# All Covering spaces of the Annulus

Consider the annulus $$A:=\{(x,y)\in\mathbb{R}^{2}\mid 1\leq x^{2}+y^{2}\leq 4\}$$ I have to find all (connected) covering up to isomorphism.

My proof so far:

First of all, we have a homeomorphism $$A\cong S^{1}\times [1,2]$$. Therefore, we find $$\pi_{1}(A)\cong \pi_{1}(S^{1}\times [1,2])\cong \pi_{1}(S^{1})=\mathbb{Z}$$. All the subgroups of $$\mathbb{Z}$$ are of the form $$\mathbb{Z}n$$ for $$n\in\mathbb{N}$$ and since $$\mathbb{Z}$$ is abelian, all of them define distinct conjugacy classes of subgroups.

Now there is a theorem, which says that,if $$A$$ admits an universl cover, the then there are for each $$n$$ a covering $$(\widetilde{A},p)$$ such that $$p_{\ast}(\pi_{1}(\widetilde{A}))\cong n\mathbb{Z}$$ and this are all coverings up to isomorphism.

Now, I know that for $$S^{1}$$, we can define the maps $$f_{n}:z\mapsto z^{n}$$, which define covering maps of $$S^{1}$$, such that $$(f_{n})_{\ast}(\pi_{1}(S^{1}))\cong n\mathbb{Z}$$. So therefore, my idea was to define the maps $$g_{n}:S^{1}\times [1,2]\to A\cong S^{1}\times [1,2]$$ exactly by $$g_{n}(z,\lambda):=(f_{n}(z),\lambda)$$. This should then be a covering with the claimed property $$(g_{n})_{\ast}(\pi_{1}(S^{1}\times [1,2]))\cong n\mathbb{Z}$$.

Now to my question: Does this look right so far? and furthermore, the annulus $$A$$ is not simply connected, therefore all of my constructed coverings above are not universal. So I still have to find a universal cover, in order to show that I can apply the theorem above.

• You know the universal cover of $S^1$, and you observed that $A$ is homotopy equivalent to $S^1$, so the universal cover of $A$ should be similar... Jan 5, 2021 at 15:32
• Consider $p:\Bbb R\times [1,2]\longrightarrow A$ defined as $p(\theta,r)=re^{2\pi i\theta}$ Jan 5, 2021 at 15:40
• Ahh I see thanks.... And the rest of my idea is right? Jan 5, 2021 at 15:41

(1) The universal cover $$p:\mathbb{R}\times [1,2] \to A, \\ (t,\lambda)\mapsto (\lambda \cos(2\pi t),\lambda\sin(2\pi t))$$
(2) For each $$n\in\mathbb{N}$$ the cover $$g_{n}:S^{1}\times [1,2] \to A, \\ (e^{i\theta},\lambda)\mapsto (\lambda \cos(\theta n ),\lambda\sin(\theta n ))$$
• I would suggest writing (2) as $(e^{i\theta},\lambda)\mapsto \lambda e^{in\theta}$ where we identify $\mathbb R^2 = \mathbb C$, just because otherwise it reads as "select a point on the unit circle with coordinate $z = e^{i\theta}$, extract the angle $\theta$, and feed the angle to trig functions" which is messy. Otherwise excellent! Jan 6, 2021 at 2:11