Global Theory of Singularities from the Viewpoint of Homotopy Theory

流形与其奇点之间的可微映射的研究始于世纪中期的Whitney，Thom等人，至今已有了很大的发展。特别地，局部性质已经被研究，并且已经为这样的研究建立了一些复杂的理论。然而，与流形结构密切相关的映射的整体性质尽管重要，却没有得到太多的研究。在本研究计划中，我们的目标是在更大的框架下，从同伦理论的角度来解决整体奇点理论中的各种重要的开放问题。更准确地说，我们进行了以下研究，得到了一些结果，下面将解释：（a）高阶障碍，（B）奇异纤维空间的分类，（c）流形的可微结构与映射的奇异性之间的关系。关于（a），Saeki和Iwase研究了正则纤维由球面组成的可微映射，并得到了允许这类映射的流形的一些同伦性质。此外，已经澄清的是，这些特性与这种地图存在的更高障碍有关。此外，Saeki和Sakuma研究了折叠映射的存在性问题，发现某些Postnikov不变量表现为更高的障碍。关于（B），Saeki利用曲面上函数的奇异纤维理论构造了曲面丛的特征类。此外，Saeki和Ohmoto成功地构造了一个奇异纤维的分类空间。至于（c），佐伯和佐久间收集和安排已知的结果之间的关系的奇异性的微分地图和微分结构的流形，和佐伯表明，在某些情况下消除明确的折叠奇点是可能的独立的微分结构的流形。

The study of differentiable maps between manifolds and their singularities was begun by Whitney, Thom etc. in the middle of the 20th century and has developed a lot since then. In particular, local properties have been studied and some sophisticated theories have been established for such studies. However, global properties of maps that are essentially related to the structures of manifolds have not been studied so much in spite of their importance. In this research project, we aimed at solving various important open problems in the global theory of singularities from the viewpoint of homotopy theory in a larger framework. More precisely, we performed the following studies and obtained some results, which will be explained below :(a) Higher obstructions,(b) Classifying space of singular fibers,(c) Relationship between the differentiable structures of manifolds and singularities of maps.As to (a), Saeki and Iwase studied differentiable maps whose regular fibers consist of spheres, and obtained some homotopical properties of those manifolds which admit such maps. Furthermore, it has been clarified that these properties are related to higher obstructions to the existence of such maps. Moreover, Saeki and Sakuma studied the existence problem of fold maps and found that certain Postnikov invariants appear as higher obstructions. As to (b), Saeki constructed characteristic classes of surface bundles by using the theory of singular fibers of functions on surfaces. Furthermore, Saeki and Ohmoto succeeded in constructing a classifying space of singular fibers. As to (c), Saeki and Sakuma collected and arranged the known results about the relationship between the singularities of differentiable maps and differentiable structures of manifolds, and Saeki showed that in certain cases the elimination of definite fold singularities is possible independently of the differentiable structures of manifolds.