Lagrangian cobordism in Lefschetz fibrations

TL;DR

This paper investigates Lagrangian cobordisms within Lefschetz fibrations, establishing a generation result in the Fukaya category and unifying known relations among Lagrangian submanifolds.

Contribution

It provides a new generation theorem for Lagrangian cobordisms in Lefschetz fibrations and unifies previous relations induced by Dehn twists and trivial fibrations.

Findings

Proves a generation result in the derived Fukaya category.

Analyzes relations among Lagrangian submanifolds induced by cobordisms.

Unifies and generalizes existing relations from Dehn twists and trivial 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.