On the nonexistence of certain branched covers

Pankka, Pekka¹ ; Souto, Juan²

¹ Department of Mathematics and Statistics, University of Helsinki, PO Box 68 (Gustaf Hällströmin katu 2b), FI-00014 Helsinki, Finland
² Mathematics Department, University of British Columbia, 1984 Mathematics Road, Vancouver BC V6T 1Z2, Canada

Geometry & topology, Tome 16 (2012) no. 3, pp. 1321-1344.

Résumé

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.

This article is incorrect.  Text of retraction received 14 February 2019

Détail

DOI : 10.2140/gt.2012.16.1321

Classification : 57M12, 30C65, 57R19
Keywords: branched cover, quasiregularly elliptic manifold

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