Floer Mini-Max Theory, the Cerf Diagram, and the Spectral Invariants

Abstract

Abstract. The author previously defined the spectral invariants, denoted by ρ(H; a), of a Hamiltonian function H as the mini-max value of the action functional A H over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant ρ(H; a) states that the mini-max value is a critical value of the action functional A H . The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, ω). We also prove that the spectral invariant function ρa : H → ρ(H; a) can be pushed down to a continuous function defined on the universal (étale) covering space ]Ham(M, ω) of the group Ham(M, ω) of Hamiltonian diffeomorphisms on general (M, ω). For a certain generic homotopy, which we call a Cerf homotopy H = {H s } 0≤s≤1 of Hamiltonians, the function ρa • H : s → ρ(H s ; a) is piecewise smooth away from a countable subset of [0, 1] for each non-zero quantum cohomology class a.The proof of this nondegenerate spectrality relies on several new ingredients in the chain level Floer theory, which have their own independent interest: a structure theorem on the Cerf bifurcation diagram of the critical values of the action functionals associated to a generic one-parameter family of Hamiltonian functions, a general structure theorem and the handle sliding lemma of Novikov Floer cycles over such a family and a family version of new transversality statements involving the Floer chain map, and many others. We call this chain level Floer theory as a whole the Floer mini-max theory.

The Seiberg–Witten equations and the Weinstein conjecture

“…/ in the context of Floer homology of symplectomorphisms and proves an analogous continuity theorem. The author also recently learned of Oh's article [20] which contains similar constructions.…”

“…Meanwhile, Lemma 6.2 implies that r and Ä can be chosen so that (6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) …”

Loops in the Hamiltonian group: a survey

“…These invariants are defined by looking at the filtered Floer complex 9 CF * (H, J) of the generating (normalized) Hamiltonian H, and by [47,61] turn out to be particular critical values of the corresponding action functional A H .…”

“…However the proofs for spheres (due to Polterovich [53]) and other Riemann surfaces (due to Lalonde-McDuff [25]) are very different. In fact, as noted by Ostrover [49] one can use the spectral invariants of Schwarz [57] and Oh [46,47] (see also Usher [61]) to show that the universal cover Ham of Ham always has infinite diameter with respect to the induced (pseudo)metric. Therefore the question becomes: when does this result transfer down to Ham?…”

Floer homology in symplectic geometry and in mirror symmetry

“…The axioms 1 and 7 already hold at the level of cycles or for λ H , and follow immediately from its definition. All other axioms are proved in [Oh8] except the homotopy invariance for the irrational symplectic manifolds which is proven in [Oh10]. The additive triangle inequality was explicitly used by Entov and Polterovich in their construction of some quasimorphisms on Ham(M, ω) [EnP].…”