Geometry of singularities of mapping II

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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