Kirby Problem 3.44

Definitions and mathematical statement

An abstract open book consists of a compact connected oriented surface $F$ with nonempty boundary and a mapping class $\phi$ represented by a diffeomorphism equal to the identity near $\partial F$. Its three-manifold is formed from the mapping torus of $\phi$ by filling each boundary torus with $\partial F\times D^2$. Giroux's correspondence assigns a contact structure $\xi$ up to isotopy: a supporting contact form is positive on the binding and has positive exterior derivative on each page.

Encode $F$ by finite triangulation data and $\phi$ by a finite word in explicitly specified Dehn twists. Is there an algorithm which halts on every such input and decides whether the supported contact structure is tight? Tight means that there is no embedded overtwisted disk, namely a disk with Legendrian boundary and zero twisting of its contact framing relative to the disk.

Ask separately for decision algorithms for fillability, specifying the chosen meaning. A weak filling is a compact symplectic four-manifold with boundary $Y$ and $\omega|_\xi>0$. A strong filling additionally has an outward Liouville vector field near its boundary inducing $\xi$. An exact filling requires a global primitive of $\omega$ compatible with that boundary. A Stein filling comes from a compact strictly pseudoconvex Stein domain inducing $\xi$.

These are distinct decision problems. The source leaves "fillable" unspecified, so no equivalence among these meanings is assumed. The input fixes a contact structure, rather than asking only whether its underlying manifold supports some fillable structure.

Short English statement

Can tightness and the separate notions of fillability be decided from a finite open-book description?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2842 (KP-3.44), 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 3.44. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.