Finite approximations of $p$-local compact groups

TL;DR

This paper demonstrates that the classifying space of a $p$-local compact group can be approximated by a sequence of classifying spaces of $p$-local finite groups, leading to new insights in homotopy theory.

Contribution

It introduces a method to approximate $p$-local compact groups using telescopes of $p$-local finite groups, with implications for stable elements and mapping spaces.

Findings

Classifying space of $p$-local compact group approximated by telescope of finite groups

Stable Elements Theorem extended to $p$-local compact groups

Homotopy types of certain mapping spaces characterized

Abstract

We show that the classifying space of a $p$-local compact group is approximated by a telescope of classifying spaces of $p$-local finite groups. This result has numerous implications, like a Stable Elements Theorem for $p$-local compact groups, or the description of the homotopy type of certain mapping spaces.