Dimension Theory and Continuous Selections

这项研究涉及几何和一般拓扑学中的主要问题，如维度理论、齐次紧的同调和上同调性质、集值映射和拓扑群的选择。多年来，我研究齐次紧的动机是著名的Bing-Borsuk猜想，即每个有限维度量齐次绝对邻域收缩(ANR)都是欧氏流形。虽然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类似于具有相同维度(Valov)的欧几里德流形的局部结构。关于这样的契约，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收缩(Bing-Borsuk，1965)。这个问题与Bing-Borsuk关于齐次ANR紧性的另外两个问题密切相关。此外，是否有任何齐次度量ANR紧致在维度上是全值的仍是未知的。这在一维和二维都是正确的，我最近也证明了这在三维也是正确的。请注意，齐次度量无量纲全值ANR紧致的存在为Bryant的广义流形猜想提供了否定的答案。开发解决上述问题的方法是这项建议的目标之一。*我的研究计划的另一个目标是进一步扩展我对骨架生成空间的研究，我将其介绍为众所周知的公开生成空间的类比。这类空间可以用两个博弈者之间的拓扑博弈来刻画。其中一个问题是是否存在骨架生成空间的拟k-度量刻画(这种刻画是为骨架生成的紧致空间建立的)。我们还与我的同事Kucharski和Plewik一起引入并研究了骨架Dugundji空间类，作为骨架生成空间的一个真子类。具有拓扑群结构的骨架生成空间提供了一类具有非常好性质的重要群(Kozlov-Valov)。这一领域的主要问题是骨架生成的群是否不同于骨架生成的Dugundji群，以及骨架生成的群是否恰好是具有可数胞性并允许准k度量的拓扑群。

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.***********