2021/09/23 by Zemke, Ian · 1 citation
#57K18 #57K31 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2109.11520
In this paper, 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_∞-tensor product over our algebra K. Topologically, this connected sum formula may be viewed as a formula for gluing along torus boundary components. We compute several important examples. We show that the dual knot formula of Hedden--Levine and Eftekhary may be interpreted as the DA-bimodule for a particular diffeomorphism of the torus. As another example, if K1 and K2 are knots in S3, and Y is obtained by gluing the complements of K1 and K2 together using an orientation reversing diffeomorphism of their boundaries, then our theory may be used to compute CF-(Y) from CFK^∞(K1) and CFK^∞(K2). We additionally compute the type-D modules for rationally framed solid tori. Our theory also computes the Heegaard Floer homology of all 3-manifolds which bound a plumbing of a tree of disk bundles over 2-spheres. In a subsequent article, we use this work to verify Némethi's conjecture about lattice homology.