Research of invariants of singularities of smooth mappings

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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.