3-manifolds and Floer homologies

本课题主要研究拓扑学领域中的问题，主要是利用3-流形拓扑和辛拓扑中的不变量和方法研究规范理论/辛Floer同调的基本和重要性质。研究了瞬子(规范理论)Floer同调的半无穷性和(同调)三球的单极(Seiberg-Witten-Floer)同调的内在依赖性。本项目的主要内容是研究辛Floer上同调与经典三维流形之间的相互作用关系，以及Floer上同调与半无限上同调之间的关系，以及将瞬子和单极结果交织在一起的Seiberg-Witten-Floer理论。本项目的另一部分是研究更大类三球面不变量的内蕴性质，其基本目的是研究新不变量在某些拓扑运算下的变化。该项目将整合瞬子理论、单极理论、辛Floer理论和无限维李代数的半无限上同调之间的相互作用。三维流形是一个空间，在这个空间中，近视者可以看到标准的三维空间无处不在。(同调)三球是那些人们不能用通常的拓扑工具从标准三球中区分出来的三维流形。辛流形是具有特殊(辛)结构的偶数维空间。例如，力学系统的相空间是辛空间。这种辛结构蕴含着丰富的数学和物理知识，在任何邻近区域都是规范的。这显示了我们生活的世界的微妙和复杂。任何局部信息都不再有用，全局行为和全局(拓扑)不变量是支点。本项目中研究的精化不变量旨在从数学物理的角度区分流形，并从数学的角度将各种量子场理论相互交织在一起。因此，研究辛Floer上同调与半无限上同调之间的关系，以及瞬子上同调与单极同调之间的相互作用关系是非常有价值的。这个项目解决了这个主题中的一些最基本的问题。

The problems addressed in this project are in the area of Topology.The main theme is to study the fundamental and important properties of the gauge-theoretic/symplectic Floer homology, using invariants and methods from 3-manifold topology and symplectic topology. The investigator studied the semi-infinity of the instanton (gauge theoretic) Floer homology and the intrinsic dependence of the monopole(Seiberg-Witten-Floer) homology of (homology)three-spheres. The major part of the project is to study the interactive relation between the symplectic Floer homology and the classical 3-manifolds, and therelation between the Floer cohomology and the semi-infinite cohomology, and the Seiberg-Witten-Floer theory intertwining the instanton and monopole results. The other part of this project is to study intrinsic properties for the invariants of largerclasses of three-spheres.It is a fundamental aim to investigate the change of the newinvariants under certain topological operations. The project will integrate the interactions among the instanton theory, the monopole theory, the symplectic Floertheory and the semi-infinite cohomology of infinite-dimensional Lie algebras.A three-dimensional manifold is a space where a nearsighted person sees a standard three-dimensional space everywhere.(Homology) three-spheres are those three-manifolds that one cannot tell from the standard three-sphere by using the usual topological tools. A symplectic manifold is an even-dimensional space with a special(symplectic) structure. For instance, the phase space of a mechanical system is a symplectic space. Such a symplectic structure is rich in mathematics and physics, and is canonical from any nearby region. This shows the subtlety and the complexity of the world we live in. Any local information is no longer useful, the global behavior and the global (topological) invariants are the pivot. The refined invariants studied in this project are intended to distinguish manifolds from the aspects of mathematical physics, and to intertwine various quantum field theories from the aspect of mathematics. Thus it is extremely valuable to investigate the relation between the symplectic Floer cohomology and the semi-infinite cohomology, as well as the interactive link between the instanton homology and the monopole homology. This project addresses some of the most fundamental problems in this subject.