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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 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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