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Transactions of the American Mathematical Society

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



An algebraic property of the Čech cohomology groups which prevents local connectivity and movability

Author: James Keesling
Journal: Trans. Amer. Math. Soc. 190 (1974), 151-162
MSC: Primary 55B05
MathSciNet review: 0367973
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Abstract: 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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Keywords: Shape functor, Čech cohomology, compact space, compact connected topological group, local connectivity, movability, property L, continuous homomorphism
Article copyright: © Copyright 1974 American Mathematical Society