HF=HM III: Holomorphic curves and the differential for the ech/Heegaard Floer correspondence

TL;DR

This paper details the relationship between differentials and endomorphisms in embedded contact homology and Heegaard Floer homology, advancing the construction of an isomorphism between Seiberg-Witten Floer and Heegaard Floer homologies for 3-manifolds.

Contribution

It establishes the relationship between differentials and canonical endomorphisms in embedded contact and Heegaard Floer homologies, completing a key step in the isomorphism construction.

Findings

Relationship between ECH and Heegaard Floer differentials clarified

Canonical endomorphisms correspondence established

Progress towards the isomorphism between Seiberg-Witten and Heegaard Floer homologies

Abstract

This is the third 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 the relationship between the differential on the embedded contact homology chain complex and the differential on the Heegaard Floer chain complex. The paper also describes the relationship between the various canonical endomorphisms that act on the homology groups of these two complexes.