Classification of overtwisted contact structures on 3-manifolds. (English) Zbl 0684.57012

Eliashberg, Y.

Invent. Math. 98, No. 3, 623-637 (1989).

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.)

Keywords:

overtwisted contact structure; oriented connected 3-manifold; overtwisted disc

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