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

研究了闭4流形M^4与一般出现的4流形到R^3的光滑映射的奇异性之间的关系。这样的一般奇点有以下四种类型：一个确定的褶皱，一个无限的褶皱，一个尖点和一个燕尾。只有确定的折叠奇点的光滑映射称为特殊的一般映射。我们已经看到，我们可以刻画一个闭4流形，它允许一个特殊的泛映射作为这种4流形的微分同构型的充分必要条件。因此，这就产生了一个重要的问题，即人们是否可以在上述四种类型中移除哪些类型的奇点，或者找到移除这些奇点的一些障碍。我们知道，如果M^4是可定向的（安藤定理），燕尾总是可以去除的。因此，我们的问题是考虑尖点奇点和无限折叠奇点的可移性。一般情况下，不确定的折叠奇点是不能消除的，这种不可能性来源于固定源4流形的差异。似乎很难确定这样的障碍，不幸的是，我们无法澄清它是在哪里定义的以及如何计算的。一方面，我们证明了具有2阶Z_2贝数为1的闭的、定向的4流形M^4，每一个稳定映射f: M^4→R^3都有尖点奇点。唯一已知的结果是复射影平面上具有同构同调群的封闭4流形。因此我们的结果是对这个结果的直接推广。例如，我们看到S^1×S^3#CP^2不承认只有折叠奇点的光滑映射。

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.