Compute the fundamental group of the (surface of a) sphere when three points on it are removed.
Solution
Let the three removed points be \[ p_1,p_2,p_3\in S^2. \]
1. The three-times punctured sphere is homeomorphic to the plane with two points removed.
Stereographic projection from \(p_3\) is a homeomorphism \[ S^2\setminus\{p_3\}\longrightarrow\mathbb R^2. \] The distinct points \(p_1,p_2\) have distinct images, say \(a,b\in\mathbb R^2\). Restricting stereographic projection therefore gives a homeomorphism \[ S^2\setminus\{p_1,p_2,p_3\} \cong \mathbb R^2\setminus\{a,b\}. \]
2. The space \(\mathbb R^2\setminus\{a,b\}\) deformation retracts onto a graph homeomorphic to two circles joined by an arc.
Choose disjoint small closed disks \(D_a,D_b\) centered at \(a,b\). Choose an embedded arc \(J\) joining one point of \(\partial D_a\) to one point of \(\partial D_b\), with interior disjoint from the two disks. Set \[ \Gamma=\partial D_a\cup J\cup\partial D_b. \]
Take a sufficiently small closed regular neighborhood \(N\) of \(\Gamma\) in \[ \mathbb R^2\setminus\{a,b\}. \] The set \(N\) is a pair of pants: its complementary components are a punctured neighborhood of \(a\), a punctured neighborhood of \(b\), and the unbounded exterior component. Each of these three complementary annular ends can be pushed along its collar onto the corresponding boundary component of \(N\). Doing these three collar deformations, whose supports are disjoint, gives a deformation retraction \[ \mathbb R^2\setminus\{a,b\}\longrightarrow N. \]
By construction, a regular neighborhood of a graph collapses along its interval fibers onto the graph itself, so \(N\) deformation retracts onto \(\Gamma\). Composing the two deformation retractions gives \[ \mathbb R^2\setminus\{a,b\}\simeq\Gamma. \] The graph \(\Gamma\) consists of two circles connected by the contractible arc \(J\). Contracting \(J\) to a point gives \[ \Gamma\simeq S^1\vee S^1. \]
3. The fundamental group of \(S^1\vee S^1\) is the free group on two generators.
Let \(U\) and \(V\) be open neighborhoods in \(S^1\vee S^1\) of the first and second circles, enlarged slightly across the wedge point so that \[ U\cap V \] is contractible. Then \[ U\simeq S^1, \qquad V\simeq S^1, \qquad U\cap V\simeq\{*\}. \] The Seifert–van Kampen theorem therefore gives \[ \pi_1(S^1\vee S^1) \cong \pi_1(S^1)*\pi_1(S^1) \cong \mathbb Z*\mathbb Z. \] This is the free group \(F_2\) on two generators.
4. Therefore\[ \boxed{\pi_1\bigl(S^2\setminus\{p_1,p_2,p_3\}\bigr)\cong F_2\cong\mathbb Z*\mathbb Z.} \]
By 1, the punctured sphere is homeomorphic to the twice-punctured plane. By 2, that plane has the homotopy type of \(S^1\vee S^1\). Fundamental groups are invariant under homeomorphism and homotopy equivalence, and 3 computes the latter group as \(F_2\).