Let \(m\geq 2\), \(n \geq 1\) and let \(\ell_1, \ldots, \ell_n\) be integers coprime to \(m\). Let \(\ZZ/m\) act on the unit sphere \(S^{2n-1}\subseteq \CC^n\) with the generator acting by \[ (z_1, \ldots, z_n) \mapsto \qty{ e^{2\pi i \ell_1/m} z_1, \ldots, e^{2\pi i \ell_n /m} z_n } .\] The lens space is the orbit space \[ L_m(\ell_1, \ldots, \ell_n) \coloneqq S^{2n-1}/\qty{\ZZ/m} \] [@Hat02].
The action is free, so \(S^{2n-1}\to L_m(\ell_1,\ldots,\ell_n)\) is a covering space [@Hat02], and \(L_m(\ell_1,\ldots,\ell_n)\) is a closed \((2n-1)\)-manifold. For \(n\geq 2\) the sphere \(S^{2n-1}\) is simply connected, so it is the universal cover and \(\pi_1(L_m(\ell_1,\ldots,\ell_n)) \cong \ZZ/m\) [@Hat02]. For \(m=2\) the lens space is \(\RP^{2n-1}\).