# Algebraic properties of quasi-finite complexes

M. Cencelj; J. Dydak; J. Smrekar; A. Vavpetič; Ž. Virk

Fundamenta Mathematicae (2007)

- Volume: 197, Issue: 1, page 67-80
- ISSN: 0016-2736

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topM. 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 -

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