We construct four-dimensional symplectic cobordisms between contact
three-manifolds generalizing an example of Eliashberg. One key feature is
that any handlebody decomposition of one of these cobordisms must involve
three-handles. The other key feature is that these cobordisms contain chains of
symplectically embedded two-spheres of square zero. This, together with
standard gauge theory, is used to show that any contact three-manifold of
non-zero torsion (in the sense of Giroux) cannot be strongly symplectically
fillable. John Etnyre pointed out to the author that the same argument
together with compactness results for pseudo-holomorphic curves implies
that any contact three-manifold of non-zero torsion satisfies the Weinstein
conjecture. We also get examples of weakly symplectically fillable contact
three-manifolds which are (strongly) symplectically cobordant to overtwisted
contact three-manifolds, shedding new light on the structure of the set of
contact three-manifolds equipped with the strong symplectic cobordism partial
order.
