Multisections of Lefschetz fibrations and topology of symplectic 4-manifolds

TL;DR

This paper develops a framework using positive multisections of Lefschetz fibrations to study symplectic 4-manifolds, enabling new insights and constructions in their topology and classification.

Contribution

It introduces methods to analyze symplectic surfaces via positive factorizations, producing novel examples and counterexamples in the topology of symplectic 4-manifolds.

Findings

Constructed new counter-examples to Stipsicz's conjecture

Produced non-isomorphic Lefschetz pencils on the same manifolds

Presented the first examples of exotic Lefschetz pencils

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.