A bound on the cohomology of quasiregularly elliptic manifolds

Abstract

We show that a closed, connected and orientable Riemannian manifold $M$ of dimension $d$ that admits a nonconstant quasiregular mapping from $\mathbb {R}^d$ must have bounded dimension of the cohomology 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 \hbox $d$-dimensional, simply connected manifold that does not admit a quasiregular mapping from $\mathbb {R}^d$. Our result gives an affirmative answer to this question.