Relative group cohomology and the orbit category

TL;DR

This paper investigates the existence of periodic relative projective resolutions in group cohomology, providing a negative answer through explicit calculations for specific groups and exploring spectral sequences related to the orbit category.

Contribution

It introduces a method to compute relative group cohomology via orbit category ext-groups and constructs a spectral sequence connecting group cohomology with relative cohomology.

Findings

Negative result on the existence of certain periodic resolutions for specific groups

Explicit calculation of relative cohomology for G=Z/2×Z/2 with cyclic subgroups

Development of a spectral sequence linking group and relative cohomology

Abstract

Let $G$ be a finite group and $\cF$ be a family of subgroups of $G$ closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative $\cF$-projective resolution for $\bbZ$ when $\cF$ is the family of all subgroups $H \leq G$ with $\rk H \leq \rk G-1$. We answer this question negatively by calculating the relative group cohomology $\cF H^* (G, \bbF_2)$ where $G=\bbZ/2\times \bbZ /2$ and $\cF$ is the family of cyclic subgroups of $G$. To do this calculation we first observe that the relative group cohomology $\cF H^*(G, M)$ can be calculated using the ext-groups over the orbit category of $G$ restricted to the family $\cF$. In second part of the paper, we discuss the construction of a spectral sequence that converges to the cohomology of a group $G$ and whose horizontal line at $E_2$ page is isomorphic to the relative group cohomology of $G$.