distinct) subgroups of infinite index in
G and M a Z2G-module. In [4] the authors defined a homological
invariant E∗(G,S,M), which is “dual” to the cohomological invari-
ant E(G,S,M), defined in [1]. In this paper we present a more
general treatment of the invariant E∗(G,S,M) obtaining results
and properties, under a homological point of view, which are dual to
those obtained by Andrade and Fanti with the invariant E(G,S,M).
We analyze, through the invariant E∗(G,S,M), properties about
groups that satisfy certain finiteness conditions such as Poincaré
duality for groups and pairs.
Introduction
Based in the theory of cohomology of groups, Andrade and Fanti
in [1], defined a cohomological invariant denoted by E(G,S,M), where
G is a group and S = {Si, i ∈ I} is a family of subgroups of G with
[G : Si] = ∞, ∀i ∈ I. Through this invariant they proved results in
duality for groups and pairs and splitting of groups (see Andrade and
∗The first author was partially supported by FAPESP, grant no. 2012/24454-8 and
the second and third authors were supported by CAPES.
2010 MSC: 20J05, 20J06, 57P10.
Key words and phrases: (co)homology of groups, duality groups, duality pairs,
homological invariant.
“adm-n2” — 2018/7/24 — 22:26 — page 178 — #16
178 On certain homological invariant
Fanti in [1], [3] and Andrade et al in [2]). In [4], Andrade and Gazon
defined a homological invariant, denoted by E∗(G,S,M), which is “dual"
to the cohomological invariant defined in [1], and they obtained results
when M = Z2. In this work we study this invariant under a more general
point of view, obtaining results about groups that satisfy certain finiteness
conditions, such as duality conditions for groups and group pairs .
The results presented in this paper provide an alternative way of
obtaining applications and properties in the duality theory of groups and
pairs of groups, working with greater emphasis in the homology of groups,
instead of cohomology.
We assume that the reader is familiar with the theory of absolute and
relative (co)homology of groups. For details see [6] and [7]. We recall here
some definitions and results which will be useful in this paper.
LetG be a group,T a subgroup ofG andM a Z2T -module. Consider the
Z2G-modules IndGT M = Z2G⊗Z2TM and CoindGT M = HomZ2T (Z2G,M)
([7], III.5). We have the following result.
Proposition 1 ([7],III.5). Let G be a group, T a subgroup of G and
M a Z2G-module. Consider the additive group Hom(Z2(G/T ),M) with
the diagonal G-action given by (g.f)(α) = g.f(g−1α), for all g ∈ G and
α ∈ Z2(G/T ). Then we have the Z2G-isomorphism
CoindGT M ≃ Hom(Z2(G/T ),M).
Definition 1. Let G be a group and M a Z2G-module. The group of
coinvariants of M , denoted by MG, is defined by MG = M/A, where
A = 〈g ·m−m; g ∈ G and m ∈M〉 is an additive subgroup of M .
Remark 1. (1) Let G be a group and M a Z2G-module. We have
H0(G;M) = MG ([7], III.1). Hence, if M is a Z2G-module with triv-
ial G-action then H0(G;M) =M .
(2) If G is a finitely generated group, S is a subgroup of G with
[G : S] = ∞ and M is a Z2S-module, then (CoindGS M)G = 0 ([7], III.5).
Proposition 2 (Proposition 1.1, [6]). Let G be a group, S = {Si, i ∈ I}
a family of subgroups of G and M a Z2G-module. Denote
⊕
i∈I H∗(Si;M)
by H∗(S;M). We have the following long exact sequence:
· · · → H1(S;M)
corG
S−−−→ H1(G;M)
J
−→ H1(G,S;M)
δ
−→
→ H0(S;M) → H0(G;M) → 0,
which is natural in the module M and the group pair (G,S). �
“adm-n2” — 2018/7/24 — 22:26 — page 179 — #17
M. G. C. Andrade, A. B. Gazon, A. F. Lima 179
Remark 2. Let (G,S) a group pair with S = {Si, i ∈ I} and M a
Z2G-module. For each i ∈ I, the corestriction map
corGSi
: H1(Si;M) −→ H1(G;M)
is induced in homology by the inclusion Si →֒ G. The map
corGS :
⊕
i∈I
H1(Si;M) −→ H1(G;M),
which appears in the long exact sequence of the Proposition 2, is defined
by:
corGS ((αi)i∈I) =
∑
i∈I
corGSi
(αi),
where (αi)i∈I ∈
⊕
i∈I H1(Si;M), with αi = 0 for almost all i, that is,
except possible finitely many i.
