Composition, type-D structures, and links in S^2xS^1
Bordered Floer homology is a suite of smooth invariants which associates to each 3-manifold with (parametrized) boundary a collection of modules of various types. In its simplest incarnation, the object CFD(Y) associated to a manifold Y with connected boundary can be thought of as a dg-module. A pairing theorem of Lipshitz–Ozsváth–Thurston tells us that the complex Mor(CFD(Y_1),CFD(Y_2)) of module homomorphisms between two of these objects is homotopy equivalent to the Heegaard Floer complex of the manifold Y obtained by gluing Y_1 and Y_2 along their common boundary. In this talk, I will outline a topological realization of composition of such module homomorphisms as the map induced on Heegaard Floer complexes by a particular type of cobordism. I will then use this construction to briefly describe work-in-progress establishing the existence of a spectral sequence relating the Khovanov homology for links in S^2xS^1 due to Rozansky and Willis and an invariant defined using bordered Floer homology.