Global Theory of Singularities from the Viewpoint of Homotopy Theory

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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.