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arXiv · 2008.07718

Local-to-global Urysohn width estimates

Alexey Balitskiy·Aleksandr Berdnikov
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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.

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BibTeXRIS

Alexey Balitskiy, Aleksandr Berdnikov. 2020-08-18. Local-to-global Urysohn width estimates. https://doi.org/10.1515/crelle-2021-0047

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