Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 21F (Algebraic Topology)

The Seifert-van Kampen theorem says that if a path-connected space is the union of path-connected open sets $U,V$ with path-connected intersection, thenUsing the usual polygonal cell decompositions, the torus attaches a two-cell to $S^1\vee S^1$ along $ab{a}^{-1}{b}^{-1}$, while the projective plane attaches one along $a^2$. Hence

A 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.

Label the four vertices cyclically by $\mathbb{Z}/4$. Reading the arrows in the diagram, an $a$-edge moves one step in one direction and a $b$-edge one step in the other. Thus the monodromy homomorphism is(up to simultaneously reversing the chosen numbering). By the Galois correspondence for covering spaces, the subgroup associated with the chosen vertex above $x_0$ is