# Fundamental group of the configuration space in a plane [closed]

Let $C_n$ be the set of unordered distinct $n$-tuples in $\mathbb{R}^2$: $$C_n= \left\{ (x_1,\ldots,x_n)\in (\mathbb{R}^2)^n \mid x_i\neq x_j \text{ for }i\neq j \right\}/S_n .$$

I wish to prove that its fundamental group $\pi_1(C_n)=B_n$, where $B_n$ is the Artin braid group on $n$ strands.

While there are probably various standard techniques for computing fundamental group, I do not know which I should look out for. Therefore, I am looking for a sketch of the proof, or a list of key concepts/theorems I should read up on.

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## closed as too localized by Gjergji Zaimi, Loop Space, Bruce Westbury, Mark Sapir, Mark GrantMay 9 '12 at 17:28

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This is, more or less, by the definition of the braid group: If you have a braid (say, going from a plane at height 0 to a plane at height 1), you can intersect it with parallel planes at height t, to get a closed path in the configuration space of the plane. Depending on the exact definition of a braid, this even defines a homeomorphism from the space of closed loops (with given starting point) in the configuration space of the plane to the space of braids. In particular, we get a bijection of the sets of components - the fundamental group of the configuration space and the braid group. –  Lennart Meier May 9 '12 at 10:06
Look at Birman, "Braids, Links, and Mapping Class Groups" –  Lee Mosher May 9 '12 at 12:50
This seems more appropriate for math.stackexchange.com. –  Sean Tilson May 9 '12 at 16:35

One application is given there, namely computing the fundamental group of the symmetric square of a space $X$ (it is the fundamental group of $X$ made abelian, which of course is known through other routes). A question arising from this work is whether there are applications of these methods to braid groups, and configuration spaces. I'd like to see a really novel application!