# 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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## 3 Answers

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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For ordered configuration spaces, see Fred Cohen's paper(s), including On configuration spaces, their homology, and Lie algebras ☆ F.R. Cohen E-mail the corresponding author Department of Mathematics, University of Rochester, Rochester, NY 14627, USA (J of pure and applied algebrac, 1995, availabe for free on line).

For getting from ordered to unordered see this question.

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The paper you mention deals with configurations in the Euclidean space, not the torus, or am I wrong? – Vladimir Dotsenko Aug 22 '13 at 15:21
@VladimirDotsenko You are wrong :) That's what the abstract says, but if you read further, he talks about surfaces. – Igor Rivin Aug 22 '13 at 15:30
Well, I read that paper, in fact :) and while it provides a recipe in a sense, it is a rather vague one, I think. The OP asked for a reference, and this particular paper never felt like a reference for this particular question to me, however lovely a paper it is ;) – Vladimir Dotsenko Aug 22 '13 at 15:35

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 '13 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 '13 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 '13 at 21:55