Define the configuration space of $n$ points in a general manifold $M$, where $\dim M=m$, as $K=(M^n-D)/S_n$ where $S_n$ is the permutation group and $D=\{(x_1,\cdots,x_n)| \exists i,j\ s.t. x_i=x_j \}$.

Then my question is

(1) How to prove $M=\mathbb{R}^m$ then $\pi_1(K)=S_n$ for $m>2$ and $\pi_1(K)=B_n$ for $m=2$ where $B_n$ is the Braid group. And what is the $\pi_1(K)$ when $m=1$?

(2) In general when $M$ is non-simply connected then what is $\pi_1(K)$ and how to calculate it? For example, $M=T^2$.

same question: https://math.stackexchange.com/questions/1748136/how-to-calculate-the-fundamental-group-of-general-configuration-space

braid groups of manifolds. Key to calculations are the so-calledFadell-Neuwirth fibrations. You could probably find an answer for the torus by searching using these terms. $\endgroup$