Volume 20, issue 4 (2016)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 21
Issue 6, 3191–3810
Issue 5, 2557–3190
Issue 4, 1931–2555
Issue 3, 1285–1930
Issue 2, 647–1283
Issue 1, 1–645

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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