Research for low-dimensional manifolds with various geometric structures

从Seiberg-Witten理论出发，利用某些不变量研究了3,4流形的微分同胚型的约束。狄拉克算子的指标对他先前研究的v4流形孤立奇点的贡献是球面3流形及其自旋结构对的一个整数值不变量，它给出了Rochlin不变量（确定模16）的积分提升，这与Neumann-Siebenmann不变量一致。他考虑了某球面3-流形通过在一个结上进行手术得到的情况，并根据上述不变量给出了其类型的一些约束条件，同时给出了在一个普通结上同时进行手术得到的球面3-流形不变量之间的某些关系。他还将结果推广到一般的Seifert 3-流形的情况，并给出了一些约束条件，这些约束条件可以用Neumann-Siebenmann不变量在结上进行手术得到。最近，利用Ozsvath-Szabo的Floer同调给出了在结上手术得到的Seifert 3-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继承了用高斯超几何函数研究三维双曲锥流形的局部变换。Imanishi利用关于区间的Lipschitz同胚群的几个结果，继承了Lipschitz同胚群的上同调的研究，保持了余维数为1的可微叶。

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.