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

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

Extension properties of orbit spaces for proper actions revisited

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.Comment: arXiv admin note: substantial text overlap with arXiv:2308.1223

Homotopy characterization of G-ANR\u27s

Let G be a compact Lie group. We prove that if each point x X of a G-space X admits a Gx-invariant neighborhood U which is a Gx-ANE then X is a G-ANE, where Gx stands for the stabilizer of x. This result is further applied to give two equivariant homotopy characterizations of G-ANR\u27s. One of them sounds as follows: a metrizable G-space Y is a G-ANR iff Y is locally G-contractible and every metrizable closed G-pair (X, A) has the G-equivariant homotopy extension property with respect to Y. In the same terms we also characterize G-ANR subsets of a given G-ANR space

Fiberwise retraction and shape properties of the orbit space

From the point of view of retracts and shape theory, the category G-TOPB of G-spaces over a G-space B, where G is a compact group, is investigated. In particular, we prove that if B has only one orbit type and E is a metric G-ANR over B, then the orbit space E/G is an ANR over B/G. As an application we construct a fiberwise G-orbit functor μ : G-SHB → SHB/G on shape level