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

Let M 0 µ denote S² × S² endowed with a split symplectic form µσ⊕σ normalized so that µ ≥ 1 and σ(S 2) = 1. Given a symplectic embedding ι: Bc ֒ → M 0 µ of the standard ball of capacity c ∈ (0,1) into M 0 µ, consider the corresponding symplectic blow-up f M 0 µ,c. In this paper, we study the homotopy type of the symplectomorphism group Symp ( f M 0 µ,c) and that of the space ℑEmb(Bc, M 0 µ) of unparametrized symplectic embeddings of Bc into M 0 µ. 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 0 µ), 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 nontrivial bundle CP 2 # CP 2 obtained by blowing down f M 0 µ,c. It follows that for c < λ, the space ℑEmb(Bc, M 0 µ) 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 CP 2 # CP 2. By contrast, we show that the embedding spaces ℑEmb(Bc, CP 2) and ℑEmb(Bc1 ⊔Bc2, CP 2), if non empty, are always homotopy equivalent to the spaces of ordered configurations F(CP 2,1) ≃ CP 2 and F(CP 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-McDuff.

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