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?
Best Answer
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 $(x,y+1)_{\sim2}$.