Let $\Sigma$ be the doubled torus (a compact oriented) surface of genus 2) and let $T$ be the torus. Suppose $f: \Sigma \rightarrow T$. Prove that $f$ is not a local homeomorphism.
Attempt at solution: Suppose $f$ was a local homeomorphism. Then by this exercise A Local Homeomorphism Between Compact Connected Hausdorff Topological Spaces it would be a covering map. Covering maps induce an injection from the fundamental group of the covering to that of the base. However, for the doubled torus, the fundamental group is given by $\langle a,b,c,d\mid aba^{-1}b^{-1}cdc^{-1}d^{-1}\rangle$ while that of the torus is $\langle a,b\mid aba^{-1}b^{-1}\rangle$, hence no such injection can exist.
If you have the machinery of the Euler characteristic available, you can use the fact that if $f$ were an $n$-fold covering map, the Euler characteristic of $\Sigma$ would have to be $n$ times the Euler characteristic of $T$.