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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The Dror-Whitehead theorem in prohomotopy and shape theories
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by S. Singh PDF
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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Additional Information
  • © Copyright 1981 American Mathematical Society
  • 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