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On the failure of the Urysohn-Menger sum formula for cohomological dimension


Authors: A. N. Dranišnikov, D. Repovš and E. V. Ščepin
Journal: Proc. Amer. Math. Soc. 120 (1994), 1267-1270
MSC: Primary 55M10; Secondary 54D35, 54F45, 54G20
DOI: https://doi.org/10.1090/S0002-9939-1994-1205488-1
MathSciNet review: 1205488
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Abstract: We prove that the classical Urysohn-Menger sum formula, $ \dim (A \cup B) \leqslant \dim A + \dim B + 1$, which is also known to be true for cohomological dimension over the integers (and some other abelian groups), does not hold for cohomological dimension over an arbitrary abelian group of coefficients. In particular, we prove that there exist subsets $ A,\;B \subset {\mathbb{R}^4}$ such that $ 4 = {\dim _{\mathbb{Q}/\mathbb{Z}}}(A \cup B) > {\dim _{\mathbb{Q}/\mathbb{Z}}}A + {\dim _{\mathbb{Q}/\mathbb{Z}}}B + 1 = 3$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1205488-1
Keywords: Urysohn-Menger sum formula, cohomological dimension, Boltyanskii compacta, Bockstein inequalities, dimension type, compactification, completion
Article copyright: © Copyright 1994 American Mathematical Society

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