In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, 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, 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 $$ (\alpha_k)_*:H_n(F(M,k)/\Sigma_k)\to H_n(\Gamma_k(M;S^0))?) $$


1 Answer 1


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 $i \leq k/2$.


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