  In this paper we study the topology of the space $\I_\omega$ of complex
structures compatible with a fixed symplectic form $\omega$, using the
framework of Donaldson. By comparing our analysis of the space $\I_\omega$ with
results of McDuff on the space $\cat J_\omega$ of compatible almost complex
structures on rational ruled surfaces, we find that $\I_\omega$ is contractible
in this case. We then apply this result to study the topology of the
symplectomorphism group of a rational ruled surface, extending results of Abreu
and McDuff.
