Fukaya A_\infty structures associated to Lefschetz fibrations. I

Details Fukaya A_\infty structures associated to Lefschetz fibrations. I Fukaya A_\infty structures associated to Lefschetz fibrations. I

2009-12-19

arxiv.org/abs/0912.3932v2

Mathematics

Symplectic Geometry

v2: added comments

Scientific paper

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.

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