Slow quasiregular mappings and universal coverings

Abstract

We define slow quasiregular mappings and study cohomology and universal coverings of closed manifolds receiving slow quasiregular mappings. We show that closed manifolds receiving a slow quasiregular mapping from a punctured ball have the de Rham cohomology type of either $\mathbb{S}^n$ or $\mathbb{S}^{n-1}\times \mathbb{S}^1$. We also show that in the case of manifolds of the cohomology type of $\mathbb{S}^{n-1}\times \mathbb{S}^1$, the universal covering of the manifold has exactly two ends, and the lift of the slow mapping into the universal covering has a removable singularity at the point of punctuation. We also obtain exact growth bounds and a global homeomorphism–type theorem for slow quasiregular mappings into the manifolds of the cohomology type $\mathbb{S}^{n-1}\times \mathbb{S}^1$