Study on the diffeomorphism types of 4-manifolds via a generalization of Morse theory

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The purpose of the research is to study the relation between a closed 4-manifold M^4 and singularities of a smooth map of the 4-manifolds into R^3 which generically appeared. Such generic singularities are the following four types : a definite fold, an indefinite fold, a cusp and a swallowtail. A smooth map with only definite fold singularities is called a special generic map. We have seen that we can characterize a closed 4-manifold which admits a special generic map as the necessary and sufficient condition on the diffeomorphism types of such a 4-manifold. Therefore, it arises an important question whether one can remove which types of singularities in the above four types or find some obstructions for removing those singularities. It is known that swallowtails can be always removed if M^4 is orientable (Ando's theorem). Hence our problem is to consider the removability of cusp singularities and indefinite fold singularities. In general, indefinite fold singularities cannot be removed and the impossibility derives from the difference of a fixed source 4-manifold. It seems very difficult to determine such an obstruction and unfortunately we cannot clarify where it is defined and how it is calculated. On one hand, we have proved that for a closed, oriented 4-manifold M^4 with 2-nd Z_2 betti number 1, every stable map f : M^4→R^3 has cusp singularities. Only known result is for a closed 4-manifold with isomorphic homology groups of the complex projective plane. Hence our result is a direct generalization this result. For example, we see that S^1×S^3#CP^2 does not admit a smooth map with only fold singularities.