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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 Grant May 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 – Sean Tilson May 9 '12 at 16:35

I'd like to follow this up with a seemingly related question. In my book "Topology and groupoids" (2006) there is a whole Chapter 11 on Orbit Spaces, Orbit Groupoids, giving joint work with P.J. Higgins showing that for discontinuous actions of a discrete group on a Hausdorff space which has a universal cover, the

fundamental groupoid of the orbit space is the orbit groupoid of the fundamental groupoid.

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!

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