Finite groups acting on spheres

Abstract

This paper classifies smooth actions of a finite group G acting on a sphere of sufficiently high dimension with precisely two fixed points (and the action free elsewhere) by relating the group action to an element of the group's Whitehead group. Essentially, if that element is zero, the action corresponds to a free action on a sphere of dimension one less. If not, what we call the double of the action is. An example is constructed to show that the various cases can occur.