On the uniqueness of the contact structure approximating a foliation

Vogel, Thomas

doi:10.2140/gt.2016.20.2439

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On the uniqueness of the contact structure approximating a foliation

Thomas Vogel

Geometry & Topology 20 (2016) 2439–2573

DOI: 10.2140/gt.2016.20.2439

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

References

Publication

Received: 31 July 2013

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

Volume 20, issue 5 (2016)