  We show that wave maps from Minkowski space $R^{1+n}$ to a sphere are
globally smooth if the initial data is smooth and has small norm in the
critical Sobolev space $\dot H^{n/2}$ in the high dimensional case $n \geq 5$.
A major difficulty, not present in the earlier results, is that the $\dot
H^{n/2}$ norm barely fails to control $L^\infty$, potentially causing a
logarithmic divergence in the nonlinearity; however, this can be overcome by
using co-ordinate frames adapted to the wave map by approximate parallel
transport. In the sequel of this paper we address the more interesting
two-dimensional case, which is energy-critical.
