On finite-dimensional resolutions
by
Sergei Ageev
Belarus State University, Belarys

The Dranishnikov's resolution, a special type of a (k-1)-soft map between the k-dimensional Menger compactum and the Hilbert cube Q, is an important technique of geometric topology revealing the wide analogy between Q-manifold and Menger manifold theories.

On the other hand, by its properties, the Dranishnikov's resolution is a finite-dimensional analogue of the trivial Q-fibration over the Hilbert cube with the exception of being k-soft. It is clear that the more properties of Dranishnikov's resolution will be found the more convenient instrument it will become.

The other bridge between the infinite dimensional and finite dimensional topologies (in fact, between Nöbeling and Hilbert (l₂-) manifold theories) - Chigogidze's resolution c_k:n^k→ l₂, possesses properties which are every bit as remarkable as those of Dranishnikov's resolution. The opinion to consider c_k as a finite-dimensional analogue of the trivial l₂-fibration over the Hilbert space is justified to a greater degree than even in the case of Dranishnikov's resolution.

We intend to focus our attention on the intimate interrelation between Dranishnikov's and Chigogidze's resolutions and to make a definite advance toward the investigation of these finite-dimensional resolutions.

Date received: May 1, 2008