Quantitative nullhomotopy and rational homotopy type
Open Access Publications from the University of California

## Quantitative nullhomotopy and rational homotopy type

• Author(s): Chambers, Gregory R
• Manin, Fedor
• Weinberger, Shmuel
• et al.

## Published Web Location

https://doi.org/10.1007/s00039-018-0450-2
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.

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