Splitting homotopy equivalences: Applications, calculations, foundations

分裂同伦等价：应用、计算、基础

• 批准号: 0904276
• 负责人: C. Bruce Hughes
• 金额: $ 10.35万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0904276&HistoricalAwards=false
• 关键词: Splitting; homotopy; equivalences; Applications; calculations

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The goal of this project is to deepen and enhance our understanding of geometric topology, that is, the homeomorphism classification of closed manifolds (of dimension greater than three) within a given homotopy type. This project focuses on splitting homotopy equivalences along a two-sided codimension-one submanifold. The obstructions to this ideal situation are given by the Waldhausen Nil-groups and the Cappell UNil-groups. Hence, from an algebraic viewpoint, one theme of the project is the calculation and foundation of the splitting obstruction groups in L-theory.On the other hand, from a topological viewpoint, splitting is a special case of the coarser decomposition into big, non-simply connected pieces. The principal investigator proposes the use of these non-simply connected metablocks to study the Four-dimensional Surgery Conjecture. These metablocks allow for the more flexible notion of classifying space BsubF/subΓ for families F of small subgroups of Γ. This approach would hybridize the techniques of three isolated communities that have sprung up in the past 25 years: controlled surgery, surgery on 4-manifolds with small fundamental group, and assembly maps for families.A manifold is a smooth shape, without any sharp edges or singularities. For example, a one-dimensional manifold is a curve, and a two-dimensional manifold is a surface. Three and four-dimensional manifolds occur in physics, such as general relativity. Higher-dimensional manifolds occur in string theory and also as configuration spaces for robot motion planning. The mathematical classification of manifolds is fundamental to understanding both the global structure of our universe and the hidden routes through which a closed physical system is connected to itself.

该奖项是根据2009年《美国复苏与再投资法》（公法111-5）资助的。该项目的目的是加深和增强我们对几何拓扑的理解，即给定同义类型中的封闭歧管（大于三个）的同态分类。该项目着重于沿双面的一定次数submanifold拆分同质验证。 Waldhausen nil groups和Cappell Unil-Groups给出了这种理想情况的障碍。因此，从代数角度来看，该项目的一个主题是计算和基础，在L理论中，从拓扑角度来看，分裂是一种特殊的情况，是将较粗略分解为大型，非轻微连接的零件。主要研究者提议使用这些非紧密连接的甲库岩研究四维手术猜想。这些metablocks允许对空间bsubf/sub＆gamma的更灵活的概念进行分类。适用于＆伽马小组的家庭f。这种方法将杂交过去25年中三个孤立的社区的技术：受控手术，4个具有小基本组的4个manifolds手术以及家庭的组装图。A歧管是平稳的形状，没有任何尖锐的边缘或奇异性。例如，一维歧管是曲线，二维歧管是表面。物理中发生了三维和四维流形，例如一般相对论。较高的歧管出现在字符串理论中，也是机器人运动计划的配置空间。歧管的数学分类对于理解我们宇宙的全局结构和封闭物理系统与自身连接的隐藏途径至关重要。