Let $X$ be a topological space. For $n \ge 2$, the group $S_n$ acts naturally on $X^n$ as $\sigma . (x_1,...,x_n)=(\sigma(x_1),...,\sigma (x_n)), \forall (x_1,...,x_n) \in X^n$ . So we can consider the quotient space $X^n /S_n$.
Now let $X$ be a Hausdorff, contractible, locally path connected topological space. Then is the quotient space $X^2/S_2$ also contractible ? If it is not in general, then is the fundamental group at least trivial ? ($X^2/S_2$ is obviously path connected ) What about $X^n/S_n$ in general ?