Intermediaries in Bredon (Co)homology and Classifying Spaces

TL;DR

This paper develops a spectral sequence for F-Bredon (co)homology of groups acting on certain complexes, providing new tools to analyze the cohomological dimensions of classifying spaces and introducing a new class of groups with finite (co)homological dimension.

Contribution

It constructs a spectral sequence for F-Bredon (co)homology and introduces a hierarchically defined class of groups with finite (co)homological dimension.

Findings

Spectral sequence converges to F-Bredon cohomology.

Finite (co)homological dimension characterized by jump (co)homology.

Introduces a new class of groups including all countable elementary amenable and linear groups.

Abstract

For certain contractible G-CW-complexes and F a family of subgroups of G, we construct a spectral sequence converging to the F-Bredon cohomology of G with E1-terms given by the F-Bredon cohomology of the stabilizer subgroups. As applications, we obtain several corollaries concerning the cohomological and geometric dimensions of the classifying space for the family F. We also introduce a hierarchically defined class of groups which contains all countable elementary amenable groups and countable linear groups of characteristic zero, and show that if a group G is in this class, then G has finite F-Bredon (co)homological dimension if and only if G has jump F-Bredon (co)homology.