Actions on Sn

AbstractIt is the purpose of this paper to construct some examples of Z2, S1, and SO(k) actions on Sn. Two constructions are given. The first proves the existence of S1 and Z2 actions on Sn, the fixed point set of the S1 action will be an (n − 2)-sphere, the fixed point set of the Z2 action will be an (n − 1)-sphere, the actions are free on the complement of the fixed point set, and in both cases the orbit space fails to be a manifold with boundary. The second construction will prove the existence of SO(k) actions on Sn with somewhat similar properties. Related results may be found in [1], [4], [9], [10], [13], [14] and [15]. The author wishes to thank Professor D. Montgomery for bringing these problems to his attention.The second construction is the “standard” construction used in the differentiable and generalized cohomology manifold cases to classify actions of this type. It is shown in this paper that topological actions of SO(k) on Sn can be constructed by this method which are not equivalent to differentiable actions. It follows from the first part of the paper that there exist examples of S1 (and Z2) acting on Sn which can not be constructed using this “standard” construction.It is known ([5], Chapter XV) that two topological actions of the type studied in this paper are classified by the topological type of the orbit space