Dimension Theory and Continuous Selections

本研究项目涉及几何和一般拓扑学中的主要问题，如维数理论，齐次群的同调和上同调性质，集值映射和拓扑群的选择。多年来，我的动机研究齐次多项式是著名的宾博苏克猜想，每一个有限维度量齐次绝对邻域收缩（ANR）是一个欧几里得流形。虽然Bryant-Ferry最近宣布存在一个反例，但任何齐次度量ANR-紧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。关于这一点，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）。这个问题是紧密相连的另两个问题的Bing-Borsuk有关齐次ANR ta。此外，它仍然是未知的，如果任何齐次度量ANR紧维满值。这在一维和二维中是正确的，我最近证明了这在三维中也是正确的。请注意，齐次度量无维全值ANR紧致的存在为Bryant的广义流形猜想提供了否定的答案。 制定解决上述问题的方法是本提案的目标之一。我的研究计划的另一个目标是进一步扩展我的研究，我介绍了作为一个类似的著名的公开生成的空间的空间生成。这类空间可以用两个玩家之间的拓扑博弈来描述。其中一个问题是，是否存在一个关于拟k-度量的代数生成空间的特征（这样的特征是为代数生成空间建立的）。与我的同事Kucharski和Plewik一起，我们还介绍并研究了作为实生空间的真子类的实生Dugundji空间类。允许拓扑群结构的拓扑生成空间提供了一类具有非常好的性质的重要群（Kozlov-Valov）。 这一领域的主要问题是：代数生成群是否不同于代数Dugundji群，代数生成群是否就是具有可数胞腔且允许拟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.