The Hilbert-Smith conjecture states that, for any connected topological manifold M, any locally compact subgroup of Homeo(M) is a Lie group. We generalize basic results of Segal-Kosniowski-tomDieck, James-Segal, G. Bredon, Jaworowski-Antonyan et al, and E. Elfving. The last is our main result: for any Lie group G, any Palais-proper topological G-manifold has the G-homotopy type of a countable proper G-CW complex. Along the way, we verify an n-classifying space for principal G-bundles.

I thank Christopher Connell for various basic discussions. I am grateful to Ric Ancel and Alex Dranishnikov for email dialogue on Lemma 6.6. The referee kindly pointed out the special case Corollary 3.9 is more recently known.