  The purpose of this paper is to introduce Liouville hypersurfaces in contact
manifolds, which generalize ribbons of Legendrian graphs and pages of
supporting open books. Liouville hypersurfaces are used to define a gluing
operation for contact manifolds called the Liouville connect sum. Performing
this operation on a contact manifold $(M, \xi)$ gives an exact -- and in many
cases, Weinstein -- cobordism whose concave boundary is $(M, \xi)$ and whose
convex boundary is the surgered manifold. These cobordisms are used to
establish the existence of "fillability" and "non-vanishing contact homology"
monoids in symplectomorphism groups of Liouville domains, study the symplectic
fillability of a family of contact manifolds which fiber over the circle,
associate cobordisms to certain branched coverings of contact manifolds, and
construct exact symplectic cobordisms that do not admit Weinstein structures.
  The Liouville connect sum generalizes the Weinstein handle attachment and is
used to extend the definition of contact $(1/k)$-surgery along Legendrian knots
in contact 3-manifolds to contact $(1/k)$-surgery along Legendrian spheres in
contact manifolds of arbitrary dimension. We use contact surgery to construct
exotic contact structures on $5$- and $13$-dimensional spheres after
establishing that $S^{2}$ and $S^{6}$ are the only spheres along which
generalized Dehn twists smoothly square to the identity mapping. The exoticity
of these contact structures implies that Dehn twists along $S^{2}$ and $S^{6}$
do not symplectically square to the identity, generalizing a theorem of Seidel.
A similar argument shows that the $(2n+1)$-dimensional contact manifold
determined by an open book whose page is $(T^{*}S^{n}, -\lambda_{can})$ and
whose monodromy is any negative power of a symplectic Dehn twist is not exactly
fillable.
