Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 21F / a / Solution

The map $p$ is the quotient of ${\mathbb{R}}^2$ by integer translations, andGiven a based loop $\gamma$ at $x_0$, the path lifting theorem gives a unique lift $\tilde{\gamma }$ beginning at $0$. Its endpoint lies in ${\mathbb{Z}}^2$. The homotopy lifting property shows that this endpoint depends only on $[\gamma ]$, so defineLifting a concatenation after translating the second lift by the endpoint of the first shows that $\Phi$ is a group homomorphism. It is surjective because, for every $m\in {\mathbb{Z}}^2$, the path $r\mapsto p(rm)$ is a loop whose lift ends at $m$. If a lift ends at zero, it is a loop in the simply connected space ${\mathbb{R}}^2$ and contracts there; projecting the contraction proves that its original loop is null-homotopic. Thus $\Phi$ is injective, and the lift-endpoint description of the fundamental group of the torus gives