Shape theory and compact connected abelian topological groups

Abstract:

Let C denote the category of compact Hausdorff spaces and continuous maps. Let $S:C \to SC$ denote the functor of shape in the sense of Holsztyński from C to the shape category SC determined by the homotopy functor $H:C \to HC$ from C to the homotopy category HC. Let A, B, and D denote compact connected abelian topological groups. In this paper it is shown that if G is a morphism in the shape category from A to B, then there is a unique continuous homomorphism $g:A \to B$ such that $S(g) = G$. This theorem is used in a study of shape properties of continua which support an abelian topological group structure. The following results are shown: (1) The spaces A and B are shape equivalent if and only if $A \simeq B$. (2) The space A is movable if and only if A is locally connected. (3) The space A shape dominates $B,S(A) \geq S(B)$, if and only if there is a D such that $A \simeq B \times D$. (4) The fundamental dimension of A is the same as the dimension of $A,{\text {Sd}}(A) = \dim A$. In an Appendix it is shown that the Holsztyński approach to shape and the approach of Mardešić and Segal using ANR-systems are equivalent. Thus, the results apply to either theory and to the Borsuk theory in the metrizable case.