Hurewicz-Serre Theorem in extension theory 


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:

Theorem 1:
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>= 2.

Theorem 2:
Let X be a metrizable space such that dim(X) < infinity 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
a. H_1(L) is  finitely generated.
b. H_1(L) is a torsion group.