Degrees of maps and multiscale geometry

TL;DR

This paper investigates the degree bounds of Lipschitz maps between Riemannian manifolds, revealing how topological properties influence the growth rate of maximal degrees with respect to Lipschitz constants.

Contribution

It introduces new bounds for the degree of Lipschitz maps based on manifold topology and constructs examples illustrating these bounds.

Findings

Max degree of self-maps on certain manifolds scales as L^n with topological type.

For formal but non-scalable manifolds, degree grows like L^n times a logarithmic factor.

Non-formal manifolds have degree bounded by L^α for some α < n.

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 $\alpha < n$.