Abstract
Let C be a class of spaces. An element Z ∈ C is called universal for C if each element of C embeds in Z . It is well-known that for each n ∈ N , there exists a universal element for the class of metrizable compacta X of (covering) dimension dim X ≤ n . The situation in cohomological dimension over an abelian group G , denoted dim G , is almost the opposite. Our results will imply in contradistinction that for each nontrivial abelian group G and for n ≥ 2 , there exists no universal element for the class of metrizable compacta X with dim G X ≤ n .
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