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?**



Kunneth formula of Cartesian product modulo orders of coordinates
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](http://www.maths.ed.ac.uk/~aar/papers/milgram4.pdf) (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. 


