Let $M=\mathbb{C}P^n$ or $\mathbb{R}P^n$ with $m$ punctures, is it known what the homology of the configuration space, $H_*(C_k(M))$ is? How are cases $\mathbb{C}P^n$ and $\mathbb{R}P^n$ different?

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    $\begingroup$ I'll try to write an answer later (no time now) but the best way is probably by induction. If $\underline{m}$ is a set with $m$ elements, then there is a fiber bundle $C_k(M \setminus \underline{m+1}) \to C_{k+1}(M \setminus \underline{m}) \to M \setminus \underline{m}$. $\endgroup$ – Najib Idrissi Apr 10 '20 at 6:57

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