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

Sílvia Anjos, François Lalonde and Martin Pinsonnault

Geometry & Topology 13 (2009) 1177–1227

Let M := (M4,ω) be a 4–dimensional rational ruled symplectic manifold and denote by wM its Gromov width. Let Embω(B4(c),M) be the space of symplectic embeddings of the standard ball of radius r, B4(c) 4 (parametrized by its capacity c := πr2), into (M,ω). By the work of Lalonde and Pinsonnault [Duke Math. J. 122 (2004) 347–397], we know that there exists a critical capacity 0 < ccrit wM such that, for all 0 < c < ccrit, the embedding space Embω(B4(c),M) is homotopy equivalent to the space of symplectic frames SFr(M). We also know that the homotopy type of Embω(B4(c),M) changes when c reaches ccrit and that it remains constant for all c such that ccrit c < wM. In this paper, we compute the rational homotopy type, the minimal model and the cohomology with rational coefficients of Embω(B4(c),M) in the remaining case of c with ccrit c < wM. 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.

rational homotopy type, symplectic embeddings of balls, rational symplectic $4$–manifold, group of symplectic diffeomorphisms
Mathematical Subject Classification 2000
Primary: 53D35, 57R17, 57S05
Secondary: 55R20
Received: 7 July 2008
Accepted: 5 January 2009
Published: 5 February 2009
Proposed: Yasha Eliashberg
Seconded: Leonid Polterovich, Simon Donaldson
Sílvia Anjos
Centro de Análise Matemática, Geometria e Sistemas Dinâmicos
Departamento de Matemática
Instituto Superior Técnico
François Lalonde
Université de Montréal
Canada H3C 3J7
Martin Pinsonnault
The University of Western Ontario
Canada N6A 3K7