On the uniqueness of the contact structure approximating a foliation

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.