Autohomeomorphism groups
type: Item
instance of: scholarly article
title: Autohomeomorphism groups
zbMATH Open document ID: 1020.54026
DOI: 10.1016/S0166-8641(02)00140-2
author: Paul Gartside
author: Aneirin Glyn
published in: Topology and its Applications
publication date: 2003-05-27T00:00:00Z
review text: Let \(H(X)\) denote the group of homeomorphisms of a topological space \(X\). The authors show that if \(M\) is a metric one-dimensional manifold with or without boundary, then the compact-open topology on \(H(M)\) is the unique minimum group topology (one contained in all \(T_1\) group topologies) on \(H(M)\). They note that they were informed by the referee that this result was also obtained, using different techniques, by \textit{D. M. Hayes} in her unpublished Ph.D. dissertation [Minimality of the special linear groups, University of North Texas, Ph.D. Dissertation (1997)]. A compact, connected, locally connected, metric space is referred to as a nice space. The authors show that for every compact, zero-dimensional, metric group \(G\), there exists a continuum of pairwise non-homeomorphic nice spaces whose homeomorphism groups, each with the compact-open topology, are all topologically isomorphic to \(G\). \textit{D. Evans} and \textit{P. R. Hewitt} in [Ann. Pure Appl. Logic 46, No. 2, 201-209 (1990; Zbl 0714.03030)] constructed two infinite, compact, zero-dimensional metric groups which are algebraically isomorphic but not isomorphic as topological groups. From this and the previous result, the authors deduce that there are nice spaces \(X\) and \(Y\) such that \(H(X)\) and \(H(Y)\) are infinite, algebraically isomorphic, and homeomorphic but not topologically isomorphic.
Mathematics Subject Classification ID: 54H15
Mathematics Subject Classification ID: 57S10
Mathematics Subject Classification ID: 57S05
zbMATH DE Number: 1917825
zbMATH Keywords: autohomeomorphism group
zbMATH Keywords: manifold
zbMATH Keywords: rigid space
zbMATH Keywords: profinite group
reviewed by: jun. Kenneth D. Magill
MaRDI profile type: Publication
cites work: Q3741814
cites work: Continua which admit only the identity mapping onto non-degenerate subcontinua
cites work: Counterexamples to a conjecture on relative categoricity
cites work: Uniqueness Results for Homeomorphism Groups
cites work: On isomorphic groups and homeomorphic spaces
cites work: An algebraic characterization of stability groups
full work available at URL: s0166-8641(02)00140-2
OpenAlex ID: W4210320919
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description: scientific article; zbMATH DE number 1917825