×

zbMATH — the first resource for mathematics

Existence and classification of overtwisted contact structures in all dimensions. (English) Zbl 1344.53060
The main result is the following:
Theorem. Let \(M\) be a \((2n+1)\)-manifold, \(A\subset M\) a closed set, and \(\xi\) an almost contact structure on \(M\). If \(\xi\) is genuine on \(O_pA\subset M\), then \(\xi\) is homotopic relative to \(A\) to a genuine contact structure. In particular, any almost contact structure on a closed manifold is homotopic to a genuine contact structure.
It is also proved that on any closed manifold \(M\), any almost contact structure is homotopic to an overtwisted contact structure which is unique to isotopy.
An explicit classification of overtwisted contact structures on spheres is also given.

MSC:
53D10 Contact manifolds, general
57R17 Symplectic and contact topology in high or arbitrary dimension
53D35 Global theory of symplectic and contact manifolds
53D15 Almost contact and almost symplectic manifolds
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Albers, P.; Hofer, H., On the Weinstein conjecture in higher dimensions, Comment. Math. Helv., 84, 429-436, (2009) · Zbl 1166.53052
[2] Bennequin, D., Entrelacements et équations de Pfaff, in Third Schnepfenried Geometry Conference, Vol. 1 (Schnepfenried, 1982), Astérisque, 107, pp. 87-161. Soc. Math. France, Paris, 1983. · Zbl 0573.58022
[3] Bourgeois, F., Odd dimensional tori are contact manifolds, Int. Math. Res. Not., 30, 1571-1574, (2002) · Zbl 1021.53055
[4] Casals, R., Murphy, E. & Presas, F., Geometric criteria for overtwistedness. Preprint, 2015. arXiv:1503.06221 [math.SG]. · Zbl 1408.57026
[5] Casals, R.; Pancholi, D.; Presas, F., Almost contact 5-folds are contact, Ann. of Math., 182, 429-490, (2015) · Zbl 1333.53116
[6] Casals, R., del Pino, A. & Presas, F., \(h\)-principle for 4-dimensional contact foliations. Preprint, 2014. arXiv:1406.7328 [math.SG]. · Zbl 1401.53022
[7] Casals, R. & Presas, F., On the strong orderability of overtwisted 3-folds. Preprint, 2014. arXiv:1408.2077 [math.SG]. · Zbl 1348.53078
[8] Casals, R., Presas, F. & Sandon, S., On the non-existence of small positive loops of contactomorphisms on overtwisted contact manifolds. Preprint, 2014. arXiv:1403.0350 [math.SG]. · Zbl 1365.53072
[9] Chern, S. S., The geometry of \(G\)-structures, Bull. Amer. Math. Soc., 72, 167-219, (1966) · Zbl 0136.17804
[10] Chernov, V.; Nemirovski, S., Non-negative Legendrian isotopy in ST*\(M\), Geom. Topol., 14, 611-626, (2010) · Zbl 1194.53066
[11] Colin, V., Giroux, E. & Honda, K., On the coarse classification of tight contact structures, in Topology and Geometry of Manifolds (Athens, GA, 2001), Proc. Sympos. Pure Math., 71, pp. 109-120. Amer. Math. Soc., Providence, RI, 2003. · Zbl 1052.57036
[12] Eliashberg, Y., Classification of overtwisted contact structures on 3-manifolds, Invent. Math., 98, 623-637, (1989) · Zbl 0684.57012
[13] Eliashberg, Y., Topological characterization of Stein manifolds of dimension >2, Internat. J. Math., 1, 29-46, (1990) · Zbl 0699.58002
[14] Eliashberg, Y., Contact 3-manifolds twenty years Since J. martinet’s work, Ann. Inst. Fourier (Grenoble)., 42, 165-192, (1992) · Zbl 0756.53017
[15] Eliashberg, Y., Kim, S. S. & Polterovich, L., Geometry of contact transformations and domains: orderability versus squeezing. Geom. Topol., 10 (2006), 1635-1747; Erratum in Geom. Topol., 13 (2009), 1175-1176. · Zbl 1134.53044
[16] Eliashberg, Y.; Mishachev, N. M., Wrinkling of smooth mappings and its applications. I., Invent. Math., 130, 345-369, (1997) · Zbl 0896.58010
[17] Eliashberg, Y.; Polterovich, L., Partially ordered groups and geometry of contact transformations, Geom. Funct. Anal., 10, 1448-1476, (2000) · Zbl 0986.53036
[18] Etnyre, J. B., Contact structures on 5-manifolds. Preprint, 2012. arXiv:1210.5208 [math.SG]. · Zbl 0912.57019
[19] Geiges, H., Contact structures on 1-connected 5-manifolds, Mathematika., 38, 303-311, (1991) · Zbl 0724.57017
[20] Geiges, H., Applications of contact surgery, Topology, 36, 1193-1220, (1997) · Zbl 0912.57019
