Geometry & Topology

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

R İnanç Baykur and Kenta Hayano

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

Permanent link to this document
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

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds 57R17: Symplectic and contact topology 57R55: Differentiable structures 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]
Secondary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx] 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 57R22: Topology of vector bundles and fiber bundles [See also 55Rxx]

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

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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