A connected sum formula for embedded contact homology.(English)Zbl 07985646
Summary: Given two closed contact three-manifolds, one can form their contact connected sum via the Weinstein one-handle attachment. We study how pseudo-holomorphic curves in the symplectization behave under this operation. As a result, we give a connected sum formula for embedded contact homology.
MSC:
| 32Q65 | Pseudoholomorphic curves |
| 53D40 | Symplectic aspects of Floer homology and cohomology |
| 53D10 | Contact manifolds (general theory) |
| 53D42 | Symplectic field theory; contact homology |
Cite
References:
| [1] | F. Bourgeois, O. van Koert, Action filtration and contact homology of connected sums, Unpublished manuscript. ·Zbl 1204.53074 |
| [2] | J. Bloom, T. Mrowka, P. Ozsváth, Connected sums in monopole Floer homology, Unpublished manuscript. |
| [3] | Cieliebak, K.; Eliashberg, Y., From Stein to Weinstein and back, (Symplectic Geometry of Affine Complex Manifolds. Symplectic Geometry of Affine Complex Manifolds, American Mathematical Society Colloquium Publications, vol. 59, 2012, American Mathematical Society: American Mathematical Society Providence, RI), pp. xii+364 ·Zbl 1262.32026 |
| [4] | Colin, V.; Ghiggini, P.; Honda, K., Equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions, Proc. Natl. Acad. Sci. USA, 108, 20, 8100-8105, 2011, ISSN: 0027-8424 ·Zbl 1256.57020 |
| [5] | Colin, V.; Ghiggini, P.; Honda, K., The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus, Publ. Math. Inst. Hautes Études Sci., 139, 349-385, 2024, ISSN: 0073-8301, 1618-1913 ·Zbl 07864512 |
| [6] | Colin, V.; Ghiggini, P.; Honda, K., The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I, Publ. Math. Inst. Hautes Études Sci., 139, 13-187, 2024, ISSN: 0073-8301, 1618-1913 ·Zbl 07864510 |
| [7] | Colin, V.; Ghiggini, P.; Honda, K., The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions II, Publ. Math. Inst. Hautes Études Sci., 139, 189-348, 2024, ISSN: 0073-8301, 1618-1913 ·Zbl 07864511 |
| [8] | Christian, A.; Menke, M., A JSJ-type decomposition theorem for symplectic fillings, 2022 |
| [9] | Colin, V.; Ghiggini, P.; Honda, K.; Hutchings, M., Sutures and contact homology I, Geom. Topol., 15, 3, 1749-1842, 2011, ISSN: 1465-3060 ·Zbl 1231.57026 |
| [10] | Dragnev, D. L., Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations, Commun. Pure Appl. Math., 57, 6, 726-763, 2004, ISSN: 0010-3640 ·Zbl 1063.53086 |
| [11] | Eliashberg, Y.; Givental, A.; Hofer, H., Introduction to symplectic field theory, (Special Volume, Part II. GAFA 2000. Special Volume, Part II. GAFA 2000, Tel Aviv, 1999, 2000), 560-673 ·Zbl 0989.81114 |
| [12] | Fish, J. W.; Siefring, R., Connected sums and finite energy foliations I: contact connected sums, J. Symplectic Geom., 16, 6, 1639-1748, 2018, ISSN: 1527-5256 ·Zbl 1415.53062 |
| [13] | Hutchings, M.; Sullivan, M., Rounding corners of polygons and the embedded contact homology of \(T^3\), Geom. Topol., 10, 169-266, 2006, ISSN: 1465-3060 ·Zbl 1101.53053 |
