Extension of maps to nilpotent spaces. II
The main result of the paper is the equivalence of the following four conditions for any nilpotent CW-complex \(M\) with finitely generated fundamental group and any finite-dimensional compactum \(X\): (1) \(\dim X\leq M\); (2) \(\dim X\leq \text{SP}^\infty M\); (3) \(\dim_{H_i(M)}\leq i\) for all \(i>0\); (4) \(\dim_{\pi_i(M)}\leq i\) for all \(i>0\) (here \(\dim X\leq M\) means that each continuous map from a closed subspace of \(X\) into \(M\) admits a continuous extension onto the whole of \(X\)). In [\textit{M. Cencelj} and \textit{A. Dranishnikov}, Can. Bull. Math. 44, 266-269 (2001; Zbl 0989.55001)], this equivalence was proved under the assumption that all the homotopy groups of \(M\) are nilpotent and finitely generated while in [\textit{A. Dranishnikov}, Mat. Sb. 182, 1300-1310 (1991; Zbl 0739.55007)] the equivalence was proved under the assumption that \(M\) is simply connected. At the end of the paper the authors remark that the nilpotency of \(M\) is essential because in case of the projective plane \(M\) (which is not nilpotent) the four conditions fail to be equivalent. These and similar results are important in Cohomological Dimension Theory, see the survey of A. Dranishnikov at http://at.yorku.ca/t/a/i/c/43.pdf.
scientific article; zbMATH DE number 1879083