Diffeomorphism between $n$-fold and $m$-fold tori

Question

Consider $M_n, M_m$ which are an n-fold torus and an m-fold torus, respectively, in $\mathbb{R}^3$.

Prove that there exists a map $f:M_n\to M_m$ which is a local diffeomorphism if and only if $$m-1|n-1$$ for $m\ge1$.

Answer 1

Suppose that $f:M_n\to M_m$ is a local diffeomorphism. Since the surfaces are compact, the map is proper and then it is a covering map. It follows that the Euler characteristic of $M_m$ divides that of $M_n$, that is $$2-2m=\chi(M_m)\mid\chi(M_n)=2-2n.$$ It follows that your condition is necessary.

For sufficiency, you have to construct a map. One way to do that is to construct first a surface $M_\infty$ which is an infinite chain, like but with infinitely many links. On such a surface the group $\mathbb Z$ acts, and if $a$ and $b$ are positive integers such that $a$ divides $b$, then you can consider the quotients $M_\infty/a\mathbb Z$ and $M_\infty/b\mathbb Z$, which are surfaces and the obvious induced map $$M_\infty/a\mathbb Z\to M_\infty/b\mathbb Z.$$ This map is a covering and in this way you can realize coverings in all cases that they exist (although not all possible coverings are of this form!)