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Problems on the eliminability for smooth maps of manifolds

流形光滑映射的可消性问题

基本信息

批准号：
14540094
负责人：
SAKUMA Kazuhiro
金额：
$ 0.9万
依托单位：
Kinki University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540094/
关键词：
stable map Morin singularity Thom polynomial self-in tersection class Pontrjagin class Fold map Morin map local system 自己交叉類大域的特異点論特異点集合 Thom多項式 Morin写像第二障害類

项目摘要

If there exists a smooth map with only Morin singularities of a closed n-manifold into a p-dimensional Euclidean space, the singular set of the map is a submanifold of the source manifold. When n-p+1【greater than or equal】2k, we can define the self-intersection cohomology class of the singular set as the Poincare dual of the transversal intersection of the singular set and its small perturbation in the source manifold. If the singular set is non-orientable, one should note that the coefficient group in the cohomology group need to be taken with local system. Then it is particularly important to find whether it is expressed by the characteristic classes of the tangent bundle to the source manifold in the sense that it corresponds to the elaboration of the notion of Thorn polynomial which was defined by R. Thorn around 1950's. In our research we have found that this cohomology class coincides with Pontrjagin class. From this result we can deduce the necessary condition for the existence of fold maps in terms of cohomology classes. As an application we can also obtain non-exitence results for fold maps, which are not derived from the calculation of Thorn polynomials.

如果存在一个从闭n维流形到p维欧氏空间的光滑映射，该映射的奇异集是源流形的子流形。当n-p+1[大于或等于]2k时，我们可以定义奇异集的自交上同调类为奇异集的横截与其在源流形上的小扰动的Poincare对偶。如果奇异集是不可定向的，则应注意上同调群中的系数群需要取局部系。然后，特别重要的是要找出它是否由源流形的切丛的特征类表示，在这个意义上，它对应于由R定义的Thorn多项式概念的阐述。1950年左右的荆棘。在我们的研究中，我们发现这个上同调类与Pontrjagin类重合。从这个结果我们可以推出折叠映射存在于上同调类中的必要条件。作为应用，我们还可以得到折叠映射的不存在性结果，这些结果不是由Thorn多项式的计算得到的。