#### Volume 20, issue 5 (2016)

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On the uniqueness of the contact structure approximating a foliation

### Thomas Vogel

Geometry & Topology 20 (2016) 2439–2573
##### Abstract

According to a theorem of Eliashberg and Thurston, a ${C}^{2}$–foliation on a closed $3$–manifold can be ${C}^{0}$–approximated by contact structures unless all leaves of the foliation are spheres. Examples on the $3$–torus show that every neighbourhood of a foliation can contain nondiffeomorphic contact structures.

In this paper we show uniqueness up to isotopy of the contact structure in a small neighbourhood of the foliation when the foliation has no torus leaf and is not a foliation without holonomy on parabolic torus bundles over the circle. This allows us to associate invariants from contact topology to foliations. As an application we show that the space of taut foliations in a given homotopy class of plane fields is not connected in general.

##### Keywords
foliations, contact structures
##### Mathematical Subject Classification 2010
Primary: 53D10, 57R30, 57R17
##### Publication
Revised: 26 August 2015
Accepted: 28 September 2015
Published: 7 October 2016
Proposed: Yasha Eliashberg
Seconded: Ian Agol, Leonid Polterovich
##### Authors
 Thomas Vogel Mathematisches Institut der LMU Universität München Theresienstr. 39 D-80333 München Germany