  In this paper we prove the soliton resolution conjecture for all times, for
all solutions in the energy space, of the co-rotational wave map equation. To
our knowledge this is the first such result for all initial data in the energy
space for a wave-type equation. We also prove the corresponding results for
radial solutions, which remain bounded in the energy norm, of the cubic
(energy-critical) nonlinear wave equation in space dimension 4.