| [14] | Hutchings, M.; Taubes, C. H., Gluing pseudoholomorphic curves along branched covered cylinders. I, J. Symplectic Geom., 5, 1, 43-137, 2007, ISSN: 1527-5256 ·Zbl 1157.53047 |
| [15] | Hutchings, M.; Taubes, C. H., Gluing pseudoholomorphic curves along branched covered cylinders. II, J. Symplectic Geom., 7, 1, 29-133, 2009, ISSN: 1527-5256 ·Zbl 1193.53183 |
| [16] | Hutchings, M.; Taubes, C. H., The Weinstein conjecture for stable Hamiltonian structures, Geom. Topol., 13, 2, 901-941, 2009, ISSN: 1465-3060 ·Zbl 1169.53065 |
| [17] | Hutchings, M.; Taubes, C. H., Proof of the Arnold chord conjecture in three dimensions, II, Geom. Topol., 17, 5, 2601-2688, 2013, ISSN: 1465-3060 ·Zbl 1396.53111 |
| [18] | Hutchings, M., An index inequality for embedded pseudoholomorphic curves in symplectizations, J. Eur. Math. Soc., 4, 4, 313-361, 2002, ISSN: 1435-9855 ·Zbl 1017.58005 |
| [19] | Hutchings, M., The embedded contact homology index revisited, (New Perspectives and Challenges in Symplectic Field Theory. New Perspectives and Challenges in Symplectic Field Theory, CRM Proc. Lecture Notes., vol. 49, 2009, Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 263-297 ·Zbl 1207.57045 |
| [20] | Hutchings, M., Quantitative embedded contact homology, J. Differ. Geom., 88, 2, 231-266, 2011, ISSN: 0022-040X, 1945-743X ·Zbl 1238.53061 |
| [21] | Hutchings, M., Lecture notes on embedded contact homology, (Contact and Symplectic Topology. Contact and Symplectic Topology, Bolyai Soc. Math. Stud. János Bolyai Math., vol. 26, 2014, Soc.: Soc. Budapest), 389-484 ·Zbl 1432.53126 |
| [22] | Hofer, H.; Wysocki, K.; Zehnder, E., Properties of pseudo-holomorphic curves in symplectisations. II. Embedding controls and algebraic invariants, Geom. Funct. Anal., 5, 2, 270-328, 1995, ISSN: 1016-443 ·Zbl 0845.57027 |
| [23] | Hofer, H.; Wysocki, K.; Zehnder, E., Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics, Ann. Inst. Henri Poincaré C, Anal. Non Linéaire, 13, 3, 337-379, 1996, ISSN: 0294-1449 ·Zbl 0861.58018 |
| [24] | Kutluhan, Ç.; Lee, Y.-J.; Taubes, C., HF=HM, IV: the Sieberg-Witten Floer homology and ech correspondence, Geom. Topol., 24, 7, 3219-3469, 2020, ISSN: 1465-3060 ·Zbl 1494.57058 |
| [25] | Kutluhan, Ç.; Lee, Y.-J.; Taubes, C. H., \( \operatorname{HF} = \operatorname{HM} \), I: Heegaard Floer homology and Seiberg-Witten Floer homology, Geom. Topol., 24, 6, 2829-2854, 2020, ISSN: 1465-3060 ·Zbl 1493.57026 |
| [26] | Kutluhan, Ç.; Lee, Y.-J.; Taubes, C. H., \( \operatorname{HF} = \operatorname{HM} \), II: Reeb orbits and holomorphic curves for the ech/Heegaard Floer correspondence, Geom. Topol., 24, 6, 2855-3012, 2020, ISSN: 1465-3060 ·Zbl 1494.57056 |
| [27] | Kutluhan, Ç.; Lee, Y.-J.; Taubes, C. H., \( \operatorname{H} \operatorname{F} = \operatorname{H} \operatorname{M} \), III: holomorphic curves and the differential for the ech/Heegaard Floer correspondence, Geom. Topol., 24, 6, 3013-3218, 2020, ISSN: 1465-3060 ·Zbl 1494.57057 |
