ABSTRACT
The character of the theory of group actions on manifolds differs greatly in the case of finite groups from the case of (positive dimensional) compact Lie groups. Yet, of course, there is a connection; every Lie group has many finite subgroups so one can restrict and try to (re)construct the original action. This is not a new idea. For linear actions on the sphere, for instance, the finite subgroups detect since characters do, and characters are continuous functions while the elements of finite order are dense. Many nonlinear settings have also been studied; some rather simple examples where the finite subgroups do not detect will be given shortly. Yet, the study has been rather ad hoc. One examines the theory of G-actions on M as H ranges over the subgroups of G, and computes the forgetful map’.
