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 K3 and Enriques surfaces, and introducing monodromy substitutions tailored for generating multisections, we obtain several novel applications, allowing us to construct: new counter-examples to Stipsicz's conjecture on fiber sum indecomposable Lefschetz fibrations, non-isomorphic 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.

49 pages, lots of figures. This extends (with various applications) and replaces the earlier version

References in corpus (1)

• Topological triviality of smoothly knotted surfaces in 4-manifolds

Cited by in corpus (8)

• Positive factorizations of mapping classes
• Sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration
• Exotically knotted disks and complex curves
• Kodaira Dimension in Low Dimensional Topology
• On a class of symplectic $4$-orbifolds with vanishing canonical class
• Unchaining surgery, branched covers, and pencils on elliptic surfaces
• Universal Lefschetz fibrations and Lefschetz cobordisms
• The existence of an indecomposable minimal genus two Lefschetz fibration