Mathematics – Symplectic Geometry
Scientific paper
2012-04-14
Mathematics
Symplectic Geometry
54 pages, 12 figures
Scientific paper
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 $\Mxi$ gives an exact -- and in many cases, Stein - cobordism from $\Mxi$ to 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 Stein 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. This is used 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.
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