The study of low-dimensional manifolds with various geometric structures

首席研究员继续研究3维和4维流形的结构，特别是它们的微分同胚型。为了研究带边界的4-流形的结构，他将Fukumoto-Furuta不变量推广到有理同调3-球面上，这是在Seiberg-Witten理论的基础上，利用V-流形上Dirac算子的指标来定义的。特别地，他证明了Siefert 3-流形上的Fukumoto-Furuta不变量与Neumann-Siebenmann不变量重合，并证明了它的自旋有理同调协边不变性。他将这些结果应用于边界是Seifert流形的4-流形的交形的约束，以及Seifert 3-流形通过对3-球面上的纽结进行Dehn运算而获得的条件。用Oszvath和Szabo的Heegaard Floer同调得到的3-流形不变量，研究了4-流形与边界的交形式的约束或3-流形在纽结上的Dehn运算得到的条件。我们利用ETA不变量研究了透镜空间的Oszvath-Szabo不变量和Fukumoto-Furuta不变量之间的关系，但在更一般的情况下，它们的关系仍有待进一步的研究。Fujii发现了双曲纽结补的双曲结构的变形与椭圆曲线的有理点之间的关系。研究人员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、Gauss的优美定理，Imanishi继续了对叶层的研究。

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.