Research for low-climensional manifolds with various geometric structures

利用某些不变量，特别是源于Seiberg-Witten理论的不变量，研究了来自不同几何结构的3，4-流形的微分同胚型的约束。首先，在与古田美纪夫和福本良弘的合作中，他研究了同调3-球的W不变量，证明了在Seifert同调3-球的情况下，这个不变量与Neumann-Siebenmann不变量重合，并且在某些额外条件下它是同调余边不变量。Saveliev利用了我们关于V4-流形上Dirac算子指数估计的主要结果，推广了我们的结果。我们还确定了孤立奇点对V4-流形上Dirac算子指数的所有贡献。这一贡献本身就是一对球面三维流形及其自旋结构的不变量。作为应用，他给出了球面3-流形所界定的正定自旋4-流形的交形式的某些约束条件，并对嵌入4-流形的实射影平面的法欧拉数给出了新的估计。这些结果适用于更广泛类型的3-流形。Fujii研究了仅以单环为奇点集的三维锥形流形，并成功地给出了调和向量场的常微分方程组的高斯超几何函数形式的显式解，这是描述奇异集管状邻域上的调和1-形式的关键.同样在与Kazuhiko Fukui的合作中，Imanishi利用关于区间的Lipschitz同胚群的几个结果，确定了保持余维1的可微叶的Lipschitz同胚群的第一上同调。

Ue studied the constraints on the diffeomorphism types of 3, 4-manifolds coming from the various geometric structures by certain invariants, in particular the one originated from Seiberg-Witten theory. First in the joint work with Mikio Furuta and Yoshihiro Fukumoto, he studied the W invariants of homology 3-spheres, and showed that in case of Seifert homology 3-spheres this invariant coincides with the Neumann-Siebenmann invariant, and that it is homology cobordism invariant under certain extra conditions. Saveliev used our key result about the estimates on the index of the Dirac operator over V 4-manifolds, and extended our results. Also Ue determined all the contributions from the isolated singularities to the index of the Dirac operator over V 4-manifolds. This contribution itself is an invariant for a pair of spherical 3-manifold and its spin structure. As its application, he gave certain constraints on the intersection forms of definite spin 4-manifolds bounded by spherical 3-manifolds, and also new estimates on the normal Euler number of the real projective plane embedded in 4-manifolds. These results are applicable to a wider class of 3-manifolds. Fujii studied 3-dimension cone manifolds with only simple loops as their singlar sets, and succeeded to give explicit solutions in terms of Gaussian hypergeometric functions to the system of ordinary differential equations for the harmonic vector field, which is a key to describe the harmonic 1-form on the tubular neighborhood of the singular set. Also in the joint work with Kazuhiko Fukui, Imanishi determined the first 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.