Surgery obstructions on closed manifolds and the Inertia subgroup

The Wall surgery obstruction groups have two interesting geometrically defined subgroups, consisting of the surgery obstructions between closed manifolds, and the inertial elements. We show that the inertia group $I_{n+1}(\pi,w)$ and the closed manifold subgroup $C_{n+1}(\pi,w)$ are equal in dimensions $n+1\geq 6$, for any finitely-presented group $\pi$ and any orientation character $w\colon \pi \to \cy 2$.