Proper actions on topological groups: Applications to quotient spaces

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 $F\subset X$ such that $FG=X$ and the restriction of the quotient projection $X\to X/G$ to $F$ is a perfect map $F\to X/G$. This is a key result to prove that many topological properties (among them, paracompactness and normality) are transferred from $X$ to $X/G$, and some others are transferred from $X/G$ to $X$. Yet another application leads to the inequality $\mathrm {dim} X\le \textrm {dim} X/G + \mathrm {dim} G$ for every paracompact topological group $X$ and a locally compact subgroup $G$ of $X$ having a compact group of connected components.