Kirby Problem 2.22

Definitions and mathematical statement

A positive Lefschetz fibration over the sphere is a smooth surjection $p:X\to S^2$ from a closed oriented smooth four-manifold with connected closed oriented regular fiber $F$, finitely many critical points, and local orientation-compatible complex coordinates in which each critical point has model $(z_1,z_2)\mapsto z_1z_2$. Assume at least one critical point, as required in the source. Away from the critical values, $p$ is a smooth fiber bundle.

A section is a smooth map $s:S^2\to X$ satisfying $p\circ s=\operatorname{id}_{S^2}$. Such a map is necessarily an embedding and avoids the critical points. Does every positive Lefschetz fibration in this class admit a section?

The fiber genus and number of singular fibers may vary, and the section may be chosen freely. It need not have a prescribed self-intersection number. The assertion does not restrict to fibrations obtained by blowing up pencils, where exceptional spheres already provide sections.

For fiber genus at least two, an equivalent formulation uses monodromy: given a positive factorization of the identity into Dehn twists in $\operatorname{Mod}(F)$ describing the fibration, can the twists be lifted to positive twists in the mapping class group fixing one marked point so that their product is still the identity? A continuous multisection meeting a regular fiber several times does not by itself supply the one-point section asked for here. Achiral fibrations, allowing negative singularities, are a different class.

Short English statement

Does every positive Lefschetz fibration over the sphere with singular fibers have a section?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2770 (KP-2.22), 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 2.22. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.