On cohomological dimensions of totally disconnected locally compact groups
Ilaria Castellano, Nadia Mazza, Brita Nucinkis
TL;DR
This work extends Mackey-functor theory to totally disconnected locally compact groups by formulating Mackey systems and the Burnside Mackey functor, and then introduces cohomological dimensions in the Mackey, rational discrete, and Bredon settings. It proves a fundamental chain of inequalities ${ m cd}_{oldsymbol{ m Q}}G \,\le\, { m cd}_{{oldsymbol{ m M}}_{ rak CO}}G \,\le\, { m cd}_{oldsymbol{ m O}_{ rak CO}}G$, and shows how evaluations at compact subgroups yield compatible resolutions linking the three theories. The paper further develops the relationship between Mackey and Bredon cohomology via restriction/induction functors, and extends geometric-dimension results to the t.d.l.c. context, including explicit classifications for certain families of subgroups and examples arising from buildings and Bass–Serre theory. These results unify cohomological dimensions across Mackey, Bredon, and rational discrete frameworks and illuminate the homological behavior of t.d.l.c. groups in relation to compact open subgroups and their actions.
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.