A note on 𝑈-cyclic elements in monopole Floer homology

Edtmair and Hutchings have recently defined, using periodic Floer homology (PFH), a “ U U -cycle property” for Hamiltonian isotopy classes of area-preserving diffeomorphisms of closed surfaces. They show that every Hamiltonian isotopy class satisfying the U 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 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 work of Lee and Taubes to be isomorphic, as a U-module, to the periodic Floer homology in sufficiently high degree. We show that the analogous U U -cycle property holds for every rational Hamiltonian isotopy class on any closed surface and, more generally, for any nontorsion spin-c structure. On the other hand, we also show that a rational isotopy class may contain elements that are not U 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.