Fundamental group of the torus

For the standard longitude $a$ and meridian $b$,

For the universal coveringlift 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 ${\pi }_1(T^2)\simeq {\mathbb{Z}}^2$.