Contact structures, deformations and taut foliations

TL;DR

This paper explores the deformation space of taut foliations and contact structures, revealing their complex topology and non-connectedness, and classifies certain tight contact structures on Seifert fibered spaces.

Contribution

It introduces new examples demonstrating the non-path-connectedness of taut foliations and representation spaces, and classifies tight contact structures on Seifert fibered spaces.

Findings

The space of taut foliations in a homotopy class is generally not path connected.

Representation spaces of surface groups into Diff(S^1) are not path connected.

Classification of universally tight contact structures on Seifert fibered spaces.

Abstract

Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields is in general not path connected. Similar methods also show that the space of representations of the fundamental group of a hyperbolic surface to the group of smooth diffeomorphisms of the circle with fixed Euler class is in general not path connected. As an important step along the way we resolve the question of which universally tight contact structures on Seifert fibered spaces are deformations of taut or Reebless foliations when the genus of the base is positive or the twisting number of the contact structure in the sense of Giroux is non-negative.