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!

  • 1
    $\begingroup$ Have you tried using universal coefficient theorem? $\endgroup$
    – user43326
    Dec 31 '15 at 9:56

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, 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).

  • $\begingroup$ Thanks, Prof. Mark! Are there any results about the mod $p$ case for $p\geq 3$? $\endgroup$
    – QSH
    Jan 1 '16 at 10:39
  • $\begingroup$ The paper shows there is no $p$-torsion for $p\ge3$. $\endgroup$
    – Mark Grant
    Jan 2 '16 at 15:47

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