Countable approximation of topological $G$-manifolds, II: linear Lie groups $G$

TL;DR

This paper extends the approximation of topological G-manifolds with proper actions to linear Lie groups, generalizing previous results for locally linear manifolds and improving classical theorems for compact groups.

Contribution

It generalizes the countable G-CW complex approximation to matrix groups and enhances the Bredon--Floyd theorem for compact groups.

Findings

Topological G-manifolds with Palais-proper action have the G-homotopy type of countable G-CW complexes.

Generalization of Elfving's theorem to linear Lie groups.

Improved Bredon--Floyd theorem for compact groups.

Abstract

Let $G$ be a matrix group. Topological $G$-manifolds with Palais-proper action have the $G$-homotopy type of countable $G$-CW complexes (3.2). This generalizes E Elfving's dissertation theorem for locally linear $G$-manifolds (1996). Also we improve the Bredon--Floyd theorem from compact groups $G$ (1960).