UNIQUENESS OF THE CONTACT STRUCTURE APPROXIMATING A FOLIATION

ABSTRACT. 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 non-diffeomorphic contact structures. In this paper we show uniqueness up to isotopy of the contact struc-ture in a small neighbourhood of the foliation when the foliation has no torus leaf and is not a foliation without holonomy on parabolic torus bun-dles 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 i