In this article, we study the topology of the space $\mathcal{I}_\omega$ of complex structures compatible with a fixed symplectic form $\omega$ , using the framework of Donaldson. By comparing our analysis of the space $\mathcal{I}_\omega$ with results of McDuff on the space $\mathcal{J}_\omega$ of compatible almost complex structures on rational ruled surfaces, we find that $\mathcal{I}_\omega$ 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