On certain homological invariant and its relation with Poincaré duality pairs

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 Poincaré duality for groups and pairs.