Koszul dual bordered algebras and the wrapped Fukaya category

Abstract

By slicing the Heegaard diagram for a given 3-manifold in a particular way, it is possible to constructA-infinity bimodules, the tensor product of which retrieves the Heegaard Floer homology of the original 3-manifold. The first step in this is to construct algebras corresponding to the individual slices. In this dissertation, we construct Koszul dual A-infinity algebras A and B, called star algebras, for a particular star-shaped class of slice, called star diagrams. Using A-infinity bimodules over A and B, we then verify the Koszul duality relation, Theorem 1.0.2. The second chapter establishes an isomorphism between endomorphism algebras from the wrappedFukaya category of the star diagrams, and the star algebras constructed in the first chapter. By viewing the star algebras as A-infinity deformations of underlying associative algebras and making several calculations with Hochschild cohomology, we verify that they are unique with a given set of generators and basic A-infinity relations. We then make model calculations in order to establish that the endomorphism algebras have these generators and basic operations, so that the desired isomorphism follows.