Floer mini-max theory, the Cerf diagram, and the spectral invariants

Abstract

The author previously defined the spectral invariants, denoted by rho(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 rho(H; a) states that the mini-max value is a critical value of the action functional AH. The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, omega). We also prove that the spectral invariant function rho(a) : H (sic) rho(H; a) can be pushed down to a continuous function defined on the universal (etale) covering space (sic)(M,omega) of the group Ham(M,omega) of Hamiltonian diffeomorphisms on general (M, omega). For a certain generic homotopy, which we call a Cerf homotopy H = {H(s)}(0 <= s <= 1) of Hamiltonians, the function rho(a) circle H : s (sic) rho(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.