The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I

Abstract Given an open book decomposition (S, h) (S, h) adapted to a closed, oriented 3-manifold M M, we define a chain map Φ from a certain Heegaard Floer chain complex associated to (S, h) (S, h) to a certain embedded contact homology chain complex associated to (S, h) (S, 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 − M and the hat version of embedded contact homology of M M.