Existence and classification of overtwisted contact structures in all   dimensions
  We establish a parametric extension $h$-principle for overtwisted contact structures on manifolds of all dimensions, which is the direct generalization of the $3$-dimensional result from \cite{Eli89}. It implies, in particular, that any closed manifold admits a contact structure in any given homotopy class of almost contact structures. 
Matthew Strom Borman, Yakov Eliashberg, Emmy Murphy
math.SG math.GT