# Nonnormal covering space of Klein bottle by Torus

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: $$$$, but I'm not sure how to describe the covering map $$p \colon T \to K$$, especially how do I set a partition of the fundamental polygon of $$T$$ into the squares representing $$K$$.

• I gave an example here. Dec 3, 2020 at 4:50
• Why didn't you accept your own answer? Jan 25 at 14:08

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