Spherical isotropy representations
An old question of P. A. Smith considers a finite group G acting smoothly on a closed homotopy sphere \(\Sigma\) with \(\Sigma^ G\) consisting of precisely two points and asks whether the representations of G on the tangent spaces at the two fixed points are the same. This would be true, obviously, for a linear action on a sphere. For G abelian of odd order with at least four noncyclic Sylow subgroups this paper shows that the answer is no, providing proofs and elaboration of previously announced results. For a large family of representations one has necessary and sufficient conditions that two representations in the family occur as fixed point representation after stabilization.
finite group actions on homotopy spheres
scientific article; zbMATH DE number 3938062
scientific article