Pseudofree group actions on spheres

Abstract:

R. S. Kulkarni showed that a finite group acting pseudofreely, but not freely, preserving orientation, on an even-dimensional sphere (or suitable sphere-like space) is either a periodic group acting semifreely with two fixed points, a dihedral group acting with three singular orbits, or one of the polyhedral groups, occurring only in dimension 2. It is shown here that the dihedral group does not act pseudofreely and locally linearly on an actual $n$-sphere when $n\equiv 0\mod 4$. It is also shown that the dihedral group does act pseudofreely and locally linearly, with three singular orbits, on an $n$-manifold when $n\equiv 2\mod 4$. Orientation-reversing actions are also considered.