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. 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\in Y} rkH_1(f^{-1}(y)) measures the topological complexity of the map. 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. [ABSTRACT FROM AUTHOR]