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

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 ⁺ 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 ⁺ Φ + commutes up to homotopy with the U U -maps defined on both sides and reduces to the quasi-isomorphism Φ 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 ⁺ Φ + is a quasi-isomorphism.