Abstract
Let C denote the category of compact Hausdorff spaces and continuous maps and H: C-,.HC the homotopy functor to the homotopy category. Let S: C-..SC denote the functor of shape in the sense of Holsztynski for the projection functor H. Every continuous mapping f between spaces gives rise to a shape morphism S(f) in SC, but not every shape morphism is in the image of S. In this paper it is shown that if X is a continuum with x E X and A is a compact connected abelian topological group, then if F is a shape morphism from X to A, then there is a continuous map f:X-'.A such thatf(x)=O and S(f)=F. It is also shown that if f, g: X-A are continuous withf(x)=g(x)=O and S(f)=S(g), then fandg are homotopic. These results are then used to show that there are shape classes of continua containing no locally connected continua and no arcwise connected continua. Some other applications to shape theory are given also. Ihtroduction. Let C denote the category of compact Hausdorff spaces and continuous maps and H: C-iHC the homotopy functor to the homotopy category. Let S: C-).SC denote the functor of shape in the sense of Holsztyn'ski for the projection functor H [5]. Let X and Y be compact Hausdorff spaces. In [6] it is shown that if X and Y are associated with ANR-systems X and Y, respectively, then there is one to one correspondence between Morsc(X, Y) and the homotopy classes of maps of ANRsystems used in the approach of Mardesic and Segal [7]. Thus, our results in this paper will apply to either approach to shape. In the first part of the paper we show that if X is a continuum with x E X and A is a compact connected abelian topological group, then if Fe Morsc(X, A), then there is a continuous f: X-.A with S(f)=F and withf(x)=0. It is also shown that if X and A are as above andf, g:X-+A are continuous with f(x)=g(x)=O and with S(f)=S(g), then f and g are homotopic. These results are clearly related to the results in [6]. Received by the editors November 14, 1972. AMS (MOS) subject classfications (1970). Primary 55D99; Secondary 22B99.
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