Transitive actions of compact groups and topological dimension
type: Item
instance of: scholarly article
title: Transitive actions of compact groups and topological dimension
zbMATH Open document ID: 0972.22004
author: Sidney A. Morris
author: Karl H. Hofmann
published in: Journal of Algebra
publication date: 2001-11-04T00:00:00Z
review text: The weight \(w(X)\) of a topological space \(X\) is the smallest of all cardinals of bases for the topology. Using the Lebesgue covering dimension \(\text{cdim} X\), a new dimension function is defined: \(\dim{X}=\text{cdim} X\) if the latter is finite, and \(\dim X\) is the supremum over the weights of the connected components of \(X\), otherwise. For any weakly complete topological vector space \(V\) over the reals, this topological dimension of \(V\) equals the linear dimension of the vector space dual of \(V\). For any compact group \(G\), let \(L(G)\) denote the Lie algebra of \(G\); this is a weakly complete space [same authors, The structure of compact groups (Berlin 1998; Zbl 0919.22001)]. The authors prove the following. If a compact group \(G\) acts transitively on \(X\), with isotropy subgroup \(H\), then \(\dim X=\dim(L(G)/L(H))\). Moreover, the space \(X\) contains a cube \([0,1]^{\dim X}\), and the dimension of cubes in \(X\) is bounded by \(\dim X\). Let \(G\) be a locally compact group. Then every neighborhood \(U\) of the identity of \(G\) contains a compact subgroup \(N\) such that there exist a simply connected Lie group \(L\) and a continuous open homomorphism \(\varphi \colon N\times L\to G\) with discrete kernel such that \(\varphi (n,1)=n\) for all \(n\in N\). This generalizes a famous theorem due to \textit{K. Iwasawa} [Ann. Math., Princeton, II. Ser. 50, 507-558 (1949; Zbl 0034.01803)]. Let \(G\) be a locally compact group and \(H\) a closed subgroup of \(G\). Then \(\dim(G/H)=\dim(L(G)/L(H))\), and \(\dim (G/H)\) is maximal among the dimensions of cubes contained in \(G/H\).
Mathematics Subject Classification ID: 22C05
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 22F30
Mathematics Subject Classification ID: 57S10
zbMATH DE Number: 1565869
zbMATH Keywords: Lebesgue covering dimension
zbMATH Keywords: weakly complete topological vector space
zbMATH Keywords: topological dimension
zbMATH Keywords: compact group
zbMATH Keywords: locally compact group
reviewed by: Markus Stroppel
MaRDI profile type: Publication
full work available at URL: jabr.2000.8543
OpenAlex ID: W2016182404
cites work: Q4060144
cites work: Q4185470
cites work: Homogeneity of infinite products of manifolds with boundary
cites work: Weight and c
cites work: The structure of compact groups. A primer for the student -- a handbook for the expert
cites work: A structure theorem on compact groups
cites work: Q5545552
cites work: On some types of topological groups
DOI: 10.1006/JABR.2000.8543
Recommended article: Q4793251
Recommended article: Q3202823
Recommended article: Some aspects of dimension theory for topological groups
Recommended article: Topological groups and covering dimension
Recommended article: Q4993801
Recommended article: Q3692461
Recommended article: Locally compact subgroup actions on topological groups
Recommended article: The dimension of an LCA group in its Bohr topology
Recommended article: Universal Spaces for Locally Finite-Dimensional Tychonoff Spaces
Recommended article: Q4695734
description: scientific article; zbMATH DE number 1565869