Mathematical Sciences: The Structure of Smooth 4-Manifolds

这个项目的目标是更好地理解单连通光滑四维流形。在20世纪90年代早期，主要研究者和其他人开发并使用了一些困难的技术来证明关于唐纳森不变量的计算和结构的尖锐陈述（1984年引入）。1994年10月，Seiberg-Witten不变量的引入使得这些Donaldson不变量在光滑4流形研究中的作用变得清晰起来。现在尘埃已经开始尘埃落定，是时候回顾一下我们对光滑4流形的理解了。令人惊讶的是，1994年9月提出的与光滑4流形拓扑有关的大多数问题仍然没有解决。特别是，我们仍然不知道如何对单连通光滑4流形进行分类。该项目的第一部分是确定基本构建块，并确定在这些构建块上执行的操作，以恢复任何给定的光滑4流形。作为焦点，给定一个单连通不可约光滑4流形X，是否存在一个有限的复曲面集合，每个复曲面上都有一束曲线，X通过以下三种运算得到：(1)沿着铅笔的一般纤维求和；(2)沿零平方环面局部纤维和；(3)对零平方环面进行拓扑对数变换？这个项目的第二部分是确定Seiberg-Witten和Donaldson不变量的有效性。特别是，一个复杂表面的变形类型是否决定了它的微分同形类型？该项目将开始关注具有相同Seiberg-Witten和Donaldson不变量的显式例子（Horikawa曲面），已知是变形不等价的（即与复杂流形不同），但不知道是微分同构的。Seiberg-Witten和Donaldson不变量的一个令人不安的特征是，它们仅定义于其特征和欧拉特征之和可被4的奇数倍整除的流形。这个项目的最后一部分将集中在那些单连通的4流形上，它们的签名和欧拉特征的总和可以被4的偶数倍数整除，并且实际上对它们一无所知。在底层，这个项目的中心是对象的分类，这些对象局部建模于四维欧几里德空间，在此基础上可以对实值函数进行微分计算。这些物体就是所谓的光滑四维流形。其基本技术是从理论物理中出现的偏微分方程流形的解空间中提取代数拓扑数据，即使用规范理论技术。众所周知，不能期望简单地给出光滑4流形的完整列表。然而，人们可以期望确定可分类的基本对象和可以在这些对象上执行的操作列表，以获得任何光滑的4流形。这个项目的目标是提供这些构建块和组装规则。此外，光滑4流形的已知不变量和未来不变量在这些操作下的行为应该很容易确定。该项目的成功将在拓扑学和理论物理学之间建立另一个紧密的联系，部分原因是，如上所述，所涉及的拓扑工具来自规范理论，即量子力学，部分原因是关于4流形的任何启示都与我们对相对论的4维时空的理解有关。＊＊＊

9626330 Stern The goal of this project is better to understand simply-connected smooth four-dimensional manifolds. In the early 1990's difficult techniques were developed and used by the principal investigator and others to prove sharp statements about the computations and structure of the Donaldson invariants (which were introduced in 1984). The October 1994 introduction of the brilliantly conceived and more easily handled Seiberg-Witten invariants has made clear the role of these Donaldson invariants in the study of smooth 4-manifolds. Now that the dust has begun to settle, it is time to review our understanding of smooth 4-manifolds. Surprisingly, most of the questions and problems related to the topology of smooth 4-manifolds present in September 1994 remain open. In particular, we still do not know how to classify simply-connected smooth 4-manifolds. The first part of this project is to determine the fundamental building blocks and to determine the operations performed on these building blocks to recover any given smooth 4-manifold. As a focal point, given a simply-connected irreducible smooth 4-manifold X, does there exist a finite collection of complex surfaces, each of which carries a pencil of curves from which X is obtained by using the following three operations: (1) fiber sum along a general fiber of the pencils; (2) local fiber sum along tori of square zero; (3) performing a topological log transform on tori of square zero? The second part of this project is to determine the effectiveness of the Seiberg-Witten and Donaldson invariants. In particular, does the deformation type of a complex surface determine its diffeomorphism type? This project will begin to focus on explicit examples (the Horikawa surfaces) that have the same Seiberg-Witten and Donaldson invariants, are known to be deformation inequivalent (i.e., not the same as complex manifolds), but are not known to be diffeomorphic. One disturbing feature of the Seiberg-Witten and Donaldson invariants is that they are defined only for manifolds for which the sum of its signature and Euler characteristic is divisible by an odd multiple of 4. The final part of this project will focus on those simply-connected 4-manifolds for which the sum of the signature and Euler characteristic is divisible by an even multiple of 4, and about which virtually nothing is known. At bottom, this project centers on the classification of objects that are locally modeled on 4-dimensional Euclidean space and upon which one can do differential calculus for real-valued functions. These objects are the so-called smooth four-dimensional manifolds. The basic technique is to extract algebraic topological data from the solution space on these manifolds of partial differential equations that arise in theoretical physics, i.e., to use gauge-theoretic techniques. It is known that one cannot expect simply to give a complete list of smooth 4-manifolds. However, one can expect to determine classifiable fundamental objects and a list of operations that one can perform on these objects in order to obtain any smooth 4-manifold. It is the goal of this project to provide these building blocks and assembly rules. Further, the behavior under these operations of known and future invariants of smooth 4-manifolds should be easily determined. Success of the project will create another strong tie between topology and theoretical physics, partly because, as noted above, the topological tools involved come from gauge theory, i.e., from quantum mechanics, and partly because any light shed on 4-manifolds bears on our understanding of the 4-dimensional space-time of relativity theory. ***