  Building on the work of the fourth author in math.AG/9904074, we prove the
weak factorization conjecture for birational maps in characteristic zero: a
birational map between complete nonsingular varieties over an algebraically
closed field K of characteristic zero is a composite of blowings up and
blowings down with smooth centers. Such a factorization exists which is
functorial with respect to absolute isomorphisms, and compatible with a normal
crossings divisor. The same holds for algebraic and analytic spaces. Another
proof of the main theorem by the fourth author appeared in math.AG/9904076.
