Let $M$ be a manifold and $C\_n(M)$ the $n$-th unordered configuration space consisting of unordered $n$-tuples of distinct points in $M$. The mod $p$ homology module, $p$ prime, and the rational homology module, were given in the paper [*On the homology of configuration spaces*](http://www3.nd.edu/~taylor/papers/cocs.pdf). 


**Question.** I want to find the mod $p$ torsion part of the cohomology module
$$
H^\*(C\_p(M);\mathbb{Z})
$$ 
for any prime $p$. If I cannot find a statement for general $M$, then I want to find as many as possible examples of $M$ such that the mod $p$ torsion part of the cohomology module
$
H^\*(C\_p(M);\mathbb{Z})
$ 
 are known. I find one example $M=\mathbb{R}^m$ and
$$
\text{mod } p \text{ torsion part of }H^k(C\_p(\mathbb{R}^m);\mathbb{Z})=\mathbb{Z}\_p, \text{ if }k=2s(p-1), 1\leq s\leq [(m-1)/2],
$$


$$
\text{mod } p \text{ torsion part of }H^k(C\_p(\mathbb{R}^m);\mathbb{Z})=0, \text{ otherewise. }
$$ Are there any other examples? Thanks! 



torsion part of the cohomology module of configuration spaces of manifolds
The following paper answers your question for the case $M=\mathbb{R}P^m$ and $n=2$: 


*Carlos Domínguez, Jesús González, and Peter Landweber*, [**The integral cohomology of configuration spaces of pairs of points in real projective spaces**](http://dx.doi.org/10.1515/form.2011.145), *Forum Math.* **25** (2013), no. 6, 1217--1248.


The results and proofs are quite technical, and use the Bockstein spectral sequence. To summarise, the only torsion is 2- and 4-torsion. The authors of that paper were motivated by trying to compute a symmetrized version of Farber's topological complexity for real projective spaces. which is closely related to the embedding dimension.


The case of spheres $M=S^m$ and $n=2$ should be do-able by the same methods (and the groups $H^\*(B(S^m,2);\mathbb{Z})$ might even be known already). 


