Bundle Structure of the Homeomorphism Groups of Locally Compact Homogeneous Spaces

The space ℋ(X) of homeomorphisms on a locally compact homogeneous space X with a local cross-section is a bundle space over X. If X is separable metrizable and admits a nontrivial flow in addition, then ℋ(X) is an l2-manifold if and only if X is an ANR and ℋ(X,a) is an l2-manifold, where ℋ(X,a) is the subspace of ℋ(X) consisting of all those which leave a point α of X fixed. If X is a locally connected, compact metrizable homogeneous space that is an ANR and admits a local cross-section and a nontrivial flow, then ℋ(X) is an l2-manifold if and only if ℋ(X-a) is an l2-manifold, where ℋ(X-a) is the space of homeomorphisms on X-a (a∈X)