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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!

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1 Answer 1

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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.
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