On Thurston's Euler class one conjecture

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.

arXiv comment42 pages, 21 figures. The paper is split into two parts, and the appendix is appearing as a separate article joint with David Gabai. The results on taut foliations on sutured solid tori are generalised. A section on relative Euler class is added to address a possible oversight in the literature. Exposition is improved, and new open questions are raised. Final version to appear in Acta Mathematica