cohomology algebra of braid spaces, configuration spaces

Question

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!

Answer 1

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.

Answer 2

The homology of configuration spaces of manifolds is completely calculated in Bödigheimer-Cohen-Taylor's "On the homology of configuration spaces" in Topology, at least at the prime 2, or at odd primes (or rationally) for odd dimensional manifolds.