Heegaard-Floer homology of broken fibrations over the circle

We extend Perutz's Lagrangian matching invariants to 3-manifolds which are not necessarily fibered using the technology of holomorphic quilts. We prove an isomorphism of these invariants with Ozsváth-Szabó's Heegaard-Floer invariants for certain extremal spinc structures. As applications, we give new calculations of Heegaard-Floer homology of certain classes of 3-manifolds, and a characterization of Juhász's sutured Floer homology.