Global Theory of Singularities of Differentiable Maps and its Applications

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We studied singularities of differentiable maps between manifolds from a global point of view and obtained various new results about such singularities and differential topological properties of manifolds. For example, it is known that many 4-dimensional topological manifolds admit infinitely many differentiable structures, and we have proved that among them there is only one such structure that allows the existence of a differentiable map with the simplest singularities. We also discovered some examples of global results that can be used to study local singularities. In this manner, we have clarified the deep relationship between the singularities (or singular fibers) of maps and the cobordism classes of manifolds and maps, obtaining a lot of explicit and concrete results.