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arXiv · 2207.12347

Degrees of maps and multiscale geometry

Aleksandr Berdnikov·Larry Guth·Fedor Manin
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Abstract

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 $\alpha < n$.

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Aleksandr Berdnikov, Larry Guth, Fedor Manin. 2022-07-25. Degrees of maps and multiscale geometry. https://doi.org/10.1017/fmp.2023.33

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