HF=HM IV: The Seiberg-Witten Floer homology and ech correspondence

TL;DR

This paper constructs an isomorphism linking Seiberg-Witten Floer homology and Heegaard Floer homology for 3-manifolds, focusing on the second step in a three-part chain of isomorphisms.

Contribution

It details the second isomorphism connecting embedded contact homology and Seiberg-Witten Floer homology on an auxiliary manifold.

Findings

Established the second isomorphism in the chain

Clarified the relationship between embedded contact homology and Seiberg-Witten Floer homology

Contributed to the overall proof of the isomorphism between Floer homologies

Abstract

This is the fourth 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. The second isomorphism relates the relevant version of the embedded contact homology on the auxilliary manifold with a version of the Seiberg-Witten Floer homology on this same manifold. The third isomorphism relates the Seiberg-Witten Floer homology on the auxilliary manifold with the appropriate version of Seiberg-Witten Floer homology on the original manifold. The paper describes the second of these isomorphisms.