Locally compact subgroup actions on topological groups

Abstract

Let $X$ be a Hausdorff topological group and $G$ a locally compact subgroup of $X$. We show that $X$ admits a locally finite $\sigma$-discrete $G$-functionally open cover each member of which is $G$-homeomorphic to a twisted product $G{\times }_H{S}_i$, where $H$ is a compact large subgroup of $G$ (i.e., the quotient $G/H$ is a manifold). If, in addition, the space of connected components of $G$ is compact and $X$ is normal, then $X$ itself is $G$-homeomorphic to a twisted product $G{\times }_{K}S$, where $K$ is a maximal compact subgroup of $G$. This implies that $X$ is $K$-homeomorphic to the product $G/K\times S$, and in particular, $X$ is homeomorphic to the product ${\mathbb{R}}^n\times S$, where $n=\dim G/K$. Using these results we prove the inequality $\dim X\le \dim X/G+\dim G$ for every Hausdorff topological group $X$ and a locally compact subgroup $G$ of $X$.

12 pages