Symplectomorphism groups and embeddings of balls into rational ruled 4-manifolds

Abstract

Let $X$ be any rational ruled symplectic four-manifold. Given a symplectic embedding $ι:B_{c}\into X$ of the standard ball of capacity $c$ into $X$, consider the corresponding symplectic blow-up $\tX_ι$. In this paper, we study the homotopy type of the symplectomorphism group $\Symp(\tX_ι)$, simplifying and extending the results of math.SG/0207096. This allows us to compute the rational homotopy groups of the space $\IEmb(B_{c},X)$ of unparametrized symplectic embeddings of $B_c$ into $X$. We also show that the embedding space of one ball in $C{P}^2$, and the embedding space of two disjoint balls in $C{P}^2$, if non empty, are always homotopy equivalent to the corresponding spaces of ordered configurations. Our method relies on the theory of pseudo-holomorphic curves in 4-manifolds, on the theory of Gromov invariants, and on the inflation technique of Lalonde-McDuff.

New title, new abstract, content now agrees with the published version, small correction to the proof of Theorem 1.10. A sequel to the paper SG/0207096