Quantitative estimates for fractional Sobolev mappings in rational homotopy groups
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866910473350283264 |
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| author | Park, Woongbae Schikorra, Armin |
| author_facet | Park, Woongbae Schikorra, Armin |
| contents | Let $\mathcal{N} \subset \mathbb{R}^M$ be a smooth simply connected compact manifold without boundary. A rational homotopy subgroup of $π_{N}(\mathcal{N})$ is represented by a homomorphism \[{\rm deg}: π_{N}(\mathcal{N}) \to \mathbb{R}.\] For maps $f: \mathbb{S}^N \to \mathcal{N}$ we give a quantitative estimate of its rational homotopy group element ${\rm deg}([f]) \in \mathbb{R}$ in terms of its fractional Sobolev-norm or Hölder norm. That is, we show that for all $β\in (β_0({\rm deg}),1]$, \[
|{\rm deg}([f])|\leq C({\rm deg})\, [f]_{W^{β,\frac{N}β}(\mathbb{S}^N)}^{\frac{N+L({\rm deg})}β}, \] and \[
|{\rm deg}([f])|\leq C({\rm deg})\, [f]_{C^β(\mathbb{S}^N)}^{\frac{N+L({\rm deg})}β}. \] Here $C({\rm deg}) > 0$, $L({\rm deg}) \in \mathbb{N}$, $β_0({\rm deg}) \in (0,1)$ are computable from the rational homotopy group represented by ${\rm deg}$. This extends earlier work by Van Schaftingen and the second author on the Hopf degree to the Novikov's integral representation for rational homotopy groups as developed by Sullivan, Novikov, Hardt and Rivière. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_04207 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quantitative estimates for fractional Sobolev mappings in rational homotopy groups Park, Woongbae Schikorra, Armin Analysis of PDEs Algebraic Topology Functional Analysis Let $\mathcal{N} \subset \mathbb{R}^M$ be a smooth simply connected compact manifold without boundary. A rational homotopy subgroup of $π_{N}(\mathcal{N})$ is represented by a homomorphism \[{\rm deg}: π_{N}(\mathcal{N}) \to \mathbb{R}.\] For maps $f: \mathbb{S}^N \to \mathcal{N}$ we give a quantitative estimate of its rational homotopy group element ${\rm deg}([f]) \in \mathbb{R}$ in terms of its fractional Sobolev-norm or Hölder norm. That is, we show that for all $β\in (β_0({\rm deg}),1]$, \[ |{\rm deg}([f])|\leq C({\rm deg})\, [f]_{W^{β,\frac{N}β}(\mathbb{S}^N)}^{\frac{N+L({\rm deg})}β}, \] and \[ |{\rm deg}([f])|\leq C({\rm deg})\, [f]_{C^β(\mathbb{S}^N)}^{\frac{N+L({\rm deg})}β}. \] Here $C({\rm deg}) > 0$, $L({\rm deg}) \in \mathbb{N}$, $β_0({\rm deg}) \in (0,1)$ are computable from the rational homotopy group represented by ${\rm deg}$. This extends earlier work by Van Schaftingen and the second author on the Hopf degree to the Novikov's integral representation for rational homotopy groups as developed by Sullivan, Novikov, Hardt and Rivière. |
| title | Quantitative estimates for fractional Sobolev mappings in rational homotopy groups |
| topic | Analysis of PDEs Algebraic Topology Functional Analysis |
| url | https://arxiv.org/abs/2207.04207 |