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Quantitative null-cobordism

Authors: Gregory R. Chambers, Dominic Dotterrer, Fedor Manin and Shmuel Weinberger; with an appendix by Fedor Manin; Shmuel Weinberger
Journal: J. Amer. Math. Soc. 31 (2018), 1165-1203
MSC (2010): Primary 53C23; Secondary 57R75
Published electronically: July 10, 2018
MathSciNet review: 3836564
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Abstract: 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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Additional Information

Gregory R. Chambers
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005

Dominic Dotterrer
Affiliation: Department of Computer Science, Stanford University, Stanford, California 94305

Fedor Manin
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210

Shmuel Weinberger
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637

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.
Article copyright: © Copyright 2018 American Mathematical Society