Let K be the Klein bottle obtained by the quotient of $[0, 1] × [0; 1]$
by the equivalence relation $(x, 0) ∼ (1 − x, 1)$ and $(0, y) ∼ (1, y)$.
Is there a non trivial covering of $K$ by $K$?
The universal cover of $K$ is $\Bbb R^2$ and I know the torus can also be a cover of $K$, but I don't know where to start.
Thank you for any hints and help.
Best Answer
One way you can envision the two-fold cover of $K$ by the torus by placing two copies of the given square next to each other such that the $(x,0)$ side of one is touching the $(x,1)$ side of the other. To check that this translates to a well-defined map $T\to K$ is fairly straightforward.
This can be extended to a 3-fold cover of $K$ by itself if you place three such squares next to each other (or more generally for any odd $n$).