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Broken Lefschetz fibrations and mapping class groups

R İnanç Baykur and Kenta Hayano

Geometry & Topology Monographs 19 (2015) 269–290

arXiv: 1410.5531


The purpose of this note is to explain a combinatorial description of closed smooth oriented 4–manifolds in terms of positive Dehn twist factorizations of surface mapping classes, and further explore these connections. This is obtained via monodromy representations of simplified broken Lefschetz fibrations on 4–manifolds, for which we provide an extension of Hurwitz moves that allows us to uniquely determine the isomorphism class of a broken Lefschetz fibration. We furthermore discuss broken Lefschetz fibrations whose monodromies are contained in special subgroups of the mapping class group; namely, the hyperelliptic mapping class group and in the Torelli group, respectively, and present various results on them which extend or contrast with those known to hold for honest Lefschetz fibrations. Lastly, we observe that there are infinitely many pairwise nonisomorphic broken Lefschetz fibrations with smoothly isotopic regular fibers.

broken Lefschetz fibration, Hurwitz system
Mathematical Subject Classification 2010
Primary: 30F99, 57R45
Received: 3 June 2015
Accepted: 3 June 2015
Published: 29 December 2015
R İnanç Baykur
Department of Mathematics and Statistics
University of Massachusetts
Amherst, MA 01003-9305
Kenta Hayano
Department of Mathematics
Graduate School of Science
Hokkaido University
Sapporo, Hokkaido 060-0810