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For a manifold $M$ and a positive integer $n$, the unordered configuration space $B(M,n)$ is the space consisting of all unordered collections of $n$ distinct points on $M$. Precisely, $$ B(M,n)=\{(m_1,\cdots,m_n)\in M^n\mid m_i\neq m_j \text { if } i\neq j\}/\Sigma_n $$ where $\Sigma_n$ is the symmetric group on $n$-letters acting on $M^n$ by permuting the order of coordinates.

In the paper on the cohomology of configuration spaces on surfaces, the cell-structure of $B(S^2,n)$ is given in section 2.2. And the cell-structure of $B(\mathbb{R}P^2,n)$ is given in section 2.6.

Question. I want to know the cohomology rings $$ H^*(B(S^2,n);\mathbb{Z}_2) $$ and $$ H^*(B(\mathbb{R}P^2,n);\mathbb{Z}_2). $$ How could these ring structures be derived from the cell-structures given in section 2.2 and section 2.6, on the cohomology of configuration spaces on surfaces? How could I derive the explicit solution or find the explicit solution in any references?

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