One-fixed-point actions on spheres and smith sets

Let G be a finite group. The Smith equivalence for real G-modules of finite dimension gives a subset of real representation ring, called the primary Smith set. Since the primary Smith set is not additively closed in general, it is an interesting problem to find a subset which is additively closed in the real representation ring and occupies a large portion of the primary Smith set. In this paper we introduce an additively closed subset of the primary Smith set by means of smooth one-fixed-point G-actions on spheres, and we give evidences that the subset occupies a large portion of the primary Smith set if G is an Oliver group.