Relative group homology theories with coefficients and the comparison homomorphism

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 denition of the relative  group homology theories of the pair (G;H) dened by Adamson and Takasu to have coecients in an Or(G)-module. There is a canonical comparison homomorphism dened by Cisneros- Molina and Arciniega-Nevrez from Takasu's theory to Adamson's theory. We give a  necessary and sucient condition on the subgroup H for which the comparison homomorphism is an isomorphism for all coecients. We also use the Luck-Wiermann construction to introduce a long exact sequence for Adamson homology. Finally, we provide some examples of explicit computations for the comparison homomorphism.