# Configuration spaces of the torus

I would like a reference that calculates the rational homology of the unordered configuration spaces of the torus.

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The calculation for even-dimensional manifolds, and in particular the torus, is carried out by Felix-Thomas in their paper "Rational Betti numbers of configuration spaces."

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I would suggest the two following references:

C.-F. Bödigheimer, F.R. Cohen, Rational cohomology of configuration spaces of surfaces. Algebraic Topology and Transformation Groups, Springer LNM 1361 (1987), 7-13. (http://www.math.uni-bonn.de/people/cfb/PUBLICATIONS/rational-cohomology-of-configuration-spaces-of-surfaces.pdf)

Bezrukavnikov, R. Koszul DG-algebras arising from configuration spaces. Geom. Funct. Anal. 4 (1994), no. 2, 119–135. (http://link.springer.com/content/pdf/10.1007%2FBF01895836.pdf)

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I don't think either of these references give a complete solution to the question. The former deals only with surfaces with a puncture, and the latter with configurations of ordered points. I wonder if there is a reference in the literature which really solves this problem in a satisfactory way. –  Dan Petersen Aug 22 at 16:29
@Dan: these two are the closest I am aware of. But passing to unordered is not a big deal, so maybe one should not be too upset :) –  Vladimir Dotsenko Aug 22 at 21:49
In principle I agree, you just take $S_n$-invariants. And I believe you can (say) write a computer program to compute the cohomology together with its $S_n$-action from the PBW basis in Bezrukavnikov's paper. But it would be nice with something more explicit. –  Dan Petersen Aug 22 at 21:55