Equivariant periodicity for compact group actions

For a manifold M, the structure set S(M, rel partial derivative) is the collection of manifolds homotopy equivalent to M relative to the boundary. Siebenmann [3] showed that in the topological category, the structure set is 4-periodic: S(M, rel partial derivative) similar or equal to S(M x D-4, rel partial derivative) up to a copy of Z. The periodicity has been extended to topological manifolds with hornotopically stratified group actions for various representations in place of D-4, including twice any complex representation of a compact abelian group. In this paper, we extend the result to twice any complex representation of a compact Lie group. We also prove the bundle version of the periodicity.