On the nonexistence of certain branched covers

Abstract

We prove that while there are maps ${\mathbb{T}}^4\to {\#}^3\left({\mathbb{S}}^2\times {\mathbb{S}}^2\right)$ of arbitrarily large degree, there is no branched cover from the $4$–torus to ${\#}^3\left({\mathbb{S}}^2\times {\mathbb{S}}^2\right)$. More generally, we obtain that, as long as a closed manifold $N$ satisfies a suitable cohomological condition, any ${\pi }_1$–surjective branched cover ${\mathbb{T}}^n\to N$ is a homeomorphism.