The study of low-dimensional manifolds with various geometric structures

首席研究员继续研究3流形和4流形的结构，特别是它们的微分同构类型。为了研究带边界的4-流形的结构，他推广了Fukumoto-Furuta不变量，将其应用于基于Seiberg-Witten理论的v -流形上Dirac算子的指标所定义的有理同调3球。特别地，他证明了Siefert 3-流形的Fukumoto-Furuta不变量与Neumann-Siebenmann不变量是一致的，并证明了它的自旋理性同调协同不变量。他将这些结果应用于边界为塞弗特流形的4-流形的交点形式的约束，以及通过对3球中的结点进行Dehn手术得到塞弗特3-流形的条件。利用Oszvath和Szabo从Heegaard flower同调中导出的基于不同原理的3-流形不变量，研究了带边界的4-流形交点形式的约束条件和结点上Dehn手术得到3-流形的条件。我们通过不变量考察了透镜空间的Oszvath-Szabo不变量和Fukumoto-Furuta不变量之间的关系，但它们在更一般情况下的关系仍有待进一步研究。研究者藤井发现了双曲结补的双曲结构的变形与椭圆曲线的有理点之间的关系。Kato、Ushiki和Nishiwada分别研究了p进对称空间的表示理论、二维复杂动力系统、高斯的优雅定理，Imanishi继续研究了叶分。

The head investigator continued the research on the structures of 3 and 4-manifolds, in particular the diffeomorphism types of them. For the research of the structures of 4-manifolds with boundary, he generalized the Fukumoto-Furuta invariants to apply them to rational homology 3-spheres, which have been originally defined by using the index of the Dirac operator on V-manifolds based on the Seiberg-Witten theory. In particular he proved that the Fukumoto-Furuta invariant for Siefert 3-manifolds coincides with the Neumann-Siebenmann invariant, and also proved its spin rational homology cobordism invariance. He applied these results to the constraints for the intersection forms of 4-manifolds whose boundaries are Seifert manifolds, and to the conditions for the Seifert 3-manifolds to be obtained by Dehn surgery on knots in the 3-sphere. The constraints for the intersection forms of 4-manifolds with boundary or the conditions for the 3-manifolds to be obtained by Dehn surgery on knots also have been studied by 3-manifold invariants derived from the Heegaard Floer homology by Oszvath and Szabo, which are based on the different principle from ours. We investigated the relation between Oszvath-Szabo's invariant and the Fukumoto-Furuta invariant for the lens spaces via the eta invariant, but their relationship for more general cases is still open for the research in the future.The investigator Fujii found the relation between the deformation of the hyperbolic structure of the complement of a hyperbolic knot and the rational points of the elliptic curves. The investigator Kato, Ushiki, and Nishiwada studied the representation theory of p-adic symmetric spaces, two dimensional complex dynamical systems, Theorema elegantissimum by Gauss respectively, and Imanishi continued the study of foliations.