  The Hilbert scheme X^{[a]} of points on a complex manifold X is a
compactification of the configuration space of a-element subsets of X. The
integral cohomology of X^{[a]} is more subtle than the rational cohomology. In
this paper, we compute the mod 2 cohomology of X^{[2]} for any complex manifold
X, and the integral cohomology of X^{[2]} when X has torsion-free cohomology.
  The results of this paper are used in Voisin's work on the universal CH_0
group of cubic hypersurfaces, because the crucial point there is to study the
2-torsion in the Chow group.
