Algebraic properties of quasi-finite complexes

M. Cencelj, et al. "Algebraic properties of quasi-finite complexes." Fundamenta Mathematicae 197.1 (2007): 67-80. <http://eudml.org/doc/286250>.

@article{M2007,
abstract = { 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. },
author = {M. Cencelj, J. Dydak, J. Smrekar, A. Vavpetič, Ž. Virk},
journal = {Fundamenta Mathematicae},
keywords = {Extension dimension; cohomological dimension; ablosute extensor; universal space; quasi-finite complex; invertible map},
language = {eng},
number = {1},
pages = {67-80},
title = {Algebraic properties of quasi-finite complexes},
url = {http://eudml.org/doc/286250},
volume = {197},
year = {2007},
}

TY - JOUR
AU - M. Cencelj
AU - J. Dydak
AU - J. Smrekar
AU - A. Vavpetič
AU - Ž. Virk
TI - Algebraic properties of quasi-finite complexes
JO - Fundamenta Mathematicae
PY - 2007
VL - 197
IS - 1
SP - 67
EP - 80
AB - 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.
LA - eng
KW - Extension dimension; cohomological dimension; ablosute extensor; universal space; quasi-finite complex; invertible map
UR - http://eudml.org/doc/286250
ER -