Seiberg-Witten and Instanton Floer Homologies

点击翻译按钮获取中文摘要

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.