La suite spectrale de Leray-Serre en homologie de Floer des varietes symplectiques compactes a bord de type contact

Abstract : Unlike Floer homology groups for closed manifolds, the Floer homology groups for compact manifolds with contact type boundary have no topological correspondent. The aim of this thesis is to describe their qualitative properties when the manifold is endowed with supplementary topological structure. More specifically, we consider symplectic fibrations (including trivial ones). The first chapter is divided into two parts: the first one compares the different constructions of Floer homology and underlines its specificity for manifolds with boundary, that is the need to obtain a priori estimates on the solutions of Floer's equation. We explain the relationship between Floer homology groups and Weinstein's conjecture and we compute (using a new method) the Floer homology of a ball in a complex vector space. The second part presents an extension of the definition of Floer homology by using asymptotically linear" Hamiltonians. This extension will be used later on. We choose the framework of non-compact manifolds which are convex at infinity, that is symplectic completions of compact manifolds with contact type boundary. The second chapter proves the Künneth formula for a product of manifolds with restricted contact type boundary. This corresponds to a trivial symplectic fibration. The third chapter gives a complete description of the classical Leray-Serre spectral sequence in exclusive Morse homological terms, a simple model for Floer homology. The fourth chapter studies the existence of a spectral sequence of Leray-Serre type for a certain kind of symplectic fibrations over a closed symplectic base. The existence of the spectral sequence is proved for hermitian line bundles of negative curvature. In the general case, its existence is reduced to an energy estimate that we conjecture to be true.
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https://tel.archives-ouvertes.fr/tel-00005504
Contributor : Alexandru Oancea <>
Submitted on : Thursday, April 1, 2004 - 8:30:30 PM
Last modification on : Wednesday, March 27, 2019 - 4:08:30 PM
Long-term archiving on : Wednesday, September 12, 2012 - 2:25:18 PM

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Alexandru Oancea. La suite spectrale de Leray-Serre en homologie de Floer des varietes symplectiques compactes a bord de type contact. Mathématiques [math]. Université Paris Sud - Paris XI, 2003. Français. ⟨tel-00005504⟩

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