Getting a handle on contact manifolds

Abstract

In this thesis, we develop the details of a surgery theory for contact manifolds of arbitrary dimension via convex structures, extending the 3-dimensional theory developed by Giroux. The theory is analogous to that of Weinstein manifolds in symplectic geometry, with the key difference that the vector field does not necessarily have positive divergence everywhere. The surgery theory for contact manifolds contains the surgery theory for Weinstein manifolds via a sutured model for attaching critical points of low index. Using this sutured model, we show that the existence of convex structures on closed contact manifolds is guaranteed, a result equivalent to the existence of supporting Weinstein open book decompositions. In the final chapter, we provide a few words about how this theory is related to the Giroux correspondence between Weinstein open book decompositions and contact structures in three dimensions, as well as providing a framework for possible generalizations to higher dimensions and homotopy data.