New York Journal of Mathematics
Volume 27 (2021), 1554-1579


Qayum Khan

Countable approximation of topological G-manifolds, III: arbitrary Lie groups

view    print

Published: November 16, 2021.
Keywords: Topological manifold; proper action; Lie group; CW complex.
Subject: 57N65, 57S20, 54H15, 54C55.

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.

Author information

Qayum Khan:
Department of Mathematics
Indiana University
Bloomington, IN 47405, USA