In the paper [Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado,](http://arxiv.org/pdf/1410.2200.pdf) page 3, line from bottom 1-3, it is given that for a $m$-manifold $M$, there is a map from the labelled configuration space to the section space of a $m$-sphere bundle
$$
\alpha: C(M;S^0)\to \Gamma(M;S^0)
$$
where 
$$
C(M;S^0)=\bigsqcup\_{k\geq 0}F(M,k)/\Sigma\_k
$$
is the disjoint union of unordered configuration spaces, and 
$$
\Gamma(M;S^0)=\bigsqcup\_{q\in \mathbb{Z}} \Gamma\_q(M;S^0)
$$
is the disjoint union of sections of degree $q$.


In the paper [Configuration spaces of positive and negative particles, D. McDuff,](http://www.sciencedirect.com/science/article/pii/0040938375900385) Theorem 1.1, it is proved that 


(1.1) for a closed compact manifold $M$ and any fixed $n$, we can choose $k$ sufficiently large such that for any $t\geq k$, the map $\alpha$ restricted to $F(M,t)/\Sigma\_t$ induces an isomorphism on $n$-th homology group
$$
(\alpha\_t)\_\*:H\_n(F(M,t)/\Sigma\_t)\to H\_n(\Gamma\_t(M;S^0)).
$$


**Question:** Could the theorem 1.1 be strengthened to the statement that for some $k$,


$$
(\alpha\_k)^\*: H^\*(\Gamma\_k(M;S^0))\to H^\*(F(M,k)/\Sigma\_k)
$$
 is a ring isomorphism of cohomology rings?


(i.e., does there exist some $k$ such that for all $n$, the map $\alpha$ restricted to $F(M,k)/\Sigma\_k$ induces an isomorphism on $n$-th homology group
$$
(\alpha\_k)\_\*:H\_n(F(M,k)/\Sigma\_k)\to H\_n(\Gamma\_k(M;S^0))?)
$$



Configuration spaces of positive and negative particles
Using the homological stability results of Segal from the appendix of his paper "The topology of spaces of rational functions," one can see that the scanning map from $C\_k(M;S^0)$ to $\Gamma\_k(M;S^0)$ induces an isomorphism in homology groups $H\_i$ for $i \leq k/2$.


