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

  • 2
    $\begingroup$ The first question is a basic fact about the braid groups. This is where their name comes from too. $\endgroup$ Apr 19, 2016 at 12:02
  • 3
    $\begingroup$ These are called braid groups of manifolds. Key to calculations are the so-called Fadell-Neuwirth fibrations. You could probably find an answer for the torus by searching using these terms. $\endgroup$
    – Mark Grant
    Apr 19, 2016 at 12:30

1 Answer 1


(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

(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 Configuration Spaces and Braid Groups by Cohen & Pakianathan for examples.

Note: Fred Cohen has written many papers about the topology of configuration spaces. Here are some introductory notes I found by him hosted by NUS: Lecture 1 and Lecture 2.

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.

  • $\begingroup$ What's the Dold-Puppe Theorem? $\endgroup$
    – Dan Ramras
    Apr 22, 2016 at 4:26
  • $\begingroup$ ams.org/mathscinet/search/… $\endgroup$ Apr 22, 2016 at 5:19
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    $\begingroup$ Ah, cool. This is sort of an unstable version of the Dold-Thom theorem. For those who can't follow the above link, the paper is Ann. Inst. Fourier Grenoble 11 1961 201–312. $\endgroup$
    – Dan Ramras
    Apr 22, 2016 at 5:40
  • $\begingroup$ The link you provided has expired. $\endgroup$
    – maplemaple
    Dec 6, 2023 at 22:13
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    $\begingroup$ @maplemaple I did my best to fix it. Thanks for pointing it out. $\endgroup$ Dec 13, 2023 at 16:36

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