Countable approximation of topological $G$-manifolds: compact Lie groups $G$

TL;DR

This paper demonstrates that compact topological G-manifolds have the G-homotopy type of countable G-CW complexes, extending Elfving's theorem to a broader class of G-manifolds.

Contribution

It generalizes Elfving's theorem from locally linear G-manifolds with linear G to all compact topological G-manifolds, showing they have the G-homotopy type of countable G-CW complexes.

Findings

Compact topological G-manifolds are G-homotopy equivalent to countable G-CW complexes

Extension of Elfving's theorem to non-linear G-manifolds

Broader applicability to topological G-manifolds

Abstract

Let $G$ be a compact Lie group. (Compact) topological $G$-manifolds have the $G$-homotopy type of (finite-dimensional) countable $G$-CW complexes (2.5). This partly generalizes Elfving's theorem for locally linear $G$-manifolds [Elf96], wherein the Lie group $G$ is linear (such as compact).