Global properties of singularities

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We have studied the topology of the space consisting of all smooth or holomorphic maps with prescribed singularities. In the category of smooth manifolds and smooth maps we can expect so called homotopy principle concerning these singularities. However in the complex category we cannot expect it, In both cases it is important to study the topology of the ambiant space of the bad singularities except for them in the jet space. We have determined the topological type of these spaces for very simple singularities in the case of restricted dimensions. By the homotopy principle of the smooth maps with prescribed singularities, the existence of such maps is reduced to the homotopy-theoretic problem of the existence of sections of the associated subbundle of the Jet space. We have studied the higher obstructions of these sections such as primary and secondary ones in thecase where the singularities are of fold type and other related problems. We believe that this study will lead us to new observation of other problems in differential topology.