# How to calculate the fundamental group of general configuration space

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, $M=T^2$.

• The first question is a basic fact about the braid groups. This is where their name comes from too. Apr 19, 2016 at 12:02
• 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. Apr 19, 2016 at 12:30

(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.

• What's the Dold-Puppe Theorem? Apr 22, 2016 at 4:26
• ams.org/mathscinet/search/… Apr 22, 2016 at 5:19
• 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. Apr 22, 2016 at 5:40
• The link you provided has expired. Dec 6, 2023 at 22:13
• @maplemaple I did my best to fix it. Thanks for pointing it out. Dec 13, 2023 at 16:36