Mathematics – Symplectic Geometry
Scientific paper
2004-05-04
The breadth of symplectic and Poisson geometry, 525--570, Progr. Math., 232, Birkh\"auser Boston, Boston, MA, 2005.
Mathematics
Symplectic Geometry
43 pages, In this version, we fill a gap in the proof of spectrality axiom in the previous version and provide a complete proo
Scientific paper
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.
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