  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.
