Quasiregular values and cohomology

2025/11/05 by Heikkilä, Susanna, Kangasniemi, Ilmari
Mathematics · #Holomorphic and Operator Theory #Advanced Operator Algebra Research #Geometry and complex manifolds

paper · doi:10.48550/arxiv.2511.03514

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 ∈ C(ℝⁿ, M) ∩ W^1,n_loc(ℝⁿ, M) satisfying | Df(x) |ⁿ ≤ K J_f(x) + distⁿ(f(x), f(x₀)) Σ(x) a.e. in ℝⁿ with K ≥ 1, x₀ ∈ ℝⁿ, and Σ∈ L¹(ℝⁿ) ∩ L^1+ε_loc(ℝⁿ) for some ε > 0, then the real singular cohomology ring H^*(M; ℝ) of M embeds into the exterior algebra \wedge^* ℝⁿ 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.

Citations

• On elliptic and quasiregularly elliptic manifolds
• Quasiregular curves: Removability of singularities
• Linear distortion and rescaling for quasiregular values
• Quasiregular curves and cohomology
• De Rham algebras of closed quasiregularly elliptic manifolds are Euclidean
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• Quasiregular curves
• The Rickman-Picard Theorem
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• Finite distortion Sobolev mappings between manifolds are continuous

Related

• A Bound on the Cohomology of Quasiregularly Elliptic Manifolds
• Quasiregular curves and cohomology
• De Rham algebras of closed quasiregularly elliptic manifolds are Euclidean
• Quasiregular curves: Removability of singularities
• Quasiregular ellipticity of open and generalized manifolds