Algebraic Properties of Quasi-finite Complexes

A countable CW complexK is quasi-finite (as defined by A.Karasev [30]) if for every finite subcomplex M of K there is a finite subcomplex e(M) containing M such that any map f : A → M , A closed in a separable metric space X satisfying XτK, there is an extension g : X → e(M) of f . Levin’s [36] 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 whose Postnikov invariants are trivial or they have only finitely many nontrivial homotopy groups. Here are the main results of the paper: Theorem 0.1. Suppose K is a countable CW complex whose Postnikov invariants are trivial or it has only finitely many nontrivial homotopy groups. If π1(K) is a local group and K is quasi-finite, then K is acyclic. Theorem 0.2. Suppose K is a countable non-contractible CW complex whose Postnikov invariants are trivial or it has only finitely many nontrivial homotopy groups. If π1(K) is nilpotent and K is quasi-finite, then K is extensionally equivalent to S1. Theorem 0.3. Suppose K1 is a connected countable CW complex such that all maps f : K → ΩkL are null-homotopic if L is a finite simple connected CW complex with finite homotopy groups. Suppose K2 is a connected countable CW complex such that XτK(Hn(K2), n) for n ≥ 1 implies XτK2 for all compacta X. If K = K1 ∨ K2 is not acyclic and quasi-finite, then K is extensionally equivalent to S1. Date: August 19, 2005. 1991 Mathematics Subject Classification. Primary: 54F45; Secondary: 55M10, 54C65 .