Bordered manifolds with torus boundary and the link surgery formula

We prove a connected sum formula for Manolescu and Ozsv\'{a}th's link surgery formula. We interpret the connected sum formula as an $A_\infty$-tensor product over an associative algebra $\mathcal{K}$, which we introduce. More generally, we are able to interpret the link surgery formula as associating a type-$D$ and type-$A$ module to any bordered 3-manifold with torus boundary. Our connected sum formula gives a pairing theorem which computes the minus Heegaard Floer homology of the glued manifold. We apply our tools to give a combinatorial algorithm to compute the minus Heegaard Floer homology of 3-manifolds obtained by plumbing along a tree. We prove that for such 3-manifolds, Heegaard Floer homology is isomorphic to a deformation of lattice homology, and we give an algorithm to compute the deformation. Finally, 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 any orientation reversing diffeomorphism of their boundaries, then we give a formula which computes $\mathit{CF}^-(Y)$ from $\mathit{CFK}^\infty(K_1)$ and $\mathit{CFK}^\infty(K_2)$.