  The existence of a positive allowable Lefschetz fibration on a compact Stein
surface with boundary was established by Loi and Piergallini by using branched
covering techniques. Here we give an alternative simple proof of this fact and
construct explicitly the vanishing cycles of the Lefschetz fibration, obtaining
a direct identification of the set of compact Stein manifolds with positive
allowable Lefschetz fibrations over a 2-disk. In the process we associate to
every compact Stein manifold infinitely many nonequivalent such Lefschetz
fibrations.
