Research of invariants of singularities of smooth mappings

在本研究中，我们研究了奇点的全局性质。特别地，我们研究了以下三个主题：设P是一个连通且闭的p维光滑流形，我们用某些稳定同伦群的形式表示了n维闭流形到P的光滑映射的所有余边类组成的群，其中P具有给定的稳定同伦群，我们假设这些光滑映射在存在水平上是成立的。构造了n维闭定向流形的0次折叠映射群到n球面到球面的n次稳定同伦群的同构。作为应用，我们证明了n次稳定同伦群的元素是由n&lt；8维的某些映射的高次奇点来检测的。我们结合外科理论和一般光滑稳定映射或有限余维映射的不存在结果，研究了在n维上存在闭n流形N，P(可能N=P)的条件以及N到P的同伦等价。

In this research we have studied golobal properties of singularities. In particular we have studied the following three subjects.Let P be a connected and closed smooth manifold of dimension p. We have represented the group of all cobordism classes of smooth maps of n-dimensional closed manifolds into P with singularities of given class in terms of certain stable homotopy groups by applying the homotopy principle on the existence level which is assumed to hold for those smooth maps. We have also dealed with the oriented version.We have constructed an isomorphism of the group of all cobordism classes of fold-maps of degree 0 of n-dimensional closed oriented manifolds to the n-sphere to the n-th stable homotopy group of spheres. As an application we showed that elements of n-th stable homotopy group are detected by higher singularities of certain maps in dimensions n<8.We have studied the condition on dimension n under which there exist closed n-manifolds N,P (possibly N=P) and a homotopy equivalence N to P which is not homotopic to any smooth stable map and to any map of finite codimension by combining the surgery theory and the nonexistence result for general maps which are smooth stable or of finite codimension.