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 homology the- ories 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-Nevrez from Takasu’s theory to Adamson’s theory. 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 Lück-Wiermann construction to introduce a long exact sequence for Adamson homology. Finally, we provide some examples of explicit computations for the comparison homomorphism.

摘要

令G为群，令H为G的子群，令 Or( G ) 为轨道范畴。在本文中，我们将Adamson 和 Takasu 定义的对 ( G, H )的相关群同源理论的定义扩展为在 Or( G ) 模中具有系数。从 Takasu 的理论到 Adamson 的理论，存在 Cisneros-Molina 和 Arciniega-Nevrez 定义的规范比较同态。我们给出了子群H 的一个充要条件，其中比较同态是所有子群的同构系数。我们还使用 Lück-Wiermann 结构来为 Adamson 同源引入一个长精确序列。最后，我们提供了一些比较同态的显式计算示例。