Global properties of singularities

我们研究了由所有具有指定奇点的光滑映射或全纯映射组成的空间的拓扑。在光滑流形和光滑映射的范畴中，我们可以期待关于这些奇点的所谓同伦原理。然而，在复杂范畴中我们不能期望它，在这两种情况下，重要的是研究除了在喷流空间中的坏奇点的环境空间的拓扑。在约束维的情况下，我们确定了这些空间对于非常简单的奇点的拓扑类型。利用给定奇点光滑映射的同伦原理，将这类映射的存在性归结为Jet空间中伴随子丛的截面的存在性的同伦论问题。我们研究了折叠型奇点的初级和次级等截面的较高障碍物及其他相关问题。我们相信，这一研究将引导我们对微分拓扑学中的其他问题进行新的观察。

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.