Prove that $\pi_1(L(p,q))\cong\mathbb{Z}/p\mathbb{Z}$

Question

Let $p, q>0$ be coprime integers and

$S^3\equiv\{(z, w)\in\mathbb{C}^2||z|^2+|w|^2=1\}$

Then, an equivalence relation $\sim$ on $S^3$ as follows.

$(z, w)\sim(z', w') \overset{def} \Leftrightarrow \exists k\in\mathbb{Z}, z=e^{\frac{2\pi i k}{p}}z', w=e^{\frac{2\pi i kq}{p}}w'$

Last, $L(p, q)\equiv S^3/\sim$ (Lens space) be a quotient space and $f:S^3\longrightarrow L(p,q)$ be a quotient map.

(i) Prove that $f$ is a covering map.

(ii) Prove that $\pi_1(L(p,q))\cong\mathbb{Z}/p\mathbb{Z}$.

I know the equivalence relation is related to a group action of $\mathbb{Z}/p\mathbb{Z}$, but I don't still understand (ii) in particular. Can someone help solve this problem? I'm sorry but I'm a beginner in topology. Thanks.

Answer 1

There is a covering action of $\mathbb{Z}/p$ on $S^3$ (in the sense of Hatcher) given by $(e^{2 \pi i/p})\cdot(z,w)=(z \cdot e^{2\pi i /p},w \cdot e^{2 \pi i q/p})$. The quotient space of $S^3$ by this action is $L(p,q)$. Since $S^3$ is simply connected, Hatcher 1.40 (page 72) implies that $\mathbb Z/p$ is the fundamental group.