Lift-endpoint description of the fundamental group of the torus
= Lift-endpoint description of the fundamental group of the torus
For the universal covering
$$
p:\mathbb R^2\to S^1\times S^1,
\qquad
p(r_1,r_2)=(e^{2\pi i r_1},e^{2\pi i r_2}),
$$
lift a based loop from the origin. Its endpoint lies in the fibre $\mathbb Z^2$, depends only on its based homotopy class, and turns concatenation into addition. This gives a natural isomorphism $\pi_1(T^2)\simeq\mathbb Z^2$.