Let $X$ be any rational ruled symplectic four-manifold. Given a symplectic embedding $\iota:B_{ c}\into X$ of the standard ball of capacity $c$ into $X$, consider the corresponding symplectic blow-up $\tX_{ \iota}$. In this paper, we study the homotopy type of the symplectomorphism group $\Symp(\tX_{ \iota})$, 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 $CP^2$, and the embedding space of two disjoint balls in $CP^2$, 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.
Categories: math.SG math.DG
