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

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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.