Study on characteristic numbers of G-manifolds and its fixed points submanifolds

G流形及其不动点子流形特征数研究

• 批准号: 17540082
• 负责人: KOMIYA Katsuhiro
• 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: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540082/
• 关键词: transformation group theory; G-manifold; fixed points submanifold; characteristic number; Euler characteristic; normal representation; オイラー類; Borsuk-Ulamの定理; cut and paste; SK群

1. The cut-and-paste operation defines an equivalence relation on the set of smooth G-manifolds, G a compact Lie group. This relation is called SK-equivalence. The set of SK-equivalence classes becomes a semiring with the addition induced from the disjoint union and the product induced from the Cartesian product of two of G-manifolds. Its Grothendieck ring is called the SK-ring of G-manifolds. We obtain a necessary and sufficient condition for the decomposability in the SK-ring, if G is the cyclic group of order 2. Here the decomposability means for a given G-manifold to be Sk-equivalent to the product of two G-manifolds with a lower dimension than M. The condition is obtained in terms of the Euler characteristics of G-manifolds and its fixed points submanifolds.2. For a closed subgroup H of G, M^H denotes the fixed points submanifold of M by the restricted H-action. We already know various types of arithmetic congruences for the Euler characteristics of G-manifold and its fixed points submanoifolds. Making use of these results we obtain the following facts: if G is abelian, M is of odd-dimension, and M^G is of 0-dimension, i.e., M^G consists of finite isolated fixed points, then the number of points of M^G is even. Moreover, if the index 2 subgroup H of G is unique, then the tangential representations at fixed points are pairwisely isomorphic to each other as representations of H.

1.剪切和粘贴操作定义了光滑G-流形集合上的等价关系，G是紧李群。这种关系称为SK-等价。SK-等价类的集合成为一个半环，其加法由不交并导出，乘积由两个G-流形的笛卡尔积导出。它的Grothendieck环称为G-流形的SK-环。当G是2阶循环群时，得到了SK-环可分解的一个充要条件。这里的可分解性意味着给定的G-流形是SK-等价于两个维数低于M的G-流形的乘积。利用G-流形及其不动点子流形的欧拉特征线得到了该条件.对于G的闭子群H，M^H表示M的受限制H-作用的不动点子流形。我们已经知道G-流形及其子流形的不动点的Euler特征线的各种算术同余。利用这些结果，我们得到如下事实：如果G是阿贝尔的，M是奇维的，M^G是0维的，即，若M^G由有限个孤立不动点组成，则M^G的点数为偶数。此外，如果G的指数为2的子群H是唯一的，则不动点处的切表示作为H的表示彼此成对同构。

项目成果

Stable homotopy groups of spheres and higher singularities

球体的稳定同伦群和更高奇点

• 发表时间: 2006
• 期刊: Journal of Mathematics of Kyoto University 46
• 作者: N.Iwase;M.Mimura;T.Nishimoto;S.Okuyama;島川和久;Kohhei Yamaguchi;Kohhei Yamaguchi;Shigo Okuyama;Yoshifumi Ando;Yoshifumi Ando
• 通讯作者: Yoshifumi Ando

Nonexistence of homotopy equivalences which are C^∽ stable or of finite codimension

不存在 C^∽ 稳定或有限余维的同伦等价

• 发表时间: 2006
• 期刊: Topology and its applications 135
• 作者: N.Iwase;M.Mimura;T.Nishimoto;S.Okuyama;島川和久;Kohhei Yamaguchi;Kohhei Yamaguchi;Shigo Okuyama;Yoshifumi Ando;Yoshifumi Ando;Yasuyuki Miyazawa;Y.Ando
• 通讯作者: Y.Ando

MAGNETIC GRAPHS AND AN INVARIANT FOR VIRTUAL LINKS

• DOI: 10.1142/s0218216506005135
• 发表时间: 2006-12
• 期刊: Journal of Knot Theory and Its Ramifications
• 影响因子: 0.5
• 作者: Y. Miyazawa

Symmetric submanifolds associated with irreducible symmetric R-spaces

• DOI: 10.1007/s00208-005-0646-2
• 发表时间: 2005-04
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: J. Berndt;J. Eschenburg;H. Naitoh;K. Tsukada

Nonexistence of homotopy equivalences which are C^∞ stable or of finite codimension

不存在 C^∞ 稳定或有限余维的同伦等价

• 发表时间: 2006
• 期刊: Topology and its applications 153
• 作者: N.Iwase;M.Mimura;T.Nishimoto;S.Okuyama;島川和久;Kohhei Yamaguchi;Kohhei Yamaguchi;Shigo Okuyama;Yoshifumi Ando
• 通讯作者: Yoshifumi Ando