  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.
