On the Euler class one conjecture for fillable contact structures

TL;DR

This paper demonstrates that for hyperbolic 3-manifolds, certain boundary classes on the Thurston norm ball cannot be realized as Euler classes of fillable contact structures, providing counterexamples to Thurston's conjecture.

Contribution

It proves the existence of finite covers of hyperbolic 3-manifolds with boundary classes not realizable by fillable contact structures, challenging previous conjectures.

Findings

Counterexamples to Thurston's Euler class one conjecture

Existence of boundary classes not realizable as fillable contact Euler classes

Finite covers with specific non-realizable boundary classes

Abstract

In this paper, it is proved that every oriented closed hyperbolic $3$--manifold $N$ admits some finite cover $M$ with the following property. There exists some even lattice point $w$ on the boundary of the dual Thurston norm unit ball of $M$, such that $w$ is not the real Euler class of any weakly symplectically fillable contact structure on $M$. In particular, $w$ is not the real Euler class of any transversely oriented, taut foliation on $M$. This supplies new counter-examples to Thurston's Euler class one conjecture.