# What are all three 2-fold coverings of the Klein bottle?

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:

1. $$a\mapsto 0$$, $$b\mapsto 1$$. This one is easy to find by splitting the Klein bottle in two.
2. $$a\mapsto 1$$, $$b\mapsto 0$$. This one is also easy to find, similar to the one above but with a torus.
3. $$a\mapsto 1$$, $$b\mapsto 1$$.

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!

The third covering space can also be viewed by "splitting" the Klein bottle $$K$$, if you are careful about what that means. In each case there is a simple closed curve $$c \subset K$$ such that if you first let $$L_c = K \setminus c$$ be the compact surface-with-boundary obtained by cutting $$K$$ open along $$c$$, then the covering space is obtained by taking the disjoint union of two copies $$L_1,L_2$$ of $$L_c$$, by gluing the boundaries $$\partial L_1$$, $$\partial L_2$$ using an appropriately chosen homeomorphism $$\partial L_1 \mapsto \partial L_2$$.
Careful choice of $$c$$ for each of the three homomorphisms is what distinguishes the three cases, and the idea is to choose $$c$$ to be a simple closed curve in $$K$$ representing an appropriate element $$g_c \in \pi_1 K$$, chosen by two criteria: $$a \mapsto 0$$ if and only if $$a$$ is represented by a closed curve in $$L_c$$; and $$b \mapsto 0$$ if and only if $$b$$ is represented by a closed curve in $$L_c$$. Here are the three cases, with a description of $$L_c$$ (I worked this out with some pictures, so I do THINK it's right):
1. For $$a \mapsto 0$$, $$b \mapsto 1$$, choose $$g_c=a$$. $$L_c$$ is an annulus.
2. For $$a \mapsto 1$$, $$b \mapsto 0$$, choose $$g_c=b$$. $$L_c$$ is a Möbius band.
3. For $$a \mapsto 1$$, $$b \mapsto 1$$, choose $$g_c=aba^{-1}$$. $$L_c$$ is a Möbius band.