Canonical contact unit cotangent bundle
-
Takahiro Oba
Abstract
We describe an explicit open book decomposition adapted to the canonical contact structure on the unit cotangent bundle of a closed surface.
Acknowledgements
We would like to thank Paul Seidel for helpful correspondence. The first author would like to express his gratitude to Koç University for its hospitality during a visit while this work was mainly carried out.
Funding: The first author was partially supported by JSPS KAKENHI Grant Number 15J05214.
References
[1] N. A’Campo, Real deformations and complex topology of plane curve singularities. Ann. Fac. Sci. Toulouse Math. (6) 8 (1999), 5–23. MR1721511 Zbl 0962.3202510.5802/afst.918Search in Google Scholar
[2] D. Auroux, I. Smith, Lefschetz pencils, branched covers and symplectic invariants. In: Symplectic 4-manifolds and algebraic surfaces, volume 1938 of Lecture Notes in Math., 1–53, Springer 2008. MR2441411 Zbl 1142.1400810.1007/978-3-540-78279-7_1Search in Google Scholar
[3] K. Cieliebak, Y. Eliashberg, From Stein to Weinstein and back, volume 59 of American Mathematical Society Colloquium Publications. Amer. Math. Soc. 2012. MR3012475 Zbl 1262.3202610.1090/coll/059Search in Google Scholar
[4] P. Dehornoy, Genus-one Birkhoff sections for geodesic flows. Ergodic Theory Dynam. Systems35 (2015), 1795–1813. MR3377285 Zbl 1352.3709310.1017/etds.2014.14Search in Google Scholar
[5] Y. Eliashberg, Unique holomorphically fillable contact structure on the 3-torus. Internat. Math. Res. Notices (1996), no. 2, 77–82. MR1383953 Zbl 0852.5803410.1155/S1073792896000074Search in Google Scholar
[6] Y. Eliashberg, Symplectic geometry of plurisubharmonic functions. In: Gauge theory and symplectic geometry (Montreal, PQ, 1995), volume 488 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 49–67, Kluwer 1997. MR1461569 Zbl 0881.3201010.1007/978-94-017-1667-3_2Search in Google Scholar
[7] J. B. Etnyre, Planar open book decompositions and contact structures. Int. Math. Res. Not. (2004), no. 79, 4255–4267. MR2126827 Zbl 1069.5701610.1155/S1073792804142207Search in Google Scholar
[8] J. B. Etnyre, B. Ozbagci, Invariants of contact structures from open books. Trans. Amer. Math. Soc. 360 (2008), 3133–3151. MR2379791 Zbl 1157.5701510.1090/S0002-9947-08-04459-0Search in Google Scholar
[9] E. Ghys, Right-handed vector fields & the Lorenz attractor. Jpn. J. Math. 4 (2009), 47–61. MR2491282 Zbl 1188.3702810.1007/s11537-009-0854-8Search in Google Scholar
[10] E. Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures. In: Proc. International Congress of Mathematicians, Vol. II (Beijing, 2002), 405–414, Higher Ed. Press, Beijing 2002. MR1957051 Zbl 1015.53049Search in Google Scholar
[11] R. E. Gompf, A. I. Stipsicz, 4-manifolds and Kirby calculus, volume 20 of Graduate Studies in Mathematics. Amer. Math. Soc. 1999. MR1707327 Zbl 0933.5702010.1090/gsm/020Search in Google Scholar
[12] K. Honda, On the classification of tight contact structures. I. Geom. Topol. 4 (2000), 309–368. MR1786111 Zbl 0980.5701010.2140/gt.2000.4.309Search in Google Scholar
[13] J. Johns, Lefschetz fibrations on cotangent bundles of two-manifolds. In: Proceedings of the Gökova Geometry-Topology Conference 2011, 53–84, Int. Press, Somerville, MA 2012. MR3076043 Zbl 1360.57032Search in Google Scholar
