Embedded contact homology and open book decompositions

TL;DR

This paper establishes a key step towards proving the equivalence of Heegaard Floer homology and embedded contact homology by relating ECH on a closed contact 3-manifold to ECH on the complement of an open book's binding.

Contribution

It proves the equivalence between ECH of a closed contact 3-manifold and a version of ECH defined on the complement of an open book binding, advancing the series' goal.

Findings

Established the ECH equivalence on open book complements.

Provided a full proof of Morse-Bott gluing needed for the isomorphism.

Fixed a mistake and added coauthor in the appendix.

Abstract

This is the first of a series of papers devoted to proving the equivalence of Heegaard Floer homology and embedded contact homology (abbreviated ECH). In this paper we prove that, given a closed, oriented, contact $3$-manifold, there is an equivalence between ECH of the closed $3$-manifold and a version of ECH, defined on the complement of the binding of an adapted open book decomposition. In the appendix we give a full proof of the Morse-Bott gluing result that we need in this article and in the subsequent ones of the series proving the isomorphism between Heegaard Floer homology and ECH. V.8: we fixed a mistake in the appendix and added Yuan Yao as a coauthor.