FLOER MINI-MAX THEORY, THE CERF DIAGRAM, AND THE SPECTRAL INVARIANTS

Abstract

The author previously defined the spectral invariants, denoted by <TEX>$\rho(H;\;a)$</TEX>, of a Hamiltonian function H as the mini-max value of the action functional <TEX>${\cal{A}}_H$</TEX> over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant <TEX>$\rho(H;\;a)$</TEX> states that the mini-max value is a critical value of the action functional <TEX>${\cal{A}}_H$</TEX>. The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, <TEX>$\omega$</TEX>). We also prove that the spectral invariant function <TEX>${\rho}_a$</TEX> : <TEX>$H\;{\mapsto}\;\rho(H;\;a)$</TEX> can be pushed down to a continuous function defined on the universal (<TEX>${\acute{e}}tale$</TEX>) covering space <TEX>$\widetilde{HAM}$</TEX>(M, <TEX>$\omega$</TEX>) of the group Ham((M, <TEX>$\omega$</TEX>) of Hamiltonian diffeomorphisms on general (M, <TEX>$\omega$</TEX>). For a certain generic homotopy, which we call a Cerf homotopy <TEX>${\cal{H}}\;=\;\{H^s\}_{0{\leq}s{\leq}1}$</TEX> of Hamiltonians, the function <TEX>${\rho}_a\;{\circ}\;{\cal{H}}$</TEX> : <TEX>$s\;{\mapsto}\;{\rho}(H^s;\;a)$</TEX> 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.