Mathematical Sciences: Smooth 4-Manifolds and Their Donaldson Series

小行星9401032 本计画的主要内容是利用规范理论所得到的不变量来研究光滑单连通4-流形。 这有三个部分。 第一是理解不变量本身的结构理论。 对于简单型流形，这已经由Kronheimer和Mrowka完成，并使用完全不同的技术，由研究者和R。胸骨切开术组 研究者和R. Stern还证明了一般4-流形的唐纳森不变量的爆破公式，人们期望这个公式，连同在简单型情况下使用的技巧，将导致一个一般的结构理论。 本项目的第二部分是唐纳森不变量的计算。 调查人员打算使用他的“合理排污”技术来提供计算。 这涉及到给出一个合理排污的唐纳森不变量的一般公式，这基本上已经由研究者和R。斯特恩（一些细节仍有待检查），并导致，例如，到计算所有椭圆曲面的完全唐纳森不变量和许多其他有趣的例子。 该项目的最后一部分是产生新的例子，这些不变量可以应用于以找到一些分类理论的顺序。 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个“空间”维度和1个“时间”维度）。 这些流形在拓扑学中起着核心作用，因为在4维空间中，高维外科手术的强大技术和低维空间的更特殊和复杂的技术都失败了。 当然，这也是物理上最相关的情况，理论物理学中有大量文献与四维拓扑学有关。 多年来，人们从高维和低维两个维度研究了4-流形，但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强大的技术，这种技术源于四维几何和理论物理之间的相互作用。 唐纳森的技术产生了惊人的后果，和学校的拓扑学家和geometers已经形成了希望利用这些技术来了解四维拓扑完全。 (That这与帮助产生该理论的物理学并不太遥远，维滕等物理学家的许多贡献证明了这一点。） 特别是去年，进展非常迅速。 研究者希望能对唐纳森不变量的理解和四维流形的分类做出进一步的贡献。 ***

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. ***