The topology of broken
Lefschetz fibrations is studied by means of handle decompositions. We consider a
slight generalization of round handles and describe the handle diagrams for all that
appear in dimension four. We establish simplified handlebody and monodromy
representations for a certain subclass of broken Lefschetz fibrations and pencils,
showing that all near-symplectic closed 4-manifolds can be supported by such
objects, paralleling a result of Auroux, Donaldson and Katzarkov. Various
constructions of broken Lefschetz fibrations and a generalization of the symplectic
fiber sum operation to the near-symplectic setting are given. Extending the study of
Lefschetz fibrations, we detect certain constraints on the symplectic fiber sum
operation to result in a 4-manifold with nontrivial Seiberg–Witten invariant, as well
as the self-intersection numbers that sections of broken Lefschetz fibrations can
acquire.
