The integral cohomology of the Hilbert scheme of two points

For a complex manifold X of dimension at least 3, the Hilbert scheme of m points on X is a manifold only when m <= 3. We consider the case m = 2. The rational cohomology of the Hilbert scheme is easy to compute, but the integral or mod 2 cohomology is subtle, related to mod 2 Steenrod operations on the cohomology of X. Nonetheless, we compute the integral cohomology of the Hilbert scheme of 2 points and prove some good properties. These results 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.