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Geometric and Functional Analysis

, Volume 28, Issue 3, pp 563–588 | Cite as

Quantitative nullhomotopy and rational homotopy type

  • Gregory R. Chambers
  • Fedor Manin
  • Shmuel Weinberger
Article

Abstract

In a 2014 survey, 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 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. 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 shown to be determined by rational homotopy type.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Gregory R. Chambers
    • 1
  • Fedor Manin
    • 2
  • Shmuel Weinberger
    • 3
  1. 1.Department of MathematicsRice UniversityHoustonUSA
  2. 2.Department of MathematicsOhio State UniversityColumbusUSA
  3. 3.Department of MathematicsUniversity of ChicagoChicagoUSA

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