Open Access
2016 Multisections of Lefschetz fibrations and topology of symplectic $4$–manifolds
R İnanç Baykur, Kenta Hayano
Geom. Topol. 20(4): 2335-2395 (2016). DOI: 10.2140/gt.2016.20.2335


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.


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R İnanç Baykur. Kenta Hayano. "Multisections of Lefschetz fibrations and topology of symplectic $4$–manifolds." Geom. Topol. 20 (4) 2335 - 2395, 2016.


Received: 20 March 2015; Revised: 31 August 2015; Accepted: 5 October 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1371.57014
MathSciNet: MR3548468
Digital Object Identifier: 10.2140/gt.2016.20.2335

Primary: 57M50 , 57R17 , 57R55 , 57R57
Secondary: 20F65 , 53D35 , 57R22

Keywords: Dehn twist factorization , exotic 4-manifold , exotic embedding , fiber sum , Lefschetz fibration , Lefschetz pencil , mapping class group , multisection , nonisomorphic fibration , Seiberg-Witten invariant , symplectic 4-manifold , symplectic Calabi-Yau , symplectic Kodaira dimension

Rights: Copyright © 2016 Mathematical Sciences Publishers

Vol.20 • No. 4 • 2016
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