Minimum numbers and Wecken theorems in topological coincidence theory. I
type: Item
instance of: scholarly article
title: Minimum numbers and Wecken theorems in topological coincidence theory. I
zbMATH Open document ID: 1270.55002
author: Ulrich Koschorke
published in: Journal of Fixed Point Theory and Applications
publication date: 2011-11-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1305.1664
review text: Given a self-map \(f: X\to X\) on a compact polyhedron \(X\), the Nielsen number \(N(f)\) is a lower bound for the number of fixed point of all self-maps homotopic to \(f\). In the 1940's, Wecken proved that \(N(f)\) is sharp (i.e., there is a self-map which is homotopic to \(f\) and has \(N(f)\) fixed points) if \(X\) is a smooth manifold of dimension greater than \(3\). This result is called Wecken theorem. This paper deals with a more general setting: coincidence points of two maps \(f_1, f_2: M\to N\) between two manifolds without boundaries. The author provides several invariants which can be used to estimate from below the number of coincidence points or the number of components of the coincidence point sets, and obtains Wecken type theorems: under some assumptions these lowers bound are sharp. Although some of the invariants already appeared in previous works by the author, this paper contains a systematic treatment. All these invariants come from the path space \(E(f_1, f_2)\). The computation of the invariants is in general difficult, however the author in this paper gives many concrete and interesting examples where one can see some new approach to access these invariants. Some further theoretical applications are also provided. For instance, if \(M\) and \(N\) are spheres of special dimensions, respectively, the relations between the Kervaire invariant and the coincidence invariants are illustrated. These show the theoretical interaction among coincidence point theory and other mathematical fields.
reviewed by: Xuezhi Zhao
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 55Q25
Mathematics Subject Classification ID: 55Q40
zbMATH DE Number: 5968113
zbMATH Keywords: coincidence point
zbMATH Keywords: Nielsen number
zbMATH Keywords: Wecken condition
zbMATH Keywords: Kervaire invariant
zbMATH Keywords: Hopf invariant
MaRDI profile type: Publication
OpenAlex ID: W3100441417
arXiv ID: 1305.1664
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DOI: 10.1007/S11784-011-0058-8
description: scientific article