Every Contact Manifolds can be given a Nonfillable Contact Structure

Abstract

Recently Francisco Presas Mata constructed the first examples of closed contact manifolds of dimension larger than 3 that contain a plastikstufe, and are hence nonfillable. Using contact surgery on his examples, we create on every sphere 𝕊2n−1, n ≥ 2, an exotic contact structure ξ− that allows us to embed a plastikstufe. As a consequence, every closed contact manifold (M, ξ) (with exception of 𝕊1) can be converted by taking the connected sum (M, ξ)#(𝕊2n−1, ξ−) into a contact manifold that is not (semi-positively) fillable.