Degrees of maps and multiscale geometry

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 $\mathbb{C}{P}^2$ 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 $\alpha <n$.

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

References in corpus (3)

• Plato's cave and differential forms
• Positive weights and self-maps
• Scalable spaces