  We prove that every closed oriented smooth 4-manifold X admits a broken
Lefschetz fibration (aka singular Lefschetz fibration) over the 2-sphere. Given
any closed orientable surface F of square zero in X, we can choose the
fibration so that F is a fiber. Moreover, we can arrange it so that there is
only one Lefschetz critical point when the Euler characteristic e(X) is odd,
and none when e(X) is even. We make use of topological modifications of smooth
maps with fold and cusp singularities due to Saeki and Levine, and thus we get
alternative proofs of previous existence results. Also shown is the existence
of broken Lefschetz pencils with connected fibers on any near-symplectic
4-manifold.
