A classical result gives that if there exists a holomorphic mapping $f\colon \mathbb C \to M$, then $M$ is homeomorphic to $S^2$ or $S^1\times S^1$, where $M$ is a compact Riemann surface. I will discuss a generalization of this problem to higher dimensions. I will show that if $M$ is an $n$-dimensional, closed, connected, orientable Riemannian manifold that admits a quasiregular mapping from $\mathbb R^n$, then the dimension of the degree $l$ de Rham cohomology of $M$ is bounded above by $\binom{n}{l}$. This is a sharp upper bound that proves a conjecture by Bonk and Heinonen. A corollary of this theorem answers an open problem posed by Gromov. He asked whether there exists a simply connected manifold that does not admit a quasiregular map from $\mathbb R^n$. The result gives an affirmative answer to this question.