Homogeneous spaces and dimension theory

齐次空间和维度理论

• 批准号: RGPIN-2015-06200
• 负责人: Karassev, Alexandre
• 金额: $ 0.8万
• 依托单位: Nipissing University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635490
• 关键词: Homogeneous; spaces; dimension; theory

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.

同质性是一个非常自然的概念，存在于各个科学领域。在拓扑中，如果一个空间的每两个点都存在同胚，将其中一个点发送到另一个点，则该空间被称为齐次空间。直观上，这意味着空间的每个点周围看起来都是一样的。许多模拟现实生活现象的拓扑空间是同质的，例如欧几里得流形、无限维流形、螺线管、门格尔紧致体等。其中许多空间也表现出类似分形的行为。然而，并非所有分形都是齐次的。尽管这个概念很重要，但拓扑同质性仍然没有得到很好的理解。我的项目的目标之一是研究同质空间。我特别对“好的”同质空间的结构感兴趣，即那些绝对邻域收缩（ANR）的空间。请注意，所有流形以及更一般的多面体都是 ANR。然而，有许多 ANR 的奇异例子与流形有很大不同。我想要研究的核心问题之一是 Bing-Borsuk 猜想：每个有限维齐次 ANR 紧致都是流形吗？ Jakobsche 在 1980 年表明，Bing-Borsuk 猜想蕴含着著名的庞加莱猜想，最近由 Grigori Perelman 证明。另一个目标是开发一种统一的方法来构造同质空间。