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

Let Mμ0 denote S2×S2 endowed with a split symplectic form $\mu \sigma \oplus \sigma $ normalized so that μ≥1 and σ(S2)=1. Given a symplectic embedding $\iota :B_{c}\hookrightarrow M^0_{\mu }$ of the standard ball of capacity c∈(0,1) into Mμ0, consider the corresponding symplectic blow-up $\widetilde {M}^0_{\mu ,c}$. In this paper, we study the homotopy type of the symplectomorphism group ${\mathrm {Symp}}(\widetilde {M}^0_{\mu ,c})$ and that of the space $\Im {\mathrm {Emb}}(B_{c},M^0_{\mu })$ of unparametrized symplectic embeddings of Bc into Mμ0. Writing ℓ for the largest integer strictly smaller than μ, and λ∈(0,1] for the difference μ−ℓ, we show that the symplectomorphism group of a blow-up of ‘small’ capacity c<λ is homotopically equivalent to the stabilizer of a point in Symp(Mμ0), while that of a blow-up of ‘large’ capacity c≥λ is homotopically equivalent to the stabilizer of a point in the symplectomorphism group of a non-trivial bundle $\mathbb {C}P^2\#\,\overline {\mathbb {C}P^2}$ obtained by blowing down $\widetilde {M}^0_{\mu ,c}$. It follows that, for c<λ, the space $\Im {\mathrm {Emb}}(B_{c},M^0_{\mu })$ is homotopy equivalent to S2 ×S2, while, for c≥λ, it is not homotopy equivalent to any finite CW-complex. A similar result holds for symplectic ruled manifolds diffeomorphic to $\mathbb {C}P^2\#\,\overline {\mathbb {C}P^2}$. By contrast, we show that the embedding spaces $\Im {\mathrm {Emb}}(B_{c},\mathbb {C}P^{2})$ and $\Im {\mathrm {Emb}}(B_{c_{1}}\sqcup B_{c_{2}},\mathbb {C}P^{2})$, if non-empty, are always homotopy equivalent to the spaces of ordered configurations $F(\mathbb {C}P^{2},1)\simeq \mathbb {C}P^{2}$ and $F(\mathbb {C}P^{2},2)$. Our method relies on the theory of pseudo-holomorphic curves in 4 -manifolds, on the computation of Gromov invariants in rational 4 -manifolds, and on the inflation technique of Lalonde and McDuff.