局部到全局Urysohn宽度估计
Local-to-global Urysohn width estimates
摘要：
Urysohn $d$ -宽度的概念测量了度量空间在多大程度上可以被$d$ -维单纯复形近似。研究了黎曼流形上的局部Urysohn宽度界对流形整体宽度的影响。本文利用黎曼流形的第一同调和单位球的极大宽度，给出了它的$1$ -宽度的界。回答Larry Guth的一个问题，我们给出了具有相当大的$(n-1)$ -宽度的$n$ -流形的例子，其中所有的单位球都具有任意小的$1$ -宽度.我们还给出了局部近低维的拓扑简单流形的例子。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
