I want to find all (good enough) covers of $S^1$. So I'm thinking like.
$$\pi_1{(S^1)}=\mathbb Z$$
It's abelian, that's why all subgroups (which are exactly $\mathbb Z/n\mathbb Z$) are normal. As I understood from Hatcher book, for each normal subgroup $H$ there is one (up to isomorphism) covering space $X$ s.t. $\pi_1{(X)} = H$.
So for each $n$ there should be unique (up to isomorphism) covering space $X$ s.t. $\pi_1{(X)} = Z/nZ$. Am I right? How can I construct these spaces?
Thanks.