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Let $M$ be a manifold. Then $F(M,k)/\Sigma_k$, the unordered configuration space of $k$ points, is obtained as a quotient of $F(M,k)$, the ordered configuration space of $k$ points, by the group action of $\Sigma_k$, the symmetric group on $k$ letters.

Let $p$ be an odd prime. What is the relation between

$H^*(F(M,p);\mathbb{Z}/p\mathbb{Z})$ and $H^*(F(M,p)/\Sigma_p;\mathbb{Z}/p\mathbb{Z})$?

Suppose the cohomology algebra $H^*(F(M,p);\mathbb{Z}/p\mathbb{Z})$ is known. Is there any method, including spectral sequence, to get the cohomology algebra $H^*(F(M,p)/\Sigma_p;\mathbb{Z}/p\mathbb{Z})$? Could you give some references?

Thanks.

Note: when the coefficient field is of characteristic $0$ or of characteristic $q$, for $q$ not dividing $o(\Sigma_k)=k!$, the question is answered at Relation between cohomology of ordered and unordered configuration spaces?

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The group $\Sigma_k$ acts freely on $F(M,k)$, this implies that the orbits coincide with the homotopy orbits. In general, if a group $G$ acts on a topological space $X$ you have a spectral sequence

$$H_*(G,H_*(X,K))\implies H_*(X_{hG},K)$$

where $K$ is any commutative field. The left hand side is the homology of the group $G$ with coefficients in the $G$ module $H_*(X,K)$. One way to construct this spectral sequence is to realize that you have a fiber sequence : $$X\to X_{hG}\to BG$$ in which the action of $G\simeq \Omega BG$ on the fiber $X$ is the action you started with. Then you can run a Leray-Serre spectral sequence.

Note that if $G$ is finite and $K$ has characteristic $0$, the left hand side collapses to $H_*(X,K)/G$ as mentioned in the question you are citing.

A possible reference for this is Hatcher's book project on spectral sequences http://www.math.cornell.edu/~hatcher/SSAT/SSch3.pdf

Your question is about cohomology rather than homology but since you are working over a field, assuming that $M$ has finitely generated homology, then the cohomology of $F(M,k)/\Sigma_k$ in a given degree is the dual of the homology of $F(M,k)/\Sigma_k$ in that degree.

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  • $\begingroup$ Dear Prof. Geoffroy, Serre spectral sequence requires that the fundamental group acts trivially on the homology of the fibre. Here it may not satisfy hence we cannot apply. $\endgroup$ – QSH Jan 13 '15 at 3:26
  • $\begingroup$ There is a version of Serre spectral sequence when the fundamental group acts non-trivially. In that case you just replace ordinary homology by homology in the local coefficient system given by the homology of the fiber. In the case of $BG$ a local coefficient system is just a $G$-module and homology with coefficients in that local coefficient system is group homology. $\endgroup$ – Geoffroy Horel Jan 13 '15 at 7:14

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