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.