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

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

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 ho...

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Bibliographic Details
Main Authors: Arciniega-Nevárez, José Antonio, Cisneros-Molina, José Luis, Saldaña, Luis Jorge Sánchez
Format: Journal Article
Language:English
Published: 04-09-2018
Subjects:
Mathematics - Algebraic Topology
Mathematics - K-Theory and Homology
Online Access:Get full text
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Summary: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.
DOI:10.48550/arxiv.1809.01209