A class of non-fillable contact structures

Abstract

A geometric obstruction, the so called “PS–structure”, for a contact structure to not being fillable has been found by Niederkrüger. This generalizes somehow the concept of overtwisted structure to dimensions higher than $3$. This paper elaborates on the theory showing a big number of closed contact manifolds having a "PS–structure". So, they are the first examples of non-fillable high dimensional closed contact manifolds. In particular we show that $S^3\times {\prod }_j{\Sigma }_j$, for $g\left({\Sigma }_j\right)\ge 2$, possesses this kind of contact structure and so any connected sum with those manifolds also does it.