Bordered Heegaard Floer Homology: Invariance and Pairing

We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented two-manifold a di erential graded algebra. For a three-manifold with speci ed boundary, the invariant comes in two di erent versions, one of which (type D) is a module over the algebra and the other of which (type A) is an A∞ module. Both are well-de ned up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A∞ tensor product of the type D module of one piece and the type A module from the other piece is ĤF of the glued manifold. As a special case of the construction, we specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for ĤF . We relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a lling.