A question concerning fundamental groups The following is a qualifying exam problem.  I am having great difficulty and have no clue where to start.  I believe it is a legitimately difficult problem.  Any help is greatly appreciated.  Thank you very much.
Let $n$ be a positive integer. Let $X$ be the space of all $n$-tuples $(x_1, ..., x_n)$ of points of $\mathbb{R}^4$
such that for $i \neq j$, $x_i \neq x_j$
, with the topology induced by the inclusion into $\mathbb{R}^{4n}$
. Let $\sim$  be the equivalence relation on $X$ given by
$(x_1, ..., x_n) \sim  (x_{\sigma(1)}, ..., x_{\sigma(n)})$
for every permutation  of $1, ..., n$. Compute $\pi_1(X)$ and $\pi_{1}(X/\sim )$.
 A: I'm not really sure what tools you're allowed to use, but here goes.  I'm going to use the notation $X_n$ instead of $X$.
Claim 1:  $\pi_1(X_n)$ is trivial.  
Proof:  By induction on $n$.  When $n=1$, $X_1 = \mathbb{R}^4$ so is simply connected.  Now, assume $X_{n-1}$ is simply connected.
Let $\pi:X_n\rightarrow X_{n-1}$ be the projection onto the first $n-1$ coordinates.  Then $\pi$ gives $X_n$ the structure of a fiber bundle with fiber $\mathbb{R}^4\setminus\{x_1,...,x_n\}$.  From this we get a long exact sequence in homotopy groups $$\ldots\rightarrow \pi_1\left(\mathbb{R}^4\setminus\{x_1,...,x_n\}\right)\rightarrow \pi_1(X_n)\rightarrow \pi_1(X_{n-1})\rightarrow\pi_0\left(\mathbb{R}^4\setminus\{x_1,...,x_n\}\right)\ldots $$
Note that $\mathbb{R}^4\setminus\{x_1,...,x_n\}$ is both connected and simply connected and further, by assumption $\pi_1(X_{n-1}) = 0$.  Then, exactness implies $\pi_1(X_n) = 0$ so $X_n$ is simply connected as well.
Claim 2:  The space $X_n/\sim$ is nothing but the orbit space $X_n/S_n$ where $S_n$, the symmetric group, acts freely on $X_n$ by permuting coordinates.  In particular, $X_n\rightarrow X_n/S_n$ is a covering and so $\pi_1(X/\sim) = \ldots$
