On the Nielsen numbers of slide homeomorphisms on 3-manifolds.
type: Item
instance of: scholarly article
title: On the Nielsen numbers of slide homeomorphisms on 3-manifolds.
zbMATH Open document ID: 1042.55002
DOI: 10.1016/S0166-8641(03)00218-9
author: Xuezhi Zhao
published in: Topology and its Applications
publication date: 2004-02-03T00:00:00Z
review text: Every closed oriented \(3\)-manifold can be written uniquely as a connected sum of prime manifolds. Following the work of \textit{D. McCullough} [Geometric and algebraic topology, Banach Cent. Publ. 18, 61--76 (1986; Zbl 0637.57009)], every self-homeomorphism on a reducible \(3\)-manifold \(M\) is a composition of homeomorphisms of four types, namely, homeomorphisms preserving summands, those interchanging summands, spins of \(S^1 \times S^2\), and slide homeomorphisms. Write \(M=W \cup (\bigcup {M_i}')\) where \(W\) is a punctured \(3\)-cell, either \(M_i'=M_i- \text{int}(D_i)\) where \(M_i\) is irreducible and \(D_i\) is a closed \(3\)-disk, or \(M_i'\) is homeomorphic to \(S^2 \times [0,1]\). Let \(S\) be an essential \(2\)-sphere in \(M\), which is isotopic to a boundary component of \(W\) and \(\alpha :[0,1] \to M\) be a path such that \(\alpha ((0,\varepsilon))\cup \alpha ((1-\varepsilon,1))\subset W-\partial W\) and \(\alpha ([0,1])\cap S=\{\alpha (0), \alpha (1)\}\) for some \(\varepsilon >0\). Take two regular neighborhoods \(N_1, N_2\) of \(S \cup \alpha ([0,1])\) in \(M\) such that \(N_1 \subset \text{int}(N_2)\). Then \(\text{int}(N_2-N_1)\) has two components, one of which, denoted by \(T(S,\alpha)\), is homeomorphic to \(T^2\times (0,1)\) and the other to \(S^2 \times (0,1)\). Let \(c:T(S,\alpha) \to T^2\times (0,1)\) be the homeomorphism. By a slide homeomorphism, we mean a map \(s:M\to M\) such that \(s(x)=c^{-1}(\theta +2\pi t, \varphi, t)\) if \(x=c^{-1}(\theta +2\pi t, \varphi, t)\in T(S,\alpha)\) and \(s(x)=x\) otherwise. The purpose of the paper under review is to study the fixed point set of composites of slide homeomorphisms, from the point of view of Nielsen fixed point theory. First, the author shows that the Nielsen number of a slide homeomorphism is always zero while that of a composite of two slide homeomorphisms need not be even though such composites must have zero Lefschetz number. Furthermore, he shows, by means of an example, that th…
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 2037251
zbMATH Keywords: Nielsen number
zbMATH Keywords: fixed point index
zbMATH Keywords: reducible 3-manifolds
zbMATH Keywords: slide homeomorphisms
reviewed by: Peter Wong
MaRDI profile type: Publication
cites work: Q5619504
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description: scientific article; zbMATH DE number 2037251