  A geometric obstruction, the so called "plastikstufe", for a contact
structure to not being fillable has been found by K. Niederkruger. 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 with a "plastikstufe". They are the first examples of
non-fillable contact closed high-dimensional manifolds. In particular we show
that $S^3 \times \prod_j \Sigma_{j}$, for $g(\Sigma_j)\geq 2$, possesses this
kind of contact structure and so any connected sum with those manifolds also
does it.
