2004/05/31 by Yong-Geun Oh · 2 citations
Mathematics · #math.SG #msc:53D35 #msc:53D40
published as The breadth of symplectic and Poisson geometry, 525--570, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005. · 43 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
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,ω). To each given time dependent Hamiltonian function H and quantum cohomology class 0 ≠ a ∈ QH^*(M), we associate an invariant ρ(H;a) which varies continuously over H in the C0-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 Ω0(M) of the contractible loop space Ω0(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.