#### Volume 22, issue 1 (2022)

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Macroscopic band width inequalities

### Daniel Räde

Algebraic & Geometric Topology 22 (2022) 405–432
##### Abstract

Inspired by Gromov’s work on Metric inequalities with scalar curvature we establish band width inequalities for Riemannian bands of the form $\left(V=M×\left[0,1\right],g\right)$, 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 $\left(V,g\right)$ have volume less than a constant $\frac{1}{2}{𝜀}_{n}$, then $\mathrm{width}\left(V,g\right)\le 1$. 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.

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