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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