Let $M$ be a Riemannian manifold. For any $\epsilon\geq 0$, we define the $k$-th ($k=1,2,\cdots$) "$\epsilon$-configuration space" as $$ F(M,k,\epsilon)=\{(x_1,\cdots,x_k)\in M^k\mid d(x_i, x_j)>\epsilon \text{ whenever }i\neq j\} $$ and the $k$-th ($k=1,2,\cdots$) "unordered $\epsilon$-configuration space" as $$ B(M,k,\epsilon)=F(M,k,\epsilon)/S_k=\{(x_1,\cdots,x_k)\in M^k\mid d(x_i, x_j)>\epsilon\text{ whenever }i\neq j\}/S_k $$ where $S_k$ is the symmetric group of order $k$ acting on $M^k$ by permuting the order of coordinates.
We notice that $F(\mathbb{R}^n,k,\epsilon)=F(\mathbb{R}^n,k,0)$, but $F(S^n,k,\epsilon)\neq F(S^n,k,0)$ for $S^n$ the $n$-sphere.
Question: are there any references for the cohomology ring $$ H^*(B(M,k,\epsilon);\mathbb{Z}_2)? $$
