Lagrangian cobordism in Lefschetz fibrations

Abstract

Given a symplectic manifold $(M^{2n},\omega )$ we study Lagrangian cobordisms $V\subset E$ where $E$ is the total space of a Lefschetz fibration having $M$ as generic fiber. We prove a generation result for these cobordisms in the appropriate derived Fukaya category. As a corollary, we analyze the relations among the Lagrangian submanifolds $L\subset M$ that are induced by these cobordisms. This leads to a unified treatment - and a generalization - of the two types of relations among Lagrangian submanifolds of $M$ that were previously identified in the literature: those associated to Dehn twists that were discovered by Seidel and the relations induced by cobordisms in trivial symplectic fibrations described in our previous work.

118 pages, many figures; changes in this version concern the presentation (a few arguments are made more explicit, are corrected many typos, some references are given in a more precise way)

References in corpus (2)

• Real elements in the mapping class group of $T^2$
• Categorification of Seidel's representation

Cited by in corpus (1)

• The moduli space of real vector bundles of rank two over a real hyperelliptic curve