Abstract
The following problem plays an important role in shape theory: find conditions that guarantee that a shape morphism F:X ↦ Y of a topological space X to a topological space Y is generated by a continuous mapping f:X ↦ Y. In the present paper, we study this problem in equivariant shape theory and give a solution for shape-equivariant morphisms to transitive G-spaces, where G is a compact group with countable base. As a corollary, we prove a sufficient condition for equivariant shapes of a G-space X to be equal to the group G itself. We also prove some statements concerning equivariant bundles that play the key role in the proof of the main results and are of interest on their own.
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REFERENCES
R. S. Palais, “The classification of G-spaces,” Mem. Amer. Math. Soc. (1960), no. 36, 1-72.
G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York-London, 1972.
A. Gleason, “Spaces with a compact Lie group transformations,” Proc. Amer. Math. Soc., 1 (1950), no. 1, 35-43.
L. S. Pontryagin, Topological Groups, Gordon and Breach, New York-London-Paris, 1966.
W. Holsztynski, “Continuity of Borsuk's shape functor,” Bull. Acad. Polon. Ski., 19 (1971), no. 12, 1105-1108.
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Gevorkyan, P.S. Shape Morphisms to Transitive G-Spaces. Mathematical Notes 72, 757–762 (2002). https://doi.org/10.1023/A:1021429627291
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DOI: https://doi.org/10.1023/A:1021429627291