Example of a nontrivial finite covering map A covering map $p:C\to X$ is called finite when for each $x\in X$ the fiber of $x$ is finite. I have to prove something about such covering maps, but I have never seen a nontrivial example of one. Could you give me some simple and preferably well-behaved examples? What I think I mean by a trivial covering map is one where $C=\bigsqcup\{X_i\}_{i\in I}$ with homeomorphisms $p_i:X_i\to X$ and $p(x_i)=p_i(x_i)$ for $x_i\in X_i.$
I had a couple of ideas but none worked. For example I tried to take the function $p:\Bbb R\to S_1$, $p(t)=(\cos t,\sin t)$ and restrict it to some bounded intervals (open or not), but that doesn't seem to be right. I also folded a plane and tried projecting it onto a half-plane parallel to it, but this doesn't work either, unless I'm wrong of course.
 A: There are, of course, a lot of examples of finite covering maps.
A (slightly) less common class of examples that comes to mind are finite coverings of the torus. Consider, for instance, the discrete lattice $\Gamma \colon=\mathbb{Z}+\mathbb{Z}i\subsetneq\mathbb{C}$, i.e. the the Gaussian integers. Take the complex number $\alpha\colon=1+i \in \mathbb{C}\setminus\{0\}$. Next, consider the lattice $\Gamma'\colon=\alpha\Gamma$. Then the map
$$p\colon\mathbb{C}/\Gamma\longrightarrow\mathbb{C}/\Gamma'; [z]_\Gamma\longmapsto [\alpha\cdot z]_{\Gamma'}$$ is a covering map. This can be seen by showing that it is a local homeomorphism (hint: $p$ induces a map $\mathbb{C}\rightarrow \mathbb{C}$) and that it satisfies the curve lifting property (for every curve $u\colon [0,1]\rightarrow \mathbb{C}/\Gamma'$ and every point $y\in \mathbb{C}/\Gamma$ with $u(0)=p(y)$, there exists a lifting $\hat{u}\colon [0,1]\rightarrow\mathbb{C}/\Gamma$ of $u$ such that $\hat{u}(0)=y$). Since $\mathbb{C}/\Gamma'$ and $\mathbb{C}/\Gamma$ are manifolds, this implies that $p$ is a covering map, as proven in Theorem 4.19 in Forster's Lectures on Riemann Surfaces.
By one of the isomorphism theorem of groups, the degree of the covering map $p$ is equal to the index of the subgroup $\alpha\Gamma$ in $\Gamma$. In our example, the index is equal to $\vert \alpha \vert^2=2$.
A: The Möbius band $M$ can be obtained as follows. Let the group $\mathbb{Z}$ act on the strip $S=\mathbb{R}\times[-1,1]$ as
$$n\cdot(x,t)=(x+n,(-1)^nt).$$
Then $M=S/\mathbb{Z}$.
So, if you consider the subgroup $H=2\mathbb{Z}$ the quotient $C=M/H$, in fact a cylinder, is a finite cover of $M$.
More generally every time you have a group $\Gamma$ acting "nicely enough" on a space $X$ and $\Gamma^\prime$ is a subgroup of $\Gamma$ of finite index then the natural map
$$
X/\Gamma^\prime\longrightarrow X/\Gamma
$$
is a finite cover. This generalizes all the examples given so far.
