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I've been trying to construct a non-normal covering space for a Klein bottle $K$ by some torus $T$. I've found some non-normal subgroups of $\pi_{1}(K)= \langle a,b \mid a b a b^{-1}=e \rangle$ that should correspond to Torus covering spaces, e.g: $<a^{3},a^{2}b^{2}>$, but I'm not sure how to describe the covering map $p \colon T \to K$, especially how do I set a partition of the fundamental polygon of $T$ into the squares representing $K$.

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  • $\begingroup$ I gave an example here. $\endgroup$
    – Steve D
    Dec 3, 2020 at 4:50
  • $\begingroup$ Why didn't you accept your own answer? $\endgroup$
    – Ramanujan
    Jan 25 at 14:08

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Ok, I got it.

Let $a$, $b$ span a lattice $\mathbb{Z}^{2}$ in $\mathbb{R}^{2}$. If we are considering the covering space corresponding to $\langle a^{i}b^{j},a^{k}b^{l} \rangle \le \pi_{1}(K)$, we can choose the parallelogram $P$ made by vectors $ia+jb$ and $ka+lb$, then cut it into squares along the lattice. An edge of $P$ within each square can be mapped homotopically onto the boundary of the square, and the composition of these boundaries should give back $a^{i}b^{j}$ or $a^{k}b^{l}$ that corresponds to the edge.

Take the case $\langle a^{3},a^{2}b^{2} \rangle$ as an example: enter image description here

In the parallelogram, the edge $c_{1}$, $c_{2}$ are homotopic to $ab$, $a^{-1}b$, so the edge $c_{1}c_{2}$ is presented by $aba^{-1}b=a^{2}b^{2}$.

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