Quantitative null-cobordism

Chambers, Gregory; Dotterrer, Dominic; Manin, Fedor; Weinberger, Shmuel

doi:10.1090/jams/903

Quantitative null-cobordism
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by Gregory R. Chambers, Dominic Dotterrer, Fedor Manin and Shmuel Weinberger; with an appendix by Fedor Manin and Shmuel Weinberger

J. Amer. Math. Soc. 31 (2018), 1165-1203

DOI: https://doi.org/10.1090/jams/903

Published electronically: July 10, 2018

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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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