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年美国复苏和再投资法案（公法111-5）资助的。这个项目的目标是加深和增强我们对几何拓扑的理解，即给定同伦类型内闭流形（维数大于3）的同胚分类。这个计画的重点是沿着沿着一个双侧余维1子流形分裂同伦等价。Waldhausen Nil-群和Cappell Unil-群给这一理想情形设置了障碍。因此，从代数的角度来看，该项目的一个主题是L-理论中分裂障碍群的计算和基础。另一方面，从拓扑的角度来看，分裂是粗分解为大的非单连通块的特殊情况。主要研究者建议使用这些非单连通元块来研究四维手术猜想。这些元块允许更灵活的概念分类空间BsubF/sub Gamma;为家庭F的小子群的Gamma;。这种方法将混合在过去25年中出现的三个孤立社区的技术：控制手术，具有小基本群的4-流形上的手术，以及家庭的装配映射。例如，一维流形是曲线，二维流形是曲面。三维和四维流形出现在物理学中，如广义相对论。高维流形出现在弦理论中，也作为机器人运动规划的配置空间。流形的数学分类对于理解我们宇宙的整体结构和封闭物理系统与自身相连的隐藏路线都是至关重要的。