A Bound on the Cohomology of Quasiregularly Elliptic Manifolds

TL;DR

This paper proves a sharp upper bound on the cohomological dimension of manifolds admitting quasiregular maps from Euclidean space, confirming a conjecture and answering an open problem in geometric analysis.

Contribution

It establishes a universal cohomological bound for quasiregularly elliptic manifolds, resolving the Bonk-Heinonen conjecture and Gromov's open question.

Findings

Bound on cohomology dimension is independent of map distortion

Dimension of degree l cohomology is at most binomial coefficient of d and l

Existence of simply connected manifolds without quasiregular maps from R^d confirmed

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 $\binom{d}{l}$. 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.