Abstract
Given an open book decomposition \((S,\mathfrak{h} )\) adapted to a closed, oriented 3-manifold \(M\), we define a chain map \(\Phi \) from a certain Heegaard Floer chain complex associated to \((S,\mathfrak{h} )\) to a certain embedded contact homology chain complex associated to \((S,\mathfrak{h} )\), as defined in (Colin et al. in Geom. Topol., 2024), and prove that it induces an isomorphism on the level of homology. This implies the isomorphism between the hat version of Heegaard Floer homology of \(-M\) and the hat version of embedded contact homology of \(M\).
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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 Denis Auroux, 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. KH also thanks the Simons Center for Geometry and Physics for their hospitality during his visit in May 2011. Finally we thank the referees for a large number of corrections and suggestions.
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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 via open book decompositions I. Publ.math.IHES (2024). https://doi.org/10.1007/s10240-024-00145-x
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DOI: https://doi.org/10.1007/s10240-024-00145-x