Every Contact Manifold Can Be given a Non-fillable Contact Structure

Recently Francisco Presas Mata constructed the first examples of closed contact manifolds of dimension larger than 3 that contain a plastikstufe, and hence are non-fillable. Using contact surgery on his examples we create on every sphere S, n ≥ 2, an exotic contact structure ξ − that also contains a plastikstufe. As a consequence, every closed contact manifold (M, ξ) (except S) can be converted into a contact manifold that is not (semi-positively) fillable by taking the connected sum (M, ξ)#(S, ξ − ). Most of the natural examples of contact manifolds can be realized as convex boundaries of symplectic manifolds. These manifolds are called symplectically fillable. An important class of contact manifolds that do not fall into this category are so-called overtwisted manifolds ([Eli88], [Gro85]). Unfortunately, the notion of overtwistedness is only defined for 3–manifolds. A manifold is overtwisted if one finds an embedded disk D such that TD ∣∣ ∂D ⊂ ξ, an overtwisted disk DOT. This topological definition gives an effective way to find many examples of contact 3–manifolds that are non-fillable. Until recently no example of a non-fillable contact manifold in higher dimension was known, but Francisco Presas Mata recently discovered a construction that allowed him to build many non-fillable contact manifolds of arbitrary dimension ([Pre06]). He showed that after performing this construction on certain manifolds, they admit the embedding of a plastikstufe. Roughly speaking, a plastikstufe PS(S) can be thought of as a disk-bundle over a closed (n−2)–dimensional submanifold S, where each fiber looks like an overtwisted disk. As shown in [Nie06b], the existence of such an object in a contact manifold (M, ξ) excludes the existence of a symplectic filling. In this paper we extend Presas’ results to a much larger class of contact manifolds. The idea is to start with one of his examples and to use contact surgery to simplify the topology and convert it into a contact sphere SE that admits the embedding a plastikstufe. If (M, ξ) is any other contact manifold, then (M, ξ)#SE ∼= M carries a contact structure that also has an embedded plastikstufe and is hence non-fillable. Definition. Let (M,α) be a cooriented (2n − 1)–dimensional contact manifold, and let S be a closed (n − 2)–dimensional manifold. A plastikstufe PS(S) with singular set S in M is an embedding of the n–dimensional manifold ι : D × S →֒ M that carries a (singular) Legendrian foliation given by the 1–form β := ια satisfying: • The boundary ∂PS(S) of the plastikstufe is the only closed leaf. • There is an elliptic singular set at {0} × S. • The rest of the plastikstufe is foliated by an S–family of stripes, each one diffeomorphic to (0, 1)×S, which are spanned between the singular set on one end and approach ∂PS(S) on the other side asymptotically. The importance of the plastikstufe lies in the following theorem. Theorem 1. Let (M,α) be a contact manifold containing an embedded plastikstufe. Then M does not have a semipositive strong symplectic filling. In particular, if dimM ≤ 5, then M does not have any strong symplectic filling at all. 1In the scope of this article a contact sphere will be a smooth sphere carrying a contact structure.