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.

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)