Relative group (co)homology theories with coefficients and the comparison homomorphism
arXiv:1809.01209
Abstract
Let be a group, let be a subgroup of and let $\Or(G)$ be the orbit category. In this paper we extend the definition of the relative group (co)homology theories of the pair defined by Adamson and Takasu to have coefficients in an $\Or(G)$-module. There is a canonical comparison homomorphism defined by Cisneros-Molina and Arciniega-Nevárez from Takasu's theory to Adamson's one. We give a necessary and sufficient condition on the subgroup for which the comparison homomorphism is an isomorphism for all coefficients. We also use the Lück-Wiermann construction to introduce a long exact sequence for Adamson (co)homology. Finally, we provide some examples of explicit computations for the comparison homomorphism.
arXiv commentShortened version. This version was accepted for publication in Quaestiones Mathematicae