# cohomology algebra of braid spaces, configuration spaces

In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, Chapter 5, 6, 7, 8, 9, 10, 11, the cohomology algebra $H^*(B(\mathbb{R}^{n+1},p),\mathbb{Z}_p)$, for $p$ prime and $B(\mathbb{R}^{n+1},p)=F(\mathbb{R}^{n+1},p)/\Sigma_p$, is obtained. A spectral sequence for fibration $\Sigma_p\to F(\mathbb{R}^{n+1},p)\to B(\mathbb{R}^{n+1},p)$ is used with the action of $\Sigma_p$ on $H^*(F(\mathbb{R}^{n+1},p);\mathbb{Z})$.

For other manifolds $M$ such as $S^m$ and $S^m\times \mathbb{R}^k$ ($H^*F(\mathbb{R}^{n+1},p;\mathbb{Z}_p)$ is known in these cases), are there any results for the cohomology algebra $H^*(B(M,p);\mathbb{Z}_p)$ in any references? Thanks!

For $p=2$ there are references from which you can extract the $\mathbb{Z}/2$ cohomology algebra of $B(M,2)$ for any closed manifold $M$. The answer in principle depends only on $H^\ast(M;\mathbb{Z}/2)$ as a module over the Steenrod algebra, together with the Stiefel-Whitney classes of $M$, but is usually not very pleasant (even for manifolds with nice cohomology such as $\mathbb{R}P^n$). The method consists of showing that the map $$B(M,2)\to S^\infty \times_{\mathbb{Z}/2} M\times M$$ is surjective in cohomology, and calculating its kernel. The situation is nicely summarised in Section 4 of