Macroscopic band width inequalities

Räde, Daniel

doi:10.2140/agt.2022.22.405

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Macroscopic band width inequalities

Daniel Räde

Algebraic & Geometric Topology 22 (2022) 405–432

DOI: 10.2140/agt.2022.22.405

Abstract

Inspired by Gromov’s work on Metric inequalities with scalar curvature we establish band width inequalities for Riemannian bands of the form $(V = M \times [0, 1], g)$ , where $M^{n - 1}$ is a closed manifold. We introduce a new class of orientable manifolds we call filling-enlargeable and prove: If $M$ is filling-enlargeable and all unit balls in the universal cover of $(V, g)$ have volume less than a constant $\frac{1}{2} 𝜀_{n}$ , then $width (V, g) \leq 1$ . We show that if a closed orientable manifold is enlargeable or aspherical, then it is filling-enlargeable. Furthermore, we establish that whether a closed orientable manifold is filling-enlargeable or not only depends on the image of the fundamental class under the classifying map of the universal cover.

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Keywords

large manifolds, volumes of balls, systolic geometry, band width inequality

Mathematical Subject Classification

Primary: 53C23

References

Publication

Received: 3 September 2020

Revised: 9 December 2020

Accepted: 31 January 2021

Published: 26 April 2022

Authors

Daniel Räde
Institut für Mathematik Universität Augsburg Augsburg Germany

Volume 22, issue 1 (2022)