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

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