3-manifolds and Floer homologies

3-流形和Floer同源性

• 批准号: 0245323
• 负责人: Weiping Li
• 金额: $ 6.48万
• 依托单位: Oklahoma State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245323&HistoricalAwards=false
• 关键词: manifolds; Floer; homologies

The problems addressed in this project are in the area of Topology.The main theme is to study the fundamental and important properties of the gauge-theoretic/symplectic Floer homology, using invariants and methods from 3-manifold topology and symplectic topology. The investigator studied the semi-infinity of the instanton (gauge theoretic) Floer homology and the intrinsic dependence of the monopole(Seiberg-Witten-Floer) homology of (homology)three-spheres. The major part of the project is to study the interactive relation between the symplectic Floer homology and the classical 3-manifolds, and therelation between the Floer cohomology and the semi-infinite cohomology, and the Seiberg-Witten-Floer theory intertwining the instanton and monopole results. The other part of this project is to study intrinsic properties for the invariants of largerclasses of three-spheres.It is a fundamental aim to investigate the change of the newinvariants under certain topological operations. The project will integrate the interactions among the instanton theory, the monopole theory, the symplectic Floertheory and the semi-infinite cohomology of infinite-dimensional Lie algebras.A three-dimensional manifold is a space where a nearsighted person sees a standard three-dimensional space everywhere.(Homology) three-spheres are those three-manifolds that one cannot tell from the standard three-sphere by using the usual topological tools. A symplectic manifold is an even-dimensional space with a special(symplectic) structure. For instance, the phase space of a mechanical system is a symplectic space. Such a symplectic structure is rich in mathematics and physics, and is canonical from any nearby region. This shows the subtlety and the complexity of the world we live in. Any local information is no longer useful, the global behavior and the global (topological) invariants are the pivot. The refined invariants studied in this project are intended to distinguish manifolds from the aspects of mathematical physics, and to intertwine various quantum field theories from the aspect of mathematics. Thus it is extremely valuable to investigate the relation between the symplectic Floer cohomology and the semi-infinite cohomology, as well as the interactive link between the instanton homology and the monopole homology. This project addresses some of the most fundamental problems in this subject.

该项目中解决的问题是在拓扑领域中。主题是使用3个manifold拓扑和象征性拓扑结构的不变性和方法研究量规理论/符号浮子同源性的基本和重要特性。研究人员研究了激体（量规理论）的浮动浮子同源性的半含量以及单台（Seiberg-witten-loplloer）（同源性）三范围同源性的固有依赖性。该项目的主要部分是研究象征性的浮子同源性与经典的3个manifolds之间的互动关系，以及浮子共同体学与半偶然的共同体之间的交互关系，以及塞伯格 - 纽约 - 纽约 - 韦特 - 温培养徒理论结果。该项目的另一部分是研究三范围较大级别的较大级别的固有属性。这是调查在某些拓扑操作下新知识分子的变化的基本目的。该项目将在intsanton理论，单极理论，符号浮点理论和无限二维谎言代数的半偶然共同体之间整合相互作用。到处都是。（同源性）三个角度是通过使用常规拓扑工具从标准的三个球体中无法从标准的三个球体中分辨出来的三个小小的。符号歧管是具有特殊（符号）结构的均匀空间。例如，机械系统的相空间是一个符号空间。这种符号结构富含数学和物理学，并且来自附近任何地区的规范。这表明了我们生活的世界的微妙和复杂性。任何本地信息都不再有用，全球行为和全球（拓扑）不变性是枢纽。该项目中研究的精致不变性旨在区分歧管与数学物理学的各个方面，并将各种量子场理论与数学方面交织在一起。因此，研究符合性浮子的共同体与半无限共同体之间的关系以及Instanton同源性与Monopole同源性之间的互动联系非常有价值。该项目解决了该主题中一些最根本的问题。