Thurston norm and the Euler class

TL;DR

This paper explores the geometric and topological significance of the Thurston norm and Euler class in 3-manifolds, examining various structures that produce integral points in the dual norm ball.

Contribution

It discusses how multiple structures on 3-manifolds relate to Thurston's Euler class conjecture and their implications for understanding the dual norm ball boundary.

Findings

Thurston's conjecture links Euler classes of taut foliations to boundary points of the dual norm ball.

Various structures like contact structures and flows give rise to integral points in the dual unit ball.

The paper reviews what is known about Thurston's Euler class conjecture in different contexts.

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.