Liouville hypersurfaces and connect sum cobordisms

Abstract

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 $(T^\ast S^n , -\lambda_{can})$ and whose monodromy is any negative power of a symplectic Dehn twist is not exactly fillable.