On certain homological invariant and its relation with Poincaré duality pairs

Abstract

Let \(G\) be a group, \(\mathcal{S} = \{ S_i, i \in I\}\) a non empty family of (not necessarily distinct) subgroups of infinite index in \(G\) and \(M\) a \(\mathbb{Z}_2 G\)-module. In [4] the authors defined a homological invariant \(E_*(G, \mathcal{S}, M),\) which is  “dual” to the cohomological invariant \(E(G, \mathcal{S}, M),\) defined in [1]. In this paper we present a more general treatment of the invariant \(E_*(G, \mathcal{S}, M)\) obtaining results and properties, under a homological point of view, which are dual to those obtained by Andrade and Fanti with the invariant \(E(G, \mathcal{S}, M)\). We analyze, through the invariant \(E_{*}(G, S,M)\), properties about groups that satisfy certain finiteness conditions such as Poincar\'e duality for groups and pairs.