$\mathrm{HF}=\mathrm{HM}$, IV: The Sieberg–Witten Floer homology and ech correspondence

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 auxiliary manifold to the Heegaard Floer homology on the original. The second isomorphism relates the relevant version of the embedded contact homology on the auxiliary 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 auxiliary manifold with the appropriate version of Seiberg–Witten Floer homology on the original manifold. The paper describes the second of these isomorphisms.