Show \(\pi_1(S^n) = 1\) for \(n\geq 2\).
Solution
Goal: Prove that for all integers \(n \geq 2\), the fundamental group \(\pi_1(S^n, x_0)\) is trivial.
1. Cover \(S^n\) by two open subsets \(U\) and \(V\).
1.1. Let \(N = (0, \dots, 0, 1) \in S^n \subset \mathbb{R}^{n+1}\) (North pole) and \(S = (0, \dots, 0, -1) \in S^n\) (South pole).
1.2. Define \(U = S^n \setminus \{S\}\) and \(V = S^n \setminus \{N\}\).
1.3. Since \(\{S\}\) and \(\{N\}\) are closed singletons in \(S^n\), \(U\) and \(V\) are open in \(S^n\).
1.4. \(U \cup V = S^n\) because \(N \neq S\), so no point in \(S^n\) is equal to both \(N\) and \(S\).
1.5. Every point of \(S^n\) is either not equal to \(S\) (so it lies in \(U\)) or not equal to \(N\) (so it lies in \(V\)); since \(N \neq S\), the two sets cover \(S^n\).
2. \(U\) and \(V\) are contractible, hence simply connected.
2.1. Stereographic projection from \(S\) is a homeomorphism \(U \to \mathbb{R}^n\).
2.2. Stereographic projection from \(N\) is a homeomorphism \(V \to \mathbb{R}^n\).
2.3. \(\mathbb{R}^n\) is convex, hence contractible to the origin via straight-line homotopy \(H(x, t) = (1-t)x\).
2.4. Therefore, \(U \simeq * \implies \pi_1(U, x_0) = 1\) and \(V \simeq * \implies \pi_1(V, x_0) = 1\) for any basepoint \(x_0 \in U \cap V\).
2.5. A homeomorphism preserves contractibility, and a contractible space is simply connected, so \(\pi_1(U, x_0) = \pi_1(V, x_0) = 1\).
3. The intersection \(U \cap V = S^n \setminus \{N, S\}\) is path-connected for \(n \geq 2\).
3.1. Under stereographic projection \(\phi \colon U \to \mathbb{R}^n\), the pole \(N\) maps to \(0 \in \mathbb{R}^n\).
3.2. Thus \(U \cap V = U \setminus \{N\} \cong \mathbb{R}^n \setminus \{0\}\).
3.3. \(\mathbb{R}^n \setminus \{0\}\) deformation retracts to \(S^{n-1}\) via \(x \mapsto \frac{x}{\|x\|}\).
3.4. For \(n \geq 2\), the sphere \(S^{n-1}\) has dimension \(n-1 \geq 1\), which is path-connected.
3.5. Therefore, \(\mathbb{R}^n \setminus \{0\}\) is path-connected, which implies \(U \cap V\) is path-connected.
3.6. By <2>1–<2>2, \(U \cap V \cong \mathbb{R}^n \setminus \{0\}\); by <2>3–<2>4, \(\mathbb{R}^n \setminus \{0\}\) deformation retracts to the path-connected sphere \(S^{n-1}\) for \(n \ge 2\), so \(U \cap V\) is nonempty and path-connected.
4. Apply the Seifert-van Kampen theorem to the cover \(\{U, V\}\).
4.1. Let \(x_0 \in U \cap V\) be the basepoint.
4.2. The hypotheses of the Seifert-van Kampen theorem are satisfied: \(U, V\) are open, \(U \cup V = S^n\), and \(U \cap V\) is path-connected.
4.3. The theorem gives an isomorphism: \[\pi_1(S^n, x_0) \cong \pi_1(U, x_0) *_{\pi_1(U \cap V, x_0)} \pi_1(V, x_0).\]
4.4. By 2, \(\pi_1(U, x_0) = 1\) and \(\pi_1(V, x_0) = 1\).
4.5. The amalgamated free product of two trivial groups is trivial: \(1 *_{\pi_1(U \cap V, x_0)} 1 = 1\).
4.6. Substituting \(\pi_1(U, x_0) = 1\) and \(\pi_1(V, x_0) = 1\) from 2 into the van Kampen isomorphism of 3 gives \(\pi_1(S^n, x_0) \cong 1 *_{\pi_1(U \cap V, x_0)} 1 = 1\).
5. Q.E.D.
5.1. 4 establishes \(\pi_1(S^n, x_0) = 1\) for all \(n \geq 2\).