Splitting homotopy equivalences: Applications, calculations, foundations

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，即在给定同伦类型内闭流形（维数大于3）的同胚分类。本课题主要研究沿一个双边余维一子流形的同伦等价的分裂问题。这种理想情况的障碍是由Waldhausen nil群和Cappell uni群造成的。因此，从代数的角度来看，本课题的一个主题是l理论中分裂障碍群的计算和基础。另一方面，从拓扑学的观点来看，分裂是将粗分解成大的、非单连通的块的一种特殊情况。首席研究员建议使用这些非简单连接的元块来研究四维外科猜想。这些元块允许对空间BsubF/ subgamma进行更灵活的分类。对于&amp； γ；的小亚群的科F。这种方法将融合过去25年里涌现出来的三个孤立社区的技术：控制手术、对具有小基本群的4-流形的手术，以及针对家庭的组装图。流形是光滑的形状，没有任何尖锐的边缘或奇点。例如，一维流形是一条曲线，二维流形是一个曲面。三维和四维流形出现在物理学中，比如广义相对论。高维流形出现在弦理论中，也作为机器人运动规划的位形空间。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连接自身的隐藏路径至关重要。

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.