Degrees of maps and multiscale geometry
| Autor: | Berdnikov, Aleksandr, Guth, Larry, Manin, Fedor |
|---|---|
| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Forum Math. Pi, vol. 12 (2024) art. e2 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1017/fmp.2023.33 |
| Popis: | 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 $L^n (\log L)^{\theta(1)}$. And for non-formal simply connected $n$-manifolds, the maximal degree is bounded by $L^\alpha$ for some $\alpha < n$. Comment: 49 pages, 3 figures. Theorem C was proved incorrectly in v1; v2 corrects the proof and generalizes the theorem. Other corrections and clarifications are given in response to a referee report |
| Databáze: | arXiv |
| Externí odkaz: |
|