3-manifolds and Floer homologies

本项目的研究内容是拓扑学领域的问题，主要是利用三维流形拓扑和辛拓扑的不变量和方法，研究规范论/辛Floer同调的基本和重要性质。研究了瞬子（规范论）Floer同调的半无穷性和（同调）三球的Seiberg-Witten-Floer同调的内禀依赖性。本项目的主要内容是研究辛Floer同调与经典三维流形之间的相互关系，Floer上同调与半无限上同调之间的关系，以及Seiberg-Witten-Floer理论与瞬子和瞬子结果的相互联系。本项目的另一部分是研究大类三球面不变量的内在性质，其基本目的是研究新的不变量在某些拓扑操作下的变化。该项目将整合瞬子理论，量子理论，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，在那里一个近视的人看到一个标准的三维空间无处不在。（同调）三球是那些三流形，人们不能从标准的三球使用通常的拓扑工具。辛流形是具有特殊（辛）结构的偶数维空间。例如，机械系统的相空间是辛空间。这种辛结构在数学和物理学中非常丰富，并且在任何附近区域都是典型的。这显示了我们生活的世界的微妙和复杂性。任何局部信息都不再有用，全局行为和全局（拓扑）不变量是关键。本计画所研究的精化不变量，其目的在于从数学物理的角度来区分流形，并从数学的角度来解释各种量子场论。因此，研究辛Floer上同调与半无限上同调之间的关系，以及瞬子同调与半无限上同调之间的相互联系是非常有价值的.这个项目解决了这个问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.