The study of low-dimensional manifolds with various geometric structures

首席研究员继续研究3和4-流形的结构，特别是它们的同构类型。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，有理同调3-球面最初是基于Seiberg-Witten理论通过使用V-流形上的Dirac算子的指数定义的。特别是他证明了Fukumoto-Furuta不变量的Siefert 3-流形符合纽曼-Siebenmann不变量，也证明了其自旋有理同调配边不变性。他应用这些结果的限制交叉形式的4流形的边界是塞弗特流形，并为条件的塞弗特3流形获得Dehn手术结在3球。Oszvath和Szabo从Heegaard Floer同调导出的3-流形不变量，基于与我们不同的原理，研究了4-流形与边界的交形式的约束条件或3-流形通过Dehn对纽结的手术得到的条件.我们通过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，但它们在更一般情况下的关系仍有待于今后的研究。研究者Fujii发现了双曲纽结的补的双曲结构的变形与椭圆曲线的有理点之间的关系。调查员加藤，牛木，和西和田研究了代表性理论的p-adic对称空间，二维复杂的动力系统，定理elegantissimum高斯分别，和今继续研究叶。

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.