  We show how to construct broken, achiral Lefschetz fibrations on arbitrary
smooth, closed, oriented 4-manifolds. These are generalizations of Lefschetz
fibrations over the 2-sphere, where we allow Lefschetz singularities with the
non-standard orientation as well as circles of singularities corresponding to
round 1-handles. We can also arrange that a given surface of square 0 is a
fiber. The construction is easier and more explicit in the case of doubles of
4-manifolds without 3- and 4-handles, such as the homotopy 4-spheres arising
from nontrivial balanced presentations of the trivial group.
