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

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

2006-03-13

arxiv.org/abs/math/0603310v2

Compositio Math. 144 (2008) 787-810

Mathematics

Symplectic Geometry

New title, new abstract, content now agrees with the published version, small correction to the proof of Theorem 1.10. A seque

Scientific paper

10.1112/S0010437X0700334X

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.

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