The Structure of Smooth 4-Manifolds

提案：DMS-9971667 首席研究员：Ronald J. Stern 摘要：PI 和 R. Fintushel 最近构建的光滑 4 维流形表明，光滑 4 维流形的数量比之前预期的要多得多。这些例子还表明，当前的技术，即 Donaldson 和 Seiberg-Witten 不变量，不足以提供平滑 4 流形的分类。这项研究的目标是开发光滑 4 流形的新结构，希望开始出现一个总体情况。 首先，有一个（单连通）辛 4 流形的推测分类（直至爆炸）；也就是说，它们是全纯莱夫谢茨铅笔沿全纯子流形的纤维总和。令人惊讶的是，大多数新的辛 4 流形结构都可以通过这种方式获得。这个猜想的准确性将是本研究第一部分的重点。 PI 和 R. Fintushel 提出的一种非常有效的方法是“结手术构造”，它保留了光滑 4 流形 X 的同胚类型，其中包含与平凡正束的同调必要环面，但改变了其微分同胚类型。这里，给定 3-球体中的结 K，得到与 X 同胚的 4 流形 X(K)，具有以下属性：1) X(unknot)=X，并且 2) 如果 X(K) 与 X(J) 微分同胚，则 J 和 K 的亚历山大多项式相同。该提案的第二部分是为了更好地理解结的同位素类别在该结构造中所起的作用。 加上时间，我们的世界是四维的，但四维空间的大尺度结构仍然未知。根本的数学问题是提供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。事实证明，这是数学中一个重要的未解决问题。奇怪的是，维度大于 4 的流形很好理解；然而，正是我们生活和运作的维度带来了重大的数学挑战。使用分析、几何和拓扑方面的困难技术，PI 和 R. Fintushel 开发了手术技术和可行的拓扑策略，用于计算微分方程（Yang-Mills 和 Seiberg Witten 方程）的解，从而有效地区分四维流形。之前 NSF 支持的研究允许构建出乎意料的大量四流形，为所有猜想分类方案提供了反例。该项目将更仔细地研究这些技术并创建新的结构，希望开始出现总体情况。

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.