Quantitative nullhomotopy and rational homotopy type

TL;DR

This paper investigates the Lipschitz bounds of nullhomotopies for maps between spheres and more general manifolds, establishing quadratic bounds in Lipschitz constants and linking nullhomotopy size to rational homotopy type.

Contribution

It provides explicit quadratic bounds on nullhomotopy Lipschitz constants and connects nullhomotopy size to rational homotopy type for a broad class of manifolds.

Findings

Nullhomotopies have quadratic bounds in Lipschitz constant L.

Constructed nullhomotopies with controlled thickness and width.

Nullhomotopy size is determined by rational homotopy type.

Abstract

In \cite{GrOrang}, Gromov asks the following question: given a nullhomotopic map $f:S^m \to S^n$ of Lipschitz constant $L$, how does the Lipschitz constant of an optimal nullhomotopy of $f$ depend on $L$, $m$, and $n$? We establish that for fixed $m$ and $n$, the answer is at worst quadratic in $L$. More precisely, we construct a nullhomotopy whose \emph{thickness} (Lipschitz constant in the space variable) is $C(m,n)(L+1)$ and whose \emph{width} (Lipschitz constant in the time variable) is $C(m,n)(L+1)^2$. More generally, we prove a similar result for maps $f:X \to Y$ for any compact Riemannian manifold $X$ and $Y$ a compact simply connected Riemannian manifold in a class which includes complex projective spaces, Grassmannians, and all other simply connected homogeneous spaces. Moreover, for all simply connected $Y$, asymptotic restrictions on the size of nullhomotopies are determined by rational homotopy type.