paper

Convex hypersurface theory in contact topology

arXiv:1907.06025

Abstract

We lay the foundations of convex hypersurface theory in contact topology, extending the work of Giroux in dimension three. Specifically, we prove that any closed hypersurface in a contact manifold can be $C^{0}$ -approximated by a convex one. We also prove that a $C^{0}$ -generic family of mutually disjoint closed hypersurfaces parametrized by $t \in [0, 1]$ is convex except at finitely many times $t_{1}, \dots, t_{N}$ , and that crossing each $t_{i}$ corresponds to a bypass attachment. As an application, we prove the existence of compatible (relative) open book decompositions for contact manifolds.

V4: Added two coauthors: Joseph Breen and Austin Christian; the part on contact submanifolds has been removed and will be written more carefully in a separate paper; the proof of the bypass-bifurcation correspondence has been expanded and the paper is now essentially self-contained with a 24-page appendix on bypasses in higher dimensions