Let $K$ be a Klein bottle. I know that $$\pi_1(K)\cong \langle a,b | abab^{-1}\rangle.$$ All 2-fold coverings of $K$ will correspond to index $2$ subgroups of $\pi_1(K)$. Thus up to isomorphism, 2-fold coverings of $K$ correspond to surjective homomorphisms $\pi_1(K)\rightarrow \mathbb{Z}/2\mathbb{Z}$.
There are three such homomorphisms:
- $a\mapsto 0$, $b\mapsto 1$. This one is easy to find by splitting the Klein bottle in two.
- $a\mapsto 1$, $b\mapsto 0$. This one is also easy to find, similar to the one above but with a torus.
- $a\mapsto 1$, $b\mapsto 1$.
This third one I can't seem to find the corresponding covering space. From the question I got it from I know that it should be another Klein bottle. It's also clear that the paths corresponding to $a$ and $b$ should not be loops in this covering space. I have been stuck on this for a bit, and also didn't find any clear answer online. I have tried many sorts of shapes and ways to partition the Klein bottle but can't seem to find it.
I'd appreciate any help!