Some Problems in Geometric Topology

几何拓扑中的一些问题

• 批准号: 9971296
• 负责人: Steven Ferry
• 金额: $ 8.38万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971296&HistoricalAwards=false
• 关键词: Some; Problems; Geometric; Topology

Proposal: DMS-9971296PI: Steve FerryAbstract: Our main effort will go into using the methods of controlled surgery theory to study of the geometric properties of the exotic homology manifolds discovered by Bryant, Mio, Weinberger, and the author. The most pressing questions in this area concern the existence of exotic homology manifolds which are homogeneous and the existence of exotic homology manifolds which have contractible universal covers. Construction of an exotic homogeneous homology manifold would give a counterexample to the well-known Bing-Borsuk conjecture, while an exotic aspherical homology manifold would give a counterexample to either the integral Novikov conjecture or the well-known conjecture that every Poincare duality group is the fundamental group of some aspherical manifold.We also plan to use methods from controlled topology to study the Gromov-Lawson conjecture and to study the classification of high-dimensional wild topological embeddings in codimension two.

摘要：我们的主要工作将是利用控制手术理论的方法来研究Bryant、Mio、Weinberger和作者发现的奇异同调流形的几何性质。这一领域中最紧迫的问题是齐次的奇异同调流形的存在性和具有可收缩泛盖的奇异同调流形的存在性。异域齐次同调流形的构造可以给出众所周知的宾-博苏克猜想的反例，而异域非球同调流形则可以给出积分诺维科夫猜想或众所周知的“每一个庞加莱对偶群都是某些非球流形的基群”猜想的反例。我们还计划使用控制拓扑的方法来研究Gromov-Lawson猜想，并研究高维野生拓扑嵌入在余维2中的分类。