Let $\mathbb{C}^{\times}=\mathbb{C} \setminus \{0\}$. I'm trying to find out all covering spaces of this space.
Let's start. (I'm using Massey's book: Algebraic Topology, an introduction)
First of all I know that $\mathbb{C}^{\times}$ admits universal covering space, for example $(\mathbb{C}, p=exp(z))$, so I know that Aut$_{\mathbb{C}^{\times}}(\mathbb{C})$ (=group of automorphisms of the covering sapce $\mathbb{C}$) is isomorphic to $\pi_1(\mathbb{C}^{\times})$. However $\mathbb{C}^{\times} \cong \mathbb{R}^2 \setminus \{0\}$ then $\pi_1(\mathbb{C}^{\times}) \cong \mathbb{Z} \cong $ Aut$_{\mathbb{C}^{\times}}(\mathbb{C})$. Now the problems begin.
The main idea is retracing the proof of Lemma 10.1,chapter 5 on Massey's book. For example, let $G=3\mathbb{Z}$ subgroup of $\mathbb{Z}$. Now there exists a subgroup of Aut$_{\mathbb{C}^{\times}}(\mathbb{C})$ called A$_G$ which is isomorphic to $G$. Let $\tilde X = \mathbb{C}/$A$_G$; I know that $(\mathbb{C},r)$ (with $r \colon \mathbb{C} \to \tilde X$ the natural projection) is a covering sapce of $\tilde X$ and so $(\tilde X, q)$, with $q$ such that $p=q \circ r$, is also a covering space of $\mathbb{C}^{\times}$. Finally, $G$ is normal in $\pi_1(\mathbb{C}^{\times})$ and so, by using Corollary 7.4, I see that Aut$_{\mathbb{C}^{\times}}(\tilde X) \cong \mathbb{Z}/ \mathbb{3Z}$ and $\pi_1(\tilde X) \cong \mathbb{3Z}$.
Ok, that's great, but
I would like to know how the cover map $r$ actually works; how Aut$_{\mathbb{C}^{\times}}(\mathbb{C})$ actually is (its elements); how I can imagine $\tilde X$ (maybe $\mathbb{S}^1 \times \mathbb{R}$ ? Is this isomorphic to $\mathbb{C}^{\times}$? Right?). Can I suppose $q \colon z \mapsto z^3$? But why?
Aut$_{\mathbb{C}^{\times}}(\tilde X) \cong \mathbb{Z}/ \mathbb{3Z}$ and $\pi_1(\tilde X) \cong \mathbb{3Z}$ are usefull in order to answer the previous questions or are they something extra?
Extra question: Let $(\tilde X,p)$ a covering space of $X$. I have studied the action of the group $\pi_1(X,x)$ on the fiber $p^{-1}(x)$. Is this topic
concretely used on computing covering spaces or I can see it like theory employed in order to demonstrate the Existence Theorem of covering spaces (Lemma 10.1 and Theorem 10.2)?
Best Answer
The universal cover of $\mathbb{C} - \{0\}$ is $\mathbb{C} $. The covering map is the exponential so it's fundamental group is $\mathbb{Z} $. The coverings correspond to quotient of $\mathbb{C}$ by subgroups of $\mathbb{Z}$.