Cell-like resolutions preserving cohomological dimensions

Abstract

We prove that for every compactum $X$ with ${dim}_{\mathbb{Z}}X\le n\ge 2$ there is a cell-like resolution $r:Z\to X$ from a compactum $Z$ onto $X$ such that $dimZ\le n$ and for every integer $k$ and every abelian group $G$ such that ${dim}_{G}X\le k\ge 2$ we have ${dim}_{G}Z\le k$. The latter property implies that for every simply connected CW–complex $K$ such that $e-edimX\le K$ we also have $e-dimZ\le K$.