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>



How to calculate the fundamental group of general configuration space
(1a) Assume $n\geq 2$ since otherwise the quotient is trivial. Since the action of $S\_n$ is free, the quotient map is a covering map. Since $m\geq 3$, and $M$ is simply-connected, $M^n-D$ is simply-connected by transversality since the codimension of $D$ is greater than or equal to 3 (so the inclusion map is 2-connected). From covering space theory, $\pi\_1(K)=S\_n$. 


(1 b and c) See the comments [here](https://math.stackexchange.com/questions/172717/braid-groups-and-the-fundamental-group-of-the-configuration-space-of-n-points)


(2) For non-simply connected $M$, use the idea in (1a) to get a quotient of $\pi\_1(K)$ by the image of the fundamental group of $M^n-D$ via the covering map. Use that to analyze $\pi\_1(K)$. See [here](https://www.math.rochester.edu/people/faculty/jonpak/newbraid.pdf) for examples. 


Note: [here](http://dauns.math.tulane.edu/~cbanda/lectures/fred/Lecture.I.pdf),[here](http://dauns.math.tulane.edu/~cbanda/lectures/fred/Lecture.II.pdf) and [here](http://dauns.math.tulane.edu/~cbanda/lectures/fred/Lecture.III.pdf) are some great introductory slides on this topic by Fred Cohen, who has written many papers about the topology of configuration spaces.


Note: By the Dold-Puppe Theorem, the fundamental group of the singular space $M^n/S\_n$ is simply connected when $M$ is simply connected even though $(M^n-D)/S\_n$ is not simply connected.


