Low-Dimensional Topology and Fundamental Groups

低维拓扑和基本群

• 批准号: 0604310
• 负责人: Paul Kirk
• 金额: $ 24.81万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604310&HistoricalAwards=false
• 关键词: Low; Dimensional; Topology; Fundamental; Groups

The PI will study problems in low-dimensional topology centered around the impact of the fundamental group on the geometric properties of a low dimensional manifold. Three general directions of investigation will be taken: an asymptotic geography problem for symplectic and smooth 4-manifolds with prescribed fundamental groups, surgery properties of the SU(3) Casson invariant, and the homotopy properties of the Atiyah-Patodi-Singer invariant associated to a unitary representation of the fundamental group of a manifold.The subject of topology is the study of properties of space that are unchanged under deformation or stretching. The fundamental group is a simple to define but very powerful mathematical object that measured how circles can be deformed in different spaces. Themost challenging spaces to study are the "low dimensional manifolds", i.e. those which have dimension 3 or 4, like the universe we live in. The PI proposes to study these using the fundamental group and its relation to the geometry and analysis of of a space.

PI将研究低维拓扑中的问题， 关于基本群对低维流形几何性质的影响。将采取三个一般调查方向：具有预定基本群的辛和光滑4-流形的渐近地理问题，SU（3）Casson不变量的外科手术性质，以及Atiyah-Patodi-与流形的基本群的酉表示有关的辛格不变量。拓扑学的主题是研究空间的不变性质在变形或拉伸下。基本群是一个简单的定义，但非常强大的数学对象，测量如何圆可以在不同的空间变形。最具挑战性的空间研究是“低维流形”，即那些有3或4维，就像我们生活的宇宙。 PI建议使用基本群及其与空间的几何和分析的关系来研究这些。