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I am wondering if there is a covering map of the Torus by the Klein Bottle. I know that there is a covering map of the Klein Bottle by Torus given here Two-sheeted covering of the Klein bottle by the torus.

I also do not need to know what the covering is, just if there is such a covering.

Any help would be greatly appreciated.

Bernard
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Mike
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  • hint: a basic result of covering spaces is that if $E\to B$ is a covering map of path-connected spaces then the induced homomorphism $\pi_1(E) \to \pi_1(B)$ is injective – William Jan 15 '20 at 21:19

1 Answers1

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There can be no such covering projection. This follows from standard results on how the fundamental groups are related in a covering projection. (If $K$ is the Klein bottle and $T$ is the torus, then the fundamental group, $\pi_1(K)$ of the Klein bottle is isomorphic to the non-abelian group $\langle x, y \mid x^2 = y^2 \rangle$ while $\pi_1(T)$ is isomorphic to the abelian group $\Bbb{Z} \oplus \Bbb{Z}$. The existence of a covering projection of $K$ onto $T$ would imply that $\pi_1(K)$ was isomorphic to a subgroup of $\pi_1(T)$, which is impossible.)

Rob Arthan
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