Waists of maps measured via Urysohn width

Abstract: Urysohn d-width is a measure for estimating how close a metric space $X$ is to being $d$-dimensional. Specifically, $UW_d(X)$ is the lower bound for the largest fiber of a projection of $X$ to a $d$-dimensional complex. However, the dimension estimated in such a way is less well-behaved than the usual dimension. We explore this discrepancy getting results like the following:

1. The topological projection $B^{f}\times B^m\to B^m$ with $f\sim mk$ can have a metric, such that $UW_{m+k}(F)<\varepsilon$ for all fibers $F$, and yet the total space has $UW_{f-1}=O(1)$ (``almost $m+k$ dimensional fibers over $m$-base build a $\sim mk$-space'').

2. On the other hand, for a map $X\to Y^m$ the $UW_{m+1}(X)$ is bounded by $UW_1$ and $rk (H_1(\cdot,\mathbb{Z}/2))$ of the fibers.

Joint with Alexey Balitskiy.

algebraic topologydifferential geometrygeometric topologymetric geometry

Audience: researchers in the topic

Comments: https://utoronto.zoom.us/j/82235760196

University of Toronto Geometry & Topology seminar