Strongly overtwisted contact 3-manifolds

TL;DR

The paper introduces strongly overtwisted contact 3-manifolds and proves a complete h-principle for their contact structures, removing the need to fix an overtwisted disk and enabling a homotopy-type description via the Gauss map. It simultaneously yields a weak homotopy equivalence for the contactomorphism group, linking genuine and formal contactomorphisms in this flexible setting. By developing an h-principle for strongly overtwisted surfaces and employing the Microfibration trick, the authors extend Eliashberg’s OT principle to a broader, boundary-relative and boundary-free context, with explicit applications to overtwisted disks and Legendrian submanifolds. These results imply strong flexibility phenomena for embeddings and Legendrian isotopies in strongly overtwisted manifolds and offer parametric h-principles for OT disks in general overtwisted manifolds. Overall, the work provides a robust, obstruction-reduced framework for understanding the homotopy types of OT structures, the contactomorphism group, and Legendrian submanifolds in a broad class of 3-manifolds.

Abstract

We prove the existence of a subclass of overtwisted contact structures, called strongly overtwisted, on a 3-manifold that satisfy a complete h-principle without prescribing the contact structures over any subset of the 3-manifold. As a consequence, the homotopy type of the space of overtwisted disk embeddings into a strongly overtwisted contact 3-manifold is determined. A complete h-principle for a subclass of loose Legendrians is also derived from the main result. In general, the method allows us to deduce an h-principle for overtwisted disks that are fixed near the boundary in an arbitrary overtwisted contact 3-manifold.

Theorems & Definitions (35)

• Definition 1.2
• Theorem 1.3: Bennequin-Eliashberg Inequalities
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Theorem 1.8
• Corollary 1.9
• Remark 1.10
• Theorem 1.11