In this memoir, we develop a theory of bordered HF using the link surgery formula of Manolescu and Ozsváth.We interpret their link surgery complexes as type-D modules over an associative algebra K, which we introduce.We prove a connected sum formula, which we interpret as an A 1 -tensor product over our algebra K. Topologically, this connected sum formula may be viewed as a formula for gluing along torus boundary components.We discuss several important examples.As a first example, if K 1 and K 2 are knots in S 3 , and Y is obtained by gluing the complements of K 1 and K 2 together using an orientation reversing diffeomorphism of their boundaries, then our theory may be used to compute CF .Y / from CFK 1 .K 1 / and CFK 1 .K 2 /.By computing the type-D modules for rationally framed solid tori, our theory gives a version of the link surgery formula for rationally framed links.As a final example, we use our theory to derive the Heegaard Floer homology of all 3-manifolds which bound the plumbing of a tree of disk bundles over 2-spheres.
Ian Zemke (Wed,) studied this question.