Pseudo-Anosov representatives of stable Hamiltonian structures

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2025年9月8日

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Jonathan Zung

摘要

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 $S^3$ $S^{3}$ for a slope p/q satisfying $q\ge 6$ $q \geq 6$ , $p\ne 0, \pm 1, \pm 2 {{\,\textrm{mod}\,}}q$ $p \neq 0, \pm 1, \pm 2 mod q$ . The proof of the main theorem generalizes an argument of Barthelmé–Bowden–Mann.