Now, we present the definition of E∗(G,S,M).
Definition 2. Let (G,S) be a group pair, S = {Si, i ∈ I} a family of
subgroups of G with [G : Si] = ∞, for all i ∈ I, and M a Z2G-module.
We define:
E∗(G,S,M) = 1 + dim coker(corGS ),
with coker(corGS ) = H1(G;M)/ Im(corGS ).
Consider now the category C whose objects are pairs ((G,S);M) where
G is a group, S = {Si, i ∈ I} is a family of subgroups of G and M is a
Z2G-module, and whose morphisms are maps:
ψ : ((G,S);M) −→ ((L,R = {Rj , j ∈ J});N)
consisting of
(a) a homomorphism of groups α : G −→ L;
(b) a map π : I −→ J such that α(Si) ⊂ Rπ(i);
(c) a map φ : M −→ N such that φ(g ·m) = α(g) · φ(m), i.e., φ is a
Z2G-homomorphism via α : G −→ L.
A morphism ψ is an isomorphism in C when α is an isomorphism of
groups, π is a bijection and φ is a Z2G-isomorphism.
Theorem 1. (Theorem 1, [4]) If in the category C, ((G,S);M) and
((L,R);N) are isomorphic then
E∗(G,S,M) = E∗(L,R, N),
Hence E∗(G,S,M) is an invariant in C. �
“adm-n2” — 2018/7/24 — 22:26 — page 180 — #18
180 On certain homological invariant
In the following we present a characterization of the invariant
E∗(G,S,M) in terms of “partial Euler characteristic”.
Proposition 3 (Proposition 1, [4]). Let (G,S) be a group pair, with
[G : S] = ∞, ∀S ∈ S, and M a Z2G-module. If the homology groups
H0(G;M), H0(S;M) =
⊕
S∈S H0(S;M) and H1(G,S;M) have finite
dimension as Z2-vector spaces then:
E∗(G,S,M) = 1 + dimH0(G;M)− dimH0(S;M) + dimH1(G,S;M).
We introduce now some notations which will be used in this pa-
per. For the sake of simplicity, we denote the Z2G-module CoindGS Z2 =
HomZ2S(Z2G,Z2) ≃ Hom(Z2(G/S),Z2) by Z2(G/S) and, for a family
S = {Si, i ∈ I} of subgroups of G, we denote
⊕
i∈I Coind
G
Si
Z2 by
Z2(G/S). If S = {S}, we denote E∗(G,S,M) by E∗(G,S,M). The in-
variant E∗(G,S,M), when M is the particular module Z2(G/S), will be
denoted by E∗(G,S). In the particular case in which S = {S}, we denote
E∗(G,S,Z2(G/S)) by E∗(G,S).
1. Some properties of the invariant E∗(G,S,M)
In this section we present some general properties of the invariant
E∗(G,S,M). We also present some properties for the particular invariant
E∗(G,S). We begin with some remarks.
Remark 3. Let (G,S) be a group pair with S = {Si, i ∈ I} and
[G : Si] = ∞, for all i ∈ I, and let M be a Z2G-module. It is easy to see
that:
(1) If S ′ = {Sik , ik ∈ I ′ ⊆ I} is a subfamily of S, then E∗(G,S,M) 6
E∗(G,S
′,M).
(2) If S ′ = {Hi, i ∈ I} is a family of subgroups of G such that Hi 6 Si,
for all i ∈ I, then E∗(G,S,M) 6 E∗(G,S
′,M).
In particular, if H,S and T are subgroups of G satisfying H 6 S 6
T 6 G with [G : T ] = ∞ then
E∗(G, T,Z2(G/S)) 6 E∗(G,S) 6 E∗(G,H,Z2(G/S)).
Remark 4. Let G be a group, T a subgroup of G and M a Z2T -module.
