Topology of symplectomorphism groups of rational ruled surfaces

Let $M$ be either $S^2\times S^2$ or the one point blow-up \cp# \bcp of $\text{cp}$. In both cases $M$ carries a family of symplectic forms ${\text{om}}_{\text{la}}$, where $\text{la}>-1$ determines the cohomology class $[{\text{om}}_{\text{la}}]$. This paper calculates the rational (co)homology of the group $G_{\text{la}}$ of symplectomorphisms of $(M,{\text{om}}_{\text{la}})$ as well as the rational homotopy type of its classifying space $B{G}_{\text{la}}$. It turns out that each group $G_{\text{la}}$ contains a finite collection $K_k,k=0,\mathrm{.}\mathrm{.}\mathrm{.},\mathrm{\ell }=\mathrm{\ell }(\text{la})$, of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups “asymptotically commute”, i.e. all the higher Whitehead products that they generate vanish as $\la\to \infty$\mathrm{.}However,foreachfixed$\la$ there is essentially one nonvanishing product that gives rise to a “jumping generator” $w_{\text{la}}$ in $H^{\ast }(G_{\text{la}})$ and to a single relation in the rational cohomology ring $H^{\ast }(B{G}_{\text{la}})$. An analog of this generator $w_{\text{la}}$ was also seen by Kronheimer in his study of families of symplectic forms on 4-manifolds using Seiberg–Witten theory. Our methods involve a close study of the space of ${\text{om}}_{\text{la}}$-compatible almost complex structures on $M$.