Gauge Theory and Geometry in Dimensions Three and Four

摘要奖：DMS-0100771主要研究人员：Peter B.Kronheimer这个项目的目的是将规范理论技术应用到三维流形的研究中。这位主要的调查者建议研究Floer同调和几何中密切相关的领域，并希望阐明规范理论对三维拓扑问题的适用性。特别地，希望在单极子方程定义的三维流形的Floer同调与瞬子方程定义的三维流形的Floer同调之间建立一种关系。(这些方程在四维中分别导致了四维流形的Seiberg-Wittenant不变量和Donaldson不变量，并在过去的二十年里在四维微分拓扑学中产生了大量的结果。)第一个目的是证明，如果第一个数为1的流形的瞬子Floer同调在强意义下是平凡的，即基本群的所有三个表示都可以通过完整摄动消失，那么单极Floer同调群也是平凡的。(对于瞬子群，相关的表示是SO(3)中具有非平凡Stiefel-Whitney类的表示。)通过应用单极Floer同调的一个非零化定理和Floer的精确三角形，这将导致“性质猜想”的一个证明。这个项目的一个相关目标是发展基于有限维近似技术的Floer同调的新结构(这已经在四维不变量的研究中得到了令人信服的应用)。拓扑学是对空间及其连通性的定性研究。上个世纪之交，法国数学家庞卡罗根据牛顿定律研究支配地球、月球和太阳运动的三体系统的运动规律时，认识到了拓扑学的重要性。在过去的二十年里，拓扑学在蛋白质和DNA的打结问题以及现代高能物理理论中得到了广泛的应用。与高维空间相比，三维空间的拓扑学具有特殊的微妙之处。通过这一项目，希望为解决三维拓扑学中的突出问题带来新的技术。这些技术--规范理论和Seiberg-Witten方程--起源于物理学，它们有可能应用于夸克禁闭等基本问题。它们已经成为研究四维空间(如我们的时空)的有效工具。现在的目标是将同样的技术应用于第三维度的问题。

AbstractAward: DMS-0100771.Principal Investigator: Peter B. KronheimerThe aim of this project is to apply gauge-theory techniques tothe study of three-dimensional manifolds. The principalinvestigator proposes to investigate Floer homology and closelyrelated areas of geometry, and hopes to shed light on theapplicability of gauge theory to problems in three-dimensionaltopology. In particular, it is hoped that a relation can beestablished between the Floer homologies of three-manifoldsdefined on the one hand by the monopole equations on the otherhand by the instanton equations. (These are the equations which,in four-dimensions, lead respectively to the Seiberg-Witteninvariants and Donaldson invariants of four-manifolds, and whichhave led to an flood of results in four-dimensional differentialtopology in the past twenty years.) A first goal is to provethat if the instanton Floer homology of manifold with first bettinumber one is trivial in the strong sense that all therepresentations of the fundamental group can be made to disappearby a holonomy perturbation, then the monopole Floer homologygroups are trivial also. (For the instanton groups, the relevantrepresentations are the representations in SO(3) with non-trivialStiefel-Whitney class.) By an application of a non-vanishingtheorem for the monopole Floer homology and use of Floer's exacttriangle, this would lead to a proof of the "Property Pconjecture". A related goal in this project is the developmentof new constructions for Floer homology, based on the techniqueof finite-dimensional approximation (which has already seenconvincing application in the study of the four-dimensionalinvariants).Topology is the qualitative study of space and its connectedness.Its importance was recognized at the turn of the last century bythe French mathematician Poincaro, during his investigation ofthe laws of motion that govern the movement of a three-bodysystem such as the Earth, Moon and Sun moving according toNewton's laws. In the past twenty years, topology has seenapplications in questions such as the knotting of proteins andDNA, and in modern theories of high-energy physics. The topologyof three-dimensional spaces, as opposed to those of higherdimension, is of particular subtlety. Through this project, itis hoped to bring new techniques to bear on outstanding questionsin three-dimensional topology. These techniques -- gauge theoryand the Seiberg-Witten equations -- originated in physics, wherethey had potential application to fundamental questions such asquark confinement. They have been an effective tool in the studyof four-dimensional spaces (such as our space-time). The aim nowis to apply the same techniques to questions in dimension three.