## Geometry & Topology

### 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.

#### Article information

Source
Geom. Topol., Volume 20, Number 4 (2016), 2335-2395.

Dates
Revised: 31 August 2015
Accepted: 5 October 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510859026

Digital Object Identifier
doi:10.2140/gt.2016.20.2335

Mathematical Reviews number (MathSciNet)
MR3548468

Zentralblatt MATH identifier
1371.57014

#### Citation

Baykur, R İnanç; Hayano, Kenta. Multisections of Lefschetz fibrations and topology of symplectic $4$–manifolds. Geom. Topol. 20 (2016), no. 4, 2335--2395. doi:10.2140/gt.2016.20.2335. https://projecteuclid.org/euclid.gt/1510859026

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