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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

##### MSC:
 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
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