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

1.剪切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。由G-流形的不相交并诱导的加法使SK-等价类的集合成为一个半群。它的Grothendieck群称为G-流形的SK-群。当G是2阶循环群或奇数阶有限交换群时，我们得到了SK-群可除的一个充要条件。用G-流形的不动点集的欧拉特征刻画了这一条件。2.闭光滑流形与圆周上纤维丛的全空间有余界的充要条件在我们的研究中推广到等变情形。3.群G的两个线性表示称为Smith等价，如果这两个表示可以出现在具有恰好两个不动点的同伦G-球面的不动点上的切线表示上。关于哪些群有，哪些群没有非同构的Smith等价表示的问题，已经有大量的文献。他们中的一些人给出了肯定的答案，一些人给出了否定的答案。在我们的研究中，我们表明，如果我们将注意力限制在指标2的子群的受限动作上，则问题是肯定的。

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..