The stable parametrized h-cobordism theorem provides a critical link in the chain of homotopy theoretic constructions
that show up in the classification of manifolds and their diffeomorphisms. For a compact smooth manifold M it gives a
decomposition of Waldhausen's A(M) into QM_+ and a delooping of the stable h-cobordism space of M. I will talk
about joint work with Malkiewich on this story when M is a smooth compact G-manifold.