Symplectic fillings and cobordisms of lens spaces

Etnyre, John; Roy, Agniva

doi:10.1090/tran/8474

Symplectic fillings and cobordisms of lens spaces
HTML articles powered by AMS MathViewer

by John B. Etnyre and Agniva Roy PDF

Trans. Amer. Math. Soc. 374 (2021), 8813-8867 Request permission

Abstract:

We complete the classification of symplectic fillings of tight contact structures on lens spaces. In particular, we show that any symplectic filling $X$ of a virtually overtwisted contact structure on $L(p,q)$ has another symplectic structure that fills the universally tight contact structure on $L(p,q)$. Moreover, we show that the Stein filling of $L(p,q)$ with maximal second homology is given by the plumbing of disk bundles. We also consider the question of constructing symplectic cobordisms between lens spaces and report some partial results.

References

Paolo Aceto, Duncan McCoy, and JungHwan Park, Non-simply connected symplectic fillings of lens spaces, 2020, arXiv:2009.08964.
Russell Avdek, Contact surgery and supporting open books, Algebr. Geom. Topol. 13 (2013), no. 3, 1613–1660. MR 3071137, DOI 10.2140/agt.2013.13.1613
Kenneth Baker and John Etnyre, Rational linking and contact geometry, Perspectives in analysis, geometry, and topology, Progr. Math., vol. 296, Birkhäuser/Springer, New York, 2012, pp. 19–37. MR 2884030, DOI 10.1007/978-0-8176-8277-4_{2}
Kenneth L. Baker, John B. Etnyre, Hyunki Min, and Sinem Onaran, Torus knots in lens spaces, In preparation.
Apratim Chakraborty, John B. Etnyre, and Hyunki Min, Cabling Legendrian and transverse knots, 2020, arXiv:2012.12148.
Austin Christian and Youlin Li, Some applications of Menke’s JSJ decomposition for symplectic fillings, Preprint, 2020, arXiv:2006.16825.
Fan Ding and Hansjörg Geiges, Handle moves in contact surgery diagrams, J. Topol. 2 (2009), no. 1, 105–122. MR 2499439, DOI 10.1112/jtopol/jtp002
Margaret Doig and Stephan Wehrli, A combinatorial proof of the homology cobordism classification of lens spaces, 2015, arXiv:1505.06970.
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, A few remarks about symplectic filling, Geom. Topol. 8 (2004), 277–293. MR 2023279, DOI 10.2140/gt.2004.8.277
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, Knots and contact geometry. I. Torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001), no. 1, 63–120. MR 1959579
John B. Etnyre and Ko Honda, On symplectic cobordisms, Math. Ann. 323 (2002), no. 1, 31–39. MR 1906906, DOI 10.1007/s002080100292
John B. Etnyre and Ko Honda, On connected sums and Legendrian knots, Adv. Math. 179 (2003), no. 1, 59–74. MR 2004728, DOI 10.1016/S0001-8708(02)00027-0
John B. Etnyre and Bülent Tosun, Homology spheres bounding acyclic smooth manifolds and symplectic fillings, arXiv:2004.07405, to appear in Michigan Journal of Math
Edoardo Fossati, Contact surgery on the Hopf link: classification of fillings, 2019, arXiv:1905.13026.
Edoardo Fossati, Topological constraints for Stein fillings of tight structures on lens spaces, 2019, arXiv:1905.13026.
David T. Gay, Symplectic 2-handles and transverse links, Trans. Amer. Math. Soc. 354 (2002), no. 3, 1027–1047. MR 1867371, DOI 10.1090/S0002-9947-01-02890-2
Hansjörg Geiges and Sinem Onaran, Legendrian lens space surgeries, Michigan Math. J. 67 (2018), no. 2, 405–422. MR 3802259, DOI 10.1307/mmj/1522980162
Emmanuel Giroux, Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Invent. Math. 141 (2000), no. 3, 615–689 (French). MR 1779622, DOI 10.1007/s002220000082
Marco Golla and Laura Starkston, The symplectic isotopy problem for rational cuspidal curves, 2019, arXiv:1907.06787.
Robert E. Gompf, Handlebody construction of Stein surfaces, Ann. of Math. (2) 148 (1998), no. 2, 619–693. MR 1668563, DOI 10.2307/121005
R. Hind, Stein fillings of lens spaces, Commun. Contemp. Math. 5 (2003), no. 6, 967–982. MR 2030565, DOI 10.1142/S0219199703001178
Ko Honda, On the classification of tight contact structures. I, Geom. Topol. 4 (2000), 309–368. MR 1786111, DOI 10.2140/gt.2000.4.309
Amey Kaloti, Stein fillings of planar open books, 2013, arXiv:1311.0208.
YankıLekili and Maksim Maydanskiy, The symplectic topology of some rational homology balls, Comment. Math. Helv. 89 (2014), no. 3, 571–596. MR 3260842, DOI 10.4171/CMH/327
Paolo Lisca, On symplectic fillings of lens spaces, Trans. Amer. Math. Soc. 360 (2008), no. 2, 765–799. MR 2346471, DOI 10.1090/S0002-9947-07-04228-6
Dusa McDuff, The structure of rational and ruled symplectic $4$-manifolds, J. Amer. Math. Soc. 3 (1990), no. 3, 679–712. MR 1049697, DOI 10.1090/S0894-0347-1990-1049697-8
Michael Menke, A JSJ-type decomposition theorem for symplectic fillings, 2018, arXiv:1807.03420.
Olga Plamenevskaya, On Legendrian surgeries between lens spaces, J. Symplectic Geom. 10 (2012), no. 2, 165–181. MR 2926993
Olga Plamenevskaya and Jeremy Van Horn-Morris, Planar open books, monodromy factorizations and symplectic fillings, Geom. Topol. 14 (2010), no. 4, 2077–2101. MR 2740642, DOI 10.2140/gt.2010.14.2077
Oswald Riemenschneider, Deformationen von Quotientensingularitäten (nach zyklischen Gruppen), Math. Ann. 209 (1974), 211–248 (German). MR 367276, DOI 10.1007/BF01351850
Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original. MR 1277811
Stephan Schonenberger, Planar open books and symplectic fillings, ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–University of Pennsylvania. MR 2707472
Jonathan Simone, Symplectically replacing plumbings with Euler characteristic 2 4-manifolds, Journal of Symplectic Geometry 18, 2020, pages 1285-1318.
Friedhelm Waldhausen, Heegaard-Zerlegungen der $3$-Sphäre, Topology 7 (1968), 195–203 (German). MR 227992, DOI 10.1016/0040-9383(68)90027-X
Chris Wendl, Strongly fillable contact manifolds and $J$-holomorphic foliations, Duke Math. J. 151 (2010), no. 3, 337–384. MR 2605865, DOI 10.1215/00127094-2010-001