Compositio Math. 144 (2008) 787–810 doi:10.1112/S0010437X0700334X Symplectomorphism groups and embeddings of balls into rational ruled 4-manifolds Martin Pinsonnault Abstract 0 2 2 Let Mµ denote S × S endowed with a split symplectic form µσ ⊕ σ normalized so that 2 0 µ 1andσ(S ) = 1. Given a symplectic embedding ι : Bc → Mµ of the standard ball of 0 0 capacity c ∈ (0, 1) into Mµ, consider the corresponding symplectic blow-up Mµ,c.Inthis 0 paper, we study the homotopy type of the symplectomorphism group Symp(Mµ,c)and 0 0 that of the space Emb(Bc,Mµ) of unparametrized symplectic embeddings of Bc into Mµ. Writing for the largest integer strictly smaller than µ,andλ ∈ (0, 1] for the difference µ−, we show that the symplectomorphism group of a blow-up of ‘small’ capacity c<λis 0 homotopically equivalent to the stabilizer of a point in Symp(Mµ), 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 CP 2# CP 2 obtained by blowing down 0 0 2 2 Mµ,c. It follows that, for c<λ,thespaceEmb(Bc,Mµ) is homotopy equivalent to S ×S , 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 2 2 that the embedding spaces Emb(Bc, CP )andEmb(Bc1 Bc2 , CP ), 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 and McDuff. 1. Introduction Let (X, ω) be a symplectic 4-manifold and consider the space Emb(Bc,X) of symplectic embeddings 4 2 of the standard ball Bc ⊂ R of radius r and capacity c = πr into X, endowed with its natural C∞ topology. We define the space of unparametrized symplectic embeddings by setting Emb(Bc,X):=Emb(Bc,X)/Symp(Bc). The homotopy type of this space carries important information about the diffeomorphism class of the symplectic form ω. For instance, Emb(Bc,X) is non-empty if and only if c is smaller than the Gromov width of (X, ω), and two symplectic blow-ups of (X, ω) of same capacity c are isotopic if Emb(Bc,X) is connected. It is therefore natural to investigate the homotopy type of this space, at least for some amenable families of symplectic 4-manifolds. In a previous paper with Lalonde [LP04], we proposed a general framework to study the homo- topy type of Emb(Bc,X) based on the correspondence between embeddings of balls and symplectic Received 28 March 2006, revised 1 August 2007, accepted in final form 1 October 2007, published online 14 March 2008. 2000 Mathematics Subject Classification 57R17, 57S05. Keywords: symplectomorphism groups, symplectic balls, rational ruled 4-manifolds, J-holomorphic curves. This research was funded partly by NSERC grant BP-301203-2004. This journal is c Foundation Compositio Mathematica 2008. Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.219, on 27 Sep 2021 at 19:27:35, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X0700334X M. Pinsonnault blow-ups. We showed that when Emb(Bc,X) is non-empty and connected, the natural action of the symplectomorphism group Symp(X, ω)onEmb(Bc,X) is transitive and defines a fibration Symp(Xι, Σ) → Symp(X, ω) →Emb(Bc,X)(1) whose fiber over an embedding ι : Bc → X is homotopy equivalent to the subgroup Symp(Xι, Σ) of symplectomorphisms of the symplectic blow-up along ι sending the exceptional divisor Σ to itself. We used this to investigate the space of symplectic balls in the product 0 2 2 Mµ := (S × S ,µσ⊕ σ), where σ is an area form such that σ(S2) = 1 and where the parameter µ lies in the interval [1, 2]. By computing the rational homotopy invariants of the symplectomorphism group of the blow-up 0 of Mµ at a ball of capacity c ∈ (0, 1), and using the computation of the rational cohomology ring 0 of Symp(Mµ) by Abreu and McDuff [AM00], we showed that the homotopy type of the space 0 Emb(Bc,Mµ) depends on the capacity of the ball Bc in an essential way. Theorem 1.1 (Lalonde and Pinsonnault [LP04]). 0 (i) If µ =1, then, for any c ∈ (0, 1), the embedding space Emb(Bc,Mµ) is homotopy equivalent to S2 × S2. 0 2 2 (ii) If µ ∈ (1, 2] and c ∈ (0,µ− 1),thenEmb(Bc,Mµ) is homotopy equivalent to S × S . 