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 $K$.

  • $\begingroup$ 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

1 Answer 1


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 $aba^{-1}b=a^{2}b^{2}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.