Estimation of the number of fixed points on the complement
type: Item
instance of: scholarly article
title: Estimation of the number of fixed points on the complement
zbMATH Open document ID: 0713.55001
DOI: 10.1016/0166-8641(90)90024-V
author: Xuezhi Zhao
published in: Topology and its Applications
publication date: 1990-01-01T00:00:00Z
review text: Let f: (X,A)\(\to (X,A)\) be a map of a pair of compact polyhedra. The author has introduced the so-called Nielsen number of the complementary space N(f;X-A) which is a lower bound for the number of fixed points on X-A for all maps in the homotopy class of f: (X,A)\(\to (X,A)\), and is an optimal lower bound if A can be bypassed in X and if X-A has no local cut point and is not a 2-manifold [Lect. Notes Math. 1411, 189-199 (1989; Zbl 0689.55008)]. In this paper he introduces a better lower bound, namely the surplus Nielsen number SN(f;X-A) of f on X-A, which is optimal even if A cannot be by-passed in X. He shows that SN(f;X-A)\(\geq N(f;N- A)\), and gives examples in which equality does not hold. One of the examples also shows that by-passing is a necessary condition for the optimality of the relative Nielsen number N(f;X,A) which was introduced by the reviewer as a lower bound for the number of fixed points of f: (X,A)\(\to (X,A)\) on X [Pac. J. Math. 122, 459-473 (1986; Zbl 0553.55001)].
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 4173900
zbMATH Keywords: minimal fixed point sets
zbMATH Keywords: complementary space
zbMATH Keywords: cut point
zbMATH Keywords: surplus Nielsen number
zbMATH Keywords: by-passing
zbMATH Keywords: relative Nielsen number
reviewed by: Helga Schirmer
MaRDI profile type: Publication
cites work: Q5619504
cites work: Lectures on Nielsen fixed point theory
cites work: On the Least Number of Fixed Points
cites work: Fixed points and braids
cites work: A relative Nielsen number
full work available at URL: 0166-8641(90)90024-v
OpenAlex ID: W1998061890
Wikidata QID: Q126759752
description: scientific article; zbMATH DE number 4173900