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The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus
Publications Mathématiques de l'IHÉS, Volume 139 (2024), pp. 349-385
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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,𝔥) of M, we construct a chain map Φ + from a Heegaard Floer chain complex associated to (S,𝔥) to an embedded contact homology chain complex for a contact form supported by (S,𝔥). The chain map Φ + commutes up to homotopy with the 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.

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DOI: 10.1007/s10240-024-00147-9

Vincent Colin 1; Paolo Ghiggini 1; Ko Honda 1

1 
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     author = {Vincent Colin and Paolo Ghiggini and Ko Honda},
     title = {The equivalence of {Heegaard} {Floer} homology and embedded contact homology {III:} from hat to plus},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {349--385},
     year = {2024},
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     doi = {10.1007/s10240-024-00147-9},
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Vincent Colin; Paolo Ghiggini; Ko Honda. The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus. Publications Mathématiques de l'IHÉS, Volume 139 (2024), pp. 349-385. doi: 10.1007/s10240-024-00147-9

[0.] V. Colin, P. Ghiggini and K. Honda, Embedded contact homology and open book decompositions, (2024), Geom. Topol., to appear.

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[II.] V. Colin, P. Ghiggini and K. Honda, The equivalence of Heegaard Floer homology and embedded contact homology II, Publ. Math. Inst. Hautes Études Sci., (2024). | DOI | MR

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