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

Given an open book decomposition $(S,\mathfrak{h})$ $(S,\mathfrak{h} )$ adapted to a closed, oriented 3-manifold $M$ $M$ , we define a chain map $\mathrm{\Phi }$ $\Phi $ from a certain Heegaard Floer chain complex associated to $(S,\mathfrak{h})$ $(S,\mathfrak{h} )$ to a certain embedded contact homology chain complex associated to $(S,\mathfrak{h})$ $(S,\mathfrak{h} )$ , 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 $-M$ $-M$ and the hat version of embedded contact homology of $M$ $M$ .