[21] Geiges, H., An Introduction to Contact Topology. Cambridge Studies in Advanced Mathematics, 109. Cambridge Univ. Press, Cambridge, 2008. · Zbl 1153.53002
[22] Geiges, H.; Thomas, C. B., Contact topology and the structure of 5-manifolds with \({π_{1} = Z_{2}}\), Ann. Inst. Fourier (Grenoble)., 48, 1167-1188, (1998) · Zbl 0912.57020
[23] Geiges, H.; Thomas, C. B., Contact structures, equivariant spin bordism, and periodic fundamental groups, Math. Ann., 320, 685-708, (2001) · Zbl 0983.57017
[24] Giroux, E., Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Invent. Math., 141, 615-689, (2000) · Zbl 1186.53097
[25] Giroux, E., Géométrie de contact: de la dimension trois vers les dimensions supérieures, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 405-414. Higher Ed. Press, Beijing, 2002. · Zbl 1015.53049
[26] Givental, A.B., Nonlinear generalization of the Maslov index, in Theory of Singularities and its Applications, Adv. Soviet Math., 1, pp. 71-103. Amer. Math. Soc., Providence, RI, 1990. · Zbl 0728.53024
[27] Gray, J. W., Some global properties of contact structures, Ann. of Math., 69, 421-450, (1959) · Zbl 0092.39301
[28] Gromov, M. L., Stable mappings of foliations into manifolds. Izv. Akad. Nauk SSSR Ser. Mat., 33 (1969), 707-734 (Russian); English translation in Math. USSR-Izv., 3 (1969), 671-694. · Zbl 1301.57020
[29] Gromov, M. L., Smoothing and inversion of differential operators, Mat. Sb., 88, 382-441, (1972)
[30] Gromov, M. L., Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82, 307-347, (1985) · Zbl 0592.53025
[31] Gromov, M. L., Partial Differential Relations. Ergebnisse der Mathematik und ihrer Grenzgebiete, 9. Springer, Berlin-Heidelberg, 1986. · Zbl 0651.53001
[32] Harris, B., Some calculations of homotopy groups of symmetric spaces, Trans. Amer. Math. Soc., 106, 174-184, (1963) · Zbl 0117.16501
[33] Honda, K., On the classification of tight contact structures., I. Geom. Topol., 4, 309-368, (2000) · Zbl 0980.57010
[34] Honda, K., On the classification of tight contact structures. II., J. Differential Geom., 55, 83-143, (2000) · Zbl 1038.57007
[35] Kwon, M. & van Koert, O., Brieskorn manifolds in contact topology. Preprint, 2013. arXiv:1310.0343 [math.SG]. · Zbl 1336.57002
[36] Lutz, R., Structures de contact sur LES fibrés principaux en cercles de dimension trois, Ann. Inst. Fourier (Grenoble)., 27, 1-15, (1977) · Zbl 0328.53024
[37] Lutz, R., Sur la géométrie des structures de contact invariantes, Ann. Inst. Fourier (Grenoble)., 29, 283-306, (1979) · Zbl 0379.53011
[38] Martinet, J., Formes de contact sur les variétés de dimension 3, in Proceedings of Liverpool Singularities Symposium, II (1969/1970), Lecture Notes in Math., 209, pp. 142-163. Springer, Berlin-Heidelberg, 1971.
[39] Murphy, E.; Niederkrüger, K.; Plamenevskaya, O.; Stipsicz, A. I., Loose legendrians and the plastikstufe, Geom. Topol., 17, 1791-1814, (2013) · Zbl 1301.57020
[40] Niederkrüger, K., The plastikstufe — a generalization of the overtwisted disk to higher dimensions, Algebr. Geom. Topol., 6, 2473-2508, (2006) · Zbl 1129.53056
[41] Niederkrüger K., On Fillability of Contact Manifolds. Mémoire d’habilitation á diriger des recherches, Université Toulouse III Paul Sabatier, 2013.
[42] Niederkrüger, K. & van Koert, O., Every contact manifolds can be given a nonfillable contact structure. Int. Math. Res. Not. IMRN, 23 (2007), Art. ID rnm115, 22 pp. · Zbl 1133.53053
[43] Niederkrüger, K.; Presas, F., Some remarks on the size of tubular neighborhoods in contact topology and fillability, Geom. Topol., 14, 719-754, (2010) · Zbl 1186.57020
[44] Presas, F., A class of non-fillable contact structures, Geom. Topol., 11, 2203-2225, (2007) · Zbl 1132.57023
[45] Sandon, S., An integer-valued bi-invariant metric on the group of contactomorphisms of \({{\mathbb R}^{2n} × S^{1}}\), J. Topol. Anal., 2, 327-339, (2010) · Zbl 1216.53077
[46] Ustilovsky, I., Infinitely many contact structures on \({S^{4m+1}}\), Int. Math. Res. Not., 14, 781-791, (1999) · Zbl 1034.53080
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.