Smooth 4-Manifolds

低维拓扑学的中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，任何拓扑单连通4-流形，它承认一个光滑的结构，实际上是否允许无限多。PI以前的工作与R。斯特恩开发了一种称为“逆向工程”的技术，可以用来解决这个问题。这种技术依赖于存在的“模型流形”或等价的发现嵌入nullhomologous环面上可以进行手术。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.斯特恩和私家侦探已经发现了在标准四维流形中找到这些环面的技术。这个建议解释了如何在CP ^2的（2个或更多）爆破中做到这一点。所产生的嵌入的零同调环面可以被surgered以创建同胚但互不同胚的无穷族4-流形。下一个任务是进行Floer同源性和相对的Seiberg-Witten计算，使我们的技术可以成为有用的，在其最一般的form.The更广泛的影响，这一建议将解决数学和理论物理之间的关系，研究生和本科生的拓扑学的机会，和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，提议者将研究辛流形的地理问题，这些问题已被证明通过“超共形简单型”的概念影响物理学。任何关于Seiberg-Witten理论的一般性结果都具有与物理学界产生相互作用的前景，这将是一个基本的问题。对奇异四维流形的研究已经被理论物理学界所接受，这方面的新结果也将有助于加强这种关系。这个建议的另一个基本目标是解决数学中的“管道问题”。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研究生和高级本科生可以访问的问题的发展。它提出的问题，这将是合适的论文问题的学生和博士后研究项目在密歇根州立大学。还讨论了适合本科高年级学生的计算问题。

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.