Differential Topology and Singularities

从微分拓扑学的角度详细研究了4维流形到3维流形的稳定映射的整体性质。我们首先成功地将奇异纤维完全归类到左右等价。直到这项研究项目才知道有这样的结果，所以这是一个新的结果，我们希望这将在今后的全球地图研究中发挥重要作用。此外，我们还研究了奇性纤维之间的邻接关系，得到了它们的数量之间的一些关系。对于这些主题，我们计划利用奇异纤维的泛复概念和Thom多项式的概念对它们进行更详细的研究。我们还研究了给定的闭4维流形是否允许折叠映射进入3维流形的问题，并成功地用4维…的交形式完整地刻画了存在的充要条件更加多种多样。作为相关课题，我们还研究了一般维上的折叠映射，并证明了这种映射的奇异集的自交类可以通过Pontrjagin类来表示。利用这一结果，我们得到了关于折叠映射不存在的几个新结果。我们还从奇点理论的角度研究了三维流形在五维空间中浸入的正则同伦类，得到了一个与三维流形的自旋结构密切相关的新结果。此外，我们还从微分拓扑学的角度研究了三维流形嵌入到五维空间的调和类，得到了一个重要的结果，它表明自旋结构在这一维中起着重要的作用，而不同于高维。进一步，我们从整体奇异性理论的角度定义了仅含极小值和极大值的Morse函数的(定向)余边群，并证明了它与维度大于或等于6的同伦球面群同构

We studied the global behavior of stable maps of 4-dimensional manifolds into 3-dimensional manifolds in detail from the viewpoint of differential topology. We first succeeded in completely classifying the singular fibers up to right-left equivalence. There has been known no such result until this research project so that this is a new result, and we hope that this will play an important role in the global study of maps from now on. Furthermore, we studied the adjacency among the singular fibers in deital, and we obtained some relations among their numbers. As to these subjects, we plan to study them more in detail by using the notion of the universal complex of singular fibers and that of the Thom polynomial. We also studied the problem of whether a given closed 4-dimensional manifold admits a fold map into a 3-dimensional manifold, and we succeeded in completely describing the necessary and sufficient condition for the existence in terms of the intersection form of the 4-dimensional … More manifold. As a related subject, we also studied fold maps in general dimensions, and we showed that the self-intersection class of the singular set of such a map can be expressed through the Pontrjagin class. By using this result, we obtained several new results about the non-existence of fold maps. We also studied the regular homotopy classes of immersions of 3-dimensional manifolds into 5-space from the viewpoint of singularity theory, and we obtained a new result which shows a strong relationship to the spin structures of 3-dimensional manifolds. Moreover, we studied the concordance classes of embeddings of 3-dimensional manifolds into 5-space from the viewpoint of differential topology, and we obtained an important result which shows that the spin structure plays an important role in this dimension, differently from the higher dimensions. Furthermore, we defined the (oriented) cobordism group of Morse functions with only minima and maxima from the viewpoint of global singularity theory, and we showed that it is isomorphic to the group of homotopy spheres in dimensions greater than or equal to 6. Less