The Dror-Whitehead theorem in prohomotopy and shape theories

Singh, S.

doi:10.1090/S0002-9947-1981-0632540-7

The Dror-Whitehead theorem in prohomotopy and shape theories
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Trans. Amer. Math. Soc. 268 (1981), 487-496 Request permission

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 $\underline f :\underline X \to \underline Y$ is a morphism in the pro-homotopy category of pointed and connected topological spaces which induces isomorphisms of the integral homology pro-groups. Then $\underline f$ induces isomorphisms of the homotopy pro-groups, for instance, when $\underline X$ and $\underline Y$ are simple, nilpotent, complete, or $\underline H$-objects; these notions are well known in homotopy theory and we have naturally extended them to pro-homotopy and shape theories.

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