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*.

**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!