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 Euler class with norm equal to one is Euler class of a taut foliation. We construct the first counterexamples to this conjecture, infinitely many indeed. The counterexamples are constructed by Dehn surgeries on certain fibered hyperbolic 3-manifolds. Moreover, they are constructive in the sense that the monodromy of the fibration map is given in terms of Dehn twists and the surgery coefficient is specified. We also suggest an alternative conjecture in terms of faithful representations of the fundamental group of the 3-manifold into certain group of homeomorphisms.