Minimizing the number of fixed points for self-maps of compact surfaces
type: Item
instance of: scholarly article
title: Minimizing the number of fixed points for self-maps of compact surfaces
zbMATH Open document ID: 0571.55003
author: M. R. Kelly
published in: Pacific Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let P denote the topological space obtained by taking a closed regular neighborhood of the figure-eight in the plane. Let MF(f) denote the minimum number of fixed points achievable among maps homotopic to a given self-map f of P. We present here a formula for the value of MF(f). Note that MF(f) depends on the induced homomorphism, \(f_{\#}\), on fundamental group, so our formula concerns the two relevant words in the free group on the lettes a and b corresponding to the loops which comprise the figure eight. Special case: Let \(g_ m: P\to P\), \(m\geq 0\), be given such that \((g_ m)_{\#}(a)=(bab^{-1}a^{-1})^ mba\) and \((g_ m)_{\#}(b)=1\). It is easy to show that the Nielsen number of \(g_ m\), \(N(g_ m)\), is equal to zero. On the other hand, our formula shows that \(MF(g_ m)=2m\). Hence the difference between N(f) and MF(f) can be made arbitrarily large.
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 57N05
zbMATH DE Number: 3911283
zbMATH Keywords: regular neighborhood of the figure-eight in the plane
zbMATH Keywords: minimum number of fixed points
zbMATH Keywords: Nielsen number
MaRDI profile type: Publication
full work available at URL: pjm.1987.126.81
OpenAlex ID: W1965635160
DOI: 10.2140/PJM.1987.126.81
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description: scientific article; zbMATH DE number 3911283