Subgroup collections controlling the homotopy type of a $p$-local compact group

TL;DR

This paper proves that the homotopy type of the nerve of the linking system in a p-local compact group is determined by a specific collection of subgroups, generalizing known results for p-local finite groups.

Contribution

It extends the understanding of homotopy types in p-local compact groups by identifying subgroup collections that determine the nerve's homotopy type.

Findings

Homotopy type is determined by F-centric, F-radical subgroups.

Generalizes results from p-local finite groups.

Provides a broader framework for p-local compact groups.

Abstract

Let (S,F,L) be a p-local compact group. We prove that the (uncompleted) homotopy type of the nerve of the linking system L is determined by the collection of subgroups of S that are F-centric and F-radical. This result generalizes the result for the case of p-local finite groups, which is in the literature.