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

Anjos, Sílvia; Lalonde, François; Pinsonnault, Martin

doi:10.2140/gt.2009.13.1177

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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

DOI: 10.2140/gt.2009.13.1177

Abstract

Let $M : = (M^{4}, ω)$ be a $4$ –dimensional rational ruled symplectic manifold and denote by $w_{M}$ its Gromov width. Let ${Emb}_{ω} (B^{4} (c), M)$ be the space of symplectic embeddings of the standard ball of radius $r$ , $B^{4} (c) \subset ℝ^{4}$ (parametrized by its capacity $c : = π r^{2}$ ), 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 $0 < c_{c r i t} \leq w_{M}$ such that, for all $0 < c < c_{c r i t}$ , the embedding space ${Emb}_{ω} (B^{4} (c), M)$ is homotopy equivalent to the space of symplectic frames $SFr (M)$ . We also know that the homotopy type of ${Emb}_{ω} (B^{4} (c), M)$ changes when $c$ reaches $c_{c r i t}$ and that it remains constant for all $c$ such that $c_{c r i t} \leq c < w_{M}$ . In this paper, we compute the rational homotopy type, the minimal model and the cohomology with rational coefficients of ${Emb}_{ω} (B^{4} (c), M)$ in the remaining case of $c$ with $c_{c r i t} \leq c < w_{M}$ . 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 2000

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

Sílvia Anjos
Centro de Análise Matemática, Geometria e Sistemas Dinâmicos Departamento de Matemática Instituto Superior Técnico Lisboa Portugal
http://www.math.ist.utl.pt/~sanjos
François Lalonde
Université de Montréal Montréal Canada H3C 3J7
http://www.dms.umontreal.ca/Professeurs/lalonde
Martin Pinsonnault
The University of Western Ontario London Ontario Canada N6A 3K7
http://www.math.uwo.ca/~mpinson/Bienvenue.html

Volume 13, issue 2 (2009)