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

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.