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 $\mathbb CP^2$ 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 nonscalable simply connected n-manifolds, the maximal degree grows roughly like $L^n (\log L)^{-\theta (1)}$ . And for nonformal simply connected n-manifolds, the maximal degree is bounded by $L^\alpha $ for some $\alpha < n$ .