Quantitative null-cobordism

• Published in: arXiv.org
• Main Authors: Chambers, Gregory R, Dotterrer, Dominic, Manin, Fedor, Weinberger, Shmuel
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 24.07.2017
• Subjects: Complexity; Dependence; Lie groups; Mathematics - Differential Geometry; Mathematics - Geometric Topology; Polynomials; Riemann manifold
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1610.04888

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? In [Gro99], Gromov conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on \(n\). 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 \(f, g : X \rightarrow Y\) are homotopic via a \(CL\)-Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces \(Y\).