  We construct a gauge theoretic change of variables for the wave map from $R
\times R^n$ into a compact group or Riemannian symmetric space, prove a new
multiplication theorem for mixed Lebesgue-Besov spaces, and show the global
well-posedness of a modified wave map equation - $n \ge 4$ - for small critical
initial data. We obtain global existence and uniqueness for the Cauchy problem
of wave maps into {\it compact} Lie groups and symmetric spaces with small
critical initial data and $n \ge 4$.
