Extension properties of orbit spaces of proper actions revisited

TL;DR

This paper investigates the extension properties of orbit spaces under proper group actions, establishing conditions under which these spaces inherit the ANE property, especially for metrizable orbits and Lie group actions.

Contribution

It proves that proper G-spaces that are G-ANE with metrizable orbits have orbit spaces that are ANE, and extends results to quotients by closed normal subgroups of Lie groups.

Findings

G-ANE proper G-spaces with metrizable orbits have ANE orbit spaces

Orbit spaces of normal subgroup quotients are G/H-ANE when G is a Lie group

Extension properties are preserved under certain proper group actions

Abstract

Let $G$ be a locally compact Hausdorff group. We study orbit spaces of equivariant absolute neighborhood extensors ($G$-${\rm ANE}$'s) in the class of all proper $G$-spaces that are metrizable by a $G$-invariant metric. We prove that if a proper $G$-space $X$ is a $G$-${\rm ANE}$ and all $G $-orbits in $X$ are metrizable, then the $G$-orbit space $X/G$ is an {\rm ANE}. If $G$ is a Lie group and $H$ is a closed normal subgroup of $G$, then the $H$-orbit space $X/H$ is a $G/H$-{\rm ANE}.