Equivariant Nielsen fixed point theory for \(n\)-valued maps
type: Item
instance of: scholarly article
title: Equivariant Nielsen fixed point theory for \(n\)-valued maps
zbMATH Open document ID: 1219.55002
author: Joel Better
published in: Topology and its Applications
publication date: 2010-06-09T00:00:00Z
review text: Let \(G\) be a finite group and \(X\) a finite simplicial \(G\)-complex. In this paper, the author introduces a version of Nielsen fixed point theory for \(n\)-valued \(G\)-maps \(\varphi: X \multimap X\). This theory simultaneously extends the equivariant Nielsen fixed point theory developed in \textit{J. Better} [Topol. Methods Nonlinear Anal. 36, No.~1, 179--195 (2010; Zbl 1222.55001)] and the non-equivariant theory for \(n\)-valued maps developed in \textit{H. Schirmer} [Fundam. Math. 124, 207--219 (1984; Zbl 0543.55003)]. The two Nielsen-type numbers introduced in this paper, \(NO_G(\varphi)\) and \(N_G(\varphi)\), are defined in terms of an abstract simplicial complex \(K_\varphi\) associated to the \(n\)-valued \(G\)-map \(\varphi\). Each of \(NO_G(\varphi)\) and \(N_G(\varphi)\) is shown to be \(n\)-valued \(G\)-homotopy invariant and each satisfies a lower bound property: if there is an \(n\)-valued \(G\)-homotopy between \(\varphi\) and \(\psi\), then \(NO_G(\varphi)\leq|\{\theta\mid \theta\text{ is an orbit of fixed points of }\psi\}|\) and \(N_G(\varphi)\leq|\mathrm{Fix}(\psi)|\). In addition, the author establishes a minimality result for these lower bounds for a certain restricted class of finite simplicial \(G\)-complexes. The focus of this paper is in establishing the theory and it does not discuss methods of computing either Nielsen-type number introduced.
reviewed by: William C. Kronholm
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 57R91
Mathematics Subject Classification ID: 57Q91
zbMATH DE Number: 5718379
zbMATH Keywords: equivariant fixed point theory
zbMATH Keywords: Nielsen fixed point theory
zbMATH Keywords: \(n\)-valued map
zbMATH Keywords: Nielsen numbers
MaRDI profile type: Publication
full work available at URL: j.topol.2010.02.023
OpenAlex ID: W2057155303
cites work: Q5693500
cites work: Equivariant Nielsen fixed point theory
cites work: Nielsen numbers of \(n\)-valued fiber maps
cites work: Smooth equivariant triangulations of G-manifolds for G a finite group
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: Q3031628
DOI: 10.1016/J.TOPOL.2010.02.023
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description: scientific article; zbMATH DE number 5718379