HF=HM II: Reeb orbits and holomorphic curves for the ech/Heegaard-Floer correspondence

TL;DR

This paper constructs part of an isomorphism between Seiberg-Witten Floer homology and Heegaard Floer homology for 3-manifolds, focusing on the auxiliary manifold, its geometry, and the relationship between their chain complexes.

Contribution

It describes the auxiliary manifold used in the isomorphism and relates the generators and differentials of embedded contact homology to those of Heegaard Floer homology.

Findings

Description of the auxiliary manifold and its geometry.

Relationship between generators of embedded contact and Heegaard Floer chain complexes.

Initial steps in relating differentials of the two chain complexes.

Abstract

This is the second of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an auxillary manifold to the Heegaard Floer homology on the original. This paper describes this auxilliary manifold, its geometry, and the relationship between the generators of the embedded contact homology chain complex and those of the Heegaard Floer chain complex. The pseudoholomorphic curves that define the differential on the embedded contact homology chain complex are also described here as a first step to relate the differential on the latter complex with that on the Heegaard Floer complex.