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

Let C denote the category of compact Hausdorff spaces and H : C → H C H:C \to HC be the homotopy functor. Let S : C → S C 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 ) {H^n}(X) denote n-dimensional Čech cohomology with integer coefficients. Let A x = char H 1 ( X ) {A_x} = {\text {char}}\;{H^1}(X) be the character group of H 1 ( X ) {H^1}(X) considering H 1 ( X ) {H^1}(X) as a discrete group. In this paper it is shown that there is a shape morphism F ∈ Mor S C ( X , A X ) F \in {\text {Mor}_{SC}}(X,{A_X}) such that F ∗ : H 1 ( A X ) → H 1 ( X ) {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 → A X f:X \to {A_X} such that S ( f ) = F S(f) = F and thus that f ∗ : H 1 ( A X ) → H 1 ( X ) {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 ) {H^1}(X) has property L. Examples are given to show that X may be locally connected and H n ( X ) {H^n}(X) not have property L for n > 1 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 ) / Tor H n ( X ) {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 ) {H^n}(X) itself may not have property L even if X is a finite-dimensional movable continuum.