Lagrangian cobordisms, Lefschetz Fibrations and Quantum Invariants

Abstract

In this thesis we study Lagrangian cobordisms with the tools provided by Lagrangian quantum homology. In particular, we develop the theory for the setting of Lagrangian cobordisms or Lagrangians with cylindrical ends in a Lefschetz fibration, and put the different versions of the quantum homology groups into relation by a long exact sequence. We prove various practical relations of maps in this long exact sequence and we extract invariants that generalize the notion of discriminants to Lagrangian cobordisms in Lefschetz fibrations. We prove results on the relation of the discriminants of the ends of a cobordism and the cobordism itself. We also give examples arising from Lagrangian spheres and relate the discriminant to open Gromov Witten invariants. We show that for some configurations of Lagrangian spheres the discriminant always vanishes. We study a set of examples that arise from Lefschetz pencils of complex quadric $n+1$ hypersurfaces of $\mathbb{CP}^{n+1}$ structures and their real part are the Lagrangians of interest. Using the results established in this thesis, we compute the discriminants of all these Lagrangians by reducing the calculation to the previously established case of a real Lagrangian sphere in the quadric.