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

R İnanç Baykur and Kenta Hayano

Geometry & Topology 20 (2016) 2335–2395

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.

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
Mathematical Subject Classification 2010
Primary: 57M50, 57R17, 57R55, 57R57
Secondary: 53D35, 20F65, 57R22
Received: 20 March 2015
Revised: 31 August 2015
Accepted: 5 October 2015
Published: 15 September 2016
Proposed: Ronald Stern
Seconded: Ciprian Manolescu, Robion Kirby
R İnanç Baykur
Department of Mathematics and Statistics
University of Massachusetts
Lederle Graduate Research Tower
710 North Pleasant Street
Amherst, MA 01003-9305
United States
Kenta Hayano
Department of Mathematics
Faculty of Science and Technology
Keio University
Kanagawa 223-8522