Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds

TL;DR

This paper develops a mini-max theory for spectral invariants of Hamiltonian diffeomorphisms on closed symplectic manifolds, including non-exact and non-rational cases, using chain level Floer homology.

Contribution

It introduces a new construction of spectral invariants via mini-max values over Novikov Floer cycles on arbitrary compact symplectic manifolds.

Findings

Constructed spectral invariants for non-exact, non-rational symplectic manifolds.

Defined invariants that vary continuously in the $C^0$-topology.

Applied invariants to study Hamiltonian diffeomorphisms.

Abstract

In this paper, we develop a mini-max theory of the action functional over the semi-infinite cycles via the chain level Floer homology theory and construct spectral invariants of Hamiltonian diffeomorphisms on arbitrary, especially on {\it non-exact and non-rational}, compact symplectic manifold $(M,\omega)$. To each given time dependent Hamiltonian function $H$ and quantum cohomology class $ 0 \neq a \in QH^*(M)$, we associate an invariant $\rho(H;a)$ which varies continuously over $H$ in the $C^0$-topology. This is obtained as the mini-max value over the semi-infinite cycles whose homology class is `dual' to the given quantum cohomology class $a$ on the covering space $\widetilde \Omega_0(M)$ of the contractible loop space $\Omega_0(M)$. We call them the {\it Novikov Floer cycles}. We apply the spectral invariants to the study of Hamiltonian diffeomorphisms in sequels of this paper.