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Multisections of
Lefschetz fibrations and topology of symplectic
$4$–manifolds
R İnanç Baykur and Kenta Hayano |
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Geometry & Topology 20 (2016) 2335–2395
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Abstract |
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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. |
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
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Mathematical Subject Classification 2010
Primary: 57M50, 57R17, 57R55, 57R57
Secondary: 53D35, 20F65, 57R22
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References |
Publication
Received: 20 March 2015
Revised: 31 August 2015
Accepted: 5 October 2015
Published: 15 September 2016
Proposed: Ronald Stern
Seconded: Ciprian Manolescu, Robion Kirby
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Authors |
| 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 |
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| http://people.math.umass.edu/~baykur | |
| Kenta Hayano | |
| Department of Mathematics Faculty of Science and Technology Keio University 3-14-1 Hiyoshi Kohoku-ku Yokohama Kanagawa 223-8522 Japan |
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| http://www.math.keio.ac.jp/~k-hayano/index_en.html | |
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