Minimal Nielsen root classes and roots of liftings
type: Item
instance of: scholarly article
title: Minimal Nielsen root classes and roots of liftings
zbMATH Open document ID: 1182.55002
DOI: 10.1155/2009/346519
author: Marcio Colombo Fenille
author: Oziride Manzoli Neto
published in: Fixed Point Theory and Applications
publication date: 2009-11-10T00:00:00Z
full work available at URL: https://eudml.org/doc/45680
review text: Let \(f: X\to Y\) be a map, and \(a\) be a given point in \(Y\). A root of \(f\) at \(a\) is a point \(x\) in \(X\) such that \(f(x)=a\). In Nielsen root theory, one is interested in determining the number \(\min _{g\simeq f}\sharp g^{-1}(a)\), where \(g\) ranges over the maps homotopic to \(f\). If the target space \(Y\) is a closed manifold, this number is independent of the choice of the base point \(a\), denoted \(\mu(f)\). By using traditional methods in Nielsen theory, the root set of a map splits into a disjoint union of root classes. There is a well-defined one-to-one correspondence between the sets of root classes of two homotopic maps. A root class is said to be essential if it is always non-empty under any correspondence determined by a homotopy. The Nielsen root number \(N(f,a)\) is defined to be the number of essential root classes. Clearly, \(N(f,a)\) gives a lower bound of \(\mu(f)\). Given a root class \(\mathcal{R}\) of \(f\), \(\mu_C(f, \mathcal{R})\) is defined to be the minimal number \(\min_{g\simeq f} \sharp \mathcal{R}_{g}\), where \(g\) ranges over the maps homotopic to \(f\) and \(\mathcal{R}_{g}\) is the root class of \(g\) homotopically corresponding to the given root class \(\mathcal{R}\). It is obvious that the invariants mentioned above satisfy the relation \(N(f,a)\leq\sum_{\mathcal{R}} \mu_C(f, \mathcal{R})\leq \mu(f)\). A natural question is: under which conditions do these inequalities become equalities? \textit{D. Gonçalves} and \textit{C. Aniz} proved in [Cent. Eur. J. Math. 2, No.~1, 112--122 (2004; Zbl 1052.55002)] that \(\mu_C(f, \mathcal{R})\) is independent of the choice of the root class \(\mathcal{R}\) of a given map \(f: X\to Y\) if \(Y\) is a manifold. In this case, one may simply write \(\mu_C(f)\). Hence, one may ask for a condition which ensures that \(N(f)\mu_C(f) =\mu(f)\). It is equivalent to ask for sufficient conditions to minimize simultaneously the root numbers in all root classes. By using some examples, the au…
reviewed by: Xuezhi Zhao
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 57N65
zbMATH DE Number: 5627843
zbMATH Keywords: coincidence point
zbMATH Keywords: Lefschetz number
zbMATH Keywords: maps
MaRDI profile type: Publication
OpenAlex ID: W2160072123
Wikidata QID: Q59248557
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description: scientific article; zbMATH DE number 5627843