Homotopy theoretic study of singularities of smooth mappings

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In this research we have studied global properties of simple singularities of smooth maps, in particular, fold singularities and Morin singularities from the viewpoint of homotopy theory. We have studied the set of fold-eobordism classes of fold-maps of degree m, and established a relation between this cobordism set and the set of homotopy classes to the space of base point preserving maps of degree m of spheres. In this study a homotopy principle for Morin singularities, which has been established in our research, has played an important role.Furthermore, we have determinod the homotopy type of the subspace in 2 jet space associated with fold maps. This reduces the existence problem of fold-maps toa homotopy theoretic problem of finding sections of fiber bundles.