Pseudo-isotopies of 3-manifolds
type: Item
instance of: scholarly article
title: Pseudo-isotopies of 3-manifolds
zbMATH Open document ID: 0857.57021
DOI: 10.1016/0040-9383(95)00017-8
author: Reinhard Schultz
author: Slawomir Kwasik
published in: Topology
publication date: 1997-03-11T00:00:00Z
review text: A homotopy \(H:X \times [0,1]\to Y\) between homeomorphisms \(f_0\), \(f_1:X \to Y\) is an isotopy, if it is a homeomorphism for every \(t\in[0,1]\). A homeomorphism \(H_\sharp: X\times[0,1] \to Y\times [0,1]\) is a pseudo-isotopy between homeomorphisms \(f_0\), \(f_1: X\to Y\) if \(H_\sharp (x,0)=(f_0(x),0)\) and \(H_\sharp (x,1) = (f_1(x),1)\). If \(X\) is a locally compact Hausdorff space, then isotopic homeomorphisms are pseudo-isotopic. Therefore, for manifolds isotopy implies pseudoisotopy. It also holds in the opposite direction for 1-dimensional, 2-dimensional, some classes of irreducible 3-manifolds and simply connected manifolds of dimension greater than three. The converse implication does not hold in general since in higher dimensions there are pseudo-isotopic but not isotopic homeomorphisms, like \(n\)-dimensional torus \(T^n\), \(n\geq 4\). The present paper treats this opposite implication for 3-dimensional manifolds and finds the negative answer among reducible 3-manifolds, in particular among metacyclic prism manifolds (3-dimensional spherical space whose fundamental group is properly metacyclic i.e. nonabelian with a cyclic Sylow 2-subgroup). We only quote the result for closed 3-manifolds: If \(M^3\) and \(N^3\) are metacyclic prism manifolds, then the connected sum \(M^3 \sharp N^3\) admits homeomorphisms that are pseudo-isotopic but not isotopic to the identity. The homeomorphisms in question are examples of homeomorphisms considered in [\textit{J. L. Friedman} and \textit{D. M. Witt}, ibid. 25, 35-44 (1986; Zbl 0596.57008)].
Mathematics Subject Classification ID: 57N37
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 878547
zbMATH Keywords: isotopy
zbMATH Keywords: pseudo-isotopy
zbMATH Keywords: 3-manifolds
zbMATH Keywords: metacyclic prism manifolds
reviewed by: Ivan Ivanšić
MaRDI profile type: Publication
full work available at URL: 0040-9383(95)00017-8
OpenAlex ID: W2080611953
description: scientific article; zbMATH DE number 878547