On the space of harmonic $2$-spheres in $\mathbf{C}{P}^2$

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 $(j\ge 2)$. 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}.