# Homology of configuration space of punctured projective spaces?

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?

• 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 $C_k(M \setminus \underline{m+1}) \to C_{k+1}(M \setminus \underline{m}) \to M \setminus \underline{m}$. – Najib Idrissi Apr 10 '20 at 6:57