Contact handle decompositions

TL;DR

This paper reviews contact handle decompositions in dimension three, demonstrating how bypass attachments relate to contact handles and providing explicit decompositions of overtwisted 3-spheres, reaffirming Giroux's foundational results.

Contribution

It clarifies the structure of bypass attachments as contact handles and offers explicit decompositions of overtwisted 3-spheres, extending Giroux's work.

Findings

Bypass attachment consists of contact 1 and 2-handles

Explicit contact handle decompositions of overtwisted 3-spheres

Alternative proof that all compact contact 3-manifolds admit handle decompositions

Abstract

We review Giroux's contact handles and contact handle attachments in dimension three and show that a bypass attachment consists of a pair of contact 1 and 2-handles. As an application we describe explicit contact handle decompositions of infinitely many pairwise non-isotopic overtwisted 3-spheres. We also give an alternative proof of the fact that every compact contact 3-manifold (closed or with convex boundary) admits a contact handle decomposition, which is a result originally due to Giroux.