A <covering space>[covering map] induces an injection on fundamental groups. There is no injection $\mathbb Z^2\hookrightarrow\mathbb Z/2$, and there is no injection $\mathbb Z/2\hookrightarrow\mathbb Z^2$ because $\mathbb Z^2$ is torsion-free. Thus neither proposed covering exists.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 21F / b / Solution

 Solution (source code)