Quasiregular values and cohomology
Susanna Heikkilä, Ilmari Kangasniemi
TL;DR
The paper extends the cohomological obstruction for quasiregular ellipticity to the setting of quasiregular values, showing that a non-constant map $f:\mathbb{R}^n\to M$ with a $(K,\Sigma)$-quasiregular value in its image forces a graded embedding of $H^*(M;\mathbb{R})$ into $\wedge^* \mathbb{R}^n$ under suitable integrability of $\Sigma$, thereby yielding $\dim H^k(M;\mathbb{R})\le {n\choose k}$. The authors also develop a partial higher-dimensional analogue via quasiregular curves, combining a closed $n$-form $\omega$ with $[\omega]\in K^*(M)$ to obtain similar embeddings. The core method blends a generalized pull-back/limit framework with Gehring-type higher integrability, the Poincaré homotopy operator, and a local-to-global localization strategy inspired by Rickman and Picard results, culminating in a technical abstraction that yields a graded algebra homomorphism unless certain cohomological obstructions vanish. Consequences include infinite energy for nontrivial QR values and a unified approach that recovers and extends prior embeddings for equidimensional and asymptotically Lipschitz settings, highlighting deep links between quasiregular mappings and de Rham cohomology.
Abstract
We prove that the recently shown cohomological obstruction for quasiregular ellipticity has a generalization in the theory of quasiregular values. More specifically, if $M$ is a closed, connected, and oriented Riemannian $n$-manifold, and there exists a map $f \in C(\mathbb{R}^n, M) \cap W^{1,n}_{\mathrm{loc}}(\mathbb{R}^n, M)$ satisfying $\lvert Df(x) \rvert^n \le K J_f(x) + \operatorname{dist}^n(f(x), f(x_0)) Σ(x)$ a.e. in $\mathbb{R}^n$ with $K \ge 1$, $x_0 \in \mathbb{R}^n$, and $Σ\in L^1(\mathbb{R}^n) \cap L^{1+\varepsilon}_{\mathrm{loc}}(\mathbb{R}^n)$ for some $\varepsilon > 0$, then the real singular cohomology ring $H^*(M; \mathbb{R})$ of $M$ embeds into the exterior algebra $\wedge^* \mathbb{R}^n$ in a graded manner. We also show a partial version of our result for $M$ with dimension greater than $n$, by using a class of maps that combines properties of quasiregular values and quasiregular curves.