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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 '13 at 1:10
    
Sorry, but why doesn't Totaro's paper answer your question? –  Dan Petersen Apr 14 '13 at 2:57
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@Geoffroy: the paper you reference deals with configuration spaces of unordered points, not ordered... –  Dan Petersen Apr 14 '13 at 2:58
    
@Dan Petersen: You're right. Sorry for the confusion. –  Geoffroy Horel Apr 14 '13 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 '13 at 15:59

2 Answers 2

look here:

Bezrukavnikov, R. Koszul DG-algebras arising from configuration spaces. Geom. Funct. Anal. 4 (1994), no. 2, 119–135.

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In this paper, Bezrukavnikov cites a paper of Kohno and Oda (1987) in which they prove (among other things) an LCS formula for the Poincare polynomial in question. Assuming that one can compute the ranks of the subquotients in the lower central series of the fundamental group, this completely answers Christin's question. However, Roman points out on page 133 of his paper that there are some incorrect results in the Kohno-Oda paper. Do you know if the Kohno-Oda LCS formula is correct as stated? –  Nicholas Proudfoot Apr 14 '13 at 18:18

Some partial computations for Betti numbers of configuration spaces of torus and surfaces with higher genus can be found in “Brown, White; Homology and Morse theory of third configuration spaces”

In general it is not quite straightforward to compute Betti numbers from the DGA, see for example

“S. Ashraf, B. Berceanu, Cohomology of 3-points configuration spaces of complex projective spaces, arXiv:1212.1291 ”

where the authors compute the Poincare polynomials by using the action of the symmetric group on Totaro’s DGA.

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