# torsion part of the cohomology module of configuration spaces of manifolds

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!

• Have you tried using universal coefficient theorem? Dec 31 '15 at 9:56

The following paper answers your question for the case $M=\mathbb{R}P^m$ and $n=2$:
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).
• Thanks, Prof. Mark! Are there any results about the mod $p$ case for $p\geq 3$?
• The paper shows there is no $p$-torsion for $p\ge3$. Jan 2 '16 at 15:47