The Structure of Smooth 4-Manifolds

建议：DMS-9971667主要研究人员：Ronald J.Sternr摘要：PI和R.Fintushel最近构造的光滑4维流形表明，光滑4维流形比先前预期的要多得多。这些例子也表明，目前的技术，即Donaldson和Seiberg-Witten不变量，不足以提供光滑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，希望开始出现一个普遍的图景。首先，对(单连通)辛4-流形有一个猜想分类(直到爆破)；即它们是全纯Lefschetz铅笔沿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这项拟议研究的第一部分的重点。由Pi和R.Fintushel提出的一种非常有效的方法，它保留了光滑4-流形X的同胚型，它包含一个具有平凡法丛的同调本质环面，但改变了它的微分同胚型，即“纽结外科构造”.在这里，给定3-球面上的一个纽结K，得到一个4-流形X(K)同胚于X，且具有如下性质：1)X(非纽结)=X；2)如果X(K)与X(J)异同胚，则J和K的Alexander多项式相同。这项建议的第二部分是为了更好地理解纽结的同位素类在这个纽结构造中所起的作用。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完整的四维流形列表；即在欧几里得4-空间上局部建模的对象。这成了数学中一个重要的悬而未决的问题。奇怪的是，维度大于4的流形被很好地理解；然而，正是我们生活和操作的维度带来了重大的数学挑战。PI和R.Fintushel利用分析、几何和拓扑学中的困难技术，开发了手术技术和可行的拓扑策略，用于计算微分方程组(Yang-Mills和Seiberg Witten方程)的解，从而有效地区分四维流形。以前NSF支持的研究允许构建出人意料的大量四流形，这为所有猜想分类方案提供了反例。这个项目将更仔细地研究这些技术，并创造新的建筑，希望开始出现一个大致的图景。

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.