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

Dranišnikov, A. N.; Repovš, D.; Ščepin, E. V.

doi:10.1090/S0002-9939-1994-1205488-1

On the failure of the Urysohn-Menger sum formula for cohomological dimension
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by A. N. Dranišnikov, D. Repovš and E. V. Ščepin PDF

Proc. Amer. Math. Soc. 120 (1994), 1267-1270 Request permission

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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