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Symplectomorphism groups and embeddings of balls into rational ruled 4-manifolds
Published online by Cambridge University Press: 01 May 2008
Abstract
Let Mμ0 denote S2×S2 endowed with a split symplectic form normalized so that μ≥1 and σ(S2)=1. Given a symplectic embedding
of the standard ball of capacity c∈(0,1) into Mμ0, consider the corresponding symplectic blow-up
. In this paper, we study the homotopy type of the symplectomorphism group
and that of the space
of unparametrized symplectic embeddings of Bc into Mμ0. 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(Mμ0), 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
obtained by blowing down
. It follows that, for c<λ, the space
is homotopy equivalent to S2 ×S2, while, for c≥λ, it is not homotopy equivalent to any finite CW-complex. A similar result holds for symplectic ruled manifolds diffeomorphic to
. By contrast, we show that the embedding spaces
and
, if non-empty, are always homotopy equivalent to the spaces of ordered configurations
and
. 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.
MSC classification
- Type
- Research Article
- Information
- Copyright
- Copyright © Foundation Compositio Mathematica 2008
References
This research was funded partly by NSERC grant BP-301203-2004.
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