Extension properties of orbit spaces of proper actions revisited

Abstract

Let $G$ be a locally compact Hausdorff group. We study orbit spaces of equivariant absolute neighborhood extensors ($G$-$\mathrm{ANE}$'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 $G$-$\mathrm{ANE}$ 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}.