  We prove uniqueness of solutions to the wave map equation in the natural
class, namely $ (u, \partial_t u) \in C([0,T); \dot{H}^{d/2})\times C^1([0,T);
\dot{H}^{d/2-1})$ in dimensions $d\geq 4$. This is achieved through estimating
the difference of two solutions at a lower regularity level. In order to reduce
to the Coulomb gauge, one has to localize the gauge change in suitable cones as
well as estimate the difference between the frames and connections associated
to each solutions and take advantage of the assumption that the target manifold
has bounded curvature.
