Topology of symplectomorphism groups of rational ruled surfaces

Abstract:

Let $M$ be either $S^2\times S^2$ or the one point blow-up ${\mathbb {C}}P^2\#\overline {{\mathbb {C}}P}^2$ of ${\mathbb {C}}P^2$. In both cases $M$ carries a family of symplectic forms $\omega _{\lambda }$, where $\lambda > -1$ determines the cohomology class $[\omega _\lambda ]$. This paper calculates the rational (co)homology of the group $G_\lambda$ of symplectomorphisms of $(M,\omega _\lambda )$ as well as the rational homotopy type of its classifying space $BG_\lambda$. It turns out that each group $G_\lambda$ contains a finite collection $K_k, k = 0,\dots ,\ell = \ell (\lambda )$, 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 $\lambda \to \infty$. However, for each fixed $\lambda$ there is essentially one nonvanishing product that gives rise to a “jumping generator" $w_\lambda$ in $H^*(G_\lambda )$ and to a single relation in the rational cohomology ring $H^*(BG_\lambda )$. An analog of this generator $w_\lambda$ 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 $\omega _\lambda$-compatible almost complex structures on $M$.