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I am studying for a qualifying exam and it asks for all the covering spaces of the Klein Bottle. Does anyone have any suggestions? I am not sure how to go about finding ALL of them.

dunkindonuts
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  • You should use the one-one correspondence between covers and subgroups. So to classify all the covers you need to describe subgroups of $\pi_1(K)$. – Sasha Patotski Jul 24 '13 at 04:03
  • @SashaPatotski: That is very hard, isn't it? Think about Euler characteristic. – Ted Shifrin Jul 24 '13 at 04:05
  • yeah but how do I know the covering must be compact? Is this an assumption that follows from something? – dunkindonuts Jul 24 '13 at 04:10
  • The universal cover is $\mathbb{R}^2$ --- certainly not compact! – Neal Jul 24 '13 at 04:11
  • so then how is euler characteristic useful? – dunkindonuts Jul 24 '13 at 04:12
  • I have that the torus and Klein bottle are coverings. Not sure how to see that $\mathbb{R}^2$ is a covering. – dunkindonuts Jul 24 '13 at 04:13
  • @TedShifrin Maybe I am confusing something, but I don't think it's too hard. The fundamental group has presentation $\langle a,b\mid aba^{-1}=b^{-1}\rangle$. It has normal subgroup generated by $b$, which is isomorphic to $\mathbb{Z}$. So if we take the quotient of $\pi_1$ by this subgroup, it will be isomorphic to $\mathbb{Z}$. So to classify all the subgroups of $\pi_1$, we have two cases: the subgroup maps to $0$ in the quotient or to some non-trivial subgroup of $\mathbb{Z}$. Both cases are not too difficult to describe. Am I missing something? – Sasha Patotski Jul 24 '13 at 04:15
  • You're right. It's only useful for the compact ones. You certainly should know $\mathbb R^2$ is the universal covering. @SashaPatotski: Something seems fishy there! What is the abelianization of $\pi_1$? But can you figure out the finite covering spaces with this approach? – Ted Shifrin Jul 24 '13 at 04:19

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