Seiberg-Witten and Instanton Floer Homologies

摘要建议：DMS 9802480主要研究人员：托马兹·莫罗卡和玛蒂尔德·马尔科利研究项目由三部分组成。第一个目标是构造Seiberg-Witten Floer同调的等变形式，它是潜在的三维流形的可微结构的不变量，并避免了非等变理论中出现的度量依赖问题。该项目的第二部分包括推导精确的三角形公式，以检测当通过手术修改三维流形时Seiberg-Witten Floer同调如何变化，以及适当的剪切和粘贴技术，该技术将四维流形和三维流形的Seiberg-Witten不变量联系在一起。该项目的剩余部分致力于研究Seiberg-Witten Floer同调和与Donaldson理论相关的瞬子Floer同调之间的关系。Seiberg-Witten不变量的发现，作为弦理论最新发展的结果，在低维拓扑领域产生了巨大的影响。几何学和理论物理学的丰富相互作用使人们能够更深入地理解三维和四维流形的几何结构。众所周知，三维和四维流形的拓扑和几何特别丰富的有趣现象和开放问题：高维分类方法的失败使得构造可计算的不变量特别重要，因此需要研究Seiberg-Witten不变量的性质以及它们与已知的Yang-Mills-Donaldson理论的关系。

Abstract Proposal: DMS 9802480 Principal Investigators: Tomasz Mrowka and Matilde Marcolli The research project consists of three parts. The first goal is the construction of an equivariant version of Seiberg-Witten Floer homology, which is an invariant of the differentiable structure of the underlying three-manifold and avoids the problem of metric dependence that arises in the non-equivariant theory. The second part of the project consists of deriving the exact triangles formulae that detect how the Seiberg-Witten Floer homology changes when the three-manifold is modified by surgery and a suitable cutting and pasting technique that relates the Seiberg-Witten invariants of four-manifolds and three-manifolds. The remaining part of the project is dedicated to the investigation of the relation between the Seiberg-Witten Floer homology and the instanton Floer homology associated to Donaldson theory. The discovery of the Seiberg-Witten invariants, as an outcome of recent developments in string theory, has had a tremendous impact in the field of low dimensional topology. The rich interplay of geometry and theoretical physics has allowed a deeper understanding of the geometric structure of three and four-dimensional manifolds. The topology and geometry of three and four-dimensional manifolds is known to be especially rich of interesting phenomena and open problems: the failure of the classification methods used in higher dimensions makes it particularly important to construct computable invariants, hence the need to investigate the properties of the Seiberg-Witten invariants and their relation to the previously known Yang-Mills-Donaldson theory.