Quasiregular values and cohomology
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918187327553536 |
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| author | Heikkilä, Susanna Kangasniemi, Ilmari |
| author_facet | Heikkilä, Susanna Kangasniemi, Ilmari |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03514 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasiregular values and cohomology Heikkilä, Susanna Kangasniemi, Ilmari Differential Geometry Complex Variables 30C65 (Primary) 32A30, 35R45, 53C65 (Secondary) 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. |
| title | Quasiregular values and cohomology |
| topic | Differential Geometry Complex Variables 30C65 (Primary) 32A30, 35R45, 53C65 (Secondary) |
| url | https://arxiv.org/abs/2511.03514 |