TITLE: Prove that there is a two-sheeted covering of the Klein bottle by the torus
QUESTION [37 upvotes]: Prove that there is a two-sheeted covering of the Klein bottle by the torus.
OK, so we take the the polygonal representation of the torus and draw a line in the middle as follows:

Then there are two Klein bottles in there, but how do I write down the actual covering map $q:S^1 \times S^1 \to K$?

REPLY [2 votes]: I would like to give my formula. Let us think of $T^2$ as $S^1\times S^1$ and $S^1$ as the unit circle in $\mathbb{C}$. Let $f$ act on $T^2$ by $f(z,w)=(-z, \bar{w})$, then $f^2$ is the identity. $f(z,w)\sim(z,w)$ gives an equivalence relation on $T^2$ and $T^2/\sim$ provides the desired two-fold covering map from $T^2$ to the Klein bottle, as illustraded in the figure by OP.