Bott-integrability of overtwisted contact structures

TL;DR

The paper provides a complete criterion for when an overtwisted contact structure on a closed 3-manifold is Bott-integrable, namely that $PD\left(e(\xi)\right)$ must be represented by a graph link. It develops neighbourhood and perturbation results for the Bott integral’s critical set and proves a key Euler-class relation linking critical orbits to $e(\xi)$. It then proves the ‘if’ direction for graph manifolds using Yano’s graph-link criterion, constructs $H_1$-complete Bott-integrable structures on Seifert pieces, and extends them via JSJ- and connected-sum techniques. The paper also provides explicit Bott-integrable examples on the mapping torus of Arnold’s cat map, illustrating the full classification of Bott-integrable structures on a nontrivial graph-manifold family. Collectively, these results connect Reeb dynamics, graph-manifold topology, and Eliashberg’s overtwisted classification to yield a detailed landscape of Bott-integrable overtwisted contact structures on closed 3-manifolds.

Abstract

We show that an overtwisted contact structure on a closed, oriented 3-manifold can be defined by a contact form having a Bott-integrable Reeb flow if and only if the Poincaré dual of its Euler class is represented by a graph link.

Figures (1)

• Figure 1: Introducing a Lutz component

Theorems & Definitions (29)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 2.1
• Theorem 2.2: Neighbourhood theorem for critical tori
• proof : Proof of Theorem \ref{['thm:critical-T']}
• Theorem 2.3: Neighbourhood theorem for critical Klein bottles
• proof
• Lemma 3.1
• proof