2008/07/31 by Sı́lvia Anjos, Silvia Anjos, François Lalonde +2
Mathematics · #Advanced Algebra and Geometry #Ball (mathematics) #Cohomology #Combinatorics #Embedding #Geometric and Algebraic Topology #Homotopy #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematics #Omega #Physics #Pure mathematics #Symplectic geometry #Type (biology) #math.SG #msc:53D35 #msc:55R20 #msc:57R17 #msc:57S05
paper · pdf · doi:10.2140/gt.2009.13.1177
published as Geom. Topol. 13 (2009) 1177-1227 · 38 pages; revised version
arxiv created 2008/11/03 · openalex publication_date 2009/02/05 · arxiv updated 2014/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
be a 4-dimensional rational ruled symplectic manifold and denote by w M its Gromov width. Let Emb ! .B 4 .c/; M / be the space of symplectic embeddings of the standard ball of radius r , B 4 .c/ R 4 (parametrized by its capacity c WD r 2 ), into .M; !/. By the work of Lalonde and Pinsonnault [13], we know that there exists a critical capacity c crit 2 .0; w M such that, for all c 2 .0; c crit /, the embedding space Emb ! .B 4 .c/; M / is homotopy equivalent to the space of symplectic frames SFr.M /. We also know that the homotopy type of Emb ! .B 4 .c/; M / changes when c reaches c crit and that it remains constant for all c 2 OEc crit ; w M /. In this paper, we compute the rational homotopy type, the minimal model and the cohomology with rational coefficients of Emb ! .B 4 .c/; M / in the remaining case c 2 OEc crit ; w M /. In particular, we show that it does not have the homotopy type of a finite CW-complex. Some of the key points in the argument are the calculation of the rational homotopy type of the classifying space of the symplectomorphism group of the blow up of M , its comparison with the group corresponding to M and the proof that the space of compatible integrable complex structures on the blow up is weakly contractible.