Primitivity of the fiber class

Determine whether the fiber class of every Lefschetz fibration over S^2 is primitive in the free part of H_2 of the total space.

Background

The divisibility m(f) of the fiber class controls the degrees of multisections. The authors note that the assertion that every fiber class is primitive would imply the existence of a 2-section for every Lefschetz fibration over S2. Although this primitivity assertion had previously been claimed, the cited proof contains a gap, and the authors explicitly state that its truth remains unknown.

References

In particular, it would follow from the stronger assertion that the fiber class of every Lefschetz fibration over $S2$ is primitive. This primitivity assertion was claimed in , but the proof contains a gap; see the appendix of . We do not know whether the assertion itself is true.

— Sections, pseudosections, and multisections of Lefschetz fibrations  (2610.01776 - Baykur et al., 1 Oct 2026) in Epilogue: existence of sections and 2-sections