Chromatic Homotopy Theory
《Chromatic Homotopy Theory》 · 第 97 页 · Section 3.4.10，Lens spaces（三维版本）
Let \(p\geq1\) and \(q\geq1\) be coprime integers and set \(\omega=e^{2\pi i/p}\). We identify \(\mathbb R^4\) with \(\mathbb C^2\) and see \(S^3\) as \(S^3=\{(z,w)\in\mathbb C^2\mid |z|^2+|w|^2=1\}\). The map \(f(z,w)=(\omega z,\omega^q w)\) is an isometry of \(\mathbb R^4\) because it consists of two simultaneous rotations on the coordinate planes \(w=0\) and \(z=0\). […] Therefore the group \(\Gamma=\langle f\rangle\) generated by \(f\) acts freely on \(S^3\), and is discrete because it is finite. We have constructed an elliptic manifold \(S^3/\Gamma\), called a lens space and indicated with the symbol \(L(p,q)\).