Classification of overtwisted contact structures on 3-manifolds
type: Item
instance of: scholarly article
title: Classification of overtwisted contact structures on 3-manifolds
zbMATH Open document ID: 0684.57012
DOI: 10.1007/BF01393840
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143746
review text: Consider in \({\mathbb{R}}^ 3\) the cylindric coordinates (\(\rho\),\(\phi\),z). The disc \(\Delta =\{(\rho,\phi,z):\) \(z=0\), \(\rho\leq \pi \}\) with the germ of the contact structure \(\zeta_ 1\) defined on \(\Delta\) by the equation cos \(\rho\) dz\(+\rho \sin \rho d\phi =0\) is called the standard overtwisted disc. Let M be an oriented connected 3-manifold. A contact structure \(\zeta\) on M is called overtwisted if there is a contact embedding of the standard overtwisted disc \((\Delta,\zeta_ 1)\) into (M,\(\zeta)\). Fix a point \(p\in M\) and an embedded 2-disc \(\Delta '\subset M\) centred at p. Let Distr(M) be the space of all 2-dimensional distributions on M fixed at p, equipped with the \(C^{\infty}\)-topology. Denote by Cont(M) the subspace of Distr(M) which consists of positive contact structures and by \(Cont^{ot}(M)\) the subspace of Cont(M) containing all overtwisted structures which have the disc \(\Delta '\subset M\) as the standard overtwisted disc. The author proves that the inclusion j: \(Cont^{ot}(M)\to Distr(M)\) is a homotopy equivalence.
Mathematics Subject Classification ID: 57R15
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 4121041
zbMATH Keywords: overtwisted contact structure
zbMATH Keywords: oriented connected 3-manifold
zbMATH Keywords: overtwisted disc
author: Yakov Eliashberg
MaRDI profile type: Publication
cites work: Q3692516
cites work: Q3357032
cites work: Q3184975
cites work: Some global properties of contact structures
cites work: Pseudo holomorphic curves in symplectic manifolds
cites work: Structures de contact sur les fibrés principaux en cercles de dimension trois
description: scientific article; zbMATH DE number 4121041