Proper actions on topological groups: Applications to quotient spaces

TL;DR

This paper demonstrates that the natural action of a locally compact subgroup on a Hausdorff topological group is proper, leading to significant results on the transfer of topological properties and dimension inequalities between the group and its quotient.

Contribution

It establishes the properness of subgroup actions and derives new results on property transfer and dimension inequalities in topological groups.

Findings

Existence of a closed set F with FG=X and a perfect restriction map

Transfer of properties like paracompactness and normality from X to X/G

Dimension inequality: dim X <= dim X/G + dim G

Abstract

Let X be a Hausdorff topological group and G a locally compact subgroup of X. We show that the natural action of G on X is proper in the sense of R. Palais. This is applied to prove that there exists a closed set F of X such that FG=X and the restriction of the quotient projection X -> X/G to F is a perfect map F -> X/G. This is a key result to prove that many topological properties (among them, paracompactness and normality) are transferred from X to ferred from X/G to X. Yet another application leads to the inequality dim X<= dim X/G + dim G for every paracompact group X and its locally compact subgroup G.