Kirby Problem 3.43

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A positive cooriented contact structure on an oriented closed $3$-manifold $Y$ is $\xi=\ker\alpha$, with $\alpha\wedge d\alpha>0$. It is tight if it has no overtwisted disk. A symplectic cobordism from $(Y_-,\xi_-)$ to $(Y_+,\xi_+)$ is a compact symplectic $4$-manifold with boundary $-Y_-\sqcup Y_+$. It is exact when $\omega=d\lambda$ and the Liouville vector field defined by $\iota_V\omega=\lambda$ points inward at $Y_-$, outward at $Y_+$, and induces the specified contact structures. A weak symplectic cobordism satisfies positivity of $\omega|_{\xi_\pm}$ on the oriented contact planes. A weak filling is connected, has only the positive boundary and that positivity condition. The support genus $\operatorname{sg}(Y,\xi)$ is the minimum page genus among open books supporting $\xi$. An open book supports $\xi$ when some contact form is positive on its oriented binding and its differential is positive on page interiors. The Giroux torsion $\operatorname{GT}(Y,\xi)$ is the supremum of integers $n\ge1$ for which \[ \left(T^2\times[0,1],\ker(\cos(2\pi nt)\,dx+\sin(2\pi nt)\,dy)\right) \] contact-embeds in $(Y,\xi)$, with value $0$ if none and $\infty$ if unbounded. Legendrian surgery is contact $(-1)$ surgery on a knot tangent to $\xi$, with framing one less than the contact framing.

Statement recorded in the supplied source.

Determine how contact properties change under symplectic cobordisms, in particular:

1. If $\xi_-$ is tight and an exact symplectic cobordism runs from $(Y_-,\xi_-)$ to $(Y_+,\xi_+)$, must $\xi_+$ be tight?
2. If $\xi_-$ is weakly fillable and there is a weak symplectic cobordism to $(Y_+,\xi_+)$, must $\xi_+$ be weakly fillable?
3. Determine the changes in support genus and Giroux torsion under Legendrian surgery; in particular, must support genus be nondecreasing and Giroux torsion nonincreasing?

[S2]

Short English statement

Does exact symplectic cobordism preserve tightness, does weak cobordism preserve weak fillability, and how does Legendrian surgery change support genus and Giroux torsion?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2841 (KP-3.43). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (editors), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), Problem 3.43, p. 162 and following remarks; original author version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.