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There is a reference or a methode which by it we can calculate the cohomology of a configuration space of a compact manifold simply connected? It is possible to find a spectral sequence converging to this cohomology (but the Cohen-Taylor spectral sequence)?

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  • $\begingroup$ What kind of configuration spaces do you have in mind? For example, the configuration space of k points in the 2-dimensional disc is the classifying space of the k-strand braid group, and there is a rather large literature on its properties including its cohomology. Is that an example of what you are asking about? $\endgroup$
    – Lee Mosher
    Feb 23 '13 at 20:04
  • $\begingroup$ I mean how to calculate the rational cohomology of the configuration space of a compact manifold simply connected in general, or if it is possible determinate a model fot the configuration space, i know that Kriz and Totaro gave a model for the configuration spaces $F(M,k)$ when $M$ is a complex projective manifold, but in general case, it is possible to use the same technics to determinate it? $\endgroup$ Feb 23 '13 at 22:05
  • $\begingroup$ Let me rephrase my question. What definition of a configuration space are you using? For example, is the space of k-element subsets of the 2-dimensional disc an example of your type of configuration space? The terminology "configuration space" is not completely standard, hence the need to state what definition you are using. $\endgroup$
    – Lee Mosher
    Feb 24 '13 at 5:14
  • $\begingroup$ @Lee Mosher: here configuration space means $F(M,k) = M^k \setminus \Delta$ where $\Delta$ is the "big diagonal". $\endgroup$ Feb 24 '13 at 8:18
  • $\begingroup$ I use the classical the following definition of ordered configuration space: Let $M$ an $m-$dimensional manifold. The space of ordered configurations of $k$ pointsis the space $$F(M,k)=\{(x_1,...,x_k)\in M^k ;x_i\neq x_j for i\neq j \}$$, and we ask to find the rational cohomology of this space. $\endgroup$ Feb 24 '13 at 9:57
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Shameless promotion ahead. If you are interested in cohomology with base field $\mathbb{R}$, then:

  • For simply connected manifolds without boundary you have my paper and a paper of Campos and Willwacher. Both papers give so-called "real models" for the configuration spaces; these models are commutative differential graded algebras, and in particular the cohomology of the model is the cohomology of the configuration space.
  • For simply connected manifolds with boundary (and interiors of such manifolds, it's the same thing), we have a paper with Campos, Lambrechts, and Willwacher, where we also give a real model (several, in fact). If $\dim M \ge 4$ then this model is fairly explicit, otherwise it's complicated.

Besides all the models we give are equipped with a symmetric group action that models the natural symmetric group action on $\operatorname{Conf}_k(M)$, so if you are interested in unordered configuration spaces then you can compute their cohomology by considering invariants.

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I would be extremely surprised if there was anything other than the Cohen-Taylor spectral sequence that you can do in this generality.

As you know, the first nontrivial page of Cohen-Taylor spectral sequence depends only on the ring $H^\bullet(M)$. I think it's written up somewhere that the higher order differentials are defined by Massey products on $M$. In particular the result you refer to in a comment that $H^\bullet(F(M,n))$ (or its associated graded) can be computed explicitly from $H^\bullet(M)$ extends to arbitrary formal manifolds $M$, not just smooth projective varieties. And if you give yourself a minimal model of $M$ you should be able to compute $H^\bullet(F(M,n))$ for all $n$ as well (but not a minimal model of $F(M,n)$ -- see the paper of Longoni and Salvatore).

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  • $\begingroup$ The paper of Longoni and Salvatore shows that configuration spaces are not homotopy invariant in the non-simply connected case. I believe the question of whether the rational homotopy type of $F(M,n)$ is an invariant of rational homotopy type of simply-connected closed $M$ is still open. See the paper eudml.org/doc/116129 of Lambrechts and Stanley, which treats the $2$-connected case. $\endgroup$
    – Mark Grant
    Feb 25 '13 at 7:29

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