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

Let $X$ be any rational ruled symplectic four-manifold. Given a symplectic embedding \iota:B_{c}\into X of the standard ball of capacity $c$ into $X$, consider the corresponding symplectic blow-up \tX_{\iota}. In this paper, we study the homotopy type of the symplectomorphism group \Symp(\tX_{\iota}), simplifying and extending the results of math.SG/0207096. This allows us to compute the rational homotopy groups of the space \IEmb(B_{c},X) of unparametrized symplectic embeddings of $B_{c}$ into $X$. We also show that the embedding space of one ball in $CP^2$, and the embedding space of two disjoint balls in $CP^2$, 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.Comment: New title, new abstract, content now agrees with the published version, small correction to the proof of Theorem 1.10. A sequel to the paper SG/020709

The topology of the space of symplectic balls in rational 4-manifolds

We study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball $B^4(c) \subset \R^4$ into 4-dimensional rational symplectic manifolds. We compute the rational homotopy groups of that space when the 4-manifold has the form $M_{\lambda}= (S^2 \times S^2, \mu \omega_0 \oplus \omega_0)$ where $\omega_0$ is the area form on the sphere with total area 1 and $\mu$ belongs to the interval $[1,2]$. We show that, when $\mu$ is 1, this space retracts to the space of symplectic frames, for any value of $c$. However, for any given $1 < \mu < 2$, the rational homotopy type of that space changes as $c$ crosses the critical parameter $c_{crit} = \mu - 1$, which is the difference of areas between the two $S^2$ factors. We prove moreover that the full homotopy type of that space changes only at that value, i.e the restriction map between these spaces is a homotopy equivalence as long as these values of $c$ remain either below or above that critical value.Comment: Typos corrected, 2 minor corrections in the text. Numbering consistant with the published versio

The homotopy type of the space of symplectic balls in rational ruled 4-manifolds

Let M:=(M^{4},\om) be a 4-dimensional rational ruled symplectic manifold and denote by w_{M} its Gromov width. Let Emb_{\omega}(B^{4}(c),M) be the space of symplectic embeddings of the standard ball B^4(c) \subset \R^4 of radius r and of capacity c:= \pi r^2 into (M,\om). By the work of Lalonde and Pinsonnault, we know that there exists a critical capacity \ccrit \in (0,w_{M}] such that, for all c\in(0,\ccrit), the embedding space Emb_{\omega}(B^{4}(c),M) is homotopy equivalent to the space of symplectic frames \SFr(M). We also know that the homotopy type of Emb_{\omega}(B^{4}(c),M) changes when c reaches \ccrit and that it remains constant for all c \in [\ccrit,w_{M}). In this paper, we compute the rational homotopy type, the minimal model, and the cohomology with rational coefficients of \Emb_{\omega}(B^{4}(c),M) in the remaining case c \in [\ccrit,w_{M}). In particular, we show that it does not have the homotopy type of a finite CW-complex.Comment: 38 pages; revised versio

Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces

Let $M=(M,\omega)$ be either the product $S^2\times S^2$ or the non-trivial $S^2$ bundle over $S^2$ endowed with any symplectic form $\omega$. Suppose a finite cyclic group $Z_n$ is acting effectively on $(M,\omega)$ through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism $Z_n\hookrightarrow Ham(M,\omega)$. In this paper, we investigate the homotopy type of the group $Symp^{Z_n}(M,\omega)$ of equivariant symplectomorphisms. We prove that for some infinite families of $Z_n$ actions satisfying certain inequalities involving the order $n$ and the symplectic cohomology class $[\omega]$, the actions extends to either one or two toric actions, and accordingly, that the centralizers are homotopically equivalent to either a finite dimensional Lie group, or to the homotopy pushout of two tori along a circle. Our results rely on $J$-holomorphic techniques, on Delzant's classification of toric actions, on Karshon's classification of Hamiltonian circle actions on $4$-manifolds, and on the Chen-Wilczy\'nski classification of smooth $Z_n$-actions on Hirzebruch surfaces.Comment: 36 pages. Initial release. Comments welcom

Embeddings of symplectic balls into the complex projective plane

We investigate spaces of symplectic embeddings of $n\leq 4$ balls into the complex projective plane. We prove that they are homotopy equivalent to explicitly described algebraic subspaces of the configuration spaces of $n$ points. We compute the rational homotopy type of these embedding spaces and their cohomology with rational coefficients. Our approach relies on the comparison of the action of $\mathrm{PGL}(3,\mathbb{C})$ on the configuration space of $n$ ordered points in $\mathbf{CP}^2$ with the action of the symplectomorphism group $\mathrm{Symp}(\mathbf{CP}^2)$ on the space of $n$ embedded symplectic balls.Comment: 47 pages. The revisions made in this second version significantly enhance the clarity and coherence of the exposition. The main results are the sam