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

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，则在不动点处的切表示作为H的表示彼此成对同构。

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.