Algebraic and Geometric Topology

代数和几何拓扑

• 批准号: 0808659
• 负责人: James Davis
• 金额: $ 14.95万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-15 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0808659&HistoricalAwards=false
• 关键词: Algebraic; Geometric; Topology

There are a web of conjectures about the surgery theoretic classification of high-dimensional manifolds, culminating in the Farrell-Jones Conjectures in K- and L-theory. The conjectures are closely related to splitting manifolds along codimension one submanifolds. This splitting problem is in turn related to nil groups in algebraic K- and L-theory. This project has several different aspects. One is to solve the connected sum problem -- when is a manifold which is homotopy equivalent to a connected sum itself a connected sum? Another is to provide a strengthening of the Farrell- Jones Conjecture in L-theory, similar to that achieved by Davis-Khan- Ranicki in K-theory. Yet another is to classify involutions on a torus, using the computation of the L-theory of the infinite dihedral group and the proof of the Farrell-Jones Conjecture in L-theory for crystallographic groups, joint with Connolly. There are some middle and low dimensions problems too, involving mapping tori of self- homotopy equivalences of lens spaces and a certain aspect of link concordance.The goal, as usual in geometric topology, is to use a variety of algebraic, geometric, and analytic techniques to find and compute invariants for classification. Geometric topology is the study of manifolds. An n-dimensional manifold is a set of points locally modeled on n-dimensional Euclidean space. For instance, a 2-manifold is a surface and looks like a plane near each point. Many physical phenomenon are represented by manifolds, and as such, understanding the global structure of a manifold, and what possible manifolds exist, is fundamental to the sciences, as well as to mathematics. Manifold theory connects with most areas of mathematics, as well as with physical phenomena such as cosmology, string theory, and classical and quantum mechanics.

关于高维歧管的手术理论分类，有一个猜想的网络，最终在K-和L理论的Farrell-Jones猜想中达到了最终。 这些猜想与沿着沿着一个子序列的分裂歧管密切相关。 这个分裂问题又与代数K和L理论中的零组有关。 该项目有几个不同的方面。 一种是解决连接的总和问题 - 何时是同型等同于连接总和本身的歧管？ 另一个是在L理论中加强Farrell-Jones的猜想，类似于Davis-Khan-Ranicki在K理论中所实现的。 另一个是，使用无限二面基团的L理论和与Connolly关节的L理论中的Farrell-Jones猜想的证据，并证明了Farrell-Jones的猜想的证明，将其分类。 也存在一些中层和低维问题，涉及镜头空间的自相同等效的映射摩托图和链接一致性的某些方面。与几何拓扑相像，该目标是使用各种代数，几何学和分析技术来查找和计算不可传剂进行分类。几何拓扑是对流形的研究。 N维歧管是一组在N维欧几里得空间上进行局部建模的点。例如，一个2个manifold是一个表面，看起来像每个点附近的平面。许多物理现象以多种形式表示，因此，了解流形的全球结构以及存在的歧管，对科学以及数学的基础是基础。 歧管理论与数学的大多数领域以及诸如宇宙学，弦理论以及经典和量子力学等物理现象相关。