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$\mathrm{HF}=\mathrm{HM}$, II: Reeb orbits and holomorphic curves for the ech/Heegaard Floer correspondence

Çağatay Kutluhan, Yi-Jen Lee and Clifford Henry Taubes

Geometry & Topology 24 (2020) 2855–3012
DOI: 10.2140/gt.2020.24.2855

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 auxiliary manifold to the Heegaard Floer homology on the original. This paper describes this auxiliary 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.

Seiberg–Witten Floer homology, Heegaard Floer homology
Mathematical Subject Classification 2010
Primary: 53C07, 53C15
Received: 17 February 2012
Revised: 3 November 2015
Accepted: 23 April 2018
Published: 29 December 2020
Proposed: Tomasz Mrowka
Seconded: András I Stipsicz, Bruce Kleiner
Çağatay Kutluhan
Department of Mathematics
University at Buffalo
Buffalo, NY
United States
Yi-Jen Lee
Department of Mathematics
The Chinese University of Hong Kong
Shatin, NT
Hong Kong
Clifford Henry Taubes
Department of Mathematics
Harvard University
Cambridge, MA
United States