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。剪切操作定义了平滑的G-manifolds集合的等效关系，g一个紧凑的谎言组。这种关系称为SK等效性。 Sk-queragence类别的集合成为了一个半度性，并从脱节联合引起的添加以及由两个G-manifolds的笛卡尔产物引起的产物。它的Grothendieck戒指称为G-Manifolds的Sk-Ring。如果G是循环第2阶的环状组，我们就会获得必要且充分的条件，以使给定的G-manifold与两个尺寸较低的G-manifolds的产物相等。与M相当。对于G的闭合子组H，M^H表示由限制的H-Action表示M的固定点子量。我们已经知道各种类型的算术一致性对于G-manifold及其固定点submanoifolds的Euler特性。利用这些结果，我们得到以下事实：如果G是Abelian，则M具有奇数，M^g为0维，即M^g由有限的隔离固定点组成，则M^G的点数均匀。此外，如果G的索引2子组H是唯一的，则固定点处的切向表示形式是彼此作为H的表示形式的。

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

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

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

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