Pseudo-Anosov representatives of stable Hamiltonian structures

Abstract

A pseudo-Anosov homeomorphism of a surface is a canonical representative of its mapping class. Conditional on the foundations of symplectic field theory, we explain that a transitive pseudo-Anosov flow is similarly a canonical representative of its stable Hamiltonian class. It follows that there are finitely many pseudo-Anosov flows admitting positive Birkhoff sections on any given rational homology 3-sphere. This result has a purely topological consequence: any 3-manifold can be obtained in at most finitely many ways as p/q surgery on a fibered hyperbolic knot in S 3 for a slope p/q satisfying q ≥ 6 , p ≠ 0 , ± 1 , ± 2 mod q . The proof of the main theorem generalizes an argument of Barthelmé–Bowden–Mann.