Smooth 4-Manifolds

Michigan State University, East Lansing MI

Investigators

Abstract

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.