## 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

## 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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## Additional Information

- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**190**(1974), 151-162 - MSC: Primary 55B05
- DOI: https://doi.org/10.1090/S0002-9947-1974-0367973-6
- MathSciNet review: 0367973