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

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.