A criterion for virtual Euler class one

TL;DR

This paper establishes a criterion linking rational cohomology classes and Euler classes of taut foliations in hyperbolic 3-manifolds, supporting Yazdi's conjecture through explicit examples and conditions involving Alexander polynomials.

Contribution

It introduces a new criterion connecting cohomology classes with Euler classes of taut foliations, advancing understanding of virtual Euler class one in hyperbolic 3-manifolds.

Findings

Constructed examples with first Betti number 2 or 3.

Provided partial examples with any first Betti number ≥ 4.

Linked Alexander polynomials to the existence of taut foliations with specific Euler classes.

Abstract

Let $M$ be an oriented closed hyperbolic $3$--manifold. Suppose that $w$ is a rational second cohomology class of $M$ with dual Thurston norm $1$. Upon the existence of certain nonvanishing Alexander polynomials, the author shows that the pullback of $w$ to some finite cover of $M$ is the real Euler class of some transversely oriented taut foliation on that cover. As application, the author constructs examples with first Betti number either $2$ or $3$, and partial examples with any first Betti number at least $4$, supporting Yazdi's virtual Euler class one conjecture.