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Classification of overtwisted contact structures on 3-manifolds. (English) Zbl 0684.57012
Consider in \({\mathbb{R}}^ 3\) the cylindric coordinates (\(\rho\),\(\phi\),z). The disc \(\Delta =\{(\rho,\phi,z):\) \(z=0\), \(\rho\leq \pi \}\) with the germ of the contact structure \(\zeta_ 1\) defined on \(\Delta\) by the equation cos \(\rho\) dz\(+\rho \sin \rho d\phi =0\) is called the standard overtwisted disc. Let M be an oriented connected 3-manifold. A contact structure \(\zeta\) on M is called overtwisted if there is a contact embedding of the standard overtwisted disc \((\Delta,\zeta_ 1)\) into (M,\(\zeta)\). Fix a point \(p\in M\) and an embedded 2-disc \(\Delta '\subset M\) centred at p. Let Distr(M) be the space of all 2-dimensional distributions on M fixed at p, equipped with the \(C^{\infty}\)-topology. Denote by Cont(M) the subspace of Distr(M) which consists of positive contact structures and by \(Cont^{ot}(M)\) the subspace of Cont(M) containing all overtwisted structures which have the disc \(\Delta '\subset M\) as the standard overtwisted disc. The author proves that the inclusion j: \(Cont^{ot}(M)\to Distr(M)\) is a homotopy equivalence.
Reviewer: A.Piatkowski

57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
MathOverflow Questions:
Tight vs. overtwisted contact structure
Full Text: DOI EuDML
[1] Bennequin, D.: Entrelacements et équations de Pfaff. Asterisque107-108, 87-161 (1983). · Zbl 0573.58022
[2] Eliashberg, Y.: The complexification of contact structures on 3-manifold. Usp. Math. nauk.6, (40) 161-162 (1985)
[3] Eliashberg, Y.: Filling by holomorphic discs and its applications. Proc. Symp. Pure Math. (to appear) · Zbl 0731.53036
[4] Eliashberg, Y.: On symplectic manifolds bounded by the standard contact sphere and exotic contact structures on spheres of dimension >3. J. Differ. Geom. (to appear)
[5] Erlandsson, T.: Geometry of contact transformations in dimension 3. Doctoral dissertation, Uppsala (1981)
[6] Gunning, R.C., Rossi, H.: Analytic functions of several complex variables. New Jersey: Prentice-Hall 1985, IX.C.4 · Zbl 1204.01045
[7] Gray, J.W.: Some global properties of contact structures. Ann. Math.69, 421-450 (1959) · Zbl 0092.39301
[8] Gromov, M.: Stable mappings of foliations into manifolds. Izv. Acad. Naur SSSR, Ser. mat.33, 1206-1209, (1969)
[9] Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math.82, 307-347 (1985) · Zbl 0592.53025
[10] Gonzalo, J., Varela, F.: Modèles globaux des varietes de contact. Actes du Schnepfenried (1982)
[11] Lutz, R.: Structures de contact sur les fibrés principaux en cercles de dimension 3. Ann. Inst. Fourier3, 1-15 (1977) · Zbl 0328.53024
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