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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    $\begingroup$ I think the paper of Bodigheimer and Cohen. ``Rational cohomology of configuration spaces of surfaces'' answers that question. $\endgroup$ Apr 14, 2013 at 1:10
  • $\begingroup$ Sorry, but why doesn't Totaro's paper answer your question? $\endgroup$ Apr 14, 2013 at 2:57
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    $\begingroup$ @Geoffroy: the paper you reference deals with configuration spaces of unordered points, not ordered... $\endgroup$ Apr 14, 2013 at 2:58
  • $\begingroup$ @Dan Petersen: You're right. Sorry for the confusion. $\endgroup$ Apr 14, 2013 at 15:26
  • $\begingroup$ @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. $\endgroup$ Apr 14, 2013 at 15:59

3 Answers 3


look here:

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

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    $\begingroup$ 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? $\endgroup$ Apr 14, 2013 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.


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.


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