I have to admit that I spent a while now thinking about the question below. I could see that the map f takes integers to integers keeping thus taking the vertices of $T^{2}$ to vertices of $R/Z$. I am thinking about Mayer vietoris but how no idea how to map $T^{2}$x$0$ to $T^{2}$x$1$ "in order to compute the homology of the quotient space"
Many thanks