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On the uniqueness of the contact structure approximating a foliation

Thomas Vogel

Geometry & Topology 20 (2016) 2439–2573

According to a theorem of Eliashberg and Thurston, a C2–foliation on a closed 3–manifold can be C0–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.

foliations, contact structures
Mathematical Subject Classification 2010
Primary: 53D10, 57R30, 57R17
Received: 31 July 2013
Revised: 26 August 2015
Accepted: 28 September 2015
Published: 7 October 2016
Proposed: Yasha Eliashberg
Seconded: Ian Agol, Leonid Polterovich
Thomas Vogel
Mathematisches Institut der LMU
Universität München
Theresienstr. 39
D-80333 München