Co-operative Research on Topology

首先，我会描述这项计划所取得的成果的一般性质，其次，我会指出一九八八至八九年度内的一些显著成就。在这方面的一个一般性的方面是，活动在低维拓扑结构是显着的。特别是，以前意想不到的和新发现的低维拓扑和理论物理之间的相互作用进行了广泛的调查。在这个方向上的最新成就之一是Kohno的新的不变量的3流形的发现。这是由共形场论激发的。这种不变量已经“建造”的维滕，但只有在物理学家的水平的严谨性。Kohno的构造在数学上是严格的。他的不变量本质上与曲面的映射类群有关。对这些群的上同调作了深入的研究B，y S.森田他还澄清了Torelli群和Casson不变量之间的关系。众所周知，四维拓扑与规范理论有着密切的联系。在这方面，广岛大学的一个小组和东京大学的另一个小组深入研究了瞬子模空间的几何和拓扑。规范理论有一个三维对应物，它产生了一个有趣的同调三维球的同调理论，即弗洛尔同调。T.吉田他的结果揭示了辛几何中同调与Maslov指数之间的一种有趣联系。Kawauchi等人通过他的“模仿”理论研究了多项式在多大程度上决定了纽结的类型。在我们的项目中还有许多其他有趣的结果。详细情况请参阅我们正在编写的报告。我们有20多个研讨会，并在静冈大学（1988年）和福岛大学（1989年）举行了两次年度研讨会。总之。我们的目标已达到令人满意的程度。

I shall firstly describe the general character of the results of this project and secondly point out some remarkable achievements in the term 1988-89. A general aspect in this term was that activities in low dimensional topology were remarkable. In particular, the formerly unexpected and newly discovered interaction between low dimensional topology and theoretical physics was extensively investigated.One of the newest achievements in this direction is Kohno's discovery of new invariants of 3-manifolds. which was motivated by conformal field theory. Such invariants had been of "constructed" by Witten but only at the physicist's level of rigor. Kohno's construction is mathematically rigorous. His invariants are essentially related to the mapping class groups of surfaces. Cohomology of these groups were deeply studied b,y S. Morita. He also clarified the relation between the Torelli group and the Casson invariant.It is well-known that 4-dimensional topology has a strong connection with gauge theory. In this respect, the geometry and topology of moduli spaces of instantons were deeply investigated by a group at Hiroshima Univ. and by another at Univ. of Tokyo. The gauge theory has a 3-dimensional counterpart, and it produces an interesting homology theory of homology 3-spheres, that is, the Floer homology. The structure of this homology was intensively calculated by T. Yoshida. His results suggest an interesting connection between the homology and the Maslov index in symplectic geometry.The Jones polynomial were extensively studied by A. Kawauchi et al. Kawauchi studied to what extent the polynomial determines the types of knots by his "imitation' theory. There were many other interesting results in our project. For details, please consult with the report we are preparing.We had more that 20 symposia, and two annual symposia at Shizuoka Univ. (1988) and at Fukushima Univ. (1989). In conclusion. we have achieved our aim to a satisfactory extent.