Integral linear automorphism of the torus

Every $A\in G{L}_2(\mathbb{Z})$ induces a based homeomorphismIts inverse is $f_{A^{-1}}$, and under the lift-endpoint description of the fundamental group of the torus, its induced map on ${\pi }_1(T^2)$ is multiplication by $A$.

Any two connected degree-two coverings of the torus fit into a commuting square with homeomorphisms of their total and base spaces. They correspond to index-two subgroups of ${\mathbb{Z}}^2$, which are kernels of the three nonzero maps ${\mathbb{Z}}^2\to \mathbb{Z}/2\mathbb{Z}$. Reduction modulo two shows that $G{L}_2(\mathbb{Z})$ acts transitively on these subgroups; the classification of connected covering spaces and the lifting criterion for a covering space then supply the homeomorphism of total spaces.