Dimension Theory and Continuous Selections

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This research project concerns major problems in geometric and general topology, like dimension theory, homological and cohomological properties of homogeneous compacta, selections of set-valued maps and topological groups. Over the years my motivation for studying homogeneous compacta was the famous Bing-Borsuk conjecture that every finite dimensional metric homogeneous absolute neighborhood retract (ANR) is an Euclidean manifold. Although Bryant-Ferry announced recently the existence of a counter-example to that conjecture, the local homological and cohomological structure of any homogeneous metric ANR-compactum is similar to the local structure of the Euclidean manifold having the same dimension (Valov). There are many other open questions about such compacta. One of the long standing problems is whether there is a non-trivial homogeneous finite-dimensional absolute retract (Bing-Borsuk, 1965). This question is tightly connected to another two problems of Bing-Borsuk concerning homogeneous ANR compacta. Moreover, it is still unknown if any homogeneous metric ANR compactum is dimensionally full-valued. This is true in dimensions one and two, and I have recently shown that this is also true in dimension three. Note that the existence of homogeneous metric non dimensionally full-valued ANR compaca provides a negative answer to Bryant's generalized manifolds conjecture.  Developing methods to address the above mentioned problems is one of the objectives of this proposal. Another objective of my program of research is further extend my study of skeletally generated spaces which I introduced as an analogue of the well known openly generated spaces. This class of spaces can be characterized using a topological game between two players. One of the problems is whether there is a characterization of skeletally generated spaces in terms of quasi k-metrics (such a characterization was established for skeletally generated compacta). Jointly with my colleagues Kucharski and Plewik we also introduced and investigated the class of skeletally Dugundji spaces as a proper subclass of the skeletally generated spaces. Skeletally generated spaces admitting a topological group structure provides an important class of groups with very nice properties (Kozlov-Valov).  The main problems in that area are whether skeletally generated groups are different from skeletally Dugundji groups and whether skeletally generated groups are exactly the topological groups having countable cellularity and admitting a quasi k-metric.