Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds

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 $0\ne a\in Q{H}^{\ast }(M)$, 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 ${\tilde{\Omega }}_0(M)$ 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.

arXiv comment43 pages, In this version, we fill a gap in the proof of spectrality axiom in the previous version and provide a complete proof of the spectraity axiom for the rational symplectic manifolds. A separate paper (math.SG/0406449) deals with the spectrality axiom for the irrational cases. To appear in the volume in honor of Alan Weinstein's 60th Birthday

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• Quasi-states, quasi-morphisms, and the moment map
• Quasi-morphisms and symplectic quasi-states for convex symplectic manifolds
• Descent and C^0-rigidity of spectral invariants on monotone symplectic manifolds
• The displaced disks problem via symplectic topology
• New energy-capacity-type inequalities and uniqueness of continuous Hamiltonians
• Quantitative Heegaard Floer cohomology and the Calabi invariant
• Unboundedness of the Lagrangian Hofer distance in the Euclidean ball