Dimension Theory and Continuous Selections

本研究项目涉及几何拓扑和一般拓扑中的主要问题，如维数理论、齐次紧的同调和上同调性质、集值映射的选择和拓扑群。多年来，我研究齐次紧的动机是著名的Bing-Borsuk猜想，即每一个有限维度量齐次绝对邻域缩回（ANR）都是一个欧氏流形。尽管Bryant-Ferry最近宣布了该猜想的一个反例的存在，但任何齐次度量anr紧致的局部同调和上同调结构都类似于具有相同维数（Valov）的欧几里得流形的局部结构。关于这种契约还有许多其他悬而未决的问题。一个长期存在的问题是是否存在非平凡的齐次有限维绝对缩回（Bing-Borsuk, 1965）。这个问题与Bing-Borsuk关于齐次ANR紧的另外两个问题紧密相连。此外，尚不清楚是否有齐次度量ANR紧实是维数全值的。这在一维和二维中是成立的，我最近证明了在三维中也是成立的。注意齐次度量无量纲全值ANR compaca的存在性为Bryant广义流形猜想提供了一个否定的答案。制定解决上述问题的方法是本提案的目标之一。我研究计划的另一个目标是进一步扩展我对骨骼生成空间的研究，我将其作为众所周知的开放生成空间的模拟。这类空间可以用两个参与者之间的拓扑博弈来表征。其中一个问题是是否存在骨架生成空间的拟k-度量表征（这样的表征是针对骨架生成紧体建立的）。与我的同事Kucharski和Plewik一起，我们还引入并研究了骨架Dugundji空间的类，作为骨架生成空间的适当子类。允许拓扑群结构的骨架生成空间提供了一类具有非常好的性质的群（Kozlov-Valov）。该领域的主要问题是骨架生成群是否与骨架Dugundji群不同，以及骨架生成群是否正是具有可数细胞并允许准k度规的拓扑群。

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.