Bordered manifolds with torus boundary and the link surgery formula

Abstract

In this paper, we develop a theory of bordered $\mathit{HF}^-$ using the link surgery formula of Manolescu and Ozsv\'{a}th. We interpret their link surgery complexes as type-$D$ modules over an associative algebra $\mathcal{K}$, which we introduce. We prove a connected sum formula, which we interpret as an $A_\infty$-tensor product over our algebra $\mathcal{K}$. 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 $K_1$ 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 $\mathit{CF}^-(Y)$ from $\mathit{CFK}^\infty(K_1)$ and $\mathit{CFK}^\infty(K_2)$. 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 N\'{e}methi's conjecture about lattice homology.