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In the lecture notes The Homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, page 225 -226, it is obtained that there is a group completion on homology $$ \alpha_n: C(\mathbb{R}^n;X)\to \Omega^n\Sigma^n X.$$ Moreover, I find that $$ \Omega^n\Sigma^nS^0=\Omega^n S^n=\bigsqcup_{q\in \mathbb{Z}}\{f\in [S^n,S^n]_*\mid \deg (f)=q\}. $$ Let $$ \{f\in [S^n,S^n]_*\mid \deg (f)=q\}=\Omega_q^nS^n$$ then $$ \Omega^n\Sigma^nS^0=\bigsqcup_{q\in\mathbb{Z}}\Omega_q^nS^n. $$ I find in some references that $$ C(\mathbb{R}^n;S^0)=\bigsqcup_{k\geq 0}F(\mathbb{R}^n,k)/\Sigma_k. $$ Let $\alpha_{n,k}$ be the restriction of $\alpha_n$ to $F(\mathbb{R}^n,k)/\Sigma_k$.

Question: I want to obtain the ring structure of $H^*(F(\mathbb{R}^n,k)/\Sigma_k;\mathbb{Z}_2)$. Is there any way to obtain it? Do we have a well-defined map $$\alpha_{n,k}: F(\mathbb{R}^n,k)/\Sigma_k\to\Omega_k^nS^n$$ that induces a ring isomorphism $$ H^*(\Omega_k^nS^n;\mathbb{Z}_2)\to H^*(F(\mathbb{R}^n,k)/\Sigma_k;\mathbb{Z}_2)? $$

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There is a a well-defined map from $F(\mathbb R^n,k)/\Sigma_k \to \Omega^n_k S^n$ but it only induces an iso on $H^*$ for $* \leq k/2$.

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