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Shape Theory and Compact Connected Abelian Topological Groups

James Keesling
Transactions of the American Mathematical Society
Vol. 194 (Jul., 1974), pp. 349-358
DOI: 10.2307/1996811
Stable URL: http://www.jstor.org/stable/1996811
Page Count: 10
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Shape Theory and Compact Connected Abelian Topological Groups
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Abstract

Let C denote the category of compact Hausdorff spaces and continuous maps. Let S: C → SC denote the functor of shape in the sense of Holsztynski from C to the shape category SC determined by the homotopy functor H: C → 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 → 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 ≃ B. (2) The space A is movable if and only if A is locally connected. (3) The space A shape dominates B, S(A) ≥ S(B), if and only if there is a D such that A ≃ B × D. (4) The fundamental dimension of A is the same as the dimension of $A, \operatorname{Sd}(A) = \dim A$. In an Appendix it is shown that the Holsztynski approach to shape and the approach of Mardesic and Segal using ANR-systems are equivalent. Thus, the results apply to either theory and to the Borsuk theory in the metrizable case.

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