We have the following maps:
(a) a canonical Z2G-monomorphism ϕ : IndGT M −→ CoindGT M defined
by
ϕ(g0 ⊗m)(g) =
{
gg0m if gg0 ∈ T
0 otherwise
,
“adm-n2” — 2018/7/24 — 22:26 — page 181 — #19
M. G. C. Andrade, A. B. Gazon, A. F. Lima 181
which is an isomorphism if [G : T ] <∞ ([7], III.5.9);
(b) a canonical Z2T -monomorphism i : M −→ IndGT M , defined by
i(m) = 1⊗m, for all m ∈M , ([7], p.67) and can be easily seen that
the isomorphism of Shapiro’s lemma H∗(T ;M)
≃
−→ H∗(G; IndGT M)
is induced by (α, i), where α : T →֒ G is the inclusion map;
(c) a canonical Z2G-isomorphism ψ : CoindGT CoindTS M −→ CoindGS M ,
where S 6 T 6 G ([7], p.64 (3.6));
(d) a Z2T -homomorphism χ : CoindTS M −→ CoindGS M , where S 6
T 6 G, given by the composition
CoindTS M
i
−→ IndGT CoindTS M
ϕ
−→ CoindGT CoindTS M
ψ
−→ CoindGS M,
i.e., χ = ψ ◦ ϕ ◦ i.
Theorem 2. Let S and T be subgroups of G with S 6 T 6 G and
M a Z2S-module. If [T : S] = ∞ and ϕ∗ : H1(G; Ind
G
T (Coind
T
S M)) →
H1(G; Coind
G
T (Coind
T
S M)) is an epimorphism, where ϕ∗ is the induced
map of the embedding ϕ : IndGT (Coind
T
S M) → CoindGT (Coind
T
S M), then
E∗(T, S,Coind
T
S M) 6 E∗(G,S,Coind
G
S M).
Proof. By considering the maps from Remark 4, we have the following
commutative diagram
H1(S; Coind
T
S M)
(idS ,χ)∗
��
corTS
// H1(T ; Coind
T
S M)
(α,χ)∗
��
H1(S; Coind
G
S M)
corGS
// H1(G; Coind
G
S M)
where the induced map (α, χ)∗ : H1(T ; Coind
T
S M) −→ H1(G; CoindGS M)
is given by (α, χ)∗ = ψ∗ ◦ ϕ∗ ◦ (α, i)∗ with ϕ∗ ≡ (id, ϕ)∗ and ψ∗ ≡
(id, ψ)∗. Since (α, i)∗ and ψ∗ are isomorphisms and ϕ∗ is an epimorphism
by hypothesis, it follows that (α, χ)∗ is an epimorphism. Furthermore,
(α, χ)∗(Im(corTS )) ⊂ Im(corGS ). In fact, by the commutative diagram,
(α, χ)∗ ◦ cor
T
S = corGS ◦(id, χ)∗. Hence,
∀y ∈ Im(corTS ) ⇒ y = corTS (x), for some x ∈ H1(S; Coind
T
S M)
⇒ (α, χ)∗(y) = (α, χ)∗(cor
T
S (x)) = corGS ◦(id, χ)∗(x) ∈ Im(corGS ).
“adm-n2” — 2018/7/24 — 22:26 — page 182 — #20
182 On certain homological invariant
Then we have a well-defined map
(α, χ)∗ :
H1(T ; Coind
T
S M)
Im(corTS )
−→
H1(G; Coind
G
S M)
Im(corGS )
given by (α, χ)∗(a+ Im(corTS )) = (α, χ)∗(a) + Im(corGS ). Hence, we have
dim coker(corGS ) > dim coker(corTS ) and therefore E∗(T, S,Coind
T
S M) 6
E∗(G,S,Coind
G
S M).
The next result provides a relation between E∗(T, S) and E∗(G,S),
with S 6 T 6 G.
Corollary 1. Let S, T be subgroups of G, satisfying S 6 T 6 G, and
M = Z2 the trivial Z2G-module. If [G : S] = ∞ and [G : T ] < ∞, then
E∗(T, S) 6 E∗(G,S).
2. E∗(G,S,M) and duality
In this section, through the invariant E∗(G,S,M) we prove some
results about groups and group pairs satisfying duality conditions.
Before proving the main results, we recall some definitions about
duality due to Bieri and Eckmann (for details see [5], [6] and [7]).
Definition 3. A group G is called a duality group of dimension n, or
simply a Dn-group, if there exist a Z2G-module C, called the dualizing
module of G, and natural isomorphisms
Hk(G;M) ≃ Hn−k(G;C ⊗M)
for all integers k and all Z2G-modules M . In the special case where C = Z2,
we say that G is a Poincaré duality group of dimension n, or simply a
PDn-group.
