  Carrying further work of T.A. Crawford, we show that each component of the
space of harmonic maps from the $2$-sphere to complex projective $2$-space of
degree $d$ and energy $4 \pi E$ is a smooth closed submanifold of the space of
all $C^j$ maps $(j \geq 2)$. We achieve this by showing that the Gauss
transform which relates them to spaces of holomorphic maps of given degree and
ramification index is {\bf smooth} and has {\bf injective differential}.
