Some Examples in Cohomological Dimension Theory

It is well-known that dimX ≤ n if and only if every map of a closed subspace of X into the n-dimensional sphere Sn can be extended over X. It is also well-known that for the cohomological dimension dimGX of X with respect to an abelian coefficient group G, dimGX ≤ n if and only if every map of a closed subspace of X into the Eilenberg-Mac Lane complex K(G,n) extends over X. These properties give rise to the notion of extensional dimension [3]. Let K be a CW complex. The extensional dimension of X does not exceed K, written e-dimX ≤ K, if every map of a closed subset of X into K extends over X. Here e-dimX > K means that e-dimX ≤ K does not hold. We write e-dimX > n if e-dimX > K for every CW-complex K which is not n-connected. Thus e-dim> n implies both dim> n and dimG > n for every group G = 0. Below are listed some remarkable examples in cohomological dimension.