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Let $M$ be a compact Riemannian manifold without boundary. Distinct $k$-points $x_1,\cdots,x_k\in M$ are called stable if the potential energy given by coulomb forces among $k$ electrons reaches minimal locally at $(x_1,\cdots,x_k)$. The $k$-th stable configuration space is defined by $$ S(M,k)=\{(x_1,\cdots,x_k)\in M^k\mid x_1,\cdots,x_k \text{ are distinct and stable}. \} $$

Let the $k$-th permutation group $\Sigma_k$ act on $S(M,k)$ by permuting coordinates. The $k$-th stable braid space is defined by $$ S(M,k)/\Sigma_k. $$

Question: Given $k\geq 2$ and $M$, what is the cohomology ring $H^*(S(M,k)/\Sigma_k)$? Some nontrivial examples & references are also wanted.

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  • $\begingroup$ What is the mathematical formulation of being stable? Is it defined in the presence of some real valued function $M^{\times k}\to\mathbb{R}$ that you wish to study the loci of its local minimum? $\endgroup$
    – user51223
    Commented Sep 18, 2015 at 18:57
  • $\begingroup$ Yes, Prof. It is given by coulomb force formula. $\endgroup$
    – Shi Q.
    Commented Sep 19, 2015 at 3:11
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    $\begingroup$ Can you please state this formula? I don't know what the formula is, and wonder how that will effect the computation of homology unless you can translate into some action of a group, or consider it as a map into some Lie algebra?! Otherwise, without the stability condition, the homology of the above configuration space, at least in some specific cases, is known. $\endgroup$
    – user51223
    Commented Sep 19, 2015 at 8:21

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