Pseudo-Anosov representatives of stable Hamiltonian structures

TL;DR

This paper establishes that transitive pseudo-Anosov flows serve as canonical representatives for stable Hamiltonian structures, leading to finiteness results for certain flows and topological classifications of 3-manifolds.

Contribution

It introduces the concept that pseudo-Anosov flows are canonical for stable Hamiltonian classes and derives finiteness and classification results for 3-manifolds.

Findings

Finitely many pseudo-Anosov flows admit positive Birkhoff sections on rational homology 3-spheres.

Any 3-manifold can be obtained in finitely many ways via certain surgeries on fibered hyperbolic knots.

The proof generalizes an existing argument by Barthelmé–Bowden–Mann.

Abstract

A pseudo-Anosov homeomorphism of a surface is a canonical representative of its mapping class. In this paper, 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 $q\geq 6$, $p\neq 0, \pm 1, \pm 2 \mod q$. The proof of the main theorem generalizes an argument of Barthelm\'e--Bowden--Mann.