Continuous functions induced by shape morphisms

Abstract:

Let $C$ denote the category of compact Hausdorff spaces and continuous maps and $H:C \to HC$ the homotopy functor to the homotopy category. Let $S:C \to SC$ denote the functor of shape in the sense of Holsztyński 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 $x \in X$ 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 $f:X \to A$ such that $f(x) = 0$ and $S(f) = F$. It is also shown that if $f,g:X \to A$ are continuous with $f(x) = g(x) = 0$ and $S(f) = S(g)$, then $f$ and $g$ 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.