The Dror-Whitehead theorem in prohomotopy and shape theories

Abstract

Many analogues of the classical Whitehead theorem from homotopy theory are now available in pro-homotopy and shape theories. E. Dror has significantly extended the homology version of the Whitehead theorem from the well-known simply connected case to the more general, for instance, nilpotent case. We prove a full analogue of Dror’s theorems in pro-homotopy and shape theories. More specifically, suppose f _ : X _ → Y _ \underline f :\underline X \to \underline Y is a morphism in the pro-homotopy category of pointed and connected topological spaces which induces isomorphisms of the integral homology pro-groups. Then f _ \underline f induces isomorphisms of the homotopy pro-groups, for instance, when X _ \underline X and Y _ \underline Y are simple, nilpotent, complete, or H _ \underline H -objects; these notions are well known in homotopy theory and we have naturally extended them to pro-homotopy and shape theories.