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Hurewicz-Serre theorem in extension theory

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Cencelj M., Dydak J., Mitra A., Vavpetič A.
Fundamenta Mathematicae
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2008
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tom 198
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nr 2
113-123
EN
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 $XτK(H_{k}(L),k)$ for all k ≥ 1, then $XτK(π_{k}(F),k)$ and $XτK(π_{k}(L),k)$ for all k ≥ $. Theorem 0.2. Let X be a metrizable space such that dim(X) < ∞ or X ∈ ANR. Suppose L is a nilpotent CW complex. If XτSP(L), then XτL in the following cases: (a) H₁(L) is finitely generated. (b) H₁(L) is a torsion group.
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Algebraic properties of quasi-finite complexes

88%
Cencelj M., Dydak J., Smrekar J., Vavpetič A., Virk Ž.
Fundamenta Mathematicae
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2007
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tom 197
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nr 1
67-80
EN
A countable CW complex K is quasi-finite (as defined by A. Karasev) if for every finite subcomplex M of K there is a finite subcomplex e(M) such that any map f: A → M, where A is closed in a separable metric space X satisfying XτK, has an extension g: X → e(M). Levin's results imply that none of the Eilenberg-MacLane spaces K(G,2) is quasi-finite if G ≠ 0. In this paper we discuss quasi-finiteness of all Eilenberg-MacLane spaces. More generally, we deal with CW complexes with finitely many nonzero Postnikov invariants. Here are the main results of the paper: Theorem 0.1. Suppose K is a countable CW complex with finitely many nonzero Postnikov invariants. If π₁(K) is a locally finite group and K is quasi-finite, then K is acyclic. Theorem 0.2. Suppose K is a countable non-contractible CW complex with finitely many nonzero Postnikov invariants. If π₁(K) is nilpotent and K is quasi-finite, then K is extensionally equivalent to S¹.
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