  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.
