Broken Lefschetz fibrations and mapping class groups

Abstract

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.