Mathematical Sciences: Studies in Geometric Topology

数学科学：几何拓扑研究

• 批准号: 9207973
• 负责人: Frank Quinn
• 金额: $ 16.47万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-01 至 1996-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9207973&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Studies; Geometric; Topology

The eventual goal of Quinn's program is to develop invariants of smooth compact 4-manifolds which, for example, could detect counterexamples to the smooth 4-dimensional Poincare conjecture. The first part of the program (this project) will be devoted primarily to study of a model problem, the Andrews-Curtis conjecture on 2-complexes. The invariants under consideration are "topological quantum field theories" in the sense of Atiyah and Witten. The Poincare conjecture asserts that any manifold that has certain simple topological properties in common with a topological sphere must actually be a topological sphere. It was originally stated for three-dimensional manifolds around the turn of the century and subsequently generalized to all higher dimensions. Curiously, all these generalizations have now been settled (in the affirmative), but the original case for three-dimensional spheres still resists all assaults. The foregoing remarks apply to topological manifolds. The situation is slightly different for smooth manifolds. Here both the three- and the four-dimensional versions of the Poincare conjecture remain open. What Quinn is hoping to do is to settle the four-dimensional case, using methods inspired by a mathematical formulation of quantum field theory.

Quinn计划的最终目标是开发光滑紧致4维流形的不变量，例如，它可以检测光滑4维Poincare猜想的反例。程序的第一部分(本项目)将主要致力于研究一个模型问题，即关于2-复形的Andrews-Curtis猜想。所考虑的不变量是Atiyah和Witten意义上的“拓扑量子场论”。Poincare猜想断言，任何具有与拓扑球相同的某些简单拓扑性质的流形实际上一定是拓扑球。它最初是在世纪之交对三维流形提出的，后来推广到所有更高的维度。奇怪的是，所有这些概括现在都得到了解决(肯定的)，但三维球体的原始情况仍然抵抗所有的攻击。上面的评论适用于拓扑流形。对于光滑的流形，情况略有不同。在这里，庞加莱猜想的三维和四维版本仍然是开放的。奎恩希望做的是解决四维问题，使用受量子场论数学公式启发的方法。