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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 theory)
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

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