Global Theory of Singularities from the Viewpoint of Homotopy Theory

同伦论视角下的全局奇点理论

基本信息

批准号：
16340018
负责人：
SAEKI Osamu
金额：
$ 4.22万
依托单位：
KYUSHU UNIVERCITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

The study of differentiable maps between manifolds and their singularities was begun by Whitney, Thom etc. in the middle of the 20th century and has developed a lot since then. In particular, local properties have been studied and some sophisticated theories have been established for such studies. However, global properties of maps that are essentially related to the structures of manifolds have not been studied so much in spite of their importance. In this research project, we aimed at solving various important open problems in the global theory of singularities from the viewpoint of homotopy theory in a larger framework. More precisely, we performed the following studies and obtained some results, which will be explained below :(a) Higher obstructions,(b) Classifying space of singular fibers,(c) Relationship between the differentiable structures of manifolds and singularities of maps.As to (a), Saeki and Iwase studied differentiable maps whose regular fibers consist of spheres, and obtained some homotopical properties of those manifolds which admit such maps. Furthermore, it has been clarified that these properties are related to higher obstructions to the existence of such maps. Moreover, Saeki and Sakuma studied the existence problem of fold maps and found that certain Postnikov invariants appear as higher obstructions. As to (b), Saeki constructed characteristic classes of surface bundles by using the theory of singular fibers of functions on surfaces. Furthermore, Saeki and Ohmoto succeeded in constructing a classifying space of singular fibers. As to (c), Saeki and Sakuma collected and arranged the known results about the relationship between the singularities of differentiable maps and differentiable structures of manifolds, and Saeki showed that in certain cases the elimination of definite fold singularities is possible independently of the differentiable structures of manifolds.

流形及其奇点之间的可微映射的研究是由Whitney、Thom等人在20世纪中叶开始的，此后得到了很大的发展。特别是，对局部特性进行了研究，并为此类研究建立了一些复杂的理论。然而，尽管本质上与流形结构相关的映射的全局属性很重要，但尚未得到如此多的研究。在这个研究项目中，我们旨在从更大框架中的同伦理论的角度解决全局奇点理论中的各种重要的开放性问题。更准确地说，我们进行了以下研究并获得了一些结果，这些结果将在下面解释：（a）更高的障碍物，（b）奇异纤维的分类空间，（c）流形的可微结构与映射的奇异性之间的关系。对于(a)，Saeki和Iwase研究了其规则纤维由球体组成的可微映射，并获得了允许此类映射的流形的一些同伦性质。此外，已经澄清这些属性与此类地图存在的较高障碍有关。此外，Saeki和Sakuma研究了折叠图的存在问题，发现某些Postnikov不变量表现为更高的障碍物。对于(b)，佐伯利用曲面函数奇异纤维理论构造了曲面丛的特征类。此外，佐伯和大本成功地构建了奇异纤维的分类空间。对于(c)，Saeki和Sakuma收集并整理了关于可微映射的奇点与流形的可微结构之间关系的已知结果，并且Saeki表明，在某些情况下，消除确定的折叠奇点是可能的，与可微结构无关流形。

项目成果

期刊论文数量（83）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Calculus

DOI：
10.1007/978-1-4471-0459-9_1
发表时间：
2020-08
期刊：
Bicycle or Unicycle?
影响因子：
0
作者：
Cheng-Shang Chang
通讯作者：
Cheng-Shang Chang

First order local invariants of apparent contours

表观轮廓的一阶局部不变量

DOI：
发表时间：
2005
期刊：
Topology (印刷中)
影响因子：
0
作者：
F.Aicardi;T.Ohmoto
通讯作者：
T.Ohmoto

L-S categories of simply-connected compact simple Lie groups of low rank

低阶单连紧单李群的 L-S 类

DOI：
发表时间：
2004
期刊：
Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001), Progr. Math. 215
影响因子：
0
作者：
L.A.Lucas;O.Saeki;Benaissa Bernoussi et al.;Osamu Saeki et al.;Walter Motta et al.;Benaissa Bernoussi et al.;Benaissa Bernoussi et al.;V.Blanl〓il et al.;Norio Iwase et al.
通讯作者：
Norio Iwase et al.

Codimension one embeddings of product of three spheres

三球体乘积的余维一嵌入

DOI：
发表时间：
2005
期刊：
Topology and its Applications 146-147
影响因子：
0
作者：
Y.;Taniguchi;山田道夫(共著);T.Ohmoto et al.;T.Ohmoto;N.Iwase et al.;L.A.Lucas et al.
通讯作者：
L.A.Lucas et al.

Concordance des n〓uds de dimension 4

维度 4 的索引

DOI：
发表时间：
2007
期刊：
Canadian Mathematical Bulletin (in press)
影响因子：
0
作者：
加藤忠明;西牧謙吾;原田正平編著;M. Yamada;Y. Taniguchi;Y. -Y. Hayashi;S. Takehiro;V.Blanl〓il et al.
通讯作者：
V.Blanl〓il et al.