Degrees of maps and multiscale geometry
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910656966426624 |
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| author | Berdnikov, Aleksandr Guth, Larry Manin, Fedor |
| author_facet | Berdnikov, Aleksandr Guth, Larry Manin, Fedor |
| contents | We study the degree of an $L$-Lipschitz map between Riemannian manifolds, proving new upper bounds and constructing new examples. For instance, if $X_k$ is the connected sum of $k$ copies of $\mathbb CP^2$ for $k \ge 4$, then we prove that the maximum degree of an $L$-Lipschitz self-map of $X_k$ is between $C_1 L^4 (\log L)^{-4}$ and $C_2 L^4 (\log L)^{-1/2}$. More generally, we divide simply connected manifolds into three topological types with three different behaviors. Each type is defined by purely topological criteria. For scalable simply connected $n$-manifolds, the maximal degree is $\sim L^n$. For formal but non-scalable simply connected $n$-manifolds, the maximal degree grows roughly like $L^n (\log L)^{θ(1)}$. And for non-formal simply connected $n$-manifolds, the maximal degree is bounded by $L^α$ for some $α< n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_12347 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Degrees of maps and multiscale geometry Berdnikov, Aleksandr Guth, Larry Manin, Fedor Metric Geometry Algebraic Topology Differential Geometry 53C23 (55P62, 42B25) We study the degree of an $L$-Lipschitz map between Riemannian manifolds, proving new upper bounds and constructing new examples. For instance, if $X_k$ is the connected sum of $k$ copies of $\mathbb CP^2$ for $k \ge 4$, then we prove that the maximum degree of an $L$-Lipschitz self-map of $X_k$ is between $C_1 L^4 (\log L)^{-4}$ and $C_2 L^4 (\log L)^{-1/2}$. More generally, we divide simply connected manifolds into three topological types with three different behaviors. Each type is defined by purely topological criteria. For scalable simply connected $n$-manifolds, the maximal degree is $\sim L^n$. For formal but non-scalable simply connected $n$-manifolds, the maximal degree grows roughly like $L^n (\log L)^{θ(1)}$. And for non-formal simply connected $n$-manifolds, the maximal degree is bounded by $L^α$ for some $α< n$. |
| title | Degrees of maps and multiscale geometry |
| topic | Metric Geometry Algebraic Topology Differential Geometry 53C23 (55P62, 42B25) |
| url | https://arxiv.org/abs/2207.12347 |