Algebraic properties of quasi-finite complexes 


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) 
containing M such that any map f:A -> M, A closed in a separable 
metric space X satisfying X\tau K, there is an extension g:X -> e(M) of f.
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 
whose Postnikov invariants are trivial or they have only finitely many 
nontrivial homotopy groups.


Here are the main results of the paper:

Theorem 1: 
Suppose K is a countable CW complex whose Postnikov invariants are trivial 
or it has only finitely many nontrivial homotopy groups. If \pi_1(K) is 
a local group and K is quasi-finite, then K is acyclic.


Theorem 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 \pi_1(K) is nilpotent and K is quasi-finite, then K is extensionally 
equivalent to S^1.


Theorem
Suppose K_1 is a connected countable CW complex such that all maps 
f:K -> Omega^k L are null-homotopic if L is a finite simple connected CW complex 
with finite homotopy groups. Suppose K_2 is a connected countable CW complex 
such that X\ tau K(H_n(K_2),n) for n\ge 1 implies X\tau K_2 for all compacta
X. If K=K_1 v K_2 is not acyclic and quasi-finite, then K is extensionally 
equivalent to S^1.