Lens space

Example

Let $p \ge 2$ and $q$ be coprime integers, and let $\zeta = e^{2\pi i/p}$. View the $3$-sphere as $S^3 = \set{(z_1, z_2) \in \C^2 : \abs{z_1}^2 + \abs{z_2}^2 = 1}$. The map $g(z_1, z_2) = (\zeta z_1, \zeta^q z_2)$ generates an action of the cyclic group $\Z/p$ on $S^3$. The lens space $L(p, q)$ is the quotient $S^3 / (\Z/p)$.

The action is free. Indeed, suppose $g^k$ fixes $(z_1, z_2)$ with $0 < k < p$. Then $\zeta^k \ne 1$ forces $z_1 = 0$, and $\zeta^{kq} \ne 1$ (because $q$ is prime to $p$) forces $z_2 = 0$, which is impossible on $S^3$. So the quotient map is a covering map and $L(p, q)$ is a closed orientable 3-manifold. Its fundamental group is $\Z/p$, since $S^3$ is simply connected.