Let $\Sigma\_k$ be the symmetric group on $k$-letters. Let $W$ be a **contractible** topological space with a **free** $\Sigma\_k$-action (from the left). Let $X$ be a $CW$-complex and let $X^k$ be the Cartesian product of $k$-copies of $X$. Let $\Sigma\_k$ act on $X^k$ from the right by permuting the order of coordinates. Then we have a space
$$
W\times \_{\Sigma\_k} X^k=W\times X^k/ (\sigma\cdot w,a)\sim (w,a\cdot \sigma)
$$
where $w\in W, a\in X^k$ and $\sigma\in \Sigma\_k$. 


**Question:** whether do we have
$$
W\times \_{\Sigma\_k} X^k\simeq X
$$
or not? What shall I do when meeting such problems?



free group actions on a contractible topological space
A couple of scattered things to address the "how to deal with" part of the question.




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If $X$ is connected, then probably so is $X^n$, and so you are looking at a fibration
$$ X^n \to X^n // \Sigma\_n \to \mathbf{B}\Sigma\_n $$
for which the Leray-Serre spectral sequences are usually called Cartan-Leray, and they look like
$$ H^{\mathrm{gp}}\_p ( \Sigma\_n ; H\_\*(X,\mathbb{k})^{\otimes n}\_q ) \Rightarrow H\_{p+q} (X^n // \Sigma\_n ; \mathbb{k}) $$




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In the very special case that $X$ is a strictly abelian group (that is, a product of Eilenberg-MacLane spaces), one can extend the power sequence $n\mapsto X^n$ to a functor on the category **FPartial** of finite partial functions: for $f : s \nrightarrow t$ and for $x : s \to X$, define $$ (f\_\* x)\_j = \sum\_{i : f(i) =j} x\_i $$
With this data one can use some [work by Stanislaw Betley](http://www.ams.org/journals/proc/2002-130-12/S0002-9939-02-06763-1/) to study what happens to particular $H\_p(X^n // \Sigma\_n)$ as $n\gg p$.


