Existence of sections for Lefschetz fibrations over the 2–sphere

Determine whether every smooth 4‑dimensional Lefschetz fibration π: M → S^2 admits a smooth section σ: S^2 → M.

Background

The paper constructs infinitely many homologically distinct sections for a broad class of smooth and symplectic Lefschetz fibrations, but the construction requires the existence of at least one section. While many examples admit sections and the authors provide criteria ensuring their abundance, the general existence of a section for an arbitrary Lefschetz fibration remains unresolved.

This question is fundamental because sections play a central role in linking families of curves to monodromy representations and in translating section counts into geometric information via tools such as the Parshin trick in the holomorphic setting. Clarifying universal existence in the smooth or symplectic category would sharpen the contrast with the holomorphic case and inform broader structural results about Lefschetz fibrations.