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

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