2002/06/10 by Yong-Geun Oh, Yong‐Geun Oh, Oh, Yong-Geun
Mathematics · #53D35 #53D40 #53D45 #70H05 #70H25 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Symplectic Geometry (math.SG) #math.QA #math.SG #msc:53D35 #msc:53D40 #msc:53D45 #msc:70H05 #msc:70H25
paper · pdf · doi:10.48550/arxiv.math/0206092
75 pages, the section 8 in the previous version is taken out and incoporated into a stronger existence theorem proven in a separate paper math.SG/0207214
openalex publication_date 2002/06/10 · arxiv created 2002/07/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we first 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 compact symplectic manifold (M,omega). To each given time dependent Hamiltonian function H and quantum cohomology class 0 not equal a element of QH^*(M), we associate an invariant rho(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 Omega0(M) of the contractible loop space Omega0(M). We call them the Novikov cycles. We then use the spectral invariants to construct a new invariant norm on the Hamiltonian diffeomorphism group and a partial order on the set of time-dependent Hamiltonian functions of arbitrary compact symplectic manifolds. As some applications, we obtain a new lower bound of the Hofer norm of non-degenerate Hamiltonian diffeomorphisms in terms of the area of certain pseudo-holomorphic curves and prove the semi-global C1-flatness of the Hofer norm.