The minimizing of the Nielsen root classes
type: Item
instance of: scholarly article
title: The minimizing of the Nielsen root classes
zbMATH Open document ID: 1052.55002
DOI: 10.2478/BF02475955
author: Claudemir Aniz
author: Daciberg Lima Gonçalves
published in: Central European Journal of Mathematics
publication date: 2004-02-23T00:00:00Z
review text: It is known that the root classes of two homotopic maps have a correspondence, they are homotopy related. Given a map \(f: X\to Y\) and an arbitrary root class \(C\), the authors define \(MR[f,C]\) to be the minimal cardinality among all root classes \(C'\), where \(C'\) is a root class of \(f\), and is \(H\)-related to \(C\) for a homotopy \(H\) between \(f'\) and \(f\). The root classes \(C\) with \(MR[f,C]>0\) are said to be the essential root classes. The sum \(\sum MR[f,C]\), where \(C\) ranges over all root classes of \(f\), is of course a lower bound for the number of roots. The authors give a sufficient and necessary condition that this lower bound can be realized for maps between surfaces. The condition is that the numbers of roots in all classes can be minimized at the same time. The authors also present maps without the above minimization from \(n\)-dimensional complexes to \(n\)-dimensional manifolds for all \(n\geq 3\).
reviewed by: Xuezhi Zhao
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 20F99
zbMATH DE Number: 2046458
zbMATH Keywords: absolute degree
zbMATH Keywords: Wecken property
zbMATH Keywords: closed surfaces
zbMATH Keywords: complex
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2046458