On cohomological dimensions of totally disconnected locally compact groups

TL;DR

This paper explores cohomological dimensions of totally disconnected locally compact groups by introducing Mackey functors, comparing various cohomological dimensions, and extending geometric dimension results.

Contribution

It defines a new cohomological dimension via Mackey functors and compares it with existing notions, extending geometric dimension results for t.d.l.c. groups.

Findings

Mackey functor-based cohomological dimension introduced

Comparison established between different cohomological dimensions

Extended results on geometric dimension of t.d.l.c. groups

Abstract

In this paper, we introduce Mackey functors for a t.d.l.c. group and define the cohomological dimension of this group over the Mackey category. We then compare this dimension to the rational discrete cohomological dimension defined by Castellano and Weigel, as well as to the Bredon cohomological dimension of that t.d.l.c. group with respect to the family of compact open subgroups. We also extend results about the geometric dimension of a t.d.l.c. group.