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Algebraic properties of quasi-finite complexes

Tom 197 / 2007

Fundamenta Mathematicae 197 (2007), 67-80 MSC: Primary 54F45; Secondary 55M10, 54C65. DOI: 10.4064/fm197-0-4

Streszczenie

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 $f:A\to M$, where $A$ is closed in a separable metric space $X$ satisfying $X\tau K$, has an extension $g:X\to e(M)$. Levin's results imply that none of the Eilenberg–MacLane spaces $K(G,2)$ is quasi-finite if $G\ne 0$. 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 $\pi_1(K)$ 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 $\pi_1(K)$ is nilpotent and $K$ is quasi-finite, then $K$ is extensionally equivalent to $S^1$.

Autorzy

• M. CenceljFakulteta za Matematiko in Fiziko
Univerza v Ljubljani
SI-1111 Ljubljana, Slovenija
e-mail
• J. DydakUniversity of Tennessee
Knoxville, TN 37996, U.S.A.
e-mail
• J. SmrekarFakulteta za Matematiko in Fiziko
Univerza v Ljubljani
SI-1111 Ljubljana, Slovenija
e-mail
• A. VavpetičFakulteta za Matematiko in Fiziko
Univerza v Ljubljani