Embedded contact homology and open book decompositions

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.Comment: First revision: A mistake in our use of Morse-Bott techniques in ECH has been fixed and results on sutured ECH have been added. Second revision: bibliography updated. Third revision: exposition improved and details added following referees' suggestions. Fourth revision: we added an appendix sketching the proof of the Morse-Bott gluing statements used in the articl

On the coarse classification of tight contact structures

We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.Comment: 12 pages, to appear in the 2001 Georgia International Topology Conference proceeding