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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm197-0-4,
author = {M. Cencelj and J. Dydak and J. Smrekar and A. Vavpeti\v c and \v Z. Virk},
title = {Algebraic properties of quasi-finite complexes},
journal = {Fundamenta Mathematicae},
volume = {193},
year = {2007},
pages = {67-80},
zbl = {1142.54013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm197-0-4}
}
M. Cencelj; J. Dydak; J. Smrekar; A. Vavpetič; Ž. Virk. Algebraic properties of quasi-finite complexes. Fundamenta Mathematicae, Tome 193 (2007) pp. 67-80. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm197-0-4/