Shape theory and compact connected abelian topological groups

Author:
James Keesling

Journal:
Trans. Amer. Math. Soc. **194** (1974), 349-358

MSC:
Primary 54C56

MathSciNet review:
0345064

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Abstract | References | Similar Articles | Additional Information

Abstract: Let *C* denote the category of compact Hausdorff spaces and continuous maps. Let denote the functor of shape in the sense of Holsztyński from *C* to the shape category *SC* determined by the homotopy functor 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 such that . 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 . (2) The space *A* is movable if and only if *A* is locally connected. (3) The space *A* shape dominates , if and only if there is a *D* such that . (4) The fundamental dimension of *A* is the same as the dimension of .

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.

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DOI:
https://doi.org/10.1090/S0002-9947-1974-0345064-8

Keywords:
Shape,
homotopy,
movable compactum,
shape domination,
fundamental dimension,
compact connected abelian topological group

Article copyright:
© Copyright 1974
American Mathematical Society