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 $H{F}^{+}(-M)$ $HF^{+} (-M)$ and the embedded contact homology group $ECH(M)$ $ECH(M)$ . Starting from an open book decomposition $(S,\mathfrak{h})$ $(S,\mathfrak{h} )$ of $M$ $M$ , we construct a chain map ${\mathrm{\Phi }}^{+}$ $\Phi ^{+}$ from a Heegaard Floer chain complex associated to $(S,\mathfrak{h})$ $(S,\mathfrak{h} )$ to an embedded contact homology chain complex for a contact form supported by $(S,\mathfrak{h})$ $(S,\mathfrak{h} )$ . The chain map ${\mathrm{\Phi }}^{+}$ $\Phi ^{+}$ commutes up to homotopy with the $U$ $U$ -maps defined on both sides and reduces to the quasi-isomorphism $\mathrm{\Phi }$ $\Phi $ 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 ${\mathrm{\Phi }}^{+}$ $\Phi ^{+}$ is a quasi-isomorphism.