On Thurston's Euler class one conjecture

TL;DR

This paper constructs counterexamples to Thurston's conjecture on the Euler class of taut foliations on hyperbolic 3-manifolds, challenging a long-standing assumption in 3-manifold topology.

Contribution

It provides the first counterexamples to Thurston's conjecture, conditional on the fully marked surface theorem, advancing understanding of taut foliations and their Euler classes.

Findings

Counterexamples to Thurston's conjecture are constructed.

The results are conditional on the fully marked surface theorem.

This work challenges previous beliefs about Euler classes of taut foliations.

Abstract

In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that conversely, any integral second cohomology class with norm equal to one is the Euler class of a taut foliation. This is the first from a series of two papers that together give a negative answer to Thurston's conjecture. Here counterexamples have been constructed conditional on the fully marked surface theorem. In the second paper, joint with David Gabai, a proof of the fully marked surface theorem is given.