Thurston norm and Euler classes of tight contact structures

TL;DR

This paper investigates the relationship between Euler classes and tight contact structures in hyperbolic 3-manifolds, showing that counterexamples for taut foliations also apply to tight contact structures, supporting a conjecture about Euler classes.

Contribution

It demonstrates that counterexamples to Thurston's Euler class conjecture for taut foliations also serve as counterexamples for tight contact structures, providing evidence for the conjecture in that context.

Findings

Counterexamples for taut foliations are realized as tight contact structures.

Supports the Euler class one conjecture for tight contact structures.

Links properties of taut foliations and tight contact structures.

Abstract

Bill Thurston proved that taut foliations of hyperbolic 3-manifolds have Euler classes of norm at most one, and conjectured that any integral second cohomology class of norm equal to one is realised as the Euler class of some taut foliation. Recent work of the second author, joint with David Gabai, has produced counterexamples to this conjecture. Since tight contact structures exist whenever taut foliations do and their Euler classes also have norm at most one, it is natural to ask whether the Euler class one conjecture might still be true for tight contact structures. In this short note, we show that the previously constructed counterexamples for Euler classes of taut foliations in [Yaz20] are in fact realised as Euler classes of tight contact structures. This provides some evidence for the Euler class one conjecture for tight contact structures.