Surfaces and 4-Manifolds

曲面和 4 流形

• 批准号: 0605280
• 负责人: Vyacheslav Krushkal
• 金额: $ 11.13万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0605280&HistoricalAwards=false
• 关键词: Surfaces; Manifolds

A major part of this research project concerns the validity of the 4-dimensional topological surgery conjecture for "large" fundamental groups. The PI plans to use his recently developed theory of link groups of 4-manifolds to formulate an obstruction in the context of the A-B slice problem, a reformulation of the surgery conjecture. Besides developing new invariants of 4-manifolds, this program will clarify the low-dimensional nature of topological 4-manifolds. Another part of the project concerns the asymptotic behavior of quantum representations of mapping class groups. The goal, in particular, is to use the representations that are provided by (2+1) dimensional TQFTs to investigate the rigidity properties of mapping class groups.This project fits in the general framework of classifying the possible large scale shapes of objects that locally look like the usual Euclidean space. The classification of three- and four-dimensional shapes is a particularly important and challenging problem. The results to date on topological classification in dimension four have paralleled the developments in higher dimension. This research project will help to clarify whether the analogy with higher dimensions works in general, for shapes with "large" fundamental groups.

这个研究项目的一个主要部分涉及的有效性的四维拓扑手术猜想为“大”的基本群体。PI计划使用他最近开发的4-流形的链接群理论，在A-B切片问题的背景下制定一个障碍，重新制定手术猜想。除了发展新的4-流形的不变量，这个程序将阐明拓扑4-流形的低维性质。该项目的另一部分涉及映射类群的量子表示的渐近行为。我们的目标是利用（2+1）维TQFTs提供的表示来研究映射类群的刚性性质，这个项目适合于对局部看起来像通常的欧氏空间的物体的可能的大尺度形状进行分类的一般框架。三维和四维形状的分类是一个特别重要和具有挑战性的问题。目前关于四维拓扑分类的研究成果已经在高维拓扑分类中得到了发展。这个研究项目将有助于澄清与更高维度的类比是否在一般情况下适用于具有“大”基本群的形状。