The paucity of universal compacta in cohomological dimension

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 dimG, 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 dimG⁡X≤n.