  We introduce a construction adding low-dimensional cells to a space that
satisfies certain low-dimensional conditions; it preserves high-dimensional
homology with appropriate coefficients. This includes as special cases
Quillen's plus construction, Bousfield's integral homology localization, the
existence of Moore spaces M(G,1) and Bousfield and Kan's partial k-completion
of spaces. We also use it to generalize counterexamples to the
zero-in-the-spectrum conjecture found by Farber and Weinberger, and by Higson,
Roe and Schick.
