A Lefschetz fibration is a smooth map from a 4-manifold to a surface with prescribed behavior near its (isolated) critical points. These fibrations play an important role in the study of symplectic 4-manifolds, due to work of Donaldson and Gompf. If however we loosen the definition of Lefschetz fibrations to allow for embedded circles of critical points, we obtain broken Lefschetz fibrations, whose connection to near-symplectic structures was established by Auroux, Donaldson, and Katzarkov.
Various techniques have been employed to construct such fibrations by different authors. In this talk I will describe some of these techniques, as well as a new method which involves expressing a piece of the 4-manifold as a branched covering of the 4-ball, whose branch locus can then be "braided". This technique is a 4-dimensional analogue of a classical construction of Alexander, which I will describe as motivation for this work.