Integral linear automorphism of the torus
= Integral linear automorphism of the torus
Every $A\in GL_2(\mathbb Z)$ induces a based homeomorphism
$$
f_A:\mathbb R^2/\mathbb Z^2\to\mathbb R^2/\mathbb Z^2,
\qquad
f_A([r])=[Ar].
$$
Its 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 $A$.