Homogeneous spaces and dimension theory

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Homogeneity is a very natural concept, existing in various areas of science. In topology, a space is called homogeneous if for every two of its points there exists a homeomorphism that sends one of these points to the other. Intuitively, it means that the space looks the same around each of its points. Many topological spaces, that model real-life phenomena, are homogeneous, e.g. Euclidean manifolds, infinite-dimensional manifolds, solenoids, Menger compacta, etc. Many of these spaces also exhibit a fractal-like behaviour. However, not all fractals are homogeneous. Despite the importance of the concept, topological homogeneity is still not well-understood. One of the goals of my program is the study of homogeneous spaces. In particular, I am interested in the structure of “nice” homogeneous spaces, i.e. those which are absolute neighbourhood retracts (ANRs). Note that all manifolds and, more generally, polyhedra are ANRs. However, there are many exotic examples of ANRs that differ substantially from manifolds. One of the central problems I would like to work on is the Bing-Borsuk conjecture: is every finite-dimensional homogeneous ANR-compactum a manifold? Jakobsche showed in 1980 that the Bing-Borsuk conjecture implies the famous Poincare conjecture, recently proved by Grigori Perelman. Another objective is to develop a unified approach to construction of homogeneous spaces.