A relative cohomological invariant for group pairs

Some properties of E(G,W,F_TG) and an application in the theory of splittings of groups

“…Invariant ends for pairs of groups in a more general set were studied by Kropholler and Roller, for example in [15]. The authors defined the invariant end e(G, S, M ) for S a subgroup of G and M, a Z 2 G-module and presented an interesting study for e(G, S) := e(G, S, P(S)), where P(S) is the power set of all subsets of S. In an attempt to obtain a cohomological formula for e(G, T ), Andrade and Fanti ( [2]) defined an invariant end, E(G, S, M ), where S is a family of subgroups of G and M is a Z 2 G-module. Afterwards, in [5] the authors adapted the definition of E(G, S, M ) by using the cohomology theory of Dicks and Dunwoody to pairs (G, W ), where W is a G-set, and defining the invariant E(G, W, M ), they obtained general properties and results about splittings and duality of groups, particularly by considering E(G, W, Z 2 ).…”

On Poincaré duality for pairs (G,W)

“…In Section 4, based in [1] and [2] and by using the notation from Dicks and Dunwoody, we present a characterization of the types of Poincaré duality pairs and, through of a generalized invariant "end", a cohomological criterion for a pair .G; W / to be a Poincaré duality pair.…”

A Dual Homological Invariant and Some Properties

“…Based on the cohomology theory of groups, Andrade and Fanti defined in By means of this invariant results in duality theory and splitting of groups were proved in Andrade and Fanti in [1], [2] and Andrade et al in [3], [4].…”