Waist of maps measured via Urysohn width

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.

arXiv comment18 pages, 3 figures; to appear in Transactions of the American Mathematical Society