Abstract. For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five), while also being obstructed by all known manifestations of "overtwistedness". We also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense. These depend on a higher dimensional analogue of Giroux torsion, which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary.

In this article, we give a first prototype-definition of overtwistedness in higher dimensions. According to this definition, a contact manifold is called overtwisted if it contains a plastikstufe, a submanifold foliated by the contact structure in a certain way. In three dimensions the definition of the plastikstufe is identical to the one of the overtwisted disk. The main justification for this definition lies in the fact that the existence of a plastikstufe implies that the contact manifold does not have a (semipositive) symplectic filling. 53D10, 57R17; 53D35The situation of contact topology can be roughly stated like this: the 3-dimensional contact manifolds can be understood very adequately by topological methods, a farreaching classification has been achieved and relations to many other fields have been established. In contrast, the world map of higher-dimensional contact geometry consists almost entirely of white spots. A powerful method for constructing such manifolds is contact surgery, the most promising technique developed so far to distinguish different contact structures is contact homology and with Giroux's open book decomposition, it is hoped that some classification results could be obtained.The first structural distinction found for contact 3-manifolds was the notion of overtwistedness. It turned out that such manifolds firstly do not allow an (even weak) symplectic filling by Eliashberg [4] and Gromov [8], and secondly can be classified in a very satisfactory way as in Eliashberg [3].In higher dimensions, surprisingly, no analogous criterion has yet been found. Giroux has proposed a definition based on his open book decomposition, which in three dimensions is completely equivalent to the standard one. In contrast, our definition is based on the existence of a plastikstufe, a direct generalization of the overtwisted disk. In Gromov's famous paper on holomorphic curves [8], a sketchy description of something, which possibly could be a plastikstufe, is given. The generalization of overtwistedness described in this article was found independently by Yuri Chekanov. Interestingly, his (unpublished) proof of Theorem 1 uses very different methods. The definition of the plastikstufe given in this paper is certainly only a preliminary version, meant as a prototype leading to a criterion for nonfillability in higher dimensions. Our definition implies the following theorem.Theorem 1 Let .M;˛/ be a contact manifold containing an embedded plastikstufe. Then M does not have any semipositive symplectic filling. If dim M Ä 5, then M does not have any symplectic filling at all.

English: We prove several results on weak symplectic fillings of contact 3-manifolds, including: (1) Every weak filling of any planar contact manifold can be deformed to a blow up of a Stein filling.(2) Contact manifolds that have fully separating planar torsion are not weakly fillable-this gives many new examples of contact manifolds without Giroux torsion that have no weak fillings. (3) Weak fillability is preserved under splicing of contact manifolds along symplectic pre-Lagrangian tori-this gives many new examples of contact manifolds without Giroux torsion that are weakly but not strongly fillable.

Abstract. We show that the presence of a plastikstufe induces a certain degree of flexibility in contact manifolds of dimension 2n + 1 > 3. More precisely, we prove that every Legendrian knot whose complement contains a "nice" plastikstufe can be destabilized (and, as a consequence, is loose). As an application, it follows in certain situations that two non-isomorphic contact structures become isomorphic after connectsumming with a manifold containing a plastikstufe.