Dimension Theory and Continuous Selections

这项研究经费涉及几何和一般拓扑学不同领域的主要问题，如维数理论、齐次紧、函数空间之间的算子和集值映射的选择。可拓理论的引入，以及粗糙几何与经典维数理论的相互作用，推动了近年来维数理论的研究。这个方向的主要成果要归功于Dranishnikov、Dydak、Levin、Schepin和他们的学生。可拓理论提供了覆盖维数与上同调维数的统一。这是一个具有挑战性的领域，需要新的技术和各种方法从几何拓扑，一般拓扑和同调理论。可拓理论中的一个主要问题是，对于给定上同调维或可拓维的所有度量紧包类是否存在一个全称元。同时，关于无限维空间的一些问题仍然是开放的。制定解决上述问题的方法是本提案的目标之一。另一个目标是继续齐次紧的研究。根据著名的Bing- Borsuk猜想，每一个齐次有限维anr紧实都是一个欧几里得流形（为了表明这个问题的重要性，让我提一下它的正解意味着庞加莱猜想）。因此，齐次anr紧包被期望具有欧几里得流形所具有的性质。最近关于广义康托流形的一些结果证明了这一期望。集值映射的连续选择及其应用也是一个建议的研究领域。维数理论中的许多重要结果都是基于选择定理的。

This research grant concerns major problems in different areas of geometric and general topology, like dimension theory, homogeneous compacta, operators between function spaces and selections of set-valued maps.The research in dimension theory over recent years is motivated by the introduction of extension theory, as well as, the interplay between coarse geometry and classical dimension theory. The main results in that direction are due to Dranishnikov, Dydak, Levin, Schepin and their students. The extension theory provides a unification of the covering and cohomological dimension. This is a challenging area requiring new techniques and variety of methods from geometric topology, general topology and homology theory. One of the major problems in extension theory is whether there exists a universal element for the class of all metric compacta of a given cohomological or extension dimension. At the same time some of the problems about infinite-dimensional spaces remain open. Developing methods to address the above mentioned problems is one of the objectives of this proposal. Another objective is continuing the study of homogeneous compacta. According to the famous Bing- Borsuk conjecture, every homogeneous finite-dimensional ANR-compactum is an Euclidean manifold (to indicate the importance of this problem let me mention that a positive solution of it implies the Poincare conjecture). So, homogeneous ANR-compacta are expected to have properties which are possessed by Euclidean manifolds. This expectation is witnessed by some recent results about generalized Cantor manifolds.Continuous selections of set-valued maps and their application is also a proposed area of research. A number of important results in dimension theory are based on selection theorems.