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.
