An almost uniquely homogeneous subgroup of \({\mathbb{R}}^ n\)
type: Item
instance of: scholarly article
title: An almost uniquely homogeneous subgroup of \({\mathbb{R}}^ n\)
zbMATH Open document ID: 0586.57019
DOI: 10.1016/0166-8641(86)90008-8
author: David C. Wilson
author: J. Keesling
published in: Topology and its Applications
publication date: 1986-01-01T00:00:00Z
review text: A topological group whose only homeomorphisms are translations has (among other) the property that is uniquely homogeneous. A known example of \textit{J. van Mill} [Trans. Am. Math. Soc. 280, 491-498 (1983; Zbl 0573.22001)] lives in \(\ell_ 2\) where \(\ell_ 2\) is endowed with a nonstandard group structure. The authors find an example of a topological subgroup of \({\mathbb{R}}^ n\) which is close to being uniquely homogeneous as much as is possible, because a subgroup \(H\subset {\mathbb{R}}^ n\) must always have automorphisms of the form \(g(x)=-x+h\), \(h\in H\). The obtained example is a connected locally connected (n-1)-dimensional subgroup X of \({\mathbb{R}}^ n\) with the property that for every homeomorphism \(g\in H(X)\) \((= space\) of homeomorphisms of X with compact open topology) there is an \(h\in X\) such that \(g(x)=x+h\) or \(g(x)=-x+h\). Also dim H(X)\(=n-1\). This represents the main result of the paper, but there are also some applications.
Mathematics Subject Classification ID: 57S05
Mathematics Subject Classification ID: 57N15
Mathematics Subject Classification ID: 22D99
Mathematics Subject Classification ID: 57S20
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 3940282
zbMATH Keywords: topological subgroup of \({\mathbb{R}}^ n\)
zbMATH Keywords: uniquely homogeneous
zbMATH Keywords: space of homeomorphisms
reviewed by: Ivan Ivanšić
MaRDI profile type: Publication
full work available at URL: 0166-8641(86)90008-8
OpenAlex ID: W2022318749
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description: scientific article; zbMATH DE number 3940282