Relative group (co)homology theories with coefficients and the comparison homomorphism

TL;DR

This paper extends relative group (co)homology theories to include coefficients in an orbit category module, compares two existing theories via a homomorphism, and characterizes when this is an isomorphism.

Contribution

It generalizes the definitions of relative (co)homology theories with coefficients and establishes criteria for the comparison homomorphism to be an isomorphism.

Findings

The comparison homomorphism is an isomorphism under specific subgroup conditions.

A long exact sequence for Adamson (co)homology is constructed.

Explicit examples of the comparison homomorphism are provided.

Abstract

Let $G$ be a group, let $H$ be a subgroup of $G$ 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 $(G,H)$ 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-Nev\'arez from Takasu's theory to Adamson's one. We give a necessary and sufficient condition on the subgroup $H$ for which the comparison homomorphism is an isomorphism for all coefficients. We also use the L\"uck-Wiermann construction to introduce a long exact sequence for Adamson (co)homology. Finally, we provide some examples of explicit computations for the comparison homomorphism.