We consider a functor from the category of groups to itself, G → Z_∞G, that we call right exact Z -completion of a group. It is connected with the pro-nilpotent completion G by the short exact sequence 1 → lim1 M_nG → Z_∞G → G → 1, where M_nG is nth Baer invariant of G. We prove that Z00(µ1X) is an invariant of homological equivalence of a space X. Moreover, we prove an analogue of Stallings' theorem: If G → G0 is a 2-connected group homomorphism, then Z_∞G ^ Z_∞G'. We give examples of 3-manifolds X and Y such that n1X ^ 7r17 but Z1^1X