A note on the existence of U-cyclic elements in periodic Floer homology

TL;DR

This paper investigates the U-cycle property in periodic Floer homology and related Seiberg-Witten-Floer cohomology, establishing conditions under which U-cyclic elements exist or do not exist in these theories.

Contribution

It proves the U-cycle property for rational Hamiltonian isotopy classes on closed surfaces and extends results to non-torsion spin-c structures, providing new insights into the U-module structure.

Findings

U-cycle property holds for rational isotopy classes on closed surfaces.

Some rational isotopy classes contain elements that are not U-cyclic.

Results are among the first to compute the U-module structure in these theories.

Abstract

Edtmair-Hutchings have recently defined, using periodic Floer homology, a U-cycle property for Hamiltonian isotopy classes of area-preserving diffeomorphisms of closed surfaces. They show that every Hamiltonian isotopy class satisfying the U-cycle property satisfies the smooth closing lemma and also satisfies a kind of Weyl law involving the actions of certain periodic points; they show that every rational isotopy class on the two-torus satisfies the U-cycle property. It seems that in general, not much is known about the U-module structure on PFH. Here we consider a version of Seiberg-Witten-Floer cohomology which is known by the work of Lee-Taubes to be isomorphic, as a U-module, to the periodic Floer homology in sufficiently high degree. We show that the analogous U-cycle property holds for every rational Hamiltonian isotopy class on any closed surface and, more generally, for any non-torsion spin-c structure. On the other hand, we also show that a rational isotopy class may contain elements that are not U-cyclic. By the Lee-Taubes isomorphism, the same results hold for PFH. Our results are some of the first computations concerning the U-module structure on these theories.