Atomic maps and the Chogoshvili-Pontrjagin claim

Levin, M.; Sternfeld, Y.

doi:10.1090/S0002-9947-98-01995-3

Atomic maps and the Chogoshvili-Pontrjagin claim
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by M. Levin and Y. Sternfeld

Trans. Amer. Math. Soc. 350 (1998), 4623-4632

DOI: https://doi.org/10.1090/S0002-9947-98-01995-3

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Abstract:

It is proved that all spaces of dimension three or more disobey the Chogoshvili-Pontrjagin claim. This is of particular interest in view of the recent proof (in Certain 2-stable embeddings, by Dobrowolski, Levin, and Rubin, Topology Appl. 80 (1997), 81–90) that two-dimensional ANRs obey the claim. The construction utilizes the properties of atomic maps which are maps whose fibers ($=$point inverses) are atoms ($=$hereditarily indecomposable continua). A construction of M. Brown is applied to prove that every finite dimensional compact space admits an atomic map with a one-dimensional range.

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