0 (iii) If µ ∈ (1, 2] and c ∈ [µ − 1, 1), then the rational homotopy groups of Emb(Bc,Mµ) vanish in 2 3 all dimensions except in dimensions 2, 3,and4,inwhichcaseswehaveπ2 = Q , π3 = Q ,and π4 = Q. Moreover, this space is not homotopically equivalent to a finite CW-complex. The main purpose of the present paper is the completion of these calculations for arbitrary rational ruled symplectic 4-manifolds, that is, for all symplectic S2-bundles over S2. As a byproduct, 2 2 we also describe the homotopy type of Emb(Bc, CP ) and that of the space Emb(Bc Bc , CP ) of two disjoint balls of capacities c and c in CP 2. 1.1 General setting and main results 1.1.1 Symplectic birational equivalences. In order to state our results, we first review some essential facts about rational ruled symplectic 4-manifolds and their blow-ups. To begin, recall that the classification theorem of Lalonde and McDuff [LM94] asserts that any two cohomologous symplectic forms on a ruled 4-manifold are symplectomorphic. It follows that a rational ruled symplectic 4-manifold is, after rescaling, symplectomorphic to either: 0 2 2 0 2 (a) the trivial bundle Mµ := (S ×S ,ωµ), where the symplectic area of a section S ×{∗} is µ 1 and the area of a fiber {∗} × S2 is 1; or 1 2 2 1 (b) the non-trivial bundle Mµ := (S ט S ,ωµ), where the symplectic area of a section of self- intersection −1isµ>0andtheareaofafiberis1. i i i We will denote by F the homology class of a fiber in Mµ, i ∈{0, 1},andbyB the class of a section of self-intersection −i. With this choice of normalization, the area of a fiber is less than or equal to the area of any embedded symplectic sphere of non-negative self-intersection. It follows i i i that the Gromov width of Mµ is always 1 = ωµ(F ), so that the embedding space Emb(Bc,Mµ) is non-empty if and only if c ∈ (0, 1). i In [McD96] McDuff showed that the symplectomorphism type of a symplectic blow-up of Mµ along an embedded ball of capacity c ∈ (0, 1) depends only on the capacity c, and not on the particular embedding used in the construction (this is often referred to as ‘the uniqueness of sym- i i i plectic blow-ups’ for Mµ). Hereafter we will write Mµ,c for the symplectic blow-up of Mµ at a ball of capacity c,Σi for the corresponding exceptional divisor, and Ei =[Σi] for its homology class. 788 Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.219, on 27 Sep 2021 at 19:27:35, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X0700334X Symplectomorphisms and embeddings Now recall that the (smooth) blow-up of M 0 = S2 × S2 can be identified with the blow-up of M 1 = S2ט S2 via a diffeomorphism which preserves the fibers away from the exceptional loci. At 0 1 the homology level, this identification induces an isomorphism H2(M ; Z) H2(M ; Z)whichacts on the classes {B0,F0,E0} and {B1,F1,E1} as follows: F 1 ←→ F 0, B1 + F 1 − E1 ←→ B0, F 1 − E1 ←→ E0, (2) B1 ←→ B0 − E0, E1 ←→ F 0 − E0. Note that M0 contains exactly three exceptional classes (that is, classes represented by smooth embedded spheres of self-intersection −1), namely E0, F 0 − E0,andB0 − E0. The blow-down of an exceptional curve representing either F 0 − E0 or B0 − E0 defines a manifold diffeomorphic to the non-trivial bundle S2ט S2 while the blow-down of a curve in class E0 yields S2 × S2.Inparticular, blowing up S2 × S2 and then blowing down along a curve representing F 0 − E0 defines a birational equivalence S2 × S2 S2ט S2 which preserves the fibers. When one considers this birational equivalence in the symplectic category, the uniqueness of 0 symplectic blow-ups implies that the blow-up of Mµ at a ball of capacity c ∈ (0, 1) is symplecto- 1 morphic to the blow-up of Mµ−c at a ball of capacity 1 − c.Conversely,ifµ c, the blow-up of 1 ∈ 0 − Mµ with capacity c (0, 1) is symplectomorphic to the blow-up of M1+µ−c with capacity 1 c.For i this reason, given a ruled manifold Mµ and a number c ∈ (0, 1) such that c µ,itisconvenientto i,c define the ‘c-related’ manifold Mµ by setting 1 i,c Mµ−c if i =0, Mµ := (3) 0 M1+µ−c if i =1, i i so that we can view the rational manifold Mµ,c either as a c-blow-up of Mµ or as a (1 − c)-blow-up i,c i i,c of Mµ .