 Solution

Let $A\in G{L}_2(\mathbb{Z})$. If $r-r^{\prime }\in {\mathbb{Z}}^2$, then $A(r-r^{\prime })\in {\mathbb{Z}}^2$, sodefines a continuous map on $X={\mathbb{R}}^2/{\mathbb{Z}}^2$. Since $\det A=\pm 1$, the integral general linear group contains $A^{-1}$ as another integer matrix, and $f_{A^{-1}}$ is the inverse of $f_A$. Hence $f_A$ is a homeomorphism, and $f_A(x_0)=x_0$.

The map $A:{\mathbb{R}}^2\to {\mathbb{R}}^2$ is the lift of $f_A$ fixing the origin. A loop whose lift ends at $m\in {\mathbb{Z}}^2$ is sent to a loop whose lift ends at $Am$. Under the isomorphism in part (a), the induced map is thereforeThis is precisely the integral linear automorphism of the torus associated with $A$.