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

Given a closed oriented 3-manifold \(M\), we establish an isomorphism between the Heegaard Floer homology group \(HF^{+} (-M)\) and the embedded contact homology group \(ECH(M)\). Starting from an open book decomposition \((S,\mathfrak{h} )\) of \(M\), we construct a chain map \(\Phi ^{+}\) from a Heegaard Floer chain complex associated to \((S,\mathfrak{h} )\) to an embedded contact homology chain complex for a contact form supported by \((S,\mathfrak{h} )\). The chain map \(\Phi ^{+}\) commutes up to homotopy with the \(U\)-maps defined on both sides and reduces to the quasi-isomorphism \(\Phi \) 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 \(\Phi ^{+}\) is a quasi-isomorphism.