Characterizing maximal compact subgroups

TL;DR

This paper characterizes maximal compact subgroups in almost connected locally compact groups by establishing their equivalence with several topological and homotopical properties of the quotient space.

Contribution

It proves the equivalence of multiple conditions characterizing maximal compact subgroups in a broad class of topological groups.

Findings

$H$ is maximal compact iff $G/H$ is contractible

$G/H$ is homeomorphic to Euclidean space iff $H$ is maximal compact

$G/H$ is an AE for paracompact spaces iff $H$ is maximal compact

Abstract

We prove that for a compact subgroup $H$ of an almost connected locally compact Hausdorff group $G$, the following properties are mutually equivalent: (1) $H$ is a maximal compact subgroup of $G$, (2) $G/H$ is contractible, (3) $G/H$ is homeomorphic to a Euclidean space, (4) $G/H$ is an AE for paracompact spaces, (5) $G/H$ is a $G$-AE for paracompact proper $G$-spaces having a paracompact orbit space.