Geometry & Topology 13 (2009) 1177–1227

DOI: 10.2140/gt.2009.13.1177

Abstract

Let M := (M⁴,ω) be a 4–dimensional rational ruled symplectic manifold and denote by w_M its Gromov width. Let Emb_ω(B⁴(c),M) be the space of symplectic embeddings of the standard ball of radius r, B⁴(c) ⊂ R⁴ (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 < c_crit ≤ w_M such that, for all 0 < c < c_crit, the embedding space Emb_ω(B⁴(c),M) is homotopy equivalent to the space of symplectic frames SFr(M). We also know that the homotopy type of Emb_ω(B⁴(c),M) changes when c reaches c_crit and that it remains constant for all c such that c_crit ≤ c < w_M. In this paper, we compute the rational homotopy type, the minimal model and the cohomology with rational coefficients of Emb_ω(B⁴(c),M) in the remaining case of c with c_crit ≤ c < 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.

Keywords

rational homotopy type, symplectic embeddings of balls, rational symplectic 4–manifold, group of symplectic diffeomorphisms

Mathematical Subject Classification

Primary: 53D35, 57R17, 57S05

Secondary: 55R20

References

Publication

Received: 7 July 2008
Accepted: 5 January 2009
Published: 5 February 2009
Proposed: Yasha Eliashberg
Seconded: Leonid Polterovich, Simon Donaldson

Authors