I'm brushing up on covering spaces for an upcoming exam, and I came across the following problems related to the Klein bottle:
- Can there exist a cover of the torus by the Klein bottle?
and
- Let $X$ be a connected and locally path connected space with $\pi_1(X)=\mathbb{Z}/3\mathbb{Z}$. Show the Klein bottle cannot be covered by $X$.
I'm mainly looking for hints here, not necessarily full solutions. For (1), should I be looking at the Galois correspondence for covers of the torus? And for (2), do I also want to look at the Galois correspondence, or would it be easier to show that $\mathbb{Z}/3\mathbb{Z}$ does not embed into the fundamental group of the Klein bottle $\langle a,b\mid abab^{-1}\rangle$?
Am I on the right track for these, or should I be doing something else?