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

Abstract Let M μ 0 denote S 2 × S 2 endowed with a split symplectic form $\mu \sigma \oplus \sigma $ normalized so that μ ≥1 and σ ( S 2 )=1. Given a symplectic embedding $\iota :B_{c}\hookrightarrow M^0_{\mu }$ of the standard ball of capacity c ∈(0,1) into M μ 0 , 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 B c into M μ 0 . 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 ( M μ 0 ), 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 S 2 × S 2 , 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.

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