Kirby Problem 3.45

Definitions and mathematical statement

A positive cooriented contact structure on a closed oriented three-manifold $Y$ is a plane field $\xi=\ker\alpha$ with $\alpha\wedge d\alpha>0$. An open book is a link $B\subset Y$ and a fibration $Y\setminus B\to S^1$ whose fibers close up to compact oriented surfaces with boundary $B$. Near the binding, this map is the angular coordinate of a normal disk. These compact surfaces are its pages.

The open book supports $\xi$ if some contact form for $\xi$ is positive on the oriented binding and its exterior derivative restricts positively to the interior of every page. Define the support genus \[ \operatorname{sg}(Y,\xi)=\min\{g(F):F\text{ is a page of a supporting open book}\}. \] The minimum ranges over all supporting open books, with any number of binding components, and exists because contact structures admit supporting open books.

(a) Is there a contact three-manifold with $\operatorname{sg}(Y,\xi)>1$? (b) Are these genera unbounded: for every integer $N\ge1$, is there a contact three-manifold with support genus at least $N$?

The manifold and contact structure may vary with $N$. Exhibiting a supporting page of high genus gives only an upper bound for this minimum. An example must exclude every supporting open book of smaller genus. In particular, stabilizing one open book repeatedly does not establish either assertion, because the original smaller-genus book remains available.

Short English statement

Does any contact structure require supporting pages of genus at least two, and can the required genus grow without bound?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2843 (KP-3.45), 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 3.45. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.