The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions. II.(English)Zbl 07864511
There are quite a few Floer-type homology theories associated to three-manifolds. The paper under review and its two sequels study the following three of them: monopole Floer homology, embedded contact homology, and Heegaard Floer homology.
In 2006, Taubes extended his correspondence between solutions of the Seiberg-Witten equations and J-holomorphic curves to the relative case, which allowed him to prove the long standing Weinstein conjecture in dimension three [C. H. Taubes, Geom. Topol. 11, 2117–2202 (2007;Zbl 1135.57015)] and to establish the equivalence between monopole Floer cohomology and embedded contact homology shortly thereafter [C. H. Taubes, Geom. Topol. 14, No. 5, 2497–2581 (2010;Zbl 1275.57037); ibid. 14, No. 5, 2583–2720 (2010;Zbl 1276.57024); ibid. 14, No. 5, 2721–2817 (2010;Zbl 1276.57025); ibid. 14, No. 5, 2819–2960 (2010;Zbl 1276.57026); ibid. 14, No. 5, 2961–3000 (2010;Zbl 1276.57027)]. A byproduct of this equivalence was the proof of Arnold’s chord conjecture in dimension three [M. Hutchings andC. H. Taubes, Math. Res. Lett. 18, No. 2, 295–313 (2011;Zbl 1263.53080); Geom. Topol. 17, No. 5, 2601–2688 (2013;Zbl 1396.53111)].
The goal of the paper under review, its predecessor and its sequel is to prove the equivalence of Heegaard Floer homology and embedded contact homology. In this paper (the second out of three) and its predecessor [V. Colin et al., Publ. Math., Inst. Hautes Étud. Sci. 139, 13–187 (2024;Zbl 07864510)], the authors establish an isomorphism between the hat versions of the Heegaard Floer homology and ECH groups associated to a closed, oriented 3-manifold \(M\). This isomorphism is compatible with the splitting of Heegaard Floer homology according to Spin\(^c\)-structures and of ECH according to first homology classes. The computations are done with \(\mathbb Z_2\) coefficients.
In 2006, Taubes extended his correspondence between solutions of the Seiberg-Witten equations and J-holomorphic curves to the relative case, which allowed him to prove the long standing Weinstein conjecture in dimension three [C. H. Taubes, Geom. Topol. 11, 2117–2202 (2007;Zbl 1135.57015)] and to establish the equivalence between monopole Floer cohomology and embedded contact homology shortly thereafter [C. H. Taubes, Geom. Topol. 14, No. 5, 2497–2581 (2010;Zbl 1275.57037); ibid. 14, No. 5, 2583–2720 (2010;Zbl 1276.57024); ibid. 14, No. 5, 2721–2817 (2010;Zbl 1276.57025); ibid. 14, No. 5, 2819–2960 (2010;Zbl 1276.57026); ibid. 14, No. 5, 2961–3000 (2010;Zbl 1276.57027)]. A byproduct of this equivalence was the proof of Arnold’s chord conjecture in dimension three [M. Hutchings andC. H. Taubes, Math. Res. Lett. 18, No. 2, 295–313 (2011;Zbl 1263.53080); Geom. Topol. 17, No. 5, 2601–2688 (2013;Zbl 1396.53111)].
The goal of the paper under review, its predecessor and its sequel is to prove the equivalence of Heegaard Floer homology and embedded contact homology. In this paper (the second out of three) and its predecessor [V. Colin et al., Publ. Math., Inst. Hautes Étud. Sci. 139, 13–187 (2024;Zbl 07864510)], the authors establish an isomorphism between the hat versions of the Heegaard Floer homology and ECH groups associated to a closed, oriented 3-manifold \(M\). This isomorphism is compatible with the splitting of Heegaard Floer homology according to Spin\(^c\)-structures and of ECH according to first homology classes. The computations are done with \(\mathbb Z_2\) coefficients.