Definition 4. A duality pair of dimension n, or simply a Dn-pair, consists
of a group pair (G,S), where S = {Si, i ∈ I} is a finite family of Dn−1-
subgroups of G, a Z2G-module C and natural isomorphisms
Hk(G;M) ≃ Hn−k(G,S;C ⊗M),
Hk(G,S;M) ≃ Hn−k(G;C ⊗M),
for all Z2G-modules M and all k ∈ Z. C is called the dualizing module of
the Dn-pair (G,S). If C = Z2 the duality pair (G,S) is called a Poincaré
duality pair, or simply a PDn-pair.
“adm-n2” — 2018/7/24 — 22:26 — page 183 — #21
M. G. C. Andrade, A. B. Gazon, A. F. Lima 183
Remark 5 ([5], [6]). (1) IfG is aDn-group, then G is finitely generated
and its cohomological dimension, cd(G), is n.
(2) If G is a Dn-group and S is subgroup of G, with [G : S] <∞, then
S is a Dn-group (with the same dualizing module).
(3) If (G,S) is a PDn-pair, then G is a Dn-group and each S ∈ S is a
PDn−1-group.
Lemma 1. Let (G,S) be a PDn-pair, with S = {Si, i = 1, . . . , r}. Then
(i) H1(G,S;
⊕r
i=1Coind
G
Si
Mi) =
⊕r
i=1(Mi)Si
, where Mi is a Z2Si-
module, for all i ∈ I. In particular, if Mi = Z2 is the trivial Z2Si-
module for i = 1, . . . , r, then H1(G,S;Z2(G/S)) =
⊕r
i=1Z2.
(ii) H1(G,S; Coind
G
S M) = MS where S is a PDn−1-subgroup of G
(which does not necessarily belong to the family S) and M is a
Z2S-module.
Proof. (i) By using Definitions 3 and 4, Remark 1 and Shapiro’s Lemma,
for (G,S) a PDn-pair, we have:
H1(G,S;
r
⊕
i=1
CoindGSi
Mi) = Hn−1(G;
r
⊕
i=1
CoindGSi
Mi)
=
r
⊕
i=1
Hn−1(G; CoindGSi
Mi) =
r
⊕
i=1
Hn−1(Si;Mi)
=
r
⊕
i=1
H0(Si;Mi) =
r
⊕
i=1
(Mi)Si
.
(ii) It is similar to (i).
Lemma 2. If G is a group and S is a subgroup of G, then (Z2(G/S))S 6= 0.
More specifically, there exists f ∈ (Z2(G/S))S such that 〈f〉 ≃ Z2.
Proof. Consider Z2(G/S) ≃ Hom(Z2(G/S),Z2) with the diagonal G-
action (see Proposition 1). Since Z2 is a trivial Z2G-module, it follows
that g · f(g−1α) = f(g−1α), for all g ∈ G and α ∈ Z2(G/S). In particular,
for α = 1 = 1 · S and s ∈ S we have
(s · f)(1) = f(s−1 · 1) = f(s−1) = f(1).
Hence,
s · f(1)− f(1) = 0, ∀f ∈ Z2(G/S), ∀s ∈ S. (∗)
Consider now the augmentation map ε : Z2(G/S) → Z2. We will show that
ε provides a non-null element ε in Z2(G/S)S = Hom(Z2(G/S),Z2)/A,
“adm-n2” — 2018/7/24 — 22:26 — page 184 — #22
184 On certain homological invariant
where A = 〈sf − f | f ∈ Z2(G/S), s ∈ S〉. For this, suppose that
ε = 0 in Z2(G/S)S . Thus, ε ∈ A and there exist s1, s2, . . . , sk ∈ S and
f1, f2, . . . , fk ∈ Z2(G/S) such that
ε = (s1f1 − f1) + (s2f2 − f2) + . . .+ (skfk − fk).
Now, for 1 ∈ Z2(G/S), one has
1 = ε(1) = (s1f1 − f1) + (s2f2 − f2) + . . .+ (skfk − fk)(1)
(∗)
= 0,
which gives us a contradiction. Hence, there exists f = ε 6= 0 in Z2(G/S))S
and 〈f〉 ≃ Z2 ⊂ Z2(G/S)S .
