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

Kutluhan, Çağatay¹ ; Lee, Yi-Jen² ; Taubes, Clifford³

¹ Department of Mathematics, University at Buffalo, Buffalo, NY, United States
² Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, NT, Hong Kong
³ Department of Mathematics, Harvard University, Cambridge, MA, United States

Geometry & topology, Tome 24 (2020) no. 7, pp. 3219-3469.

Résumé

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.

Détail

DOI : 10.2140/gt.2020.24.3219

Classification : 53C07, 52C15
Keywords: Heegaard Floer homology, Seiberg–Witten Floer homology, pseudoholomorphic curves

Bibliographie

[1] F Bourgeois, Y Eliashberg, H Hofer, K Wysocki, E Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003) 799 | DOI

[2] M Hutchings, The embedded contact homology index revisited, from: "New perspectives and challenges in symplectic field theory" (editors M Abreu, F Lalonde, L Polterovich), CRM Proc. Lecture Notes 49, Amer. Math. Soc. (2009) 263

[3] M Hutchings, Embedded contact homology and its applications, from: "Proceedings of the International Congress of Mathematicians" (editors R Bhatia, A Pal, G Rangarajan, V Srinivas, M Vanninathan, P Gastesi), Hindustan Book Agency (2010) 1022

[4] M Hutchings, Taubes’s proof of the Weinstein conjecture in dimension three, Bull. Amer. Math. Soc. 47 (2010) 73 | DOI

[5] M Hutchings, M Sullivan, Rounding corners of polygons and the embedded contact homology of T3, Geom. Topol. 10 (2006) 169 | DOI

[6] M Hutchings, C H Taubes, The Weinstein conjecture for stable Hamiltonian structures, Geom. Topol. 13 (2009) 901 | DOI

[7] P Kronheimer, T Mrowka, Monopoles and three-manifolds, 10, Cambridge Univ. Press (2007) | DOI

[8] Ç Kutluhan, Y J Lee, C H Taubes, HF = HM, I : Heegaard Floer homology and Seiberg–Witten Floer homology, Geom. Topol. 24 (2020) 2829 | DOI

[9] Ç Kutluhan, Y J Lee, C H Taubes, HF = HM, II : Reeb orbits and holomorphic curves for the ech/Heegaard–Floer correspondence, Geom. Topol. 24 (2020) 2855 | DOI

[10] Ç Kutluhan, Y J Lee, C H Taubes, HF = HM, III : Holomorphic curves and the differential for the ech/Heegaard Floer correspondence, Geom. Topol. 24 (2020) 3013 | DOI

[11] Y J Lee, C H Taubes, Periodic Floer homology and Seiberg–Witten–Floer cohomology, J. Symplectic Geom. 10 (2012) 81 | DOI

[12] P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. 159 (2004) 1027 | DOI

[13] P Ozsváth, Z Szabó, An introduction to Heegaard Floer homology, from: "Floer homology, gauge theory, and low-dimensional topology" (editors D A Ellwood, P S Ozsváth, A I Stipsicz, Z Szabó), Clay Math. Proc. 5, Amer. Math. Soc. (2006) 3

[14] C H Taubes, SW ⇒ Gr : from the Seiberg–Witten equations to pseudo-holomorphic curves, J. Amer. Math. Soc. 9 (1996) 845 | DOI

[15] C H Taubes, Gr ⇒ SW : from pseudo-holomorphic curves to Seiberg–Witten solutions, J. Differential Geom. 51 (1999) 203 | DOI

[16] C H Taubes, Asymptotic spectral flow for Dirac operators, Comm. Anal. Geom. 15 (2007) 569 | DOI

[17] C H Taubes, The Seiberg–Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007) 2117 | DOI

[18] C H Taubes, The Seiberg–Witten equations and the Weinstein conjecture, II : More closed integral curves of the Reeb vector field, Geom. Topol. 13 (2009) 1337 | DOI

[19] C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, I, Geom. Topol. 14 (2010) 2497 | DOI

[20] C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, II, Geom. Topol. 14 (2010) 2583 | DOI

[21] C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, III, Geom. Topol. 14 (2010) 2721 | DOI

[22] C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, IV, Geom. Topol. 14 (2010) 2819 | DOI

[23] C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology, V, Geom. Topol. 14 (2010) 2961 | DOI

[24] C H Taubes, Correction to a lemma in “Embedded contact homology and Seiberg–Witten Floer homology, IV”, (2018)