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Homologies lagrangiennes, symplectiques et attachement d'anse

Abstract : In this PhD thesis, I present a new construction of the wrapped Fukaya complex of a Lagrangian and of the Chekanov algebra of a Legendrian using techniques developed by Cieliebak, Ekholm and Oancea. These constructions behave well under cobordisms and thus are fit to study the symplectic handle attachment procedure. I prove that the wrapped Fukaya complex of the cocore is isomorphic to the Chekanov algebra of the attachment sphere and show that this isomorphism factors through Abouzaid’s Open-Closed map. I then give a strategy in order to deduce from these results two important theorems announced by Bourgeois, Ekholm and Eliashberg concerning the behaviour of symplectic homology under handle attachment and the generation of the Fukaya category. In the last chapter, I define following an idea of A’Campo a geodesic flow on the skeleton of a Brieskorn manifold and relate this flow to the Reeb flow on the link of the singularity in order to try to generalize Viterbo’s isomorphism between the symplectic homology of a cotangent bundle and the homology of a loop space.
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Submitted on : Thursday, June 11, 2020 - 4:42:25 PM
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Amiel Peiffer-Smadja. Homologies lagrangiennes, symplectiques et attachement d'anse. Géométrie symplectique [math.SG]. Sorbonne Université, 2018. Français. ⟨NNT : 2018SORUS370⟩. ⟨tel-02865356⟩



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