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

1. 剪切-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。由两个g流形的不相交并导出的加法和由两个g流形的笛卡尔积导出的积，使k -等价类的集合成为一个半环。它的格罗滕迪克环被称为g流形的sk环。当G是2阶循环群时，得到了sk环可分解的一个充分必要条件。这里的可分解性是指给定的g流形与两个低于m维数的g流形之积k等价。这个条件是根据g流形及其不动点子流形的欧拉特征得到的。对于G的闭子群H， M^H表示M在受限H作用下的不动点子流形。我们已经知道了g流形及其不动点子流形的欧拉特征的各种算术同余。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M^G由有限个孤立不动点组成，则M^G的点数是偶的。此外，如果G的指标2子群H是唯一的，则不动点处的切表示作为H的表示彼此是对同构的。

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.