The next result provides a necessary condition for a group pair (G,S)
to be a Poincaré duality pair (PDn-pair).
Theorem 3. Let (G,S) be a group pair with S = {Si, i = 1, . . . , r} and
[G : Si] = ∞, for all i. If E∗(G,S) > 1, then (G,S) is not a PDn-pair. In
other words, if (G,S) is a PDn-pair, then E∗(G,S) = 1.
Proof. Consider part of the exact sequence of the Proposition 2 for M =
Z2(G/S):
r
⊕
i=1
H1(Si;Z2(G/S))
corG
S−−−→ H1(G;Z2(G/S))
J
−→ H1(G,S;Z2(G/S))
δ
−→
r
⊕
i=1
H0(Si;Z2(G/S))
cor0,GS−−−−→ H0(G;Z2(G/S)) → 0.
Since (G,S) is a PDn-pair, it follows from Remark 5 that G is a Dn-
group, S is a PDn−1-subgroup and G is finitely generated. And, by using
Remark 1, we conclude that dimH0(G;Z2(G/S)) = 0. In fact
H0(G;Z2(G/S)) = H0(G;
r
⊕
i=1
Z2(G/Si)) =
r
⊕
i=1
H0(G;Z2(G/Si))
=
r
⊕
i=1
Z2(G/Si)G =
r
⊕
i=1
(CoindGSi
Z2)G = 0.
Now, it follows from Lemma 1, that H1(G,S;Z2(G/S)) =
⊕r
i=1 Z2. Thus,
dimH1(G,S;Z2(G/S)) = r. For calculating dim
⊕r
i=1H0(Si;Z2(G/S))
“adm-n2” — 2018/7/24 — 22:26 — page 185 — #23
M. G. C. Andrade, A. B. Gazon, A. F. Lima 185
observe that
r
⊕
i=1
H0(Si;Z2(G/S)) =
r
⊕
j=1
H0(Sj ;
r
⊕
i=1
Z2(G/Si))
=
r
⊕
j=1
[
H0(Sj ;Z2(G/Sj))⊕
r
⊕
i 6=j,i=1
H0(Sj ;Z2(G/Si))
]
=
r
⊕
j=1
[
(Z2(G/Sj))Sj
⊕
r
⊕
i 6=j,i=1
(Z2(G/Si))Sj
]
.
By using Lemma 2, we have that Z2 is isomorphic to a subset of
(Z2(G/Sj))Sj
, for all j = 1, . . . , r. Thereby,
r
⊕
j=1
Z2 →֒
r
⊕
j=1
(Z2(G/Sj))Sj
→֒
r
⊕
j=1
[
(Z2(G/Sj))Sj
⊕
r
⊕
i 6=j,i=1
(Z2(G/Si))Sj
]
,
and thus,
dim
r
⊕
i=1
H0(Si;Z2(G/S)) > r. (∗)
On the other hand, since H0(G;Z2(G/S)) = 0, the map δ of the exact
sequence (3.1) is surjective. It follows that,
dim
r
⊕
i=1
H0(Si;Z2(G/S)) 6 dim H1(G,S;Z2(G/S)) = r. (∗∗)
Hence, from (∗) and (∗∗), we have dim
⊕r
i=1H0(Si;Z2(G/S)) = r. There-
fore, by using Proposition 3, E∗(G,S) = 1 + 0− r + r = 1.
Example 1. Let X be a torus minus an open disc and Y the boundary
of X. We have G = π1(X) ≃ Z ∗ Z and S = π1(Y ) ≃ Z. It follows from
[6], Theorem 6.1, that the pair (G,S) is a PD2-pair and thus, by Theorem
3, E∗(G,S) = 1.. More generally, if X is a closed surface F minus k open
discs (k > 2 if F = S2) and Y = ∂X =
⋃k
i=1Yi, where Yi are the boundary
of the k-discs, consider S = {Si = π1(Yi), i = 1, . . . , k} and G = π1(X).
Then, (G,S) is a PD2-pair and, by Theorem 3, E∗(G,S) = 1.
We will see now simple computations of the particular invariant
E∗(G,S) when G and S satisfy some finiteness conditions.
Proposition 4. Let G be a PDn-group and S a subgroup of G.
“adm-n2” — 2018/7/24 — 22:26 — page 186 — #24
186 On certain homological invariant
(i) If S is a PDn−1-group then E∗(G,S) 6 2.
