Thurston norm and the Euler class

Abstract

In his influential work, Thurston introduced a norm on the second homology group of compact orientable 3-manifolds M, which by duality also determines a dual norm on the second cohomology group. A natural question, initiated by Thurston, is whether integral points on the boundary of the dual norm ball have a geometric interpretation. Thurston showed that the Euler class of the oriented tangent plane field to any taut foliation of M lies in the dual unit ball, and conjectured that, conversely, any integral point on the boundary of the dual unit ball is realised as the Euler class of a taut foliation. In this chapter, we discuss how several geometric, topological, and dynamical structures on a 3-manifold give rise to integral points in the dual unit ball of the Thurston norm, and what is known about Thurston's Euler class one conjecture in these contexts. These structures are taut foliations, tight contact structures, pseudo-Anosov flows, quasigeodesic flows, and circular orders on the fundamental group.