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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48 .

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An algebraic property of the Čech cohomology groups which prevents local connectivity and movability
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by James Keesling PDF
Trans. Amer. Math. Soc. 190 (1974), 151-162 Request permission


Let C denote the category of compact Hausdorff spaces and $H:C \to HC$ be the homotopy functor. Let $S:C \to SC$ be the functor of shape in the sense of Holsztyński for the projection functor H. Let X be a continuum and ${H^n}(X)$ denote n-dimensional Čech cohomology with integer coefficients. Let ${A_x} = {\text {char}}\;{H^1}(X)$ be the character group of ${H^1}(X)$ considering ${H^1}(X)$ as a discrete group. In this paper it is shown that there is a shape morphism $F \in {\text {Mor}_{SC}}(X,{A_X})$ such that ${F^\ast }:{H^1}({A_X}) \to {H^1}(X)$ is an isomorphism. It follows from the results of a previous paper by the author that there is a continuous mapping $f:X \to {A_X}$ such that $S(f) = F$ and thus that ${f^\ast }:{H^1}({A_X}) \to {H^1}(X)$ is an isomorphism. This result is applied to show that if X is locally connected, then ${H^1}(X)$ has property L. Examples are given to show that X may be locally connected and ${H^n}(X)$ not have property L for $n > 1$. The result is also applied to compact connected topological groups. In the last section of the paper it is shown that if X is compact and movable, then for every integer n, ${H^n}(X)/{\operatorname {Tor}}\;{H^n}(X)$ has property L. This result allows us to construct peano continua which are nonmovable. An example is given to show that ${H^n}(X)$ itself may not have property L even if X is a finite-dimensional movable continuum.
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 190 (1974), 151-162
  • MSC: Primary 55B05
  • DOI:
  • MathSciNet review: 0367973