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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

基本信息

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

项目摘要

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.剪切和粘贴操作定义了光滑G流形集合上的等价关系。这种关系称为SK-等价。SK-等价类的集合成为具有由G-流形的不交并导出的加法的半群。它的Grothendieck群被称为G-流形的SK-群。当G是2阶循环群或奇数阶有限交换群时，得到了SK-群可除的一个充要条件。2.闭光滑流形与圆上纤维丛的全空间共边的一个充要条件是众所周知的，本文利用上述结果将这个（非等变）结果推广到等变情形. 3.群G的两个线性表示称为Smith等价群。如果这两个表示可以作为切向表示出现在具有恰好两个不动点的同伦G球的不动点处。有大量的文献的问题，哪些群体和哪些群体没有非同构的史密斯等价表示。有些人给出肯定的回答，有些人给出否定的回答。在我们的研究中，我们表明，这个问题是肯定的，如果我们限制我们的注意力的指数2的子群的限制行动。