Bordered Manifolds with Torus Boundary and the Link Surgery Formula

In this memoir, we develop a theory of bordered ${\mathrm{HF}}^{-}$ using the link surgery formula of Manolescu and Osváth. We interpret their link surgery complexes as type-$\mathrm{D}$ 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 discuss several important examples. As a basic 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 ${\mathrm{CF}}^{-}(Y)$ from ${\mathrm{CFK}}^{\infty }(K_1)$ and ${\mathrm{CFK}}^{\infty }(K_2)$. By computing the type-$\mathrm{D}$ 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.