The normaliser decomposition for p-local finite groups

TL;DR

This paper develops a new homotopy-theoretic decomposition for p-local finite groups, allowing their classifying spaces to be expressed as homotopy colimits of finite group classifying spaces, enhancing understanding of their structure.

Contribution

It introduces an analogue of the normaliser decomposition tailored for p-local finite groups, expanding the toolkit for analyzing their classifying spaces.

Findings

Provides a homotopy colimit description of classifying spaces

Enables analysis of p-local finite groups via finite group classifying spaces

Extends normaliser decomposition techniques to p-local finite groups

Abstract

We construct an analogue of the normaliser decomposition for p-local finite groups (S,F,L) with respect to collections of F-centric subgroups and collections of elementary abelian subgroups of S. This enables us to describe the classifying space of a p-local finite group, before p-completion, as the homotopy colimit of a diagram of classifying spaces of finite groups whose shape is a poset and all maps are induced by group monomorphisms.