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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Quantitative null-cobordism
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by Gregory R. Chambers, Dominic Dotterrer, Fedor Manin and Shmuel Weinberger; with an appendix by Fedor Manin; with an appendix by Shmuel Weinberger
J. Amer. Math. Soc. 31 (2018), 1165-1203
Published electronically: July 10, 2018


For a given null-cobordant Riemannian $n$-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? Gromov has conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on $n$. In the appendix the bound is improved to one that is $O(L^{1+\varepsilon })$ for every $\varepsilon >0$.

This construction relies on another of independent interest. Take $X$ and $Y$ to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose $Y$ is simply connected and rationally homotopy equivalent to a product of Eilenberg–MacLane spaces, for example, any simply connected Lie group. Then two homotopic $L$-Lipschitz maps $f,g:X \to Y$ are homotopic via a $CL$-Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces $Y$.

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Bibliographic Information
  • Gregory R. Chambers
  • Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005
  • MR Author ID: 1075973
  • Email:
  • Dominic Dotterrer
  • Affiliation: Department of Computer Science, Stanford University, Stanford, California 94305
  • MR Author ID: 1008848
  • Email:
  • Fedor Manin
  • Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
  • MR Author ID: 1165933
  • ORCID: 0000-0002-4545-6998
  • Email:
  • Shmuel Weinberger
  • Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
  • MR Author ID: 181430
  • Email:
  • Received by editor(s): December 21, 2016
  • Received by editor(s) in revised form: July 24, 2017, January 21, 2018, and May 18, 2018
  • Published electronically: July 10, 2018
  • Additional Notes: The first author was partially supported by NSERC Postdoctoral Fellowship PDF-487617-2016.
    The fourth author was partially supported by NSF grant DMS-1510178.
  • © Copyright 2018 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 31 (2018), 1165-1203
  • MSC (2010): Primary 53C23; Secondary 57R75
  • DOI:
  • MathSciNet review: 3836564