Cohomology of configuration spaces

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.

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I think the paper of Bodigheimer and Cohen. Rational cohomology of configuration spaces of surfaces'' answers that question. –  Geoffroy Horel Apr 14 at 1:10
Sorry, but why doesn't Totaro's paper answer your question? –  Dan Petersen Apr 14 at 2:57
@Geoffroy: the paper you reference deals with configuration spaces of unordered points, not ordered... –  Dan Petersen Apr 14 at 2:58
@Dan Petersen: You're right. Sorry for the confusion. –  Geoffroy Horel Apr 14 at 15:26
@Dan Peterson: Totaro's paper does not give a formula for the Poincare polynomial. There's a big gap between being able to write down a DGA in terms of generators and relations and actually having a formula for its Poincare polynomial. –  Nicholas Proudfoot Apr 14 at 15:59
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