Topology of symplectomorphism groups of rational ruled surfaces

Let M be either S × S or the one point blow-up CP # CP 2 of CP . In both cases M carries a family of symplectic forms ωλ, where λ > −1 determines the cohomology class [ωλ]. This paper calculates the rational (co)homology of the group Gλ of symplectomorphisms of (M, ωλ) as well as the rational homotopy type of its classifying space BGλ. It turns out that each group Gλ contains a finite collection Kk, k = 0, . . . , ` = `(λ), 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 λ → ∞. However, for each fixed λ there is essentially one nonvanishing product that gives rise to a “jumping generator” wλ in H (Gλ) and to a single relation in the rational cohomology ring H (BGλ). An analog of this generator wλ 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 ωλ-compatible almost complex structures on M .