Dimension Theory and Continuous Selections

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This research grant concerns major problems in different areas of geometric and general topology, like dimension theory, homogeneous compacta, operators between function spaces and selections of set-valued maps. The research in dimension theory over recent years is motivated by the introduction of extension theory, as well as, the interplay between coarse geometry and classical dimension theory. The main results in that direction are due to Dranishnikov, Dydak, Levin, Schepin and their students. The extension theory provides a unification of the covering and cohomological dimension. This is a challenging area requiring new techniques and variety of methods from geometric topology, general topology and homology theory. One of the major problems in extension theory is whether there exists a universal element for the class of all metric compacta of a given cohomological or extension dimension. At the same time some of the problems about infinite-dimensional spaces remain open. Developing methods to address the above mentioned problems is one of the objectives of this proposal. Another objective is continuing the study of homogeneous compacta. According to the famous Bing- Borsuk conjecture, every homogeneous finite-dimensional ANR-compactum is an Euclidean manifold (to indicate the importance of this problem let me mention that a positive solution of it implies the Poincare conjecture). So, homogeneous ANR-compacta are expected to have properties which are possessed by Euclidean manifolds. This expectation is witnessed by some recent results about generalized Cantor manifolds. Continuous selections of set-valued maps and their application is also a proposed area of research. A number of important results in dimension theory are based on selection theorems.