Topological stability of contact structures in dimension 3
type: Item
instance of: scholarly article
title: Topological stability of contact structures in dimension 3
zbMATH Open document ID: 1034.53087
DOI: 10.1215/S0012-7094-99-09912-X
author: Vincent Colin
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: It has been known for some time that a contact structure \(\xi_1\) on a closed manifold \(V\) that is sufficiently \(C^1\)-close to a contact structure \(\xi_0\) is actually isotopic to it. This follows from the result of J. Gray that a path of contact structures \((\xi_t)_{t\in [0,1]}\) is of the form \(\xi_t = \phi_{t\ast}(\xi_0)\) for some isotopy \((\phi_t)_{t\in [0,1]}\) of \(V\) with \(\phi_0 = \text{id}_V\). In this paper, the author shows that the contact structure \(\xi_1\) is also isotopic to the contact structure \(\xi_0\) when \(V\) is a closed \(3\)-manifold and \(\xi_1\) is only \(C^0\)-close to \(\xi_0\). He also points out that the same result can not be true in dimensions higher than three. In fact, for overtwisted contact structures, the \(3\)-manifold result can be proved from Eliashberg's classification of overtwisted structures on \(3\)-manifolds. As the author notes, his result is therefore akin to a local version of this classification -- a local version that also holds for tight contact structures.
Mathematics Subject Classification ID: 53D35
Mathematics Subject Classification ID: 53D10
Mathematics Subject Classification ID: 57R17
zbMATH DE Number: 1425243
zbMATH Keywords: contact manifold
zbMATH Keywords: isotopic
zbMATH Keywords: closed 3-manifold
zbMATH Keywords: overtwisted
reviewed by: John Oprea
MaRDI profile type: Publication
cites work: Q3692516
cites work: Classification of overtwisted contact structures on 3-manifolds
cites work: Contact 3-manifolds twenty years since J. Martinet's work
cites work: Q4375270
cites work: Q4280750
cites work: Une structure de contact, même tendue, est plus ou moins tordue
cites work: Some global properties of contact structures
cites work: The classification of tight contact structures on the 3-torus
cites work: Q5618076
cites work: Asymptotic cycles
cites work: The theory of foliations of codimension greater than one
cites work: Q3709485
cites work: Q4066209
full work available at URL: s0012-7094-99-09912-x
OpenAlex ID: W2011330686
description: scientific article; zbMATH DE number 1425243