Bundles over finite 2-complexes have sections exactly when fiber inclusion is π1-injective and the fundamental-group short exact sequence splits; Lefschetz fibrations over disks admit an algebraic test for extendable boundary loops, yielding multiple sections on doubled sphere fibrations.
˙Inan¸ c Baykur, Robion Kirby, and Daniel Ruberman, editors.K3: A New Problem List in Low-Dimensional Topology, volume 295 ofMathematical Surveys and Monographs
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On sections of Lefschetz fibrations and bundles over 2-complexes
Bundles over finite 2-complexes have sections exactly when fiber inclusion is π1-injective and the fundamental-group short exact sequence splits; Lefschetz fibrations over disks admit an algebraic test for extendable boundary loops, yielding multiple sections on doubled sphere fibrations.