Equivariant extension properties of coset spaces of locally compact groups and approximate slices

TL;DR

This paper characterizes when coset spaces of locally compact groups are manifolds or have related properties, establishing their equivalence and introducing a new version of the Approximate Slice Theorem.

Contribution

It provides a comprehensive set of equivalent conditions for coset spaces to be manifolds or possess certain extension properties, along with a new version of the Approximate Slice Theorem.

Findings

Equivalence of manifold and extension properties for coset spaces

Characterization of locally contractible and finite-dimensional coset spaces

Introduction of a new version of the Approximate Slice Theorem

Abstract

We prove that for a compact subgroup $H$ of a locally compact Hausdorff group $G$, the following properties are mutually equivalent: (1) $G/H$ is a manifold, (2) $G/H$ is finite-dimensional and locally connected, (3) $G/H$ is locally contractible, (4) $G/H$ is an ANE for paracompact spaces, (5) $G/H$ is a metrizable $G$-ANE for paracompact proper $G$-spaces having a paracompact orbit space. A new version of the Approximate slice theorem is also proven in the light of these results.