Existence and classification of overtwisted contact structures in all dimensions (Q265877)

The main result is the following:NEWLINENEWLINENEWLINETheorem. 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.NEWLINENEWLINENEWLINEIt 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.NEWLINENEWLINENEWLINEAn explicit classification of overtwisted contact structures on spheres is also given.