Some properties of e(G,w,ft g) and an application in the theory of splittings of groups

Let us consider W a G-set and M a Z2G-module, where G is a group. In this paper we investigate some properties of the cohomological the theory of splittings of groups. Namely, we give a proof of the invariant E(G,W,M), defined in [5] and present related results with independence of E(G,W,M) with respect to the set of G-orbit representatives in W and properties of the invariant E(G,W,FT G) establishing a relation with the end of pairs of groups ẽ(G,T), defined by Kropphller and Holler in [15]. The main results give necessary conditions for G to split over a subgroup T, in the cases where M = Z2(G/T) or M = FT G.