Hurewicz-Serre Theorem in extension theory

Abstract

The paper is devoted to generalizations of Cencelj-Dranishnikov theorems relating extension properties of nilpotent CW complexes to its homology groups. Here are the main results of the paper: \par {\bf Theorem}. Suppose $L$ is a nilpotent CW complex and $F$ is the homotopy fiber of the inclusion $i$ of $L$ into its infinite symmetric product $SP(L)$. If $X$ is a metrizable space such that $X\tau K(H_k(L),k)$ for all $k\ge 1$, then $X\tau K({\pi }_k(F),k)$ and $X\tau K({\pi }_k(L),k)$ for all $k\ge 2$. \par {\bf Theorem}. Let $X$ be a metrizable space such that $\dim (X)<\infty$ or $X\in ANR$. Suppose $L$ is a nilpotent CW complex and $SP(L)$ is its infinite symmetric product. If $X\tau SP(L)$, then $X\tau L$ in the following cases: \begin{itemize} \item[a.] $H_1(L)$ is finitely generated. \item[b.] $H_1(L)$ is a torsion group. \end{itemize}