Smooth 4-Manifolds

低维拓扑的核心问题之一是光滑单连通4-流形的分类。在这个方向上，一个基本的问题是，是否任何允许光滑结构的拓扑单连通4-流形实际上允许无穷多个。PI之前与R·斯特恩合作开发了一种名为“逆向工程”的技术，可以用来解决这个问题。这项技术依赖于“模型流形”的存在，或者相当于发现了嵌入的空同源环面，可以在其上进行手术。直到最近，还没有明确的技术来寻找这样的环面。然而，在过去的一年里，R·斯特恩和P.I.发现了在标准4-流形中寻找这些环面的技术。这个建议解释了这是如何在(2个或更多)Cp^2的放大中实现的。所产生的嵌入的空同调环面可以被运算以创建无穷族的同胚但相互非微分同胚的4-流形。下一项任务是进行Floer同调和相对的Seiberg-Witten计算，以便我们的技术可以在最一般的形式下变得有用。这项提议的更广泛影响将解决数学和理论物理之间的关系，研究生和本科生在拓扑学方面的机会，以及博士后研究员的职业发展。四维几何和拓扑学与物理学有着非常密切的联系。例如，提出者将研究辛流形的地理问题，这些辛流形已经被证明通过“超保角简单类型”的概念影响物理。任何关于Seiberg-Witten理论的一般性结果都有可能与物理学界产生互动，这将是一个基本的担忧。寻找奇异的四维流形已经受到理论物理界的欢迎，这方面的新结果也将有助于加强这种关系。这项提案的另一个基本目标是解决数学中的“流水线问题”。我们的方法是让学生和年轻的数学家研究有趣的问题。这项建议的一个关键方面是开发研究生和高级本科生可以接触到的问题。它提出的问题将是适合学生的论文问题和密歇根州立大学博士后研究员的研究项目。并讨论了适合本科生学习的计算问题。

One of the central problems of low-dimensional topology is the classification of smooth simply connected 4-manifolds. In this direction, a basic question is whether any topological simply connected 4-manifold which admits a smooth structure in fact admits infinitely many. The PI's previous work with R. Stern developed a technique called 'reverse engineering' which can be used to attack this problem. This technique depends on the existence of 'model manifolds' or equivalently the discovery of embedded nullhomologous tori on which surgery can be performed. Until recently, there have been no explicit techniques for finding such tori. However, in the past year, R. Stern and the P.I. have discovered techniques for finding these tori in standard 4-manifolds. This proposal explains how this is done in (2 or more) blowups of CP^2. The embedded nullhomologous tori which are produced can be surgered to create infinite families of homeomorphic but mutually nondiffeomorphic 4-manifolds. The next task is to carry out Floer homology and relative Seiberg-Witten calculations so that our technique can become useful in its most general form.The broader impact of this proposal will address the relationship between mathematics and theoretical physics, opportunities for graduate and undergraduate students in topology, and career development of postdoctoral fellows. Four-dimensional geometry and topology has very close ties to physics. For example, the proposer will study geography problems for symplectic manifolds which have been shown to impact physics via the notion of 'superconformal simple type'. Any general results concerning Seiberg-Witten theory hold the prospect of engendering interaction with the physics community, and this will be a basic concern. The search for exotic 4-manifolds has been embraced by the theoretical physics community, and new results on this front will also serve to enhance this relationship. Another basic goal of this proposal is to address 'pipeline issues' in mathematics. Our approach is to to get students and young mathematicians working on interesting problems. A key aspect of this proposal is the development of problems which are accessible to graduate and advanced undergraduate students. It presents problems which will be suitable thesis problems for students and research projects for postdoctoral fellows at Michigan State. It also discusses computational problems which are suitable for advanced undergraduate students.