Bredon cohomological dimensions for proper actions and Mackey functors

TL;DR

This paper explores the relationship between Bredon cohomological dimensions for proper group actions and Mackey functors, focusing on groups with bounded chains of finite subgroups and their closure properties.

Contribution

It establishes connections between Bredon cohomological dimensions and Mackey functors, and examines the closure properties of groups with finite such dimensions.

Findings

Identifies bounds on cohomological dimensions for specific group classes

Analyzes the impact of Mackey functors on cohomological properties

Studies closure properties of groups with finite Bredon cohomological dimension

Abstract

For groups with a uniform bound on the length of chains of finite subgroups, we study the relationship between the Bredon cohomological dimension for proper actions and the notions of cohomological dimension one obtains by restricting the coefficients of Bredon cohomology to (cohomological) Mackey functors or fixed point functors. We also investigate the closure properties of the class of groups with finite Bredon cohomological dimension for Mackey functors.