Problems on the eliminability for smooth maps of manifolds

点击翻译按钮获取中文摘要

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.