Kirby Problem 4.120

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.

An achiral Lefschetz pencil on a smooth closed four-manifold $X$ consists of a finite base set $B\subset X$ and a map $f:X\setminus B\to S^2$ with finitely many isolated nodal critical points. Up to smooth coordinate changes, its local critical-point model is $(z_1,z_2)\mapsto z_1^2+z_2^2$, and at a base point its local model is $[z_1:z_2]\in\mathbb{CP}^1$. Coordinates in these models may reverse orientations, in contrast to a chiral pencil. Away from these points the map is a submersion. The source allows no critical points and/or no base points, and allows nonorientable $X$.

Statement recorded in the supplied source.

Characterize the closed smooth four-manifolds admitting an achiral Lefschetz pencil. In particular, does every closed simply connected smooth four-manifold admit such a pencil on the original manifold itself? [S2]

Short English statement

Determine which closed smooth four-manifolds support an achiral Lefschetz pencil, especially whether every simply connected one does.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2996 (KP-4.120). 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 4.120, author preliminary version. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.