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

By V. Colin, P. Ghiggini, K. Honda. Publications mathématiques (Bures-sur-Yvette), 2012

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

Subfields: Geometry and Topology, Mathematical Physics

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