TITLE: Cohomology of configuration spaces
QUESTION [13 upvotes]: Let $F(X,n)$ be the configuration space of ordered $n$-tuples of distinct points in $X$, where $X$ is a smooth manifold. Is there a procedure for computing the Poincare polynomial of $F(X,n)$? I am particularly interested in the case where $X$ is a 2-dimensional torus.
If $X$ is a smooth, projective, complex algebraic variety (for example an elliptic curve), Burt Totaro (in his paper "Configuration spaces of algebraic varieties") uses the Leray spectral sequence for the inclusion $F(X,n)\to X^n$  to find an explicit DGA whose cohomology is isomorphic to the cohomology ring of $F(X,n)$.  But it is not clear from this description how to compute the Betti numbers.

REPLY [2 votes]: It is perhaps worth pointing out that the virtual Poincare polynomial of $F(X,n)$ is easy to compute, at least when $X$ is a complex variety. This is treated in Section 2 of Fulton and MacPherson's paper on compactifications of configuration spaces, and the result is that the virtual Poincare polynomial of $F(X,n)$ is just 
$$
P(X)\cdot (P(X)-1)\cdot \dots (P(X)-i+1).
$$
Here the virtual Poincare polynomial of $X$ is given in terms of a signed sum over the weight graded pieces of cohomology of $X$ with compact supports. 
I think that this should allow some simple calculations, such as the Betti numbers for the (complex) braid arrangements being given by unsigned Stirling numbers of the first kind.