  We show that the pre-order defined on the category of contact manifolds by
arbitrary symplectic cobordisms is considerably less rigid than its
counterparts for exact or Stein cobordisms: in particular, we exhibit large new
classes of contact 3-manifolds which are symplectically cobordant to something
overtwisted, or to the tight 3-sphere, or which admit symplectic caps
containing symplectically embedded spheres with vanishing self-intersection.
These constructions imply new and simplified proofs of several recent results
involving fillability, planarity and non-separating contact type embeddings.
The cobordisms are built from generalized symplectic handles which have cores
that are arbitrary symplectic surfaces with boundary and co-cores that are
symplectic disks or annuli; these can be attached to contact 3-manifolds along
sufficiently large neighborhoods of transverse links or pre-Lagrangian tori. We
also sketch a construction of J-holomorphic foliations in these cobordisms and
formulate a conjecture regarding maps induced on Embedded Contact Homology with
twisted coefficients.
