Mathematical Sciences: The Structure of Smooth 4-Manifolds

数学科学：光滑 4 流形的结构

• 批准号: 9626330
• 负责人: Ronald Stern
• 金额: $ 6.42万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626330&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Structure; Smooth; Manifolds

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. ***

小行星9626330 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年代初，主要研究者开发并使用了困难的技术 和其他人证明尖锐的声明的计算和结构的唐纳森不变量（这是在1984年推出）。 1994年10月引进的出色构思和更容易处理塞伯格-威滕不变量已明确的作用，这些唐纳森不变量的研究顺利4流形。 现在尘埃已经开始沉淀，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，大多数的问题和有关的拓扑结构的光滑4流形目前在1994年9月仍然开放。 特别是，我们仍然不知道如何分类单连通光滑4-流形。 该项目的第一部分是确定基本的构建块，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。 本文的重点是，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，每个复曲面集上都有一束曲线，X可通过以下三种运算得到：（1）沿束的一般纤维沿着纤维和，（2）沿零平方环面沿着纤维和，（3）沿零平方环面的局部纤维和，（4）沿零平方环面的局部纤维和，（5）沿零平方环面的局部纤维和，（6）沿零平方环面的局部纤维和。（3）对零平方环面进行拓扑对数变换？ 本项目的第二部分是确定Seiberg-Witten和唐纳森不变量的有效性。 特别地，复杂曲面的变形类型决定了它的自同构类型吗？ 这个项目将开始集中在明确的例子（Horikawa表面），具有相同的Seiberg-Witten和唐纳森不变量，已知是变形不等价的（即，与复流形不同），但不知道是复纯的。 塞伯格-威滕和唐纳森不变量的一个令人不安的特征是，它们只定义在其签名和欧拉特征之和可被4的奇数倍整除的流形上。 这个项目的最后一部分将集中在那些简单连接的4-流形，其签名和欧拉特征线的总和可被4的偶数倍整除，并且几乎没有什么是已知的。 在底部，这个项目集中在4维欧几里得空间上局部建模的对象的分类上，并且可以对实值函数进行微分。 这些物体就是所谓的光滑四维流形。 其基本技术是抽取代数拓扑数据 从这些部分流形上的解空间 理论物理中出现的微分方程，即，使用规范理论技术。 众所周知，人们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对这些对象执行的操作列表，以获得任何光滑的4-流形。 本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强有力的联系，部分原因是，如上所述， 所涉及的拓扑工具来自规范理论，即，从量子 力学，部分是因为任何光洒在4流形承担 我们对相对论四维时空的理解。 ***