Fundamental groups, homology equivalences and one-sided h-cobordisms

We give a necessary and sufficient condition for the fundamental group homomorphism of a map between CW-complexes (manifolds) to induce partial homology equivalences. As applications, we obtain characterizations of fundamental groups of homology spheres and Moore manifolds. Moreover, a classification of one-sided h-cobordism of manifolds up to diffeomorphisms is obtained, based on Quillen’s plus construction with Whitehead torsions.