Nielsen coincidence theory applied to Borsuk-Ulam geometric problems
type: Item
instance of: scholarly article
title: Nielsen coincidence theory applied to Borsuk-Ulam geometric problems
zbMATH Open document ID: 1260.55003
author: Fabiana Santos Cotrim
author: Daniel Vendrúscolo
published in: Topology and its Applications
publication date: 2012-12-13T00:00:00Z
review text: The paper under review studies a Borsuk-Ulam type theorem for a triple \((X, \tau, Y)\) where \(X, Y\) are simplicial complexes and \(\tau\) is a free involution on \(X\). One of the main features of this work is that the authors begin to develop a sort of Nielsen-Borsuk-Ulam theory which has properties similar to classical Nielsen fixed point theory. More precisely, given a homotopy class of maps \([f]\in [X, Y]\) a number is defined which is expected to be a lower bound for the number of \(\tau\)-orbits which collapse for any map \(g\in [f]\). As expected, based in the experience with the usual Nielsen number, it is not an easy task to compute this new number. The main result of the paper is: {Theorem 4.7} Let \(X\) and \(Y\) be closed triangulable \(n\)-manifolds, \(\tau\) a free involution on \(X\) and \(f:X \to Y\) a continuous map. Suppose that \(BUCoin(f;\tau)\) is finite. If there exists a map \(f'\) homotopic to \(f\) such that \(BUCoin(f';\tau)=\emptyset\), then for each \(BU\)-coincidence class \(C=\{c_1, \tau(c_1),....,c_l, \tau(c_l)\}\) there exists a sequence of splits, joins and blends from \(\Sigma(C)\) that produces a null-signature. The paper leads naturally to many new questions related with the Borsuk-Ulam theorem, for example the ones which are modelled in the classical Nielsen theory.
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 55M35
zbMATH DE Number: 6115439
zbMATH Keywords: Borsuk-Ulam theorem
zbMATH Keywords: Nielsen coincidence theory
zbMATH Keywords: polyhedron
zbMATH Keywords: Hopf construction
reviewed by: Daciberg Lima Gonçalves
MaRDI profile type: Publication
full work available at URL: j.topol.2012.05.031
OpenAlex ID: W1983938828
cites work: Q5619504
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cites work: Fix-finite homotopies
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DOI: 10.1016/J.TOPOL.2012.05.031
description: scientific article; zbMATH DE number 6115439