[14] M. Korkmaz, B. Ozbagci, On sections of elliptic fibrations. Michigan Math. J. 56 (2008), 77–87. MR2433657 Zbl 1158.5703310.1307/mmj/1213972398Search in Google Scholar
[15] P. Massot, Topological methods in 3-dimensional contact geometry. In: Contact and symplectic topology, volume 26 of Bolyai Soc. Math. Stud., 27–83, János Bolyai Math. Soc., Budapest 2014. MR3220940 Zbl 1325.5300210.1007/978-3-319-02036-5_2Search in Google Scholar
[16] P. Massot, Two remarks on the support genus question. http://www.math.polytechnique.fr/perso/massot.patrick/exposition/genus.pdfSearch in Google Scholar
[17] D. McDuff, The structure of rational and ruled symplectic 4-manifolds. J. Amer. Math. Soc. 3 (1990), 679–712. MR1049697 Zbl 0723.5301910.1090/S0894-0347-1990-1049697-8Search in Google Scholar
[18] M. McLean, Symplectic homology of Lefschetz fibrations and Floer homology of the monodromy map.Selecta Math. (N.S.) 18 (2012), 473–512. MR2960024 Zbl 1253.5308410.1007/s00029-011-0079-6Search in Google Scholar
[19] J. Milnor, Lectures on the h-cobordism theorem. Princeton Univ. Press 1965. MR0190942 Zbl 0161.2030210.1515/9781400878055Search in Google Scholar
[20] B. Ozbagci, A. I. Stipsicz, Surgery on contact 3-manifolds and Stein surfaces, volume 13 of Bolyai Society Mathematical Studies. Springer 2004. MR2114165 Zbl 1067.5702410.1007/978-3-662-10167-4Search in Google Scholar
[21] P. Seidel, Fukaya categories and Picard-Lefschetz theory. European Mathematical Society, Zürich 2008. MR2441780 Zbl 1159.5300110.4171/063Search in Google Scholar
[22] J. Van Horn-Morris, Constructions of open book decompositions. PhD thesis, University of Texas at Austin, 2007.Search in Google Scholar
[23] O. Van Koert, Lecture notes on stabilization of contact open books. arXiv:1012.4359 [math.SG]Search in Google Scholar
[24] C. Wendl, Strongly fillable contact manifolds and J-holomorphic foliations. Duke Math. J. 151 (2010), 337–384. MR2605865 Zbl 1207.3202210.1215/00127094-2010-001Search in Google Scholar
Appendix: Diffeomorphism types of the total spaces of the Lefschetz fibrations
In this appendix, we verify that the total spaces of the Lefschetz fibrations πg: Wg → D2 (see Section 3.1) and πk: Mk → D2 (see Section 3.2) are diffeomorphic to DT*Σg and DT*Nk, respectively.
© 2018 Walter de Gruyter GmbH Berlin/Boston
Articles in the same Issue
- Frontmatter
- The index of symmetry of three-dimensional Lie groups with a left-invariant metric
- Canonical contact unit cotangent bundle
- On the umbilicity of generalized linear Weingarten hypersurfaces in hyperbolic spaces
- Regular polyhedra in the 3-torus
- Rational curves on Del Pezzo manifolds
- Commuting matrices and the Hilbert scheme of points on affine spaces
- On rational varieties of small rationality degree
- Skew symmetric logarithms and geodesics on On(ℝ)
- New homogeneous Einstein metrics on quaternionic Stiefel manifolds
Articles in the same Issue
- Frontmatter
- The index of symmetry of three-dimensional Lie groups with a left-invariant metric
- Canonical contact unit cotangent bundle
- On the umbilicity of generalized linear Weingarten hypersurfaces in hyperbolic spaces
- Regular polyhedra in the 3-torus
- Rational curves on Del Pezzo manifolds
- Commuting matrices and the Hilbert scheme of points on affine spaces
- On rational varieties of small rationality degree
- Skew symmetric logarithms and geodesics on On(ℝ)
- New homogeneous Einstein metrics on quaternionic Stiefel manifolds
