Uniqueness of the contact structure approximating a foliation

TL;DR

This paper proves the local uniqueness of contact structures approximating certain foliations on 3-manifolds, enabling the transfer of contact topology invariants to foliations and revealing non-connectedness in the space of taut foliations.

Contribution

It establishes the uniqueness up to isotopy of contact structures near specific foliations, expanding the understanding of contact-foliation relationships and their invariants.

Findings

Uniqueness of contact structures near foliations without torus leaves.

Invariants from contact topology can be associated to foliations.

The space of taut foliations is generally not connected.

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 non-diffeomorphic 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.