Global Singularity Theory of Mappings and Various Structures of Manifolds

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Saeki and Sakuma studied generic C^∞ maps of 4-manifolds into 3-manifolds and succeeded in characterizing those complex surfaces which admit special generic maps. Furthermore, Saeki studied necessary and sufficient condition for a given 4-manifold to admit a fold map into R^3 and obtained a very sharp result. Sakuma showed that a similar result holds even if a homological condition is not satisfied.Kanno studied 5-dimensional super-symmetric gauge theory from the viewpoint of local mirror symmetry. In particular, he considered a family of certain elliptic curves and calculated the prepotential of 5-dimensional super-symmetric gauge theory. Relating it to the 4-dimensional super-symmetric gauge theory by compactifying this result, we obtained a new approach to the Seiberg-Witten theory.Teragaito studied Dehn surgeries along hyperbolic knots in 3-manifolds and obtained some important results when the genera of the knots are small. Furthermore, he showed that every integer can be realized as the slope of some exceptionai Dehn surgery along a hyperbolic knot.Takata studied the embedding dimension of a given orientable closed manifold up to cobordism and obtained best possible results for every dimension.Kawamoto studied higher commutativity of loop spaces and obtained an almost complete classification of those spaces with such properties under the condition that the cohmology ring is finitely generated.Our next goal will be to study the obstruction to eliminating complicated singularities and to find invariants of manifolds and maps. As a result of our study, it has been clarified that, for such kind of a study, it is important to look at the behavior of singular fibers, and also the homotopy type of the jet space, which we are going to carry out in a near future.