#### Volume 16, issue 3 (2012)

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On the nonexistence of certain branched covers

### Pekka Pankka and Juan Souto

Geometry & Topology 16 (2012) 1321–1344
##### Abstract

We prove that while there are maps ${\mathbb{T}}^{4}\to {#}^{3}\left({\mathbb{S}}^{2}×{\mathbb{S}}^{2}\right)$ of arbitrarily large degree, there is no branched cover from the $4$–torus to ${#}^{3}\left({\mathbb{S}}^{2}×{\mathbb{S}}^{2}\right)$. 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.

##### Keywords
branched cover, quasiregularly elliptic manifold
##### Mathematical Subject Classification 2010
Primary: 57M12
Secondary: 30C65, 57R19