Kirby Problem 3.40

Definitions and mathematical statement

Let $M$ be a closed connected oriented hyperbolic three-manifold. A positive cooriented contact structure is a smooth two-plane field $\xi=\ker\alpha\subset TM$, where $\alpha$ is a globally defined one-form and $\alpha\wedge d\alpha$ is everywhere positive for the orientation of $M$. Multiplying $\alpha$ by a positive function preserves the contact structure.

A knot is Legendrian when its tangent line lies in $\xi$. An overtwisted disk is an embedded disk whose boundary is Legendrian and whose contact framing along that boundary agrees with the disk framing; equivalently its boundary has Thurston--Bennequin number zero relative to the disk. A contact structure is tight if it contains no overtwisted disk.

Does every closed orientable hyperbolic three-manifold admit a tight contact structure? One may choose the plane field freely; the question does not ask that a given contact structure be tight. No compatibility of the contact planes with the hyperbolic metric is imposed. In particular, the desired structure need not arise from a chosen geometric flow or foliation.

The target is tightness itself. Existence of an overtwisted contact structure does not suffice, while requiring a symplectic filling would add a stronger condition which is not part of this problem.

Short English statement

Does every closed oriented hyperbolic three-manifold support a contact plane field with no overtwisted disk?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2838 (KP-3.40), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.40. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.