Bordered Heegaard Floer homology

We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D D ) is a module over the algebra and the other of which (type A A ) is an A ∞ A_∞ module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A ∞ A_∞ tensor product of the type D D module of one piece and the type A A module from the other piece is HF ^ {HF} 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 HF ^ {HF} . We relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.