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

Çağatay Kutluhan, Yi-Jen Lee and Clifford Henry Taubes

Geometry & Topology 24 (2020) 3219–3469

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.

Heegaard Floer homology, Seiberg–Witten Floer homology, pseudoholomorphic curves
Mathematical Subject Classification 2010
Primary: 53C07
Secondary: 52C15
Received: 17 February 2012
Revised: 16 February 2019
Accepted: 14 November 2019
Published: 6 January 2021
Proposed: Tomasz Mrowka
Seconded: Tobias H Colding, Yasha Eliashberg
Çağatay Kutluhan
Department of Mathematics
University at Buffalo
Buffalo, NY
United States
Yi-Jen Lee
Institute of Mathematical Sciences
The Chinese University of Hong Kong
Shatin, NT
Hong Kong
Clifford Henry Taubes
Department of Mathematics
Harvard University
Cambridge, MA
United States