Waist of maps measured via Urysohn width

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.