Floer mini-max theory, the Cerf diagram, and the spectral invariants
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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 a : H ↦ ρ(H;a) can be pushed down to a continuous function defined on the universal (etale) covering space Ham(M, ω) of the group Ham(M, ω) of Hamiltonian diffeomorphisms on general (M, !). For a certain generic homotopy, which we call a Cerf homotopy H = {H^s}0≤s≤1 of Hamiltonians, the function ρ^a ․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.
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 a : H ↦ ρ(H;a) can be pushed down to a continuous function defined on the universal (etale) covering space Ham(M, ω) of the group Ham(M, ω) of Hamiltonian diffeomorphisms on general (M, !). For a certain generic homotopy, which we call a Cerf homotopy H = {H^s}0≤s≤1 of Hamiltonians, the function ρ^a ․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.
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