Equivariant coincidence Nielsen numbers
type: Item
instance of: scholarly article
title: Equivariant coincidence Nielsen numbers
zbMATH Open document ID: 1021.55001
DOI: 10.1016/S0166-8641(02)00130-X
author: Jianhan Guo
author: Philip R. Heath
published in: Topology and its Applications
publication date: 2003-05-20T00:00:00Z
review text: Let \(W\) be a finite group acting on closed \(n\)-manifolds \(X\) and \(Y\). This paper studies coincidences of \(W\)-maps \(f,g: X\to Y\), that is, solutions to \(f(x)= g(x)\). Nielsen coincidence theory for maps between closed manifolds of the same dimension began with work of Schirmer in the 1950's. A Nielsen fixed point theory for equivariant maps was introduced by Wong in 1991. Thus it was natural to extend equivariant Nielsen theory to the coincidence setting and Fagundes published a paper on this subject in 1997. The present paper improves substantially our understanding of the coincidence theory of equivariant maps. Note that if \(x\in X\) is a coincidence point of \(f\) and \(g\), then every point in the \(W\)-orbit of \(x\) is also a coincidence point, thus one may also consider coincidence orbits. Let \(M_WO(f,g)\) denote the minimum number of coincidence orbits among all \(W\)-maps \(\phi,\psi: X\to Y\) that are homotopic through equivariant maps to \(f\) and \(g\), respectively. Under reasonable hypotheses, the authors are able to define a Nielsen number that is a \(W\)-homotopy invariant lower bound for \(M_WO(f, g)\). Now let \(H\) be a subgroup of \(W\) and let \(X^H\) denote the subspace of \(X\) consisting of points that are fixed by all elements of \(H\). Then the restrictions \(f^H,g^H: X^H\to Y^H\) are \(WH\)-maps for \(WH\) the Weyl group of \(H\). For \(X^{(H)}\) the union of all \(X^K\), where \(K\) is conjugate to \(H\), there are the maps \(f^{(H)}, g^{(H)}: X^{(H)}\to Y^{(H)}\). Under appropriate hypotheses, the authors define Nielsen numbers that give \(W\)-homotopy invariant lower bounds for \(M_WO(f^{(H)}, g^{(H)})\) as well as for the corresponding concept regarding the actual number of coincidence. A significant feature of the present theory is that it eliminates a compactness restriction on the homotopies in Fagundes' theory, a requirement that appears in parts of Wong's work as well. This paper also presents an example that is impo…
reviewed by: Robert F. Brown
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 55P92
zbMATH DE Number: 1913874
zbMATH Keywords: coincidence Nielsen numbers
zbMATH Keywords: equivariant homotopy theory
MaRDI profile type: Publication
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full work available at URL: s0166-8641(02)00130-x
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description: scientific article; zbMATH DE number 1913874