Making cobordisms symplectic

Eliashberg, Yakov; Murphy, Emmy

doi:10.1090/jams/995

Making cobordisms symplectic
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by Yakov Eliashberg and Emmy Murphy;

J. Amer. Math. Soc. 36 (2023), 1-29

DOI: https://doi.org/10.1090/jams/995

Published electronically: December 1, 2021

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Abstract:

We establish an existence $h$-principle for symplectic cobordisms of dimension $2n>4$ with concave overtwisted contact boundary.

References

Mélanie Bertelson and Gael Meigniez, Conformal Symplectic structures, Foliations and Contact Structures, arXiv:2107.08839.
Matthew Strom Borman, Yakov Eliashberg, and Emmy Murphy, Existence and classification of overtwisted contact structures in all dimensions, Acta Math. 215 (2015), no. 2, 281–361. MR 3455235, DOI 10.1007/s11511-016-0134-4
Roger Casals, Emmy Murphy, and Francisco Presas, Geometric criteria for overtwistedness, J. Amer. Math. Soc. 32 (2019), no. 2, 563–604. MR 3904160, DOI 10.1090/jams/917
Kai Cieliebak and Yakov Eliashberg, From Stein to Weinstein and back, American Mathematical Society Colloquium Publications, vol. 59, American Mathematical Society, Providence, RI, 2012. Symplectic geometry of affine complex manifolds. MR 3012475, DOI 10.1090/coll/059
James Conway and John B. Etnyre, Contact surgery and symplectic caps, Bull. Lond. Math. Soc. 52 (2020), no. 2, 379–394. MR 4171373, DOI 10.1112/blms.12332
Y. Eliashberg, Classification of overtwisted contact structures on $3$-manifolds, Invent. Math. 98 (1989), no. 3, 623–637. MR 1022310, DOI 10.1007/BF01393840
Yakov Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, Internat. J. Math. 1 (1990), no. 1, 29–46. MR 1044658, DOI 10.1142/S0129167X90000034
Yakov Eliashberg, Filling by holomorphic discs and its applications, Geometry of low-dimensional manifolds, 2 (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 151, Cambridge Univ. Press, Cambridge, 1990, pp. 45–67. MR 1171908
Yakov Eliashberg, Contact $3$-manifolds twenty years since J. Martinet’s work, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 165–192 (English, with French summary). MR 1162559, DOI 10.5802/aif.1288
Yakov Eliashberg, Weinstein manifolds revisited, Modern geometry: a celebration of the work of Simon Donaldson, Proc. Sympos. Pure Math., vol. 99, Amer. Math. Soc., Providence, RI, 2018, pp. 59–82. MR 3838879, DOI 10.1090/pspum/099/01737
Yakov Eliashberg, A few remarks about symplectic filling, Geom. Topol. 8 (2004), 277–293. MR 2023279, DOI 10.2140/gt.2004.8.277
Yakov Eliashberg and Mikhael Gromov, Convex symplectic manifolds, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989) Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 135–162. MR 1128541, DOI 10.1090/pspum/052.2/1128541
Yakov Eliashberg and Emmy Murphy, Lagrangian caps, Geom. Funct. Anal. 23 (2013), no. 5, 1483–1514. MR 3102911, DOI 10.1007/s00039-013-0239-2
John B. Etnyre, On symplectic fillings, Algebr. Geom. Topol. 4 (2004), 73–80. MR 2023278, DOI 10.2140/agt.2004.4.73
John B. Etnyre and Ko Honda, On symplectic cobordisms, Math. Ann. 323 (2002), no. 1, 31–39. MR 1906906, DOI 10.1007/s002080100292
David T. Gay, Four-dimensional symplectic cobordisms containing three-handles, Geom. Topol. 10 (2006), 1749–1759. MR 2284049, DOI 10.2140/gt.2006.10.1749
Hansjörg Geiges, Symplectic manifolds with disconnected boundary of contact type, Internat. Math. Res. Notices 1 (1994), 23–30. MR 1255250, DOI 10.1155/S1073792894000048
John W. Gray, Some global properties of contact structures, Ann. of Math. (2) 69 (1959), 421–450. MR 112161, DOI 10.2307/1970192
M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, DOI 10.1007/BF01388806
Peter Kronheimer and Tomasz Mrowka, Monopoles and three-manifolds, New Mathematical Monographs, vol. 10, Cambridge University Press, Cambridge, 2007. MR 2388043, DOI 10.1017/CBO9780511543111
Oleg Lazarev, Maximal contact and symplectic structures, J. Topol. 13 (2020), no. 3, 1058–1083. MR 4100126, DOI 10.1112/topo.12149
Dusa McDuff, Symplectic manifolds with contact type boundaries, Invent. Math. 103 (1991), no. 3, 651–671. MR 1091622, DOI 10.1007/BF01239530
Patrick Massot, Klaus Niederkrüger, and Chris Wendl, Weak and strong fillability of higher dimensional contact manifolds, Invent. Math. 192 (2013), no. 2, 287–373. MR 3044125, DOI 10.1007/s00222-012-0412-5
Dusa McDuff, Symplectic manifolds with contact type boundaries, Invent. Math. 103 (1991), no. 3, 651–671. MR 1091622, DOI 10.1007/BF01239530
J. Milnor, On the cobordism ring $\Omega ^{\ast }$ and a complex analogue. I, Amer. J. Math. 82 (1960), 505–521. MR 119209, DOI 10.2307/2372970
Yoshihiko Mitsumatsu, Anosov flows and non-Stein symplectic manifolds, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 5, 1407–1421 (English, with English and French summaries). MR 1370752, DOI 10.5802/aif.1500
Tomasz Mrowka and Yann Rollin, Legendrian knots and monopoles, Algebr. Geom. Topol. 6 (2006), 1–69. MR 2199446, DOI 10.2140/agt.2006.6.1
Emmy Murphy, Loose Legendrian Embeddings in High Dimensional Contact Manifolds, ProQuest LLC, Ann Arbor, MI, 2012. Thesis (Ph.D.)–Stanford University. MR 4172336
S. P. Novikov, Some problems in the topology of manifolds connected with the theory of Thom spaces, Dokl. Akad. Nauk SSSR 132, 1031–1034 (Russian); English transl., Soviet Math. Dokl. 1 (1960), 717–720. MR 121815
Klaus Niederkrüger, The plastikstufe—a generalization of the overtwisted disk to higher dimensions, Algebr. Geom. Topol. 6 (2006), 2473–2508. MR 2286033, DOI 10.2140/agt.2006.6.2473
Alan Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991), no. 2, 241–251. MR 1114405, DOI 10.14492/hokmj/1381413841
Chris Wendl, Non-exact symplectic cobordisms between contact 3-manifolds, J. Differential Geom. 95 (2013), no. 1, 121–182. MR 3128981
C. Wendl, https://symplecticfieldtheorist.wordpress.com/author/lmpshd/.