  We prove that while there are maps $\bT^4\to\#^3(\bS^2\times\bS^2)$ of
arbitrarily large degree, there is no branched cover from $4$-torus to
$\#^3(\bS^2\times \bS^2)$. More generally, we obtain that, as long as $N$
satisfies a suitable cohomological condition, any $\pi_1$-surjective branched
cover $\bT^n \to N$ is a homeomorphism.
