De Rham algebras of closed quasiregularly elliptic manifolds are Euclidean

TL;DR

This paper proves that certain closed manifolds admitting quasiregular maps from Euclidean space have de Rham cohomology algebras that embed into Euclidean exterior algebras, leading to classification results for 4-manifolds.

Contribution

It establishes a link between quasiregular mappings and the algebraic structure of de Rham cohomology, providing a classification of quasiregularly elliptic 4-manifolds.

Findings

de Rham algebra embeds into Euclidean exterior algebra

classification of simply connected quasiregularly elliptic 4-manifolds

structure of cohomology influenced by quasiregular maps

Abstract

We show that, if a closed, connected, and oriented Riemannian $n$-manifold $N$ admits a non-constant quasiregular mapping from the Euclidean $n$-space $\mathbb R^n$, then the de Rham cohomology algebra $H_{\mathrm{dR}}^*(N)$ of $N$ embeds into the exterior algebra ${\bigwedge}^*\mathbb R^n$. As a consequence, we obtain a homeomorphic classification of closed simply connected quasiregularly elliptic $4$-manifolds.