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 $\mathbb{S}^{n-1}\times \mathbb{S}^1$