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

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 $\AA_H$ 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 $\AA_H$. The main purpose of the present paper is to prove this axiom for {\it nondegenerate} Hamiltonian functions in {\it irrational} symplectic manifolds $(M,\omega)$. We also prove that the spectral invariant function $\rho_a: H \mapsto \rho(H;a)$ can be pushed down to a {\it continuous} function defined on the universal ({\it \'etale}) covering space $\widetilde{Ham}(M,\omega)$ of the group $Ham(M,\omega)$ of Hamiltonian diffeomorphisms on general $(M,\omega)$. For a certain generic homotopy, which we call a {\it Cerf homotopy} $\HH = \{H^s\}_{0 \leq s\leq 1}$ of Hamiltonians, the function $\rho_a \circ \HH: s \mapsto \rho(H^s;a)$ 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 {\it 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 {\it Floer mini-max theory}.