Atomic maps and the Chogoshvili-Pontrjagin claim

It is proved that all spaces of dimension three or more disobey the Chogoshvili-Pontrjagin claim. This is of particular interest in view of the recent proof (in Certain 2-stable embeddings, by Dobrowolski, Levin, and Rubin, Topology Appl. 80 (1997), 81-90) that two-dimensional ANRs obey the claim. The construction utilizes the properties of atomic maps which are maps whose fibers (=point inverses) are atoms (=hereditarily indecomposable continua). A construction of M. Brown is applied to prove that every finite dimensional compact space admits an atomic map with a one-dimensional range.