# The double cover of Klein bottle

I try to find out all the double covers of Klein bottle. Since the Euler characteristic is multiplicative with respect to covering space, there are only two candidates, that is, torus and Klein bottle itself. It is not hard to construct a mapping from torus to Klein bottle.

My question is, is it possible to realize the Klein bottle as a double cover of itself?

• Are you familiar with the construction of the Klein bottle by identifying sides of a square? Aug 16 '12 at 5:07
• @QiaochuYuan Yes. Can you explain your question? Aug 16 '12 at 6:05
• @Hezudao: that construction can be used to write down a double cover. This can also be verified by a fundamental group computation. Aug 16 '12 at 6:24
• en.wikipedia.org/wiki/Klein_bottle Aug 16 '12 at 8:21

For any $c>0$ the relations $$(x,y)\sim (x, \,y+ k c)\quad(k\in{\mathbb Z}), \qquad (x,y)\sim \bigl(x+\ell,\,(-1)^\ell y\bigr) \quad(\ell\in{\mathbb Z})$$ define a Klein bottle $K_c$ of "length" $1$ and "width" $c$ as a quotient of the $(x,y)$-plane , and with a rectangle $[0,1]\times[0,c]$ as fundamental domain.
The identity map $\,\iota: \,{\mathbb R}^2\to{\mathbb R}^2$ realizes $K_2$ as a double cover of $K_1$: Each point $(x,y)_{\sim1}\in K_1$ has two preimages in $K_2$, namely $(x,y)_{\sim2}$ and $(x,y+1)_{\sim2}$.
The idea is to use the family of automorphisms $\tau(x,y) = (x,y+kc)$, $\sigma(x,y) = (x+l,(-1)^ly)$. This automorphisms can be seen to induce the Klein bottle on the square $[0,1] \times [0,c]$ with the help of usual theorems of topological identifications. So we would have that for each $c > 0$, the map $\pi_c:\mathbb{R}^2 \to \mathbb{R}^2/G_c$ is a covering map and as usual the group of automorphisms of these covers is $G_c$.
You may then observe that, $G_1 \le G_2$ we have a subgroup automorphisms so that there is a lemma we can reuse here that says that there exists a unique mapping from $\hat f:\mathbb{R}^2/G_2 \to \mathbb{R}^2/G_1$ given by $G_2 \cdot y \mapsto \pi_1(y)$ (where $\pi_1$ is the projection to the orbit space) and that this mapping is a covering. Therefore, if you identify each one as a Klein bottle you get your double cover since every point has two preimages.