Macroscopic band width inequalities

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 $\frac{1}{2}{\epsilon }_n$, then $width(V,g)\le 1$. 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.

Minor revisions, To appear in Algebraic and Geometric Topology

References in corpus (3)

• Band width estimates via the Dirac operator
• Linear bounds for constants in Gromov's systolic inequality and related results
• Width, Largeness and Index Theory