Getting a handle on contact manifolds

TL;DR

This paper extends contact manifold surgery theory to higher dimensions using convex structures, paralleling Weinstein manifold theory, and proves the existence of convex structures on closed contact manifolds.

Contribution

It develops a higher-dimensional surgery theory for contact manifolds based on convex structures, generalizing 3D Giroux theory and linking to Weinstein open books.

Findings

Established a sutured model for attaching critical points.

Proved existence of convex structures on closed contact manifolds.

Connected convex structures to Weinstein open book decompositions.

Abstract

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.