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!



cohomology algebra of braid spaces, configuration spaces
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


Bausum, David R. *Embeddings and immersions of manifolds in Euclidean space.* 
Trans. Amer. Math. Soc. 213 (1975), 263–303, 


where you will find references to the original articles of Haefliger and Yo Ging-Tzung.


