Research on families of fixed point sets of G-manifolds in transformation group theory

变换群理论中G流形不动点集族的研究

• 批准号: 15540079
• 负责人: KOMIYA Katsuhiro
• 金额: $ 2.3万
• 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: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540079/
• 关键词: cut-and-paste; SK-equivalence; SK-group; G-manifolds; fixed points; Euler characteristics; fibring over the circle; cobordism; Smith同値; 接表現; コボルディズム; 不動点多様体

1.The cut-and-paste operation defines an equivalence relation on the set of smooth G-manifolds. This relation is called SK-equivalence. The set of SK-equivalence classes becomes a semigroup with the addition induced from the disjoint union of G-manifolds. Its Grothendieck group is called the SK-group of G-manifolds. We obtain a necessary and sufficient condition for the divisibility in the SK-group, if G is the cyclic group of order 2, or a finite abelian group of odd order. The condition is described in terms of the Euler characteristics of fixed point sets of G-manifolds.2.A necessary and sufficient condition for a closed smooth manifold to be cobordant to the total space of fiber bundle over the circle is well-known In our research we extend this (non-equivariant) result to the equivariant case by making use of the result stated above..3.Two linear representations of a group G are called Smith equivalent., if those two representations can occur as the tangential representations at fixed points of a homotopy G-sphere with exactly two fixed points. There are vast literatures on the question of which groups do and which groups do not have non-isomorphic Smith equivalent representations. Some of them give an affirmative answer and some of them give a negative answer. In our research we show that the question is affirmative if we restrict our attention to the restricted actions of a subgroup of index 2..

1.剪切操作定义了平滑的G-manifolds集合的等价关系。这种关系称为SK等效性。 Sk-queralence类的集合成为了G-Manifolds脱节结合引起的添加的半群。它的Grothendieck集团被称为G-Manifolds的Sk-Group。如果g是2的循环群，或者是有限的奇数秩序组，则我们获得了Sk-Group中分裂性的必要条件。该疾病是根据G-Manifold的固定点集的欧拉特征来描述的。2。封闭的平滑歧管的必要和足够条件与圆圈的整个纤维束的总空间保持在我们的研究中是众所周知的。作为在同质g-Sphere的固定点上的切向表示，正好有两个固定点。关于哪个群体的问题以及哪些群体没有非属性史密斯的等效表示，有很多文献。他们中的一些人给出了肯定的答案，其中一些给出了负面答案。在我们的研究中，我们表明，如果我们将注意力限制在索引2子组的受限行动中，这个问题是肯定的。

项目成果

The divisibility in the cut-and-paste group of G-manifolds and fibring over the circle within a cobordism class

共边类中 G 流形和圆上的纤维的剪切和粘贴组的可分性

• 发表时间: 2005
• 期刊: Osaka Journal of Mathematics 42(1)(未定)
• 作者: Katsuhiro Komiya

Existence theorem of fold maps

折叠图的存在定理

• 发表时间: 2004
• 期刊: Japanese Journal of Mathematics 30
• 作者: Yoshifumi Ando;Yoshifumi Ando
• 通讯作者: Yoshifumi Ando

Katsuhiro Komiya: "Cutting and pasting of families of submanifolds modeled on Z-2 manifolds"Tokyo Journal of Mathematics. 26. 403-411 (2003)

Katsuhiro Komiya：“剪切和粘贴以 Z-2 流形为模型的子流形族”《东京数学杂志》。

Existence theorems of fold-maps

折叠图的存在定理

• 发表时间: 2004
• 期刊: Japanese Journal of Mathematics 30
• 作者: Yoshifumi Ando;Yoshifumi Ando;Yoshifumi Ando;Yoshifumi Ando;Yoshifumi Ando
• 通讯作者: Yoshifumi Ando

Cutting and pasting of manifolds into G-manifolds

将歧管剪切并粘贴到 G 歧管中

• 发表时间: 2003
• 期刊: Kodai Mathematical Journal 26
• 作者: Yoshifumi Ando;Yoshifumi Ando;Yoshifumi Ando;Yoshifumi Ando;Yoshifumi Ando;Katsuhiro Komiya
• 通讯作者: Katsuhiro Komiya