Let \(X\) be the union of two copies of \(S^2\) having a single point in common. What is the fundamental group of \(X\)? Prove that your answer is correct. [Be careful! The union of two simply connected spaces having a point in common is not necessarily simply connected. See [S], p. 59.]
Solution
1. Open cover construction:
1.1. Let \(X = S_1 \cup S_2\) with \(S_1 \cap S_2 = \{p\}\), where each \(S_i \cong S^2\).Choose points \(q_1 \in S_1 \setminus \{p\}\) and \(q_2 \in S_2 \setminus \{p\}\) (e.g. the antipodal points to \(p\) on each sphere).
each \(S_i \setminus \{p\}\) contains points distinct from \(p\).
1.2. Define the open sets in \(X\):\[ U = X \setminus \{q_1\} = (S_1 \setminus \{q_1\}) \cup S_2, \qquad V = X \setminus \{q_2\} = S_1 \cup (S_2 \setminus \{q_2\}). \]Since \(\{q_1\}\) and \(\{q_2\}\) are closed singletons in Hausdorff spaces, \(U\) and \(V\) are open in \(X\), and \(U \cup V = X \setminus \emptyset = X\).
complements of closed points in \(X\) are open.
2. Fundamental groups of \(U, V\), and \(U \cap V\):
2.1. Fundamental group of \(U\):\(S_1 \setminus \{q_1\}\) is homeomorphic to \(\mathbb{R}^2\), which deformation retracts onto the point \(p\).This radial deformation retraction extends to \(U\) by fixing \(S_2\) pointwise.Thus \(S_2\) is a deformation retract of \(U\), so:\[ \pi_1(U, p) \cong \pi_1(S_2, p) \cong \pi_1(S^2) = 0. \]
\(\pi_1(S^2) = 0\) and homotopy invariance.
2.2. Fundamental group of \(V\):Symmetrically, \(V\) deformation retracts onto \(S_1\), so:\[ \pi_1(V, p) \cong \pi_1(S_1, p) \cong \pi_1(S^2) = 0. \]
symmetry with 1.
2.3. Fundamental group of \(U \cap V\):The intersection is \(U \cap V = (S_1 \setminus \{q_1\}) \cup (S_2 \setminus \{q_2\})\).Both pieces are homeomorphic to \(\mathbb{R}^2\) and share the common point \(p\).The simultaneous radial retraction onto \(p\) on each piece shows that \(U \cap V\) deformation retracts to the point \(\{p\}\).Thus \(U \cap V\) is contractible, hence path-connected with \(\pi_1(U \cap V, p) = 0\).
union of two contractible open sets sharing a point.
3. Seifert–van Kampen Theorem:
3.1. Since \(U\) and \(V\) are open, \(U \cup V = X\), and \(U \cap V\) is path-connected, the Seifert–van Kampen Theorem gives:\[ \pi_1(X, p) \cong \pi_1(U, p) *_{\pi_1(U \cap V, p)} \pi_1(V, p) \cong 0 *_0 0 = 0. \]
Seifert–van Kampen Theorem for an open cover.
4. Conclusion:The fundamental group of the wedge sum of two 2-spheres is the trivial group: \(\pi_1(X) \cong 0\). Q.E.D.
1 through 3.