A remark on Newman's theorem
type: Item
instance of: scholarly article
title: A remark on Newman's theorem
zbMATH Open document ID: 0872.57028
DOI: 10.1007/BF02112530
author: Yu. Burago
author: B. V. Kalinin
published in: Siberian Mathematical Journal
publication date: 1997-10-08T00:00:00Z
review text: Newman's theorem says that for a topological manifold \(M\) endowed with a metric \(d\) there is an \(\varepsilon = \varepsilon (M,d)>0\) such that for any finite group \(G\) and any non-trivial action of \(G\) on \(M\) by homeomorphisms there is at least one orbit of diameter \(\geq \varepsilon\). This was generalized by Chernavskij to continuous partitions of \(M\) induced by pseudosubmersions. Here a pseudosubmersion is a continuous surjective open and closed mapping \(f:X\to Y\) between topological spaces which is not a homeomorphism and for which every fibre \(f^{-1} (y),\;y\in Y\) is finite. The authors prove the following generalization: Let \((X,d)\) be a locally compact metric space which is a union of a finite number of open subsets which are topological manifolds. Then there is a positive number \(\varepsilon=\varepsilon (X,d)\) such that for every pseudosubmersion defined on \(X\) there is a fibre with diameter \(\geq\varepsilon\).
Mathematics Subject Classification ID: 57N99
zbMATH DE Number: 923557
zbMATH Keywords: Newman's theorem
zbMATH Keywords: Newman constant
zbMATH Keywords: Alexandrov spaces
zbMATH Keywords: topological manifold
zbMATH Keywords: action
zbMATH Keywords: orbit
zbMATH Keywords: pseudosubmersion
reviewed by: Herbert Abels
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 923557