Locally compact subgroup actions on topological groups

zbMATH Open1274.22001arXiv1103.1407MaRDI QIDQ2839892

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Publication date: 15 July 2013

Published in: (Search for Journal in Brave)

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 $Gime{s}_H{S}_i$, 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 $Gime{s}_{K}S$, where $K$ is a maximal compact subgroup of $G$. This implies that $X$ is $K$-homeomorphic to the product $G/KimesS$, and in particular, $X$ is homeomorphic to the product $Bbb{R}^{n}imesS$, where $n=mdim,G/K$. Using these results we prove the inequality $mdim,Xlemdim,X/G+mdim,G$ for every Hausdorff topological group $X$ and a locally compact subgroup $G$ of $X$.

Full work available at URL: https://arxiv.org/abs/1103.1407

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