tailieunhanh - Đề tài " Lagrangian intersections and the Serre spectral sequence "

For a transversal pair of closed Lagrangian submanifolds L, L of a symplectic manifold M such that π1 (L) = π1 (L ) = 0 = c1 |π2 (M ) = ω|π2 (M ) and for a generic almost complex structure J, we construct an invariant with a high homotopical content which consists in the pages of order ≥ 2 of a spectral sequence whose differentials provide an algebraic measure of the highdimensional moduli spaces of pseudo-holomorpic strips of finite energy that join L and L . When L and L are Hamiltonian isotopic, we show that the pages. | Annals of Mathematics Lagrangian intersections and the Serre spectral sequence By Jean-Franc ois Barraud and Octav Cornea Annals of Mathematics 166 2007 657 722 Lagrangian intersections and the Serre spectral sequence By Jean-Francois Barraud and OcTAV Cornea Abstract For a transversal pair of closed Lagrangian submanifolds L L1 of a sym-plectic manifold M such that n1 L 1 L 0 Cl n2 M u n2 M and for a generic almost complex structure J we construct an invariant with a high homotopical content which consists in the pages of order 2 of a spectral sequence whose differentials provide an algebraic measure of the highdimensional moduli spaces of pseudo-holomorpic strips of finite energy that join L and L1. When L and L1 are Hamiltonian isotopic we show that the pages of the spectral sequence coincide up to a horizontal translation with the terms of the Serre spectral sequence of the path-loop fibration QL PL - L and we deduce some applications. Contents 1. Introduction . The main result . Comments on the main result . Some applications . The structure of the paper Acknowledgements 2. The spectral sequence . Recalls and notation . Construction of the spectral sequence . Proof of the main theorem I Invariance of the spectral sequence . Proof of the main theorem II Relation to the Serre spectral sequence 3. Applications . Global abundance of pseudo-holomorphic strips loop space homology . Local pervasiveness of pseudo-holomorphic strips . Nonsqueezing . Relaxing the connectivity conditions Partially supported by an NSERC Discovery grant and by a FQRNT group research grant. 658 JEAN-FRANCOIS BARRAUD AND OCTAV CORNEA Appendix A. Structure of manifolds with corners on Floer moduli spaces . Introduction . Sketch of the construction . Pre-gluing . Holomorphic perturbations of wp . Hamiltonian perturbations References 1. Introduction Consider a symplectic manifold M w which is convex at infinity together with two closed compact

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