Homotopy and isotopy in dimension three
type: Item
instance of: scholarly article
title: Homotopy and isotopy in dimension three
zbMATH Open document ID: 0805.57008
DOI: 10.1007/BF02565825
author: Joel R. Hass
author: Peter Scott
published in: Commentarii Mathematici Helvetici
publication date: 1995-01-31T00:00:00Z
full work available at URL: https://eudml.org/doc/140307
review text: The authors enlarge the class of 3-manifolds for which homotopic homeomorphisms are always isotopic. The class was known to include Haken manifolds and Seifert fiber spaces, and is conjectured to include any closed \(\mathbb{P}^ 2\)-irreducible 3-manifold. The authors prove Theorem 1.1: Let \(M\) be a closed orientable irreducible 3-manifold which is neither Haken nor a Seifert fiber space. If there is a closed orientable surface \(F\), not \(S^ 2\), and an immersion \(f: F\to M\) which injects \(\pi_ 1(F)\) and has the 3-plane and 1-line-intersection properties, then homotopic homeomorphisms of \(M\) are isotopic. The authors had previously shown in [Topology 31, No. 3, 493-517 (1992; Zbl 0771.57007)] that under the hypotheses of the theorem if \(M\) is homotopy equivalent to an irreducible 3-manifold \(N\), then \(M\) is homeomorphic to \(N\). Definitions of the 3-plane and 1-line-intersection properties are given in the paper.
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 490625
zbMATH Keywords: homotopic homeomorphisms
zbMATH Keywords: isotopic
zbMATH Keywords: 3-plane and 1-line-intersection properties
zbMATH Keywords: homotopy equivalent to an irreducible 3-manifold
reviewed by: Lee P. Neuwirth
MaRDI profile type: Publication
OpenAlex ID: W2013948205
description: scientific article; zbMATH DE number 490625