A unified approach to the plus-construction, Bousfield localization, Moore spaces and zero-in-the-spectrum examples

TL;DR

This paper presents a unified construction that generalizes several classical topological and homological localization techniques, and applies it to produce new counterexamples related to the zero-in-the-spectrum conjecture.

Contribution

It introduces a general method for adding low-dimensional cells that unifies plus-construction, localization, Moore spaces, and partial k-completion, extending their applications.

Findings

Unifies multiple localization and construction techniques in topology.

Generalizes counterexamples to the zero-in-the-spectrum conjecture.

Preserves high-dimensional homology while modifying low-dimensional structure.

Abstract

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.