A Wecken theorem for \(n\)-valued maps
type: Item
instance of: scholarly article
title: A Wecken theorem for \(n\)-valued maps
zbMATH Open document ID: 1257.55001
author: Joel Better
published in: Topology and its Applications
publication date: 2012-12-13T00:00:00Z
review text: \textit{B. Jiang} [Am. J. Math. 102, 749--763 (1980; Zbl 0455.55001)] proved that if \(X\) is a compact connected polyhedron different from a 2-manifold then the Nielsen number of an arbitrary self map coincides with the number of fixed points of some map in the homotopy class of the self map. \textit{H. Schirmer} [Fundam. Math. 126, 83--92 (1985; Zbl 0609.55001)] proved the following Theorem: Let \(X\) be a compact manifold (with or without boundary) of dimension greater than or equal to three, and let \(\varphi:X \multimap X\) be an \(n\)-valued map, then there is an \(n\)-valued homotopy between \(\varphi\) and an \(n\)-valued map with \(N(\varphi)\) many fixed points. This result is almost the correspondent of the famous result of Wecken for single valued maps. The paper under review generalizes the result of Schirmer by assuming that \(X\) satisfies a weaker hypothesis. More precisely it shows: Theorem 1 - Let \(X\) be a compact polyhedron without local cut points and such that no connected component is a two-manifold. Let \(\varphi:X \multimap X\) be an \(n\)-valued map. Then there is an \(n\)-valued homotopy between \(\varphi\) and an \(n\)-valued map with \(N(\varphi)\) many fixed points. As one sees, the hypothesis above that \(X\) must satisfy coincides with the weaker hypothesis that \(X\) must satisfy for which these type of results (also called Wecken properties) hold for the single valued case.
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 6115435
zbMATH Keywords: multivalued maps
zbMATH Keywords: fixed points
zbMATH Keywords: Nielsen number
zbMATH Keywords: polyhedron
zbMATH Keywords: PL
reviewed by: Daciberg Lima Gonçalves
MaRDI profile type: Publication
full work available at URL: j.topol.2012.08.023
OpenAlex ID: W2030986544
cites work: Q5619504
cites work: Q4278134
cites work: On the Least Number of Fixed Points
cites work: Fix-finite approximation of n-valued multifunctions
cites work: An index and a Nielsen number for n-valued multifunctions
cites work: A minimum theorem for n-valued multifunctions
cites work: Q5731383
DOI: 10.1016/J.TOPOL.2012.08.023
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description: scientific article; zbMATH DE number 6115435