Research for low-dimensional manifolds with various geometric structures

研究了源于Seiberg-Witten理论的某些不变量对3，4-流形的微分同胚型的约束。Dirac算子的指数对V4-流形孤立奇点的贡献是球面3-流形及其自旋结构对的整数值不变量，它给出了Rochlin不变量的积分升力(模16)，这与Neumann-Siebenmann不变量重合.他考虑了球面上的三维流形是由一个纽结上的外科手术得到的，并用上述不变量给出了球面上三维流形的类型的一些限制条件，并给出了当球面上的三维流形的不变量是由普通纽结上的同时外科手术得到的情况下它们之间的某种关系。他还将结果推广到一般Seifert 3-流形的情形，并利用Neumann-Siebenmann不变量给出了在纽结上进行手术所得到的一些约束条件。最近，Ozsvath-Szabo的Floer同调给出了Seifert三维流形的一些约束条件。因此，我们的下一个任务是研究Floer同调与上述不变量之间的关系。Fujii成功地研究了三维双曲锥流形的高斯超几何函数局部变换。Imanishi利用关于区间的Lipschitz同胚群的几个结果，成功地研究了保持余维1的可微叶的Lipschitz同胚群的上同调.

Ue studied the constraints on the diffeomorphism types of 3, 4-manifolds by certain invariants originated from Seiberg-Witten theory. The contribution of the index of the Dirac operator to the isolated singularities of V 4-manifolds previously studied by him is an integer valued invariant for the pair of a spherical 3-manifold and its spin structure, which gives an integral lift of the Rochlin invariant (determined modulo 16), which coincides with the Neumann-Siebenmann invariant. He considered the case when a certain spherical 3-manifold is obtained by surgery on a knot and gave some constraints on its type in terms of the above invariant and also gave certain relations between the invariants of the spherical 3-manifolds in the case that they are obtained by simultanious surgery on a common knot. He also extended the results to the case of general Seifert 3-manifolds and gave some constraints of them to be obtained by surgery on a knot in terms of the Neumann-Siebenmann invariants. Recently some constraints for the Seifert 3-manifolds to be obtained by surgery on a knot are given by Ozsvath-Szabo's Floer homology. So our next task is to investigate the relations between the Floer homology and the above invariants. Fujii suceeded the study of the local transfromations of 3-dimension hyperbolic cone manifolds in terms of Gaussian hypergeometric functions. Imanishi suceeded the study of the cohomology of the the group of Lipschitz homeomorphisms preserving the differentiable foliations of codimension 1 by utilizing several results about the group of Lipschitz homeomorphisms of the interval.