| [28] | Kutluhan, Ç.; Lee, Y.-J.; Taubes, C. H., HF=HM, V: Seiberg-Witten Floer homology and handle additions, Geom. Topol., 24, 7, 3471-3748, 2020, ISSN: 1465-3060 ·Zbl 1494.57059 |
| [29] | Kronheimer, P.; Mrowka, T., Monopoles and Three-Manifolds, New Mathematical Monographs, vol. 10, 2007, Cambridge University Press: Cambridge University Press Cambridge, pp. xii+796 ·Zbl 1158.57002 |
| [30] | Lin, F., \( \operatorname{Pin}(2)\)-monopole Floer homology, higher compositions and connected sums, J. Topol., 10, 4, 921-969, 2017, ISSN: 1753-8416 ·Zbl 1411.57028 |
| [31] | Nelson, J.; Weiler, M., Embedded contact homology of prequantization bundles, J. Symplectic Geom., 21, 6, 1077-1189, 2023, ISSN: 1527-5256, 1540-2347 ·Zbl 1543.53080 |
| [32] | Ozsváth, P.; Szabó, Z., Holomorphic disks and three-manifold invariants: properties and applications, Ann. Math. (2), 159, 3, 1159-1245, 2004, ISSN: 0003-486X ·Zbl 1081.57013 |
| [33] | Ozsváth, P.; Szabó, Z., Holomorphic disks, link invariants and the multi-variable Alexander polynomial, Algebraic Geom. Topol., 8, 2, 615-692, 2008, ISSN: 1472-2747 ·Zbl 1144.57011 |
| [34] | de Paulo, N. V.; Salomão, P. A.S., Systems of transversal sections near critical energy levels of Hamiltonian systems in \(\mathbb{R}^4\), Mem. Am. Math. Soc., 252, 1202, 2018, pp. v+105. ISSN: 0065-9266 ·Zbl 1486.53099 |
| [35] | Siefring, R., Relative asymptotic behavior of pseudoholomorphic half-cylinders, Commun. Pure Appl. Math., 61, 12, 1631-1684, 2008, ISSN: 0010-3640 ·Zbl 1158.53068 |
| [36] | Taubes, C. H., Embedded contact homology and Seiberg-Witten Floer cohomology I, Geom. Topol., 14, 5, 2497-2581, 2010, ISSN: 1465-3060 ·Zbl 1275.57037 |
| [37] | Taubes, C. H., Embedded contact homology and Seiberg-Witten Floer cohomology II, Geom. Topol., 14, 5, 2583-2720, 2010, ISSN: 1465-3060 ·Zbl 1276.57024 |
| [38] | Taubes, C. H., Embedded contact homology and Seiberg-Witten Floer cohomology III, Geom. Topol., 14, 5, 2721-2817, 2010, ISSN: 1465-3060 ·Zbl 1276.57025 |
| [39] | Taubes, C. H., Embedded contact homology and Seiberg-Witten Floer cohomology IV, Geom. Topol., 14, 5, 2819-2960, 2010, ISSN: 1465-3060 ·Zbl 1276.57026 |
| [40] | Taubes, C. H., Embedded contact homology and Seiberg-Witten Floer cohomology V, Geom. Topol., 14, 5, 2961-3000, 2010, ISSN: 1465-3060 ·Zbl 1276.57027 |
| [41] | Weinstein, A., Contact surgery and symplectic handlebodies, Hokkaido Math. J., 20, 2, 241-251, 1991, ISSN: 0385-4035 ·Zbl 0737.57012 |
| [42] | Wendl, C., Strongly fillable contact manifolds and J-holomorphic foliations, Duke Math. J., 151, 3, 337-384, 2010, ISSN: 0012-7094 ·Zbl 1207.32022 |
| [43] | Yau, M.-L., Cylindrical contact homology of subcritical Stein-fillable contact manifolds, Geom. Topol., 8, 1243-1280, 2004, ISSN: 1465-3060, 1364-0380 ·Zbl 1055.57036 |
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.
