Volume 20, issue 4 (2016)

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

R İnanç Baykur and Kenta Hayano

Geometry & Topology 20 (2016) 2335–2395
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.

Keywords
symplectic 4-manifold, exotic 4-manifold, Lefschetz fibration, Lefschetz pencil, multisection, nonisomorphic fibration, mapping class group, Dehn twist factorization, exotic embedding, symplectic Kodaira dimension, symplectic Calabi-Yau, fiber sum, Seiberg-Witten invariant
Mathematical Subject Classification 2010
Primary: 57M50, 57R17, 57R55, 57R57
Secondary: 53D35, 20F65, 57R22