Characterizing Universal Spaces

Samenvatting

Characterization theorems of universal spaces originated in
Torunczyk's celebrated work, where he showed that if a complete (compact) space is an absolute extensor and strongly universal for all complete (compact) spaces then the space is homeomorphic to Hilbert space (the Hilbert cube). The Menger compacta and the Nöbeling spaces are finite-dimensional topological analogues of the Hilbert cube respectively Hilbert space and are the classic examples of universal spaces for topological dimension. The Menger compacta were characterized by Bestvina and the Nöbeling spaces by Ageev, Levin, and Nagorko (independently of each other). The proofs of the characterization theorem for Nöbeling spaces are considerably different in many respects and they are also considerably different from the proofs of the characterization theorems by Torunczyk and Bestvina. This proposal is devoted to the project of finding a unified (and hopefully simpler) approach to cover
all the characterization theorems described above is the
project to which this proposal is devoted. To that end we
follow an approach to characterizing universal spaces based on resolutions. In addition, we expect this project to be relevant to the main focus of the Dimension Theory programme at the VU. Namely to find a superior characterization of stable complete Erdös space, specifically one that links this space to the homeomorphism groups of the Menger compacta. Stable complete Erdös space plays the same role for the class of almost zero-dimensional spaces as the Nöbeling spaces play for regular dimension.

Kenmerken

Projectnummer

040.11.266

Hoofdaanvrager

Prof. dr. J.J. Dijkstra em.

Verbonden aan

Vrije Universiteit Amsterdam, Faculteit der Exacte Wetenschappen, Afdeling Wiskunde

Looptijd

31/08/2011 tot 25/09/2012