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

TL;DR

This paper proves that any Palais-proper topological G-manifold for any Lie group G has the equivariant homotopy type of a countable proper G-CW complex, extending previous results and verifying classifying spaces.

Contribution

It generalizes key results to arbitrary Lie groups G, establishing the homotopy type of G-manifolds and verifying classifying spaces for principal G-bundles.

Findings

Palais-proper G-manifolds have the homotopy type of countable proper G-CW complexes

Extended previous results to arbitrary Lie groups G

Verified an n-classifying space for principal G-bundles

Abstract

The Hilbert-Smith conjecture states, for any connected topological manifold $M$, any locally compact subgroup of $\mathrm{Homeo}(M)$ is a Lie group. We generalize basic results of Segal-Kosniowski-tomDieck (2.6), James-Segal (2.12), G Bredon (3.7), Jaworowski-Antonyan et al. (5.5), and E Elfving (7.3). The last is our main result: for any Lie group $G$, any Palais-proper topological $G$-manifold has the equivariant homotopy type of a countable proper $G$-CW complex. Along the way, we verify an $n$-classifying space for principal $G$-bundles (5.10).