Algebraic properties of quasi-finite complexes

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 $f:A\to M$, where $A$ is closed in a separable metric space $X$ satisfying $XτK$, 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: Suppose $K$ is a countable CW complex with finitely many nonzero Postnikov invariants. If $π_1(K)$ is a locally finite group and $K$ is quasi-finite, then $K$ is acyclic. Suppose $K$ is a countable non-contractible CW complex with finitely many nonzero Postnikov invariants. If $π_1(K)$ is nilpotent and $K$ is quasi-finite, then $K$ is extensionally equivalent to $S^1$.