Kunneth formula of Cartesian product modulo orders of coordinates

Question

Let $X$ be a topological space and $F$ a field. Let the $n$-th permutation group $\Sigma_n$ act on $$ \prod_n X $$ by $$ \sigma(x_1,\cdots,x_n)=(x_{\sigma(1)},\cdots,x_{\sigma(n)}), \sigma\in \Sigma_n. $$ Then we have a quotient space $$ (\prod_n X)/\Sigma_n. $$ By Kunneth formula, $$ H^*(\prod_nX; F)=\bigotimes_nH^* ( X; F). $$

Question: does $$ H^*((\prod_n X)/\Sigma_n; F)=(\bigotimes_nH^* ( X; F))/\Sigma_n=\text{Symmetric tensor product }_nH^* ( X; F) $$ or not? For example, how to compute $$ H^*((\prod_n S^m)/\Sigma_n;\mathbb{Z}_2) $$ explicitly?

Answer 1

Firstly, your space $(\prod_n X)/\Sigma_n$ is better known as the $n$-fold symmetric product of $X$, and denoted $SP^n(X)$. Milgram (building on work of Steenrod, Cartan, Dold, Nakoaka and others) has shown how to compute $H_\ast(SP^n(X);k)$ for any field $k$ and any connected CW complex $X$ of finite type.

One of the key points is the Dold-Thom Theorem, which states that $SP(M(G,m))\simeq K(G,m)$. (Here $M(G,m)$ denotes a Moore space whose only non-trivial reduced homology is a $G$ in dimension $m$, and $K(G,m)$ the corresponding Eilenberg-Mac Lane space. $SP(X)$ stands for the infinite symmetric product, which is the direct limit of the inclusions $SP^n(X)\hookrightarrow SP^{n+1}(X)$). In particular, since $S^m=M(\mathbb{Z},m)$, the limit of the spaces $SP^n(S^m)$ as $n$ increases is a $K(\mathbb{Z},m)$. This indicates that, even for spheres, the $\mathbb{Z}/2$ (co)homology of these spaces can be rather complicated, although calculable due to classic results of Cartan and Serre.