# Orbifold fundamental group and configuration space

I'm not very familiar with (even simple examples of) orbifolds, so my first question is:

Let $$C_2$$ be $$\mathbb{C}$$ with one cone singularity at 0 of index 2. What is the fundamental group of $$C_2$$ minus $$k$$ points ?

My naive answer is: take $$\mathbb{C}^*$$ minus the same $$k$$ points. Its fundamental group is freely generated by the $$k+1$$ loops around the punctures. Now decide that you don't have a "hole" in 0 anymore, but a cone singularity, meaning that the generators corresponding to a loop around 0 is now of order 2. Then I would say that the fundamental group of $$C_2$$ minus $$k$$ points is $$\langle a_0,\dots,a_k | a_0^2=1\rangle$$, ie $$Z_2\ltimes F_{2k}$$, where $$F_{2k}$$ is generated by {$$a_i,a_0a_ia_0, i\geq 1$$}.

Now recall the following construction: take the pure braid group $$P_n$$ with its standard generators $$x_{i,j}, 1\leq i < j\leq n$$ given by taking the $$j$$th strand, letting it go behind all other strand, loop around the $$i$$th one and going back. Then it's quite easy to see that the subgroup generated by the $$x_{i,n}$$ is free: it is the subgroup of pure braids for which all but the last strand are fixed straight lines. In fact, it leads to a semidirect product decomposition $$P_n=P_{n-1}\ltimes F_{n-1}$$. This decomposition is actually a so called "almost direct" product, which is quite an important fact.

This construction has a nice geometric interpretation: let $$X_n$$ be the configuration space of $$n$$ points in $$\mathbb{C}$$, and recall that $$P_n=\pi_1(X_n)$$. Then the map $$X_n \rightarrow X_{n-1}$$ which forget the last coordinate is a locally trivial fibration with fiber $$\mathbb{C}$$ minus $$n-1$$ points. Then it induces a (split) short exact sequence of fundamental groups

$$1\rightarrow F_{n-1} \rightarrow P_n \rightarrow P_{n-1}\rightarrow 1$$

Let's try to do something similar with the "orbifold braid group" of $$C_2$$, that is the fundamental group $$P_n(C_2)$$ of $$O_n=$${$$z_1,\dots,z_n \in C_2, z_i \neq z_j$$}.

It seems to me that $$P_n(C_2)=P_{n+1}/ \langle x_{1,i}^2=1,i=2 \dots n+1 \rangle$$.

The above construction seems to work "at the algebraic level": let $$G_n$$ be the subgroup of $$P_n(C_2)$$ generated by (the images of) $$x_{i,n+1}$$. What is stated in this paper (in a slighty different form) is that $$P_n(C_2)=P_{n-1}(C_2) \ltimes G_n$$, and that it is an almost direct product too.

But $$G_n$$ satisfies some relations, for example $$x_{i,n+1}$$ and $$x_{0,n+1}x_{i,n+1}x_{0,n+1}$$ commute for a given $$i$$, hence it is not isomorphic to the fundamental group of $$C_2$$ minus $$n-1$$ points (at least if my first naive try is not wrong). While this construction strongly looks like to and shares many algebraic properties with the construction for $$P_n$$, it does not seems to come from a natural geometric construction. So my real question is:

Am I wrong somewhere ? Is there a natural interpretation of $$G_n$$ ?

Edit: Here is roughly what happen: assuming that $$n=2$$ for the sake of simplicity, it doesn't make sense to "freeze" the first strand (and its negative) and to make the second one loop around because the following relation holds:

now pushing the red loop (seen as a loop in the 2-punctured plane) to the bottom plane, we see that it has to be identified with its conjugate by a loop around the two strands at once, ie by the product of the generators of $$F_2$$. Therefore, this product has to be central, leading to the relation holding in $$G_n$$ above. So one can ask:

Is there a topological space modelled on this situation, i.e. which looks like to the "complementary in $$\mathbb{C}\times[0,1]$$ of two strands modulo homotopy". Or at least, is there a way to prove that there are no other relations than declaring that the big loop is central ?

• Hi Adrien, this is not an answer, but have a look at the comment of 11/02/2009 on pages.bangor.ac.uk/~mas010/pstacks.htm Commented Jan 10, 2012 at 3:45
• You may want to look at fundamental groups of tautological orbi-bundles whose fibers are orbifolds whose underlying space is the 2-sphere (with several cone point of order 2 and several other points marked) and whose base is the moduli space of such marked orbifolds, where all marked points and cone points have different labels. Such groups will have a normal subgroup of the type you are interested in (free products of cyclic groups of infinite order and order 2) and the quotient group a pure braid group. Commented May 22, 2012 at 14:32
• The link to springerlink.com is broken. I'm also unable to find any snapshot saved on the Wayback Machine. Commented Apr 10, 2023 at 0:27
• @TheAmplitwist It was a while ago, but I'm fairly sure this was "The $D_n$ generalized pure braid group" by Markushevich (link.springer.com/article/10.1007/BF00181653), I'll edit. Beware however, I've been told that their presentation by generators and relation of what is denoted $H_k$ (=$G_n$ in my question), hence their proof, is incorrect although the result itself is correct (see eg arxiv.org/abs/math/0502120). Commented Apr 11, 2023 at 7:05
• @TheAmplitwist For a more geometric perspective you might also want to have a look at papers by Roushon, eg arxiv.org/abs/2006.07106 Commented Apr 11, 2023 at 7:14