Reviewer: Roman Golovko (Praha)
MSC:
| 57K31 | Invariants of 3-manifolds (including skein modules, character varieties) |
| 57R58 | Floer homology |
Citations:
Zbl 1135.57015;Zbl 1275.57037;Zbl 1276.57024;Zbl 1276.57025;Zbl 1276.57026;Zbl 1276.57027;Zbl 1263.53080;Zbl 1396.53111;Zbl 07864510Cite
References:
| [1] | V. Colin, P. Ghiggini and K. Honda, Embedded contact homology and open book decompositions, Geom. Topol., to appear. ·Zbl 1256.57020 |
| [2] | V. Colin, P. Ghiggini and K. Honda, The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I, Publ. Math. Inst. Hautes Études Sci., (2024). doi:10.1007/s10240-024-00145-x. ·Zbl 1256.57020 |
| [3] | V. Colin, P. Ghiggini and K. Honda, The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions II, Publ. Math. Inst. Hautes Études Sci., (2024), this paper. doi:10.1007/s10240-024-00146-w. ·Zbl 1256.57020 |
| [4] | V. Colin, P. Ghiggini and K. Honda, The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus, Publ. Math. Inst. Hautes Études Sci., (2024). doi:10.1007/s10240-024-00147-9. ·Zbl 1473.57056 |
| [5] | Bao, E.; Honda, K., Definition of cylindrical contact homology in dimension three, J. Topol., 11, 1001-1052, 2018 ·Zbl 1409.53064 ·doi:10.1112/topo.12077 |
| [6] | Bourgeois, F., A Morse Bott approach to contact homology, from: “Symplectic and contact topology: interactions and perspectives, Fields Inst. Comm., 55-77, 2003, Providence: Am. Math. Soc., Providence ·Zbl 1046.57017 |
| [7] | F. Bourgeois, A Morse-Bott approach to contact homology, Ph.D. Thesis, 2002. ·Zbl 1046.57017 |
| [8] | Bourgeois, F.; Eliashberg, Y.; Hofer, H.; Wysocki, K.; Zehnder, E., Compactness results in symplectic field theory, Geom. Topol., 7, 799-888, 2003 ·Zbl 1131.53312 ·doi:10.2140/gt.2003.7.799 |
| [9] | Etnyre, J.; Ekholm, T.; Sullivan, M., The contact homology of Legendrian submanifolds in \(\mathbf{R}^{2n+1} \), J. Differ. Geom., 71, 177-305, 2005 ·Zbl 1103.53048 |
| [10] | Gromov, M., Pseudo holomorphic curves in symplectic manifolds, Invent. Math., 82, 307-347, 1985 ·Zbl 0592.53025 ·doi:10.1007/BF01388806 |
| [11] | Hofer, H.; Lizan, V.; Sikorav, J.-C., On genericity for holomorphic curves in four-dimensional almost complex manifolds, J. Geom. Anal., 7, 149-159, 1997 ·Zbl 0911.53014 ·doi:10.1007/BF02921708 |
| [12] | Hofer, H.; Wysocki, K.; Zehnder, E., Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 13, 337-379, 1996 ·Zbl 0861.58018 ·doi:10.1016/s0294-1449(16)30108-1 |
| [13] | Hörmander, L., The analysis of linear partial differential operators I, xii+440 pp., 1990, Berlin: Springer, Berlin ·Zbl 0712.35001 |
| [14] | K. Honda, W. Kazez and G. Matić, Contact structures, sutured Floer homology, and TQFT, preprint (2008), arXiv:0807.2431. ·Zbl 1171.57031 |
| [15] | Hutchings, M., An index inequality for embedded pseudoholomorphic curves in symplectizations, J. Eur. Math. Soc., 4, 313-361, 2002 ·Zbl 1017.58005 ·doi:10.1007/s100970100041 |
| [16] | Hutchings, M., The embedded contact homology index revisited, New perspectives and challenges in symplectic field theory, CRM Proc. Lecture Notes, 263-297, 2009, Providence: Am. Math. Soc., Providence ·Zbl 1207.57045 |
| [17] | Hutchings, M.; Taubes, C., Gluing pseudoholomorphic curves along branched covered cylinders I, J. Symplectic Geom., 5, 43-137, 2007 ·Zbl 1157.53047 ·doi:10.4310/JSG.2007.v5.n1.a5 |
| [18] | Hutchings, M.; Taubes, C., Gluing pseudoholomorphic curves along branched covered cylinders II, J. Symplectic Geom., 7, 29-133, 2009 ·Zbl 1193.53183 ·doi:10.4310/JSG.2009.v7.n1.a2 |
| [19] | Hutchings, M.; Taubes, C., Proof of the Arnold chord conjecture in three dimensions 1, Math. Res. Lett., 18, 295-313, 2011 ·Zbl 1263.53080 ·doi:10.4310/MRL.2011.v18.n2.a8 |
| [20] | Ionel, E.; Parker, T., Relative Gromov-Witten invariants, Ann. Math. (2), 157, 45-96, 2003 ·Zbl 1039.53101 ·doi:10.4007/annals.2003.157.45 |
| [21] | Ionel, E.; Parker, T., The symplectic sum formula for Gromov-Witten invariants, Ann. Math. (2), 159, 935-1025, 2004 ·Zbl 1075.53092 ·doi:10.4007/annals.2004.159.935 |
| [22] | S. Ivashkovich and V. Shevchishin, Gromov compactness theorem for \(J\)-complex curves with boundary, Int. Math. Res. Not. (2000), 1167-1206. ·Zbl 0994.53010 |
| [23] | Li, A.; Ruan, Y., Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds, Invent. Math., 145, 151-218, 2001 ·Zbl 1062.53073 ·doi:10.1007/s002220100146 |
| [24] | Lipshitz, R., A cylindrical reformulation of Heegaard Floer homology, Geom. Topol., 10, 955-1097, 2006 ·Zbl 1130.57035 ·doi:10.2140/gt.2006.10.955 |
| [25] | D. McDuff, Comparing absolute and relative Gromov-Witten invariants, preprint (2008). |
| [26] | McDuff, D.; Salamon, D., \(J\)-holomorphic curves and symplectic topology, 2004, Providence: Am. Math. Soc., Providence ·Zbl 1064.53051 |
| [27] | Parker, B., Exploded fibrations, Proceedings of Gökova Geometry-Topology Conference 2006, 52-90, Gökova Geometry/Topology Conference (GGT), 52-90, 2007 ·Zbl 1184.57020 |
| [28] | Seidel, P., A long exact sequence for symplectic Floer cohomology, Topology, 42, 1003-1063, 2003 ·Zbl 1032.57035 ·doi:10.1016/S0040-9383(02)00028-9 |
| [29] | C. Wendl, Finite energy foliations and surgery on transverse links, Ph.D. Thesis, 2005. |
| [30] | Wendl, C., Automatic transversality and orbifolds of punctured holomorphic curves in dimension four, Comment. Math. Helv., 85, 347-407, 2010 ·Zbl 1207.32021 ·doi:10.4171/cmh/199 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
