Geometry & Topology
- Geom. Topol.
- Volume 20, Number 4 (2016), 2335-2395.
Multisections of Lefschetz fibrations and topology of symplectic $4$–manifolds
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 –manifolds as multisections, such as Seiberg–Witten basic classes and exceptional classes, or branched loci of compact Stein surfaces as branched coverings of the –ball. Various problems regarding the topology of symplectic –manifolds, such as the smooth classification of symplectic Calabi–Yau –manifolds, can be translated to combinatorial problems in this manner. After producing special monodromy factorizations of Lefschetz pencils on symplectic Calabi–Yau homotopy 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 –manifolds, the very first examples of exotic Lefschetz pencils, and new exotic embeddings of surfaces.
Geom. Topol., Volume 20, Number 4 (2016), 2335-2395.
Received: 20 March 2015
Revised: 31 August 2015
Accepted: 5 October 2015
First available in Project Euclid: 16 November 2017
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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
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