Geometry of singularities of mapping II

我们考虑了喷流空间中的Bordman流形（及其Zebriki闭包）.本文讨论了Cohen-Macaulay性质（从代数几何中交理论的观点来判断它是否是Cohen-Macaulay性质是一个大问题）。它是由Ronga先生的去奇异化和构造支持σ ^{n^-p+1，1}的复形所组成的.当n= 2，3，4时，映射芽$（C ^n，0）到（C^2，0）$的稳定扰动中出现的尖点个数的一个显式计算公式. (The第一篇论文在下一页）其次，从奇异性的观点，重要点理论，讨论了一个经典的微分几何。我们还讨论了在这一背景下出现的几个微分方程。在前面的一段中，B-Boardman子流形的概念起到了分析这一问题的关键作用。舍入和展平的概念就是这样定义的。本文还讨论了在这些点是孤立点的情况下如何定义指数。我们注意到，我们可以捆绑完全相同的 ...更多信息 对于秩为1的映射g：R^2到R^3（对于这类映射g：R^2到R^3，舍入指数小于等于1），我们也得到了类似于Lowner猜想的结果.将几类二元微分方程作为主线微分方程的推广，定义了全真实的概念，研究了指标的基本性质，讨论了它们的奇异性分类。(the下一页的第三篇论文t）考虑函数对函数水平的限制是一个有趣的问题。当后一个函数的水平是可并行的时，可以自然地定义一个映射。证明了它的映射度是正点轨迹和负点轨迹的欧拉特征标之差。讨论了可并行性的充要条件。这与量子数结构和Carey结构有关。(the此外，还介绍了弧分析图的逆映射定理（下页第二篇）和Blow分析图理论的最新进展（下页第六篇）。少

We consider Thom-Bordman manifolds $Sigma^{i, j}$ (and their Zariski closure) in the jet space. We discussed the Cohen-Macaulay property (It is a big problem to decide whether it is Cohen-Macaulay or not from the view point in intersection theory in algebraic geometry.). It is composed by Mr.Ronga's desingularization and construct complecies supported $Sigma^{n^-p+1,1}$. An explicit formula counting the number of cusps which appeared in stable perturbation of the map-germ $(C^n,0) to(C^2,0)$ when n=2,3,4. (The first paper in the next page)Next, a classical differential geometry is discussed from the singular view point, significant point theory. We also discuss several differential equation appeared in this context. The notion of Thom-Boardman submanifold in the foregoing paragraph plays key rule to analyse this.. The notion of rounding and flattening are defined in this way. We also discuss to define the index if these points are isolated. We remark that we can bundle exactly the same … More way if do not the submanifold has singularities.We also obtain an analogy of Lowner's conjecture for rank 1 map $g : R^2to R^3$ (The rounding index is 1 or less for such maps number). Moreover, several differential equation of two variables was caught as generalization of the differential equation of principal line, and define the notion of totally real and investigate the fundamental properties of index, classification of their singularities are discussed. (the third paper in the next page t)It is interesting problem to consider the restriction of function to the level of function. A certain map can be naturally defined when the levels of the later function are parallelizable It is shown that its mapping degree are differences of Euler characteristics signposts of a positive point locus and negative point locus. The necessary and sufficient condition for parallelizability was also discussed. This is related with quotanion number structure and the Carey structures. (the fourth paper in next page)Additionally, we show the inverse-map theorem concerning the arc analysis map (the second paper in the next page) and the recent progress of the theory of the blow analysis map (the sixth paper in the next page). Less