State the Seifert–van Kampen Theorem and use it to show that \(\pi_1(S^n)=0\) for \(n>1\). Why doesn’t this argument work for \(n=1\)?
Solution
A standard form of Seifert–van Kampen says: if \(X=U\cup V\), with \(U,V,U\cap V\) path connected and the basepoint in \(U\cap V\), then \[ \pi_1(X)\cong \bigl(\pi_1(U)*\pi_1(V)\bigr)/N, \] where \(N\) is the normal subgroup generated by \[ i_{U*}(\omega)i_{V*}(\omega)^{-1},\qquad \omega\in\pi_1(U\cap V). \]
For \(S^n\) with \(n>1\), let \[ U=S^n\setminus\{N\},\qquad V=S^n\setminus\{S\}. \] Both \(U\) and \(V\) are homeomorphic to \(\mathbb R^n\), hence simply connected. Their intersection deformation retracts onto \(S^{n-1}\), and for \(n>1\) it is path connected. Van Kampen therefore gives \[ \pi_1(S^n)=0. \] For \(n=2\) the intersection has fundamental group \(\mathbb Z\), but both maps into the trivial groups \(\pi_1(U)\) and \(\pi_1(V)\) are trivial, so the conclusion is still \(0\).
For \(n=1\), the same cover has \[ U\cap V=S^1\setminus\{N,S\}, \] which has two components. Thus the path-connected-intersection hypothesis of this form of van Kampen fails; this is exactly why the argument does not prove \(\pi_1(S^1)=0\).