Quantitative nullhomotopy and rational homotopy type

@article{Chambers2016QuantitativeNA,
  title={Quantitative nullhomotopy and rational homotopy type},
  author={Gregory R. Chambers and F. Manin and S. Weinberger},
  journal={Geometric and Functional Analysis},
  year={2016},
  volume={28},
  pages={563-588}
}
In a 2014 survey, Gromov asks the following question: given a nullhomotopic map $${f:S^{m} \to S^{n}}$$f:Sm→Sn 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 thickness (Lipschitz constant in the space variable) is C(m,n)(L + 1) and whose width (Lipschitz constant in the time variable) is C(m,n)(L + 1)2… Expand

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