  We show that a closed, connected and orientable Riemannian manifold of
dimension $d$ that admits a quasiregular mapping from $\mathbb R^d$ must have
bounded cohomological dimension independent of the distortion of the map. The
dimension of the degree $l$ de Rham cohomology of $M$ is bounded above by
$\binom{d}{l}$. This is a sharp upper bound that proves the Bonk-Heinonen
conjecture. A corollary of this theorem answers an open problem posed by Gromov
in 1981. He asked whether there exists a $d$-dimensional, simply connected
manifold that does not admit a quasiregular map from $\mathbb R^d$. Our result
gives an affirmative answer to this question.
