Symplectomorphism groups and embeddings of balls into rational ruled 4-manifolds
Published in Compositio Mathematica, 2006
Recommended citation: Pinsonnault, M., Symplectomorphism groups and embeddings of balls into rational ruled 4-manifolds. Compos. Math. 144 (2008), no.3, 787–810. https://doi.org/10.1112/S0010437X0700334X
Let $X$ be any rational ruled symplectic four-manifold. Given a symplectic embedding $\iota:B_{c}\hookrightarrow X$ of the standard ball of capacity $c$ into $X$, consider the corresponding symplectic blow-up $\tilde X_{\iota}$. In this paper, we study the homotopy type of the symplectomorphism group $\mathrm{Symp}(tildeX_{\iota})$, simplifying and extending the results of math.SG/0207096. This allows us to compute the rational homotopy groups of the space $\Im\mathrm{Emb}(B_{c},X)$ of unparametrized symplectic embeddings of $B_{c}$ into $X$. We also show that the embedding space of one ball in $\mathbf{C}P^2$, and the embedding space of two disjoint balls in $\mathbf{C}P^2$, 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.