A note about splittings of groups and commensurability under a cohomological point of view

Let $G$ be a group, let $S$ be a subgroup with infinite index in $G$ and let $\mathcal{F}_SG$ be a certain $\mathbb Z_2G$-module. In this paper, using the cohomological invariant $E(G,S,\mathcal{F}_SG)$ or simply $\tilde{E}(G,S)$ (defined in [2]), we analyze some results about splittings of group $G$ over a commensurable with $S$ subgroup which are related with the algebraic obstruction "$\mathrm{sing}_G(S)$" defined by Kropholler and Roller [8]. We conclude that $\tilde{E}(G,S)$ can substitute the obstruction "$\mathrm{sing}_G(S)$" in more general way. We also analyze splittings of groups in the case, when $G$ and $S$ satisfy certain duality conditions.