I'm trying a lot to find easy examples of classification theorems of covering spaces of a given space. I've already read some examples here at Mathexchange such as

Classification of covering spaces of $\Bbb{R}\textrm{P}^2 \vee \Bbb{R}\textrm{P}^2$.

Why is a covering space of a torus $T$ homeomorphic either to $\mathbb{R}^2$, $S^1\times\mathbb{R}$ or $T$?

I found these very hard to prove, I would like to know if anyone knows some easy and trivial examples to begin with or if anyone knows some interesting sources to help beginners like me.

Thanks a lot

  • $\begingroup$ I believe Hatcher looks at covers of $S^1 \vee S^1$ in a good bit of detail. Try working through that, then going back to the two-dimensional examples. (If you're not familiar with the book: Allen Hatcher's Algebraic Topology is available for free on his website) $\endgroup$ – Tabes Bridges Jan 27 '13 at 23:24

Let's see two simple examples of classifications of coverings. As usual, let $p:E\rightarrow B$ be a covering map with $E$ path connected and $B$ locally path connected, path connected and semi-locally simply connected. Let $b_0 \in B$ and $x_0 \in p^{-1}(b_0)$.

Example 1

Take the trivial subgroup of $\pi_1(B,b_0)$, let's call it $0$. What is the class of coverings to be associated to it? According to our classification theorem, this class is characterized by $p_*(\pi_1(E,x_0))=0$.

This means that $\forall [\alpha]_E\in\pi_1(E,x_0)$, $p_*([\alpha]_E)=e_B$ (where $e_B$ is the equivalence class of the constant path in $b_0$). Another way of writing that equation is $p_*([\alpha]_E)=p_*(e_E)$, so by injectivity of $p_*$ we get $[\alpha]_E=e_E$ and so we conclude that the class is that of the universal convering (and all of its members are its automorphisms).

Example 2

Take $B=\Bbb{S}^1$. What is the conjugacy class of a subgroup of $\pi_1(\Bbb{S}^1,b_0)$ associated to a given covering map $p:E\rightarrow B$? Since $\pi_1(\Bbb{S}^1,b_0)=\Bbb{Z}$, and this is an abelian group, its conjugacy classes of subgroups are just each subgroup, and for the integers, they are of the form $n\Bbb{Z}, n\in\Bbb{N}$. So, to each covering is associated the (singleton) class of $n\Bbb{Z}$ and you can check that this corresponds to taking "n-times faster" loops around the helix that represents its typical convering map, $exp:\Bbb{R}\rightarrow\Bbb{S}^1$.


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