The purpose of this note is to explain a combinatorial description of closed smooth oriented
–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
–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.
