Kirby Problem 4.117

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 genus-$g$ trisection of a closed connected oriented smooth four-manifold $X$ is a decomposition $X=X_1\cup X_2\cup X_3$ such that $X_i$ is a four-dimensional $1$-handlebody (a boundary connected sum of $k_i$ copies of $S^1\times B^3$), each $X_i\cap X_j$ is a three-dimensional genus-$g$ handlebody, and $X_1\cap X_2\cap X_3=\Sigma_g$ is a closed genus-$g$ surface. Corners in these descriptions are rounded. Write $g(X)$ for the minimum of $g$ over all trisections of $X$. The Euler characteristic is $\chi(X)=\sum_j(-1)^j\dim_{\mathbb Q}H_j(X;\mathbb Q)$, and the rank $\operatorname{rk}\pi_1(X)$ is the minimum cardinality of a group generating set (not the rank of its abelianization).

Statement recorded in the supplied source.

For every closed connected oriented smooth four-manifold $X$, is \[ g(X)=\chi(X)-2+3\operatorname{rk}\pi_1(X)? \] Thus the question asks whether the known lower bound is always attained. [S2]

Short English statement

Is the standard Euler-characteristic and fundamental-group lower bound for the trisection genus always sharp?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2993 (KP-4.117). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 4.117, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.