2006/03/31 by Martin Pinsonnault · 2 citations
Mathematics · #Analytic and geometric function theory #Diffeomorphism #Geometric and Algebraic Topology #Geometry and complex manifolds #Group (periodic table) #Homotopy #Homotopy group #Symplectic geometry #Symplectic manifold #Symplectic vector space #Symplectomorphism #math.DG #math.SG #msc:53D35 #msc:55R20 #msc:57R17 #msc:57S05
paper · pdf · doi:10.1112/s0010437x0700334x
published as Compositio Math. 144 (2008) 787-810 · 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/0207096
openalex publication_date 2008/03/14 · arxiv created 2009/05/15 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Abstract Let M μ 0 denote S 2 × S 2 endowed with a split symplectic form μ σ ⊕ σ normalized so that μ ≥1 and σ ( S 2 )=1. Given a symplectic embedding ι :Bc\hookrightarrow M0μ of the standard ball of capacity c ∈(0,1) into M μ 0 , consider the corresponding symplectic blow-up \widetilde M0μ ,c . In this paper, we study the homotopy type of the symplectomorphism group \mathrm Symp(\widetilde M0μ ,c) and that of the space \Im \mathrm Emb(Bc,M0μ ) of unparametrized symplectic embeddings of B c into M μ 0 . 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 μ 0 ), 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 \mathbb CP2# \mathbb CP2 obtained by blowing down \widetilde M0μ ,c . It follows that, for c < λ , the space \Im \mathrm Emb(Bc,M0μ ) 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 \mathbb CP2# \mathbb CP2 . By contrast, we show that the embedding spaces \Im \mathrm Emb(Bc,\mathbb CP2) and \Im \mathrm Emb(B_c1\sqcup B_c2,\mathbb CP2) , if non-empty, are always homotopy equivalent to the spaces of ordered configurations F(\mathbb CP2,1)≃ \mathbb CP2 and F(\mathbb CP2,2) . 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.