The Structure of Smooth 4-Manifolds

提案：DMS-9971667主要研究者：罗纳德J.斯特恩摘要：最近由PI和R. Fintushel已经证明，有更多的光滑4-流形比以前预期的。这些例子也表明，目前的技术，即唐纳森和Seiberg-Witten不变量，是不足以提供一个分类的光滑4-流形。这项研究的目标是开发新的光滑4流形的结构，希望有一个一般的图片开始出现。 开始，有一个严格的分类（直到爆破）（单连通）辛4-流形;即它们是全纯Lefschetz束沿着全纯子流形的纤维和。令人惊讶的是，大多数辛4-流形的新构造都可以用这种方式得到。这一猜想的准确性将是本研究第一部分的重点。PI和R. Fintushel保留了光滑4-流形X的同胚类型，但改变了其同胚类型，称为“纽结手术构造”。这里，给定一个3-球面中的纽结K，则得到一个同胚于X的4-流形X（K），它具有以下性质：1）X（unknot）=X; 2）如果X（K）同胚于X（J），则J和K的亚历山大多项式相同。这个提议的第二部分是为了更好地理解节点的合痕类在这个节点构造中所起的作用。 包括时间在内，我们的世界是四维的，但我们的四维空间的大尺度结构仍然未知。基本的数学问题是提供一个四维流形的完整列表;即在欧几里得4-空间上局部建模的对象。这是数学中一个重要的未解决的问题。奇怪的是，大于4维的流形是非常好理解的;然而，正是我们生活和运作的维度提供了重大的数学挑战。使用困难的技术，从分析，几何和拓扑，PI和R。芬图舍尔已经开发了手术技术和可行的拓扑策略计数解决方案的微分方程（杨米尔斯和塞伯格维滕方程），有效地区分四维流形。在此之前，NSF支持的研究允许建设一个意想不到的大量的四流形提供反例的所有猜想分类方案。这个项目将更仔细地研究这些技术，并创造新的建筑，希望能开始出现一个总体的画面。

Proposal: DMS-9971667Principal Investigator: Ronald J. SternAbstract:The recent constructions of smooth 4 dimensional manifolds by the PI and R. Fintushel have shown out that there are many more smooth 4-manifolds than previously expected. These examples also indicate that the current technologies, i.e. the Donaldson and Seiberg-Witten invariants, are insufficient to provide a classification of smooth 4-manifolds. The goal of this research is to develop new constructions of smooth 4 manifolds in the hope that a general picture begins to emerge. To begin, there is a conjectured classification (up to blow-up) of (simply-connected) symplectic 4-manifolds; namely they are the fiber sums of holomorphic Lefschetz pencils along holomorphic submanifolds. Surprisingly, most of the new constructions of symplectic 4-manifolds can be shown to be obtained in this manner. The veracity of this conjecture will be the focus of the first part of this proposed research. A very effective method, introduced by the PI and R. Fintushel, which retains the homeomorphism type of a smooth 4-manifold X which contains a homologically essential torus with trivial normal bundle but alters its diffeomorphism type is the "knot surgery construction". Here, given a knot K in the 3-sphere, there results a 4-manifold X(K) homeomorphic to X with the following properties: 1) X(unknot)=X, and 2) if X(K) is diffeomorphic to X(J), then the Alexander polynomials of J and K are the same. The second part of this proposal is to better understand the role that the isotopy classes of knots play in this knot construction. With time included, our world is four dimensional, but the large-scale structure of our four-dimensional space is still unknown. The underlying mathematical issue is to provide a complete list of four-dimensional manifolds; i.e. objects which are locally modeled on Euclidean 4-space. This turns out to be an important unsolved problem in mathematics. Strangely enough, manifolds of dimension larger than 4 are very well understood; yet it is the dimension in which we live and operate that provides major mathematical challenges. Using difficult techniques from analysis, geometry, and topology, the PI and R. Fintushel have developed surgery techniques and workable topological strategies for counting solutions to differential equations (the Yang-Mills and Seiberg Witten equations) which effectively distinguish four-dimensional manifolds. Prior NSF supported research allowed for the construction of an unexpectedly large number of four-manifolds which provided counterexamples to all the conjecture classification schemes. This project will more closely examine these techniques and create new constructions in the hope that a general picture begins to emerge.