Waist of maps measured via Urysohn width

TL;DR

This paper investigates the Urysohn width of fibers in continuous maps from compact metric spaces to simplicial complexes, establishing lower bounds and constructing examples with specific width properties.

Contribution

It provides new lower bounds on fiber widths for piecewise linear maps and constructs maps with fibers of small width despite the domain's large width.

Findings

Any piecewise linear map from a high-dimensional cube to an m-polyhedron has a fiber with width at least a positive constant.

Existence of maps from high-dimensional manifolds to Euclidean space with fibers of arbitrarily small width.

The topological complexity of fibers influences the minimal possible width in such maps.

Abstract

We discuss various questions of the following kind: for a continuous map $X \to Y$ from a compact metric space to a simplicial complex, can one guarantee the existence of a fiber large in the sense of Urysohn width? The $d$-width measures how well a space can be approximated by a $d$-dimensional complex. The results of this paper include the following. 1) Any piecewise linear map $f: [0,1]^{m+2} \to Y^m$ from the unit euclidean $(m+2)$-cube to an $m$-polyhedron must have a fiber of $1$-width at least $\frac{1}{2\beta m +m^2 + m + 1}$, where $\beta = \sup_y \text{ rk } H_1(f^{-1}(y))$ measures the topological complexity of the map. 2) There exists a piecewise smooth map $X^{3m+1} \to \mathbb{R}^m$, with $X$ a riemannian $(3m+1)$-manifold of large $3m$-width, and with all fibers being topological $(2m+1)$-balls of arbitrarily small $(m+1)$-width.