Extension of maps into nilpotent spaces III

Let M be a nilpotent CW-complex. We give necessary and sufficient cohomological dimension theory conditions for a finite-dimensional metric compactum X so that every map A → M , where A is a closed subset of X, can be extended to a map X → M . s a generalization of a result by Dranishnikov [Mat. Sb. (1991)] where such conditions were found for simply-connected CW-complexes M, and Cencelj and Dranishnikov [Canad. Bull. Math. (2001) and Topology Appl. (2002)] where a condition of finitely generatedness was imposed on the nilpotent CW-complex M.