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

基于Morse理论推广的4流形微分同胚类型研究

基本信息

批准号：
11640096
负责人：
SAKUMA Kazuhiro
金额：
$ 0.77万
依托单位：
Kinki University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

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.

研究了闭4-流形M^4与一般出现的4-流形到R^3的光滑映射的奇点之间的关系。这种类属奇点有以下四种类型：确定的折叠、不确定的折叠、尖点和燕尾。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，我们可以刻画一个封闭的4-流形，它允许一个特殊的一般映射作为这种4-流形的微分同胚型的充要条件。因此，能否消除上述四种奇点中的哪种奇点，或找到消除这些奇点的障碍，就成了一个重要的问题。众所周知，如果M^4是可定向的(Ando定理)，则燕尾总是可以去掉的。因此，我们的问题是考虑尖点奇点和不定折叠奇点的可去性。一般情况下，不定折叠奇点是不能消除的，这种不可能性源于固定源4-流形的不同。似乎很难确定这种障碍，不幸的是，我们不能澄清它是在哪里定义的，以及它是如何计算出来的。一方面，我们证明了对于2阶Z_2Betti数为1的闭的定向4-流形M^4，每个稳定映射f：M^4→R^3都有尖点奇点。唯一已知的结果是具有复射影平面的同构同调群的闭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不允许只有折叠奇点的光滑映射。