Let $n\ge 2$. How can you prove that for every continuous $f:S^n\to T^n$, the induced map on singular homology $f_\star:H_n(S^n)\to H_n(T^n)$ is the zero map? Here, $S^n$ is the $n$ dimensional sphere, and $T^n=(S^1)^n$ is the $n$ dimensional torus.
I don't even know where to begin... This was on the final for my graduate course in algebraic topology. We've covered the fundamental group, covering spaces, and the basics for homology (simplicial complexes, Mayer-Viatoris, degree theory, etc). A hint in the right direction would be greatly appreciated.