Fukaya A_\infty structures associated to Lefschetz fibrations. I

TL;DR

This paper explores the algebraic structures of Lagrangian Floer cohomology in Lefschetz fibrations, comparing two methods of constructing bimodule homomorphisms and establishing their equivalence.

Contribution

It introduces a comparison theorem linking two approaches to constructing bimodule homomorphisms in Floer cohomology for Lefschetz fibrations.

Findings

Equivalence of two construction methods for bimodule homomorphisms.

Enhanced understanding of algebraic structures in Floer cohomology.

Clarification of the role of vanishing cycles and Lefschetz thimbles.

Abstract

This (partially expository) paper discusses Lagrangian Floer cohomology in the context of Lefschetz fibrations, with emphasis on the algebraic structures encountered there. In addition to the well-known directed A_infinity algebras which appear in this situation, one has additional information encoded in a certain bimodule homomorphism. There are two approaches to constructing this homomorphism: in terms of the (noncompact) Lefschetz thimbles in the total space, or else in terms of vanishing cycle in the fibre. We prove a comparison result, which shows that (up to a certain remaining ambiguity) the two approaches are equivalent.