Abstract
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.
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Acknowledgements
We are indebted to Michael Hutchings for many helpful conversations and for our previous collaboration which was a catalyst for the present work. We also thank Tobias Ekholm, Dusa McDuff, Ivan Smith and Jean-Yves Welschinger for illuminating exchanges. Part of this work was done while KH and PG visited MSRI during the academic year 2009–2010. We are extremely grateful to MSRI and the organizers of the “Symplectic and Contact Geometry and Topology” and the “Homology Theories of Knots and Links” programs for their hospitality; this work probably would never have seen the light of day without the large amount of free time which was made possible by the visit to MSRI.
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VC supported by the Institut Universitaire de France, ANR Symplexe, ANR Floer Power, and ERC Geodycon.
PG supported by ANR Floer Power and ANR TCGD.
KH supported by NSF Grants DMS-0805352, DMS-1105432, and DMS-1406564.
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Colin, V., Ghiggini, P. & Honda, K. The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus. Publ.math.IHES (2024). https://doi.org/10.1007/s10240-024-00147-9
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DOI: https://doi.org/10.1007/s10240-024-00147-9