An infinity of tight contact structures on an infinity of manifolds
type: Item
instance of: scholarly article
title: An infinity of tight contact structures on an infinity of manifolds
zbMATH Open document ID: 0969.53044
author: Emmanuel Giroux
published in: Inventiones Mathematicae
publication date: 2001-09-24T00:00:00Z
review text: It is well known that, in dimension 3, contact structures are classified in two classes: tight contact structures and overtwisted contact structures. A contact structure \(\xi\) is given as the kernel of a global 1-form \(\alpha\) with \(\alpha\wedge d\alpha\not=0\). We will say that a contact structure is tight if there is no embedded disk \(D\) which is tangent to \(\xi\) everywhere along the points of \(\delta D\) and it is said overtwisted otherwise. \textit{Y. Eliashberg} [Ann. Inst. Fourier 42, No. 1-2, 165-192 (1992; Zbl 0756.53017)] classified the tight contact structures on the 3-ball \(D^3\), the 3-sphere \(S^3\), \(S^2\times S^1\) and \(\mathbb R^3\). In this paper, the author proves the following interesting result: there exists an infinity of tight structures on \(T^2\)-bundles over \(S^1\).
Mathematics Subject Classification ID: 53D10
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1337387
zbMATH Keywords: tight contact structure
zbMATH Keywords: \(T^2\)-bundles over \(S^1\)
reviewed by: David Martín de Diego
MaRDI profile type: Publication
full work available at URL: s002220050301
OpenAlex ID: W2032415882
DOI: 10.1007/S002220050301
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description: scientific article; zbMATH DE number 1337387