Global Singularity Theory of Mappings and Various Structures of Manifolds

佐伯和佐久间研究了4-流形到3-流形的一般C^∞映射，并成功地刻画了那些允许特殊一般映射的复杂曲面。此外，Saeki研究了给定的4-流形允许折叠映射进入R^3的充要条件，得到了一个非常尖锐的结果。佐久间指出，即使不满足同调条件，类似的结果也成立。菅野从局域镜像对称的观点研究了5维超对称规范理论。特别是，他考虑了一系列特定的椭圆曲线，并计算了5维超对称规范理论的前势。Teragaito研究了三维流形中沿着双曲纽结的Dehn运算，并在纽结的亏格较小时得到了一些重要结果。此外，委员会认为，Takata研究了可定向闭流形的嵌入维数，得到了每一个维数的最佳可能结果; Kawamoto研究了loop空间的高交换性，并在loop空间满足条件下，得到了具有这种性质的空间的几乎完全分类。我们的下一个目标将是研究消除复杂奇点的障碍，并找到流形和映射的不变量。作为我们研究的结果，已经澄清，对于这样的研究，重要的是要看看奇异纤维的行为，以及我们将在不久的将来进行的喷流空间的同伦类型。

Saeki and Sakuma studied generic C^∞ maps of 4-manifolds into 3-manifolds and succeeded in characterizing those complex surfaces which admit special generic maps. Furthermore, Saeki studied necessary and sufficient condition for a given 4-manifold to admit a fold map into R^3 and obtained a very sharp result. Sakuma showed that a similar result holds even if a homological condition is not satisfied.Kanno studied 5-dimensional super-symmetric gauge theory from the viewpoint of local mirror symmetry. In particular, he considered a family of certain elliptic curves and calculated the prepotential of 5-dimensional super-symmetric gauge theory. Relating it to the 4-dimensional super-symmetric gauge theory by compactifying this result, we obtained a new approach to the Seiberg-Witten theory.Teragaito studied Dehn surgeries along hyperbolic knots in 3-manifolds and obtained some important results when the genera of the knots are small. Furthermore, he showed that every integer can be realized as the slope of some exceptionai Dehn surgery along a hyperbolic knot.Takata studied the embedding dimension of a given orientable closed manifold up to cobordism and obtained best possible results for every dimension.Kawamoto studied higher commutativity of loop spaces and obtained an almost complete classification of those spaces with such properties under the condition that the cohmology ring is finitely generated.Our next goal will be to study the obstruction to eliminating complicated singularities and to find invariants of manifolds and maps. As a result of our study, it has been clarified that, for such kind of a study, it is important to look at the behavior of singular fibers, and also the homotopy type of the jet space, which we are going to carry out in a near future.