Extension properties of orbit spaces for proper actions revisited

TL;DR

This paper investigates the extension properties of orbit spaces under proper group actions, establishing conditions under which these spaces are absolute neighborhood extensors (ANEs) and exploring the behavior of orbit spaces for various group types.

Contribution

It proves that orbit spaces of G-ANE spaces are ANEs under certain conditions and extends these results to orbit spaces of normal subgroups for Lie and almost connected groups.

Findings

Orbit spaces of G-ANE spaces are ANEs if all orbits are metrizable.

H-orbit spaces are G/H-ANE when G is Lie or almost connected and all H-orbits are metrizable.

Extension properties are preserved under specific group actions and subgroup conditions.

Abstract

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