The fully marked surface theorem

TL;DR

This paper proves a converse to Thurston's surface Euler characteristic theorem for taut foliations on hyperbolic 3-manifolds, showing the existence of foliations with prescribed Euler class properties, and provides counterexamples to Thurston's conjecture.

Contribution

It establishes a converse to Thurston's theorem for taut foliations and constructs counterexamples to Thurston's conjecture on Euler classes.

Findings

A converse to Thurston's surface theorem for taut foliations is proven.

Counterexamples to Thurston's conjecture are constructed.

Taut foliations with specific Euler class properties are shown to exist.

Abstract

In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation $\mathcal{F}$ evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation $\mathcal{F'}$ such that $S$ is homologous to a union of leaves and such that the plane field of $\mathcal{F'}$ is homotopic to that of $\mathcal{F}$. In particular, $\mathcal{F}$ and $\mathcal{F'}$ have the same Euler class. In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston's conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper.