Mathematical Sciences: Some Problems in Geometric Topology

数学科学：几何拓扑中的一些问题

• 批准号: 9305758
• 负责人: Steven Ferry
• 金额: $ 11.05万
• 依托单位: SUNY at Binghamton
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-06-01 至 1996-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9305758&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Some; Problems; Geometric

The investigator intends to study a number of topics in controlled topology and differential geometry. The most pressing of these concern the nonresolvable homology manifolds discovered by the investigator, J. Bryant, W. Mio, and S. Weinberger. Simply put, he wishes to study these spaces and see to what extent the classical theory of manifolds carries over to this new category of spaces. This study is closely related to the well-known Borel and Bing-Borsuk Conjectures. Other problems involved include applications of controlled Gamma-surgery to the study of topological embeddings in codimension two, the Borel and integral Novikov Conjectures for certain classes of groups, and applications of controlled topology to differential geometry. There are a number of potential applications for this work. The new spaces being studied have the global properties of manifolds modelled on euclidean space, but have very different small-scale properties. Such spaces could be of interest to workers trying to reconcile large-scale and small-scale models of the universe. The Borel and Novikov Conjectures are conjectures involving the rigidity of certain classes of manifolds. Typical results say that manifolds which are "close together" in some topological sense are the same. This appears to be a naturally occurring instance of quantization in geometry and topology.

研究人员打算研究受控拓扑和差异几何学的许多主题。 这些最大的紧迫性关注的是研究人员J. Bryant，W。Mio和S. Weinberger发现的不可辨别的同源歧管。 简而言之，他希望研究这些空间，并了解歧管的经典理论在多大程度上延续到了这一新的空间。 这项研究与众所周知的Borel和Bing-Borsuk猜想密切相关。 涉及的其他问题包括将受控的伽玛手术应用于编成二拟构态二类，对某些类别组的Novikov猜想的拓扑嵌入的研究，以及将受控拓扑的应用于差异几何形状。 这项工作有许多潜在的应用程序。 所研究的新空间具有在欧几里得空间上建模的歧管的全局特性，但具有非常不同的小规模特性。 试图调和宇宙大规模和小规模模型的工人可能会感兴趣。 Borel和Novikov的猜想是涉及某些歧管类的刚性的猜想。 典型的结果表明，在某种拓扑意义上“近距离”的多种流形是相同的。 这似乎是几何和拓扑中量化的天然实例。