The Extension Dimension Problem 

The concept of “extension dimension” was first formalized in work of A. Dranishnikov and J. Dydak in 1996. They defined a certain equivalence relation on the class of CW-complexes. For each CW-complex K, its equivalence class under this relation was denoted [K]. They showed that for each compact Hausdorff space X, a uniquely determined equivalence class [K], called the extension dimension of X exists. They then asked whether for each compact metrizable space X, its extension dimension could be represented by a countable CW-complex K. Very little progress has been made on this question. It is even conceivable that the extension dimension of any compact Hausdorff space has a countable representative!

In this talk, we shall present the definitions needed to define extension dimension. We shall also make note of the fact that each [K] can be represented by a triangulated polyhedron, that is, there exists a simplicial complex L such that [|L|] = [K]. One might expect that such an L could be
replaced either by a countable subcomplex or be modified into a countable simplicial complex whose polyhedron would represent the extension dimension of X.