Journal ArticleDegrees of maps and multiscale geometryBerdnikov, Aleksandr; Guth, Larry; Manin, Fedor<title>Abstract</title> <p>We study the degree of an <italic>L</italic>-Lipschitz map between Riemannian manifolds, proving new upper bounds and constructing new examples. For instance, if <inline-formula><alternatives><inline-graphic href='S2050508623000331_inline1.png' mime-subtype='png'/><tex-math>$X_k$</tex-math></alternatives></inline-formula> is the connected sum of <italic>k</italic> copies of <inline-formula><alternatives><inline-graphic href='S2050508623000331_inline2.png' mime-subtype='png'/><tex-math>$\mathbb CP^2$</tex-math></alternatives></inline-formula>for<inline-formula><alternatives><inline-graphic href='S2050508623000331_inline3.png' mime-subtype='png'/><tex-math>$k \ge 4$</tex-math></alternatives></inline-formula>, then we prove that the maximum degree of an <italic>L</italic>-Lipschitz self-map of <inline-formula><alternatives><inline-graphic href='S2050508623000331_inline4.png' mime-subtype='png'/><tex-math>$X_k$</tex-math></alternatives></inline-formula> is between <inline-formula><alternatives><inline-graphic href='S2050508623000331_inline5.png' mime-subtype='png'/><tex-math>$C_1 L^4 (\log L)^{-4}$</tex-math></alternatives></inline-formula> and <inline-formula><alternatives><inline-graphic href='S2050508623000331_inline6.png' mime-subtype='png'/><tex-math>$C_2 L^4 (\log L)^{-1/2}$</tex-math></alternatives></inline-formula>. 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 <italic>n</italic>-manifolds, the maximal degree is <inline-formula><alternatives><inline-graphic href='S2050508623000331_inline7.png' mime-subtype='png'/><tex-math>$\sim L^n$</tex-math></alternatives></inline-formula>. For formal but nonscalable simply connected<italic>n</italic>-manifolds, the maximal degree grows roughly like <inline-formula><alternatives><inline-graphic href='S2050508623000331_inline8.png' mime-subtype='png'/><tex-math>$L^n (\log L)^{-\theta (1)}$</tex-math></alternatives></inline-formula>. And for nonformal simply connected <italic>n</italic>-manifolds, the maximal degree is bounded by <inline-formula><alternatives><inline-graphic href='S2050508623000331_inline9.png' mime-subtype='png'/><tex-math>$L^\alpha $</tex-math></alternatives></inline-formula> for some <inline-formula><alternatives><inline-graphic href='S2050508623000331_inline10.png' mime-subtype='png'/><tex-math>$\alpha < n$</tex-math></alternatives></inline-formula>.</p>Cambridge University Press2024-01-0110562505Forum of Mathematics, Pi12e22050-5086https://doi.org/10.1017/fmp.2023.332204001National Science Foundation