Compute the fundamental group \(\pi_1(S^n, x_0)\) for \(n \ge 2\) using the Seifert–van Kampen Theorem by decomposing the \(n\)-sphere \(S^n\) into the complements of the north and south poles.
Solution
1. Let \(N,S\in S^n\) be the north and south poles and set\[ U=S^n\setminus\{S\},\qquad V=S^n\setminus\{N\}. \]By stereographic projection, both \(U\) and \(V\) are homeomorphic to \(\mathbb R^n\), hence\[ \pi_1(U)=\pi_1(V)=1. \]
2. Their intersection is\[ U\cap V=S^n\setminus\{N,S\}\cong\mathbb R^n\setminus\{0\}. \]This deformation retracts onto \(S^{n-1}\), which is path-connected for \(n\ge2\). Thus \(U\cap V\) is path-connected.
3. Choose \(x_0\in U\cap V\). Seifert–van Kampen identifies \(\pi_1(S^n,x_0)\) with the pushout\[ \pi_1(U,x_0)*_{\pi_1(U\cap V,x_0)}\pi_1(V,x_0). \]Both maps from \(\pi_1(U\cap V,x_0)\) land in the trivial group, so this pushout is trivial. Therefore\[ \pi_1(S^n,x_0)=1 \qquad(n\ge2). \]