Bypass moves in convex hypersurface theory

TL;DR

This paper introduces higher-dimensional bypass attachments in contact and Weinstein geometry, providing tools to relate Weinstein domains and establish an h-principle for convex hypersurfaces.

Contribution

It constructs higher-dimensional bypass attachments, characterizes their role in Weinstein handle decompositions, and proves their necessity and sufficiency for Weinstein domain equivalences.

Findings

Bypass attachments enable transformations between Weinstein hypersurfaces.

They provide a complete set of moves for Weinstein domain equivalence.

The construction recovers an existence h-principle for Weinstein hypersurfaces.

Abstract

We construct bypass attachments in higher dimensional contact manifolds that, when attached to a neighborhood of a Weinstein hypersurface, yield a neighborhood of a new Weinstein hypersurface, obtained via local modifications to the Weinstein handle decomposition of the first. For context, we give $3$-dimensional analogues of these bypass attachments and discuss their appearance in nature. We then show that our bypass attachments give a necessary and sufficient set of moves relating any two Weinstein domains which become almost symplectomorphic after one stabilization. Finally, we use our construction to produce several examples of interesting convex hypersurfaces and recover an existence $h$-principle for Weinstein hypersurfaces.