The paucity of universal compacta in cohomological dimension

Let C be a class of spaces. An element Z is an element of C is called universal for C if each element of C embeds in Z. It is well-known that for each n is an element of 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 dime X <= n. (C) 2017 Elsevier B.V. All rights reserved.