How to compute the fundamental group of $S^2 / A$, where $A$ is a finite set of points?

Question

Along the way to a much simpler solution to a homology problem, I thought about computing the fundamental group of $S^2 / A$. I quickly ran into trouble, so I want to know if there is there a slick way to do this. (For nontrivial |A|, of course.)

I don't think that Van Kampen's theorem works, since the intersection will not be path connected for $|A| \geq 2$. (At least for the obvious covers.)

Answer 1

Let us do first the following simple computation. Let $D$ be the closed unit disc in the plane and let $A$ be a finite subset of the interior of $D$. What is $\pi_1(D/A)$?

Let $n=|A|$. It does not matter which $n$ points are in $A$, only the cardinal of the set, so we may assume that the points of $A$ are the $n$ vertices of a regular polygon centerd at the origin. Let us take basepoint in $D/A$ to be the (image of) the origin. There is an obvious retract of the space $D/A$ to the space obtaned from a regular $n$-gon by identifing its vertices. This looks like a parachute.

This space can clearly be constructed as a CW-complex as follows: start with one vertex. Next add $n$ $1$-cells in the form of loops, and now glue a $2$-cell in the obvious way. The fundamental group of this space is then $\langle x_1,\dots,x_n:x_1\cdots x_n=1\rangle$. This is easily seen to be a free group in $n-1$ generators.

Now your sphere can be covered with two open sets. One of which is an open disc and the other an open set which deformation retracts to my $D/A$. The intersection is a (thick) circle. Now use van Kampen.

Answer 2

It seems standard texts fail to take into account the algebraic modelling by the theory of groupoids.

There is a nice construction introduced by Philip Higgins, see his downloadable book Categories and Groupoids, Chapter 8, in which you start with a groupoid $G$ and a function $f: Ob(G) \to Y$ where $Y$ is any set. Then there is a groupoid which he writes $U_f(G)$ with object set $Y$ and with a universal property for morphisms $G \to H$ whose function on objects factors through $f$. Thus $U_f(G)$ is obtained from $G$ by "identifying certain objects of $G$", and perhaps adding some. This construction includes that of free groups, and of free products of groups.

The topological interpretation in terms of the fundamental groupoid $\pi_1(Z,C)$ of a space $Z$ for a set $C$ of base points is given on p.343, 9.1.2 (Corollary 3) of Topology and Groupoids, and was in the 1968 edition of this book.

This answers the special case of the question, as given by Mariano.

Later: I think also that my and other, answers to this mathoverflow question on many base points and groupoids are relevant.