Universal compacta and quasi-finite complexes
by
Alex Karassev
Nipissing University
Coauthors: Vesko Valov

The following problem is considered: characterize countable CW complexes P such that there exists a P-invertible map f: X® I^w where X is a metrizable compactum with P Î AE (X). We introduce the concept of a quasi-finite complex and show that quasi-finite complexes provide required characterization. We also construct an example of a quasi-finite complex L such that its extension type [L] does not contain a finitely dominated complex. This answers in negative the following question in extension theory: let L be a countable CW complex such that the class of metrizable compacta {X | L Î AE(X)} has a universal space. Is it true that the extension type [L] of this complex contains a finitely dominated complex? We show also that if the class of all metrizable compacta of extension dimension £ [L] contains a universal element which is an absolute extensor in dimension [L], then L is quasi-finite.

Date received: February 9, 2005