Kirby Problem 3.42

Definitions and mathematical statement

Let $(Y,\xi)$ be a closed oriented three-manifold with a positive cooriented contact structure, so $\xi=\ker\alpha$ for a one-form satisfying $\alpha\wedge d\alpha>0$. It is tight if it contains no overtwisted disk: an embedded disk with Legendrian boundary whose contact framing agrees with its disk framing. Legendrian means that the tangent line of the boundary lies in $\xi$.

For each positive integer $n$, give $T^2\times[0,1]$, with circle coordinates $x,y\in\mathbb R/\mathbb Z$, the contact structure \[ \xi_n=\ker\left(\cos(2\pi nt)\,dx+\sin(2\pi nt)\,dy\right). \] Choose its contact orientation. Define the Giroux torsion $\operatorname{GT}(Y,\xi)$ as the supremum of those integers $n$ for which this contact manifold embeds in $(Y,\xi)$, setting the value to zero if no positive $n$ occurs and to infinity if they are unbounded. A contact embedding carries the displayed plane field to $\xi$.

Must $\operatorname{GT}(Y,\xi)$ be finite whenever $\xi$ is tight? The proposed bound may depend on the entire contact manifold; no universal numerical bound across all tight structures is asked for.

The embedding is allowed to vary with $n$. Therefore, bounding the twisting in one chosen torus neighborhood is not enough: one must rule out all embeddings of layers of arbitrarily large integral torsion. The full $2\pi$-turn convention fixes the normalization.

Short English statement

Can one tight contact structure contain torus layers with arbitrarily many full turns of contact twisting?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2840 (KP-3.42), 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 3.42. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.