A Bound on the Cohomology of Quasiregularly Elliptic Manifolds

Abstract

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 $\left(\genfrac{}{}{0ex}{}{d}{l}\right)$. 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.