Differential Topology and Singularities

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We studied the global behavior of stable maps of 4-dimensional manifolds into 3-dimensional manifolds in detail from the viewpoint of differential topology. We first succeeded in completely classifying the singular fibers up to right-left equivalence. There has been known no such result until this research project so that this is a new result, and we hope that this will play an important role in the global study of maps from now on. Furthermore, we studied the adjacency among the singular fibers in deital, and we obtained some relations among their numbers. As to these subjects, we plan to study them more in detail by using the notion of the universal complex of singular fibers and that of the Thom polynomial. We also studied the problem of whether a given closed 4-dimensional manifold admits a fold map into a 3-dimensional manifold, and we succeeded in completely describing the necessary and sufficient condition for the existence in terms of the intersection form of the 4-dimensional … More manifold. As a related subject, we also studied fold maps in general dimensions, and we showed that the self-intersection class of the singular set of such a map can be expressed through the Pontrjagin class. By using this result, we obtained several new results about the non-existence of fold maps. We also studied the regular homotopy classes of immersions of 3-dimensional manifolds into 5-space from the viewpoint of singularity theory, and we obtained a new result which shows a strong relationship to the spin structures of 3-dimensional manifolds. Moreover, we studied the concordance classes of embeddings of 3-dimensional manifolds into 5-space from the viewpoint of differential topology, and we obtained an important result which shows that the spin structure plays an important role in this dimension, differently from the higher dimensions. Furthermore, we defined the (oriented) cobordism group of Morse functions with only minima and maxima from the viewpoint of global singularity theory, and we showed that it is isomorphic to the group of homotopy spheres in dimensions greater than or equal to 6. Less