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

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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..