Let $mathcal{C}$ be a class of compact

metrizable spaces. An element $Zinmathcal{C}$ is called {it universal} for

$mathcal{C}$ if each element of $mathcal{C}$ embeds

topologically in $Z$. It is a well-known, classical result of dimension theory that for each

$ninmathbb{N}$, there exists a universal element for the class

of metrizable compacta $X$ of (covering) dimension $dim Xleq n$.

Modern techniques involving the Stone-v Cech compactification and

the Mardev si' c factorization theorem yield relatively easy,

albeit abstract, proofs of this result.

There is a parallel theory of dimension called cohomological

dimension; indeed there is one such theory for each abelian

group $G$. Although these theories concur with dimension in

many ways, they do not in general agree with it. One may ask

about the existence of universal compacta for this type of

dimension. But it turns out that not all the techniques that work

for $dim$ apply to cohomological dimension; in

particular one cannot use the Stone-v Cech compactification

at what would be a critical point of such a proof. It has thus been

speculated that in most cases there do not exist universal

compacta in the theory of cohomological dimension.

We are going to speak about techniques that should lead to a proof

of this nonexistence conjecture.