Convex hypersurface theory in contact topology

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.

arXiv commentV4: 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

References in corpus (6)

• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• Sutures and contact homology I
• The topology of Stein fillable manifolds in high dimensions I
• Braided embeddings of contact 3-manifolds in the standard contact 5-sphere
• Contact structures on 5-manifolds
• Bypass attachments in higher-dimensional contact topology

Cited by in corpus (6)

• The Legendrian Whitney trick
• Spaces of tight contact structures in dimension 3
• Applications of higher-dimensional Heegaard Floer homology to contact topology
• Existence of coLegendrians in contact $5$-manifolds
• Homological invariants of codimension 2 contact submanifolds
• A dynamical construction of Liouville domains