Uniform cohomological expansion of uniformly quasiregular mappings

TL;DR

This paper proves that uniformly quasiregular self-mappings on compact manifolds induce cohomology maps with eigenvalues of specific moduli, revealing a uniform cohomological expansion property.

Contribution

It establishes the diagonalizability and eigenvalue moduli of the induced cohomology maps for uniformly quasiregular mappings, a novel result in geometric analysis.

Findings

Eigenvalues of induced cohomology maps have modulus (deg f)^{k/n}

Induced homomorphisms are complex diagonalizable

Degree restrictions for such mappings on closed manifolds

Abstract

Let $f\colon M \to M$ be a uniformly quasiregular self-mapping of a compact, connected, and oriented Riemannian $n$-manifold $M$ without boundary, $n\ge 2$. We show that, for $k \in \{0,\ldots, n\}$, the induced homomorphism $f^* \colon H^k(M;\mathbb{R}) \to H^k(M;\mathbb{R})$, where $H^k(M;\mathbb{R})$ is the $k$:th singular cohomology of $M$, is complex diagonalizable and the eigenvalues of $f^*$ have modulus $(\mathrm{deg}\ f)^{k/n}$. As an application, we obtain a degree restriction for uniformly quasiregular self-mappings of closed manifolds. In the proof of the main theorem, we use a Sobolev--de Rham cohomology based on conformally invariant differential forms and an induced push-forward operator.