Mathematical Sciences: Smooth 4-Manifolds and Their Donaldson Series

数学科学：光滑 4 流形及其唐纳森级数

• 批准号: 9401032
• 负责人: Ronald Fintushel
• 金额: $ 11.7万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1998-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401032&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Smooth; Manifolds; Their

9401032 Fintushel The main concern of this project is to study smooth simply connected 4-manifolds by using invariants obtained from gauge theory. This has three parts. The first is to understand the structure theory of the invariants themselves. For manifolds of simple type, this has been accomplished by Kronheimer and Mrowka and, using completely different techniques, by the investigator and R. Stern. The investigator and R. Stern have also proved a blowup formula for the Donaldson invariant of a general 4-manifold, and one expects that this formula, together with the techniques used in the simple-type case, will lead to a general structure theory. The second part of this project is the calculation of Donaldson invariants. The investigator intends to use his technique of 'rational blowdowns' to provide calculations. This involves giving a general formula for the Donaldson invariant of a rational blowdown, which has essentially been done by the investigator and R. Stern (some details remain to be checked) and which leads, e.g., to the calculation of complete Donaldson invariants for all elliptic surfaces and many other interesting examples. The final part of the project is to produce new examples to which these invariants can be applied in order to find some order in the classification theory. Four-dimensional manifolds are spaces which locally have the structure of 4-dimensional Euclidean space (e.g. 3 'spatial' and one 'time' dimension). These manifolds play a central role in topology, for in 4 dimensions both the powerful techniques of higher dimensional surgery and the more special and intricate techniques of lower dimensions break down. This is, of course, also the most physically relevant situation, and there is a vast literature in theoretical physics related to 4-dimensional topology. For many years 4-manifolds were studied by applying techniques from both higher and lower dimensions, and the results were mostly unsatisfactory. In 1 984, Simon Donaldson introduced a powerful technique stemming from the interplay between 4-dimensional geometry and theoretical physics via the Yang-Mills equation. Donaldson's techniques have had startling consequences, and a school of topologists and geometers has formed in the hope of using these techniques to understand 4-dimensional topology completely. (That this is not too far removed from the physics which helped spawn the theory is evidenced by the many contributions by physicists such as Witten.) In the last year, especially, progress has been very rapid. The investigator hopes to make further contributions both to the understanding of the Donaldson invariant and to the classification of 4-dimensional manifolds. ***

本课题的主要目的是利用规范论的不变量研究光滑单连通4流形。这有三个部分。首先是理解不变量本身的结构理论。对于简单型流形，Kronheimer和Mrowka已经完成了这一点，研究者和R. Stern使用了完全不同的技术。研究者和R. Stern还证明了一般4流形的Donaldson不变量的放大公式，人们期望该公式与在简单类型情况下使用的技术一起，将导致一般结构理论。本课题的第二部分是唐纳森不变量的计算。研究者打算使用他的“理性吹落”技术来提供计算。这涉及到给出一个理性下倾的Donaldson不变量的一般公式，这基本上是由研究者和R. Stern完成的（一些细节仍有待检查），并且导致，例如，计算所有椭圆曲面的完全Donaldson不变量和许多其他有趣的例子。项目的最后一部分是产生新的例子，这些不变量可以应用到这些例子中，以便在分类理论中找到一些顺序。四维流形是局部具有四维欧几里得空间结构的空间（例如3维“空间”和1维“时间”）。这些流形在拓扑学中起着核心作用，因为在4维中，高维手术的强大技术和更特殊、更复杂的低维技术都失效了。当然，这也是物理上最相关的情况，在理论物理中有大量与四维拓扑相关的文献。多年来，人们从高维和低维两方面对四流形进行了研究，结果大多不令人满意。1984年，西蒙·唐纳森通过杨-米尔斯方程引入了一种源于四维几何和理论物理相互作用的强大技术。唐纳森的技术产生了惊人的结果，一个由拓扑学家和几何学家组成的学派已经形成，希望利用这些技术来完全理解四维拓扑。（这与催生这一理论的物理学并没有太大的差距，这一点可以从威滕等物理学家的许多贡献中得到证明。）特别是去年，进展非常迅速。研究者希望对Donaldson不变量的理解和四维流形的分类做出进一步的贡献。＊＊＊