The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus

Abstract Given a closed oriented 3-manifold M M, we establish an isomorphism between the Heegaard Floer homology group HF^+ (-M) H F + (− M) and the embedded contact homology group ECH (M) E C H (M). Starting from an open book decomposition (S, h) (S, h) of M M, we construct a chain map ^+ Φ + from a Heegaard Floer chain complex associated to (S, h) (S, h) to an embedded contact homology chain complex for a contact form supported by (S, h) (S, h). The chain map ^+ Φ + commutes up to homotopy with the U U -maps defined on both sides and reduces to the quasi-isomorphism Φ from (Colin et al. in Publ. Math. Inst. Hautes Études Sci. , 2024a, 2024b) on subcomplexes defining the hat versions. Algebraic considerations then imply that the map ^+ Φ + is a quasi-isomorphism.