Hurewicz–Serre theorem in extension theory
Volume 198 / 2008
Fundamenta Mathematicae 198 (2008), 113-123
MSC: Primary 54F45; Secondary 55M10, 54C65.
DOI: 10.4064/fm198-2-2
Abstract
The paper is devoted to generalizations of
the Cencelj–Dranishnikov theorems
relating extension properties of nilpotent CW complexes to their homology
groups.
Here are the main results of the paper:
Theorem 0.1.
Let $L$ be a nilpotent CW complex and $F$
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 $k\ge 2$.
Theorem 0.2.
Let $X$ be a metrizable space such that ${\mathop{\rm dim}}(X) < \infty$ or $X\in ANR$.
Suppose $L$ is a nilpotent CW complex.
If $X\tau SP(L)$,
then $X\tau L$ in the following cases$:$
(a) $H_1(L)$ is finitely generated.
(b) $H_1(L)$ is a torsion group.
