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微分几何中的许多主题。 其中最紧迫的是研究者 J. Bryant、W. Mio 和 S. Weinberger 发现的不可解析的同源流形。 简而言之，他希望研究这些空间，看看经典流形理论在多大程度上适用于这种新的空间类别。 这项研究与著名的 Borel 和 Bing-Borsuk 猜想密切相关。 涉及的其他问题包括受控伽玛手术在余维二拓扑嵌入研究中的应用、某些群类的波雷尔猜想和积分诺维科夫猜想，以及受控拓扑在微分几何中的应用。 这项工作有许多潜在的应用。 正在研究的新空间具有以欧几里得空间为模型的流形的全局属性，但具有非常不同的小尺度属性。 试图协调宇宙大尺度和小尺度模型的工作者可能会对这样的空间感兴趣。 博雷尔猜想和诺维科夫猜想是涉及某些流形类别的刚性的猜想。 典型的结果表明，在某种拓扑意义上“靠近”的流形是相同的。 这似乎是几何和拓扑中自然发生的量化实例。