Uniform cohomological expansion of uniformly quasiregular mappings

Abstract

Let $f:M\to M$ 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 $k\in \{0,\dots ,n\}$, the induced homomorphism $f^{\ast }:H^k(M;\mathbb{R})\to H^k(M;\mathbb{R})$, where $H^k(M;\mathbb{R})$ is the $k$:th singular cohomology of $M$, is complex diagonalizable and the eigenvalues of $f^{\ast }$ 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.

The presented version is the author's accepted manuscript version, with an additional note referring to the published version

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