Homotopy theory on singularities of differentiable maps and K-invariant spaces of the jet spaces

首先证明了定向n维闭流形到给定的定向闭流形P的0次折叠映射的定向配边类群与P到已知空间F的同伦群的同构。球的n阶稳定同伦群的元素可以通过相应的折叠映射的某些扩张的奇点来检测。我们试图在更高的维度上解决这个问题。4 q-流形之间的映射的两类奇异性可以检测J-映象的元素，其中4 q-流形的Thom多项式的整系数为首Pontrjagin类Pq的非零系数.

We first proved that there exists an isomorphism of the group of oriented cobordism classes of fold maps of closed oriented n-manifolds to the given oriented closed manifold P of degree$ 0$ to the homotopy group of P to the well-known space F. An element of the n-th stable homotopy group of spheres can be possibly detected by singularities of some extension of a corresponding fold map. We tried to solve this problem in the higher dimensions. Two types of singularities of maps between 4q-manifolds whose Thom polynomials with integer coefficients have nonvanishing coefficients of the leading Pontrjagin class P_{ q} can detect elements of J-images.