(ii) If cdS 6 n− 2 then E∗(G,S) = 1.
Proof. Since G is PDn-group, by the hypothesis of (i) or (ii) and Remark 5,
it follows that [G : S] = ∞. Then, E∗(G,S) can be defined. Now, by using
duality and Shapiro’s Lemma, we have:
H1(G;Z2(G/S)) ≃ Hn−1(G;Z2(G/S)) ≃ Hn−1(S;Z2).
(i) If S is a PDn−1-group, then Hn−1(S;Z2) ≃ Z2 and thus,
coker(corGS ) =
H1(G;Z2(G/S))
Im(corGS )
=
Z2
Im(corGS )
can only be {0} or Z2. Therefore, E∗(G,S) 6 2.
(ii) If cdS 6 n− 2 we have Hn−1(S,Z2) = {0}. Hence coker(corGS ) =
{0} and we have E∗(G,S) = 1.
Example 2. Consider G = Z
n and S ≃ Z
r a subgroup of G with n >
r > 2. Note that G is a PDn-group and S is a PDr-group. If r = n− 1,
then E∗(G,S) 6 2 and if r 6 n− 2, then E∗(G,S) = 1.
Finally, we prove a necessary condition for (G,S) to be not a Poincaré
duality pair.
Theorem 4. If (G,S) is a group pair with [G : S] = ∞ and S a normal
subgroup of G, then (G,S) is not a Poincaré duality pair.
Proof. If (G,S) is a PDn-pair, then, by using the technique of the proof
of Theorem 3, we can prove that dimH0(S;Z2(G/S)) = 1 and thus
H0(S;Z2(G/S)) = Z2. (∗)
Since S is normal in G, the S-action on Z2(G/S) is trivial. In fact,
∀f ∈ Z2(G/S) and s ∈ S,
(s · f)(g) = f(s−1g) (diagonal action)
= f(s−1g)
= f(s−1 · g) (S normal in G)
= f(g).
Therefore, sf = f . Hence,
H0(S;Z2(G/S)) = Z2(G/S). (∗∗)
Since [G : S] = ∞ we have, from (∗) and (∗∗), a contradiction.
“adm-n2” — 2018/7/24 — 22:26 — page 187 — #25
M. G. C. Andrade, A. B. Gazon, A. F. Lima 187
Acknowledgment
The authors would like to thank the referee for useful remarks and
suggestions.
References
[1] M.G.C. Andrade, E.L.C. Fanti, A relative cohomological invariant for pairs of
groups. Manuscripta Math., 83 (1994), 1-18.
[2] M.G.C. Andrade, E.L.C. Fanti and J.A. Daccach, On certain relative cohomological
invariant. International Journal of Pure and Applied Mathematics, 21 (2005), 335-
351.
[3] M.G.C. Andrade, E.L.C. Fanti, A note about splittings of groups and commensura-
bility under a cohomological point of view. Algebra and Discrete Mathematics, 9
(2), (2010), 1-10.
[4] M.G.C. Andrade; A. B. Gazon, A dual homological invariant and some properties,
International Journal of Applied Mathematics, 27 (1), (2014), 13-20.
[5] R. Bieri, Homological Dimension of Discrete Groups, Queen Mary College Math.
Notes, Londres, (1976).
[6] R. Bieri, B. Eckmann, Relative homology and Poincaré duality for Group Pairs.
Journal of Pure and Applied Algebra, 13 (1978), 277-319.
[7] K.S. Brown, Cohomology of Groups. New York: Springer-Verlag, (1982).
Contact information
M. G. C. Andrade UNESP - Universidade Estadual Paulista
Departamento de Matemática
Rua Cristovão Colombo, 2265
15054-000, São José do Rio Preto - SP, Brazil
E-Mail(s): gorete@ibilce.unesp.br
Web-page(s): www.ibilce.unesp.br/#!
/departamentos/matematica
/docentes/gorete
A. B. Gazon,
A. F. Lima
UFSCAR - Universidade Federal de São Carlos
Centro de Ciências Exatas e Tecnologia
Departamento de Estatística
Rodovia Washington Luís, km 235
13565-905- São Carlos - SP, Brazil
E-Mail(s): mandybg@hotmail.com,
mandinha−lima10@hotmail.com
Received by the editors: 19.08.2016
and in final form 23.06.2017.
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