Local-to-global Urysohn width estimates

TL;DR

This paper explores how local geometric bounds influence the global Urysohn width of Riemannian manifolds, providing bounds, examples, and answering a question posed by Larry Guth.

Contribution

It establishes bounds on the 1-width of Riemannian manifolds based on local properties and constructs examples addressing Guth's question about local and global width relationships.

Findings

Bound the 1-width using first homology and unit ball widths.

Construct examples of manifolds with high (n-1)-width despite small unit ball widths.

Answer Guth's question with explicit examples.

Abstract

The notion of the Urysohn $d$-width measures to what extent a metric space can be approximated by a $d$-dimensional simplicial complex. We investigate how local Urysohn width bounds on a riemannian manifold affect its global width. We bound the $1$-width of a Riemannian manifold in terms of its first homology and the supremal width of its unit balls. Answering a question of Larry Guth, we give examples of $n$-manifolds of considerable $(n-1)$-width in which all unit balls have arbitrarily small $1$-width. We also give examples of topologically simple manifolds that are locally nearly low-dimensional.