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Transactions of the American Mathematical Society

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The Dror-Whitehead theorem in prohomotopy and shape theories

Author: S. Singh
Journal: Trans. Amer. Math. Soc. 268 (1981), 487-496
MSC: Primary 55P10; Secondary 55P55
MathSciNet review: 632540
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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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Keywords: Pro-homotopy, shape, nilpotent spaces, homology pro-groups, homotopy pro-groups, $ s$-nilpotent continua, $ \underline H $-structures
Article copyright: © Copyright 1981 American Mathematical Society