Embedded contact homology and open book decompositions

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.

114 page, 8 figures. With an appendix, written in collaboration with Yuan Yao, containing a complete proof of the Morse-Bott gluing results used in the article. To appear in Geometry & Topology

References in corpus (8)

• HF=HM I : Heegaard Floer homology and Seiberg--Witten Floer homology
• Proof of the Arnold chord conjecture in three dimensions II
• HF=HM III: Holomorphic curves and the differential for the ech/Heegaard Floer correspondence
• HF=HM IV: The Seiberg-Witten Floer homology and ech correspondence
• Sutures and contact homology I
• A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds
• HF=HM II: Reeb orbits and holomorphic curves for the ech/Heegaard-Floer correspondence
• Sutured ECH is a natural invariant

Cited by in corpus (10)

• The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions II
• A combinatorial spanning tree model for knot Floer homology
• Liouville hypersurfaces and connect sum cobordisms
• A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds
• Connected sums and involutive knot Floer homology
• Graph cobordisms and Heegaard Floer homology
• An absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields
• Duality and mapping tori in Heegaard Floer homology
• Dimensionally-reduced sutured Floer homology as a string homology
• On finite energy monopoles on $\mathbb{C}\times \Sigma$