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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-1981-0632540-7
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, <IMG WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$s$">-nilpotent continua, <!– MATH $\underline H$ –> <IMG WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="$\underline H$">-structures
Article copyright: © Copyright 1981 American Mathematical Society