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Let C denote the category of compact Hausdorff spaces and H: C -* HC be the homotopy functor. Let S: C SC be the functor of shape in the sense of Holsztynski for the projection functor H. Let X be a continuum and Hn(X) denote n-dimensional Cech cohomology with integer coefficients. Let A =char H I(X) be the character group of H 1(X) considering H1(X) as a disx crete group. In this paper it is shown that there is a shape morphism F e Morsc(Xs Ax) such that F*: HI(Ax) Ht(X) is an isomorphism. It follows from the results of a previous paper by the author that there is a continuous mapping f : X Ax such that S(f =F and thus that f*: H(A )+HI(X) is an isomorphism. This result is applied to show that if X is locally connected, then H1(X) has property L. Examples are given to show that X may be locally connected and Hn(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, Hn(X)/Tor Hn(X) has property L. This result allows us to construct peano continua which are nonmovable. An example is given to show that Hn(X) itself may not have property L even if X is a finitedimensional movable continuum. Introduction. Let C denote the category of compact Hausdorff spaces and H: C -+ HC be the homotopy functor. Let S: C -. SC be the functor of shape in the sense of HolsztyAski [3] for the projection functor H. Let X be a continuum and let Hn(X) denote n-dimensional Cech cohomology with integer coefficients. Let Ax be the character group of HW(X) considering H1(X) as a discrete group. Note that H1(X) is torsion free and thus Ax is a compact connected abelian topological group. In this paper we show that if B is a compact connected abelian topological group and h: H1(B) -. H1(X) is a homomorphism, then there is a shape morphism F e Morsc (X, B) such that F* h. By the results in [6], there is a continuous function /: x B with S(f) = F and thus with h* Received by the editors November 28, 1972. AMS (MOS) subject classifications (1970). Primary 55D99; Secondary 55BO5, 22B99.

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