On the non-existence of certain branched covers

TL;DR

This paper proves the non-existence of certain branched covers from 4-torus to specific 4-manifolds, showing that under certain conditions, such covers must be homeomorphisms, highlighting topological constraints.

Contribution

It establishes new non-existence results for branched covers from tori to certain 4-manifolds under cohomological conditions, extending understanding of topological mappings.

Findings

No branched cover from 4-torus to (S^2S^2) exists.

Existence of arbitrarily large degree maps from -torus to (S^2S^2).

Under cohomological conditions, (S^2S^2) cannot be covered non-trivially by -torus.

Abstract

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.