Multisections of Lefschetz fibrations and topology of symplectic $4$–manifolds

Abstract

We initiate a study of positive multisections of Lefschetz fibrations via positive factorizations in framed mapping class groups of surfaces. Using our methods, one can effectively capture various interesting symplectic surfaces in symplectic $4$–manifolds as multisections, such as Seiberg–Witten basic classes and exceptional classes, or branched loci of compact Stein surfaces as branched coverings of the $4$–ball. Various problems regarding the topology of symplectic $4$–manifolds, such as the smooth classification of symplectic Calabi–Yau $4$–manifolds, can be translated to combinatorial problems in this manner. After producing special monodromy factorizations of Lefschetz pencils on symplectic Calabi–Yau homotopy $K3$ and Enriques surfaces, and introducing monodromy substitutions tailored for generating multisections, we obtain several novel applications, allowing us to construct: new counterexamples to Stipsicz’s conjecture on fiber sum indecomposable Lefschetz fibrations, nonisomorphic Lefschetz pencils of the same genera on the same new symplectic $4$–manifolds, the very first examples of exotic Lefschetz pencils, and new exotic embeddings of surfaces.