Topology of symplectomorphism groups of rational ruled surfaces

Abstract

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