[Paper Review] Floer mini-max theory, the Cerf diagram, and the spectral invariants

Yong‐Geun Oh|arXiv (Cornell University)|Jun 23, 2004

Geometric and Algebraic Topology11 references4 citations

TL;DR

This paper establishes the spectrality axiom for spectral invariants in Floer homology on irrational symplectic manifolds by developing a new chain-level Floer mini-max theory. It proves that spectral invariants $ρ(H;a)$ are critical values of the action functional for nondegenerate Hamiltonians, using a Cerf homotopy framework, handle sliding lemmas, and transversality techniques, and shows that the spectral invariant function descends continuously to the universal cover of the Hamiltonian group, $χ\tilde{Ham}(M,\omega)$.

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

Motivation & Objective

• To prove the spectrality axiom, stating that spectral invariants $\rho(H;a)$ are critical values of the action functional $\mathcal{A}_H$, for nondegenerate Hamiltonians on irrational symplectic manifolds.
• To extend the spectral invariant function $\rho_a: H \mapsto \rho(H;a)$ to a continuous function on the universal covering space $\widetilde{Ham}(M,\omega)$ of the Hamiltonian group.
• To establish the piecewise smoothness of $\rho_a \circ \mathcal{H}$ along generic Cerf homotopies $\mathcal{H} = \{H^s\}_{0 \leq s \leq 1}$, away from a countable set of parameters.
• To develop a new chain-level Floer theory—termed 'Floer mini-max theory'—with novel transversality and bifurcation structure theorems for Novikov Floer cycles under one-parameter families of Hamiltonians.
• To prove a handle sliding lemma for Novikov Floer cycles under Cerf families, enabling control over cycle behavior during bifurcations.

Proposed method

• Constructs a structure theorem for the Cerf bifurcation diagram of critical values of $\mathcal{A}_H$ under generic one-parameter families of Hamiltonians.
• Introduces a general structure theorem and handle sliding lemma for Novikov Floer cycles over such families, ensuring topological control during bifurcations.
• Applies a family version of transversality statements involving the Floer chain map, ensuring regularity and stability of the chain complex under perturbations.
• Uses a parametric stability argument for 'tightness' of Novikov Floer cycles to control convergence and bubbling in pseudo-holomorphic curves.
• Employs a contradiction argument with energy decay and compactness to prove that non-canonical Floer trajectories must be homotopic to canonical cylinders.
• Applies a $C^2$-perturbation argument to show that Floer trajectories with small energy and bounded energy difference must converge to stationary solutions, leading to contradiction if not canonical.

Experimental results

Research questions

• RQ1 Is the spectral invariant $\rho(H;a)$ a critical value of the action functional $\mathcal{A}_H$ for nondegenerate Hamiltonians on irrational symplectic manifolds?
• RQ2 Can the spectral invariant function $\rho_a$ be continuously extended to the universal covering space $\widetilde{Ham}(M,\omega)$ of the Hamiltonian group?
• RQ3 How does the spectral invariant $\rho(H^s;a)$ behave along a generic one-parameter family of Hamiltonians (Cerf homotopy)?
• RQ4 What is the structure of Novikov Floer cycles under a generic one-parameter family of Hamiltonians, and how do they undergo bifurcations?
• RQ5 Can a handle sliding lemma be established for Novikov Floer cycles in the context of Cerf families, and what does it imply for the chain-level Floer theory?

Key findings

• The spectrality axiom holds for nondegenerate Hamiltonians on irrational symplectic manifolds: $\rho(H;a)$ is a critical value of $\mathcal{A}_H$.
• The spectral invariant function $\rho_a$ descends to a well-defined continuous function on the universal covering space $\widetilde{Ham}(M,\omega)$.
• For any non-zero quantum cohomology class $a$, the function $s \mapsto \rho(H^s;a)$ is piecewise smooth on $[0,1]$ away from a countable subset of parameters.
• The Cerf bifurcation diagram of critical values of $\mathcal{A}_H$ under a generic one-parameter family exhibits a well-structured, finite-type singular set.
• A handle sliding lemma for Novikov Floer cycles is established, showing that non-canonical Floer trajectories must be homotopic to canonical cylinders under small energy and perturbation conditions.
• Energy decay and compactness arguments imply that any sequence of Floer trajectories with energy tending to zero and bounded energy difference must converge to a stationary solution, leading to contradiction unless the trajectory is canonical.