Locally compact subgroup actions on topological groups

TL;DR

This paper investigates the structure of topological groups with locally compact subgroups, showing they can be decomposed into twisted products and establishing a dimension inequality relating the group, subgroup, and quotient.

Contribution

It introduces a new decomposition of topological groups with locally compact subgroups into twisted products, extending understanding of their topological and geometric structure.

Findings

Existence of locally finite $\sigma$-discrete $G$-functionally open covers with twisted product members.

Under certain conditions, the entire space is $G$-homeomorphic to a twisted product $G imes_K S$.

Proves the inequality $ ext{dim}\, X \, extless=\, ext{dim}\, X/G + ext{dim}\, G$ for 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 $G\times_KS$, 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 $\Bbb R^n\times S$, where $n={\rm dim\,} G/K$. Using these results we prove the inequality $ {\rm dim}\, X\le {\rm dim}\, X/G + {\rm dim}\, G$ for every Hausdorff topological group $X$ and a locally compact subgroup $G$ of $X$.