Periodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms

Abstract

Spectral invariants arising from twisted periodic Floer homology have recently played a key role in resolving various open problems in two-dimensional dynamics. We resolve in great generality a conjecture of Hutchings regarding the relationship between the asymptotics of the twisted PFH spectral invariants and the Calabi homomorphism for area-preserving disk maps. Our result, by an argument similar to that of Irie in the setting of Reeb flows, allows us to prove the smooth closing lemma for area-preserving diffeomorphisms of a closed surface. DAN CRISTOFARO-GARDINER, ROHIL PRASAD, AND BOYU ZHANG 5. The relationship between twisted SWF and twisted PFH 72 5.1. Lee-Taubes' isomorphism in the untwisted setting 72 5.2. Lee-Taubes' isomorphism in the twisted setting 73 5.3. The twisted isomorphism and the relative Z-grading 76 5.4. Twisted SWF spectral invariants recover twisted PFH spectral invariants 85 6. Properties of twisted PFH spectral invariants 89 6.1. PFH cobordism maps 89 6.2. The pseudoholomorphic curve axiom 91 6.3. Spectrality 93 6.4. Hofer continuity 94 7. Exact triangles, vanishing and non-vanishing 98 8. Seiberg-Witten-Floer max-min families 8.1. Estimates on continuation instantons 8.2. Max-min for r ą ´2πρ 8.3. Max-min for r " ´2πρ 8.4. Differential equations 8.5. Recovering SWF spectral invariants 9. Proof of Hutchings' conjecture 10. Proof of the closing lemma References

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

“…Proof. We compute H ˚pT; Γ R ξ 1 q following the argument of [3, Page 688] (see also [1,Theorem 7.9]). Recall that 2N is the divisibility of c 1 psq and ξ 1 is defined by (3).…”

“…They proved a Weyl asymptotic formula for the PFH spectral invariants for rational Hamiltonian isotopy classes satisfying the U-cycle property, and used it to prove the closing lemma for such Hamiltonian isotopy classes. A Weyl law for the PFH spectral invariants 1 without any U-cyclic condition was proved independently in [1] for all rational isotopy classes of area-preserving maps, and the closing lemma for general area-preserving diffeomorphisms was proved. The main result of this note is the following proposition: Proposition 1.…”

“…The map φ is called monotone in[1] if and only if its Hamiltonian isotopy class is rational in the sense of[2].…”

Strong closing property of contact forms and action selecting functors

“…[1] proved, as an application of [13], a C ∞ closing lemma for Hamiltonian diffeomorphisms of closed symplectic surfaces. Recently, [5] proved an analogue of the volume theorem for spectral invariants defined from periodic Floer homology (PFH), and deduced a C ∞ closing lemma for area-preserving diffeomorphisms of closed symplectic surfaces. [6] independently proved an analogue of the volume theorem for PFH spectral invariants and C ∞ closing lemmas for area-preserving diffeomorphisms under certain conditions (rationality and U-cycle property of Hamiltonian isotopy classes).…”

An elementary alternative to PFH spectral invariants

“…Following the philosophy of [21] and [16], we define an elementary replacement for spectral invariants coming from periodic Floer homology (PFH), whose construction is highly non-trivial and in particular relies on gauge theory. PFH spectral invariants have recently found remarkable applications to C 0 symplectic geometry, Hofer geometry and symplectic dynamics in dimension two (see [5], [6], [8], [9]). We provide elementary proofs of some of these applications and significantly reduce the amount of sophisticated machinery used in the background.…”