Hurewicz–Serre theorem in extension theory

M. Cencelj ¹   ; J. Dydak ²   ; A. Mitra ²   ; A. Vavpetič ³

¹ IMFM Univerza v Ljubljani Jadranska ulica 19 SI-1111 Ljubljana, Slovenija
² University of Tennessee Knoxville, TN 37996, U.S.A.
³ Fakulteta za Matematiko in Fiziko Univerza v Ljubljani Jadranska ulica 19 SI-1111 Ljubljana, Slovenija

Fundamenta Mathematicae, Tome 198 (2008) no. 2, pp. 113-123

Résumé

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.

Détail

DOI : 10.4064/fm198-2-2

Keywords: paper devoted generalizations cencelj dranishnikov theorems relating extension properties nilpotent complexes their homology groups here main results paper theorem nilpotent complex homotopy fiber inclusion its infinite symmetric product metrizable space tau l tau tau theorem metrizable space mathop dim infty anr suppose nilpotent complex tau tau following cases finitely generated torsion group