Homology equivalences of manifolds and zero-in-the-spectrum examples

Abstract

Working with group homomorphisms, a construction of manifolds is introduced, which preserves homology groups. The construction gives as special cases Quillen’s plus construction with handles obtained by Hausmann, the existence of the one-sided $h$–cobordism of Guilbault and Tinsley, and the existence of homology spheres and higher-dimensional knots proved by Kervaire. We also use it to recover counter-examples to the zero-in-the-spectrum conjecture found by Farber and Weinberger, and by Higson, Roe and Schick.