  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.
