A NOTE ON THE EXISTENCE OF U-CYCLIC ELEMENTS IN PERIODIC FLOER HOMOLOGY

The paper proves that every rational Hamiltonian isotopy class on a surface satisfies the U-cycle property by: (i) producing, for any 3-manifold Y and non-torsion spin-c structure s with divisibility 2N of c1(s), a nonzero class σ with U^Nσ=σ in HM*(Y,s,c_b;R) (Proposition 2); (ii) transporting this via the Lee–Taubes isomorphism (U-compatible) to PFH for mapping tori of sufficiently large degree; and (iii) using ⟨c1(s_Γ),[Σ]⟩=2(d−g+1) so N divides d−g+1, hence U^{d−g+1} fixes a nonzero PFH class, yielding the U-cycle property for rational classes (Proposition 1). These steps and even the notational choices mirror the candidate’s solution outline, including the use of balanced vs monotone perturbations and the large-degree identification that preserves U (see the statement of Proposition 1 and its setup, the PFH–HM identifications preserving U, and the final deduction of U^{d−g+1}-periodicity in PFH: ; the existence of U^N-fixed classes in HM is Proposition 2 and its proof via coupled Morse homology: ).