Homotopy theoretic study of singularities of smooth mappings

光滑映射奇点的同伦理论研究

• 批准号: 11640081
• 负责人: ANDO Yoshifumi
• 金额: $ 2.18万
• 依托单位: Yamaguchi 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640081/
• 关键词: smooth maps; singularity; fold map; Morin map; jet space; homotopy; singularity; 折り目特異点; 安定ホモトピー; ジェット空間; 微分可能写像; ホモトピー類; 障害類

In this research we have studied global properties of simple singularities of smooth maps, in particular, fold singularities and Morin singularities from the viewpoint of homotopy theory. We have studied the set of fold-eobordism classes of fold-maps of degree m, and established a relation between this cobordism set and the set of homotopy classes to the space of base point preserving maps of degree m of spheres. In this study a homotopy principle for Morin singularities, which has been established in our research, has played an important role.Furthermore, we have determinod the homotopy type of the subspace in 2 jet space associated with fold maps. This reduces the existence problem of fold-maps toa homotopy theoretic problem of finding sections of fiber bundles.

在这项研究中，我们从同伦理论的角度研究了平滑映射的简单奇点的全局性质，特别是折叠奇点和莫林奇点。我们研究了m度折叠图的折叠共棱类集合，并建立了该共棱集合和同伦类集合与m度球面基点保留映射空间之间的关系。在本研究中，我们的研究中建立的莫林奇点的同伦原理发挥了重要作用。此外，我们还确定了与折叠图相关的2射流空间中的子空间的同伦类型。这将折叠图的存在问题简化为寻找纤维束截面的同伦理论问题。

项目成果

Tadashi Watanabe: "Approximate resolution of uniform spaces"Topology and its applications. (to appear). (2000)

Tadashi Watanabe：“均匀空间的近似分辨率”拓扑及其应用。

Y.Miyazawa: "The second and the third terms of the Homfly polynomial of link"Kobe J.Math.. 16. 147-159 (1999)

Y.Miyazawa：“Homfly 链接多项式的第二项和第三项”Kobe J.Math.. 16. 147-159 (1999)

Y.Ando: "On holomorphic maps with only fold singularities"Nagoya Math.J. (印刷中).

Y.Ando：“仅具有折叠奇点的全纯映射”Nagoya Math.J（正在出版）。

Y.Miyazawa(T.Kanenobu): "The second and third terms of Homfly polynomial of a link"Kobe Journal Math.. 16(2)(to appear).

Y.Miyazawa(T.Kanenobu)：“链接的 Homfly 多项式的第二项和第三项”Kobe Journal Math.. 16(2)（待出现）。

Y.Ando: "Folding maps and the surgery theory on manifolds"J.Math.Soc.Japan. 53 (to appear). (2001)

Y.Ando：“折叠地图和流形手术理论”J.Math.Soc.Japan。