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Transactions of the American Mathematical Society

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



The Čech cohomology of movable and $ n$-movable spaces

Author: James Keesling
Journal: Trans. Amer. Math. Soc. 219 (1976), 149-167
MSC: Primary 55B05; Secondary 54C56
MathSciNet review: 0407829
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Abstract: In this paper the Čech cohomology of movable and n-movable spaces is studied. Let X be a space and let $ {H^k}(X)$ denote the k-dimensional Čech cohomology of X with integer coefficients based on the numerable covers of X. Then if X is movable, there is a subgroup E of $ {H^k}(X)$ which is the union of all the algebraically compact subgroups of $ {H^k}(X)$. Furthermore, $ {H^k}(X)/E$ is an $ {\aleph _1}$-free abelian group. If X is an n-movable space, then it is shown that this structure holds for $ {H^k}(X)$ for $ 0 \leqslant k \leqslant n$ and may be false for $ k \geqslant n + 1$. If X is an $ {\text{LC}^{n - 1}}$ paracompactum, then X is known to be n-movable. However, in this case and in the case that X is an $ {\text{LC}^{n - 1}}$ compactum a stronger structure theorem is proved for $ {H^k}(X)$ for $ 0 \leqslant k \leqslant n - 1$ than that stated above. Using these results examples are given of n-movable continua that are not shape equivalent to any $ {\text{LC}^{n - 1}}$ paracompactum.

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Keywords: Shape theory, movable space, n-movable space, $ {\text{LC}^{n - 1}}$ paracompactum, Čech cohomology
Article copyright: © Copyright 1976 American Mathematical Society