Normalizer decompositions of p-local compact groups

TL;DR

This paper introduces a new normalizer decomposition framework for p-local compact groups, generalizing previous results for finite and Lie groups, and applies it to specific examples like U(p) and SU(p).

Contribution

It provides a unified normalizer decomposition for p-local compact groups that extends prior work on finite and Lie groups, avoiding topological complexities.

Findings

Derived a homotopy colimit description of |L| for p-local compact groups.

Computed normalizer decompositions for p-completed classifying spaces of U(p) and SU(p).

Extended the framework to p-compact groups of Aguade and Zabrodsky.

Abstract

We give a normalizer decomposition for a p-local compact group (S, F, L) that describes |L| as a homotopy colimit indexed over a finite poset. Our work generalizes the normalizer decompositions for finite groups due to Dwyer, for p-local finite groups due to Libman, and for compact Lie groups in separate work due to Libman. Our approach gives a result in the Lie group case that avoids topological subtleties with Quillen's Theorem A, because we work with discrete groups. We compute the normalizer decomposition for the p-completed classifying spaces of U(p) and SU(p) and for the p-compact groups of Aguade and Zabrodsky.