Degrees of maps and multiscale geometry

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 $C_2{L}^4(\log L{)}^{-1\mathrm{/}2}$. 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 $\alpha <n$.