Definition 20.3 (Generalized lens spaces):

Let $q_1, q_2, \dots, q_n$ be relatively prime to $p$. Define the action of the cyclic group $\mathbb{Z}_p$ on $S^{2n+1}$ by

$$(z_0, z_1,\dots, z_n)\mapsto \Big(z_0\exp\Big(\frac{2\pi i}{p}\Bi... ...p\Big(\frac{2\pi iq_1}{p}\Big),\dots, \exp\Big(\frac{2\pi iq_n}{p}\Big)\Big),$$

where $S^{2n+1} = \{(z, w_1, w_2, \dots, w_n)\in \mathbb{C}^{n+1} / \vert z\vert^2 + \vert w_1\vert^2 + \dots + \vert w_n\vert^2 = 1\}$. The resulting orbit space is denoted by $L(p, q_1, q_2, \dots, q_n)$ and its fundamental group is $\mathbb{Z}_p$ since the action is properly discontinuous.