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

Anjos, Sílvia¹ ; Lalonde, François² ; Pinsonnault, Martin³

¹ Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Lisboa, Portugal
² Université de Montréal, Montréal, Canada H3C 3J7
³ The University of Western Ontario, London, Ontario, Canada N6A 3K7

Geometry & topology, Tome 13 (2009) no. 2, pp. 1177-1227.

Résumé

Let $M:=\left(M^4,\omega \right)$ be a $4$–dimensional rational ruled symplectic manifold and denote by $w_M$ its Gromov width. Let ${Emb}_{\omega }\left(B^4\left(c\right),M\right)$ be the space of symplectic embeddings of the standard ball of radius $r$, $B^4\left(c\right)\subset {ℝ}^4$ (parametrized by its capacity $c:=\pi r^2$), into $\left(M,\omega \right)$. By the work of Lalonde and Pinsonnault [Duke Math. J. 122 (2004) 347–397], we know that there exists a critical capacity $0<c_{crit}\le w_M$ such that, for all $0<c<c_{crit}$, the embedding space ${Emb}_{\omega }\left(B^4\left(c\right),M\right)$ is homotopy equivalent to the space of symplectic frames $SFr\left(M\right)$. We also know that the homotopy type of ${Emb}_{\omega }\left(B^4\left(c\right),M\right)$ changes when $c$ reaches $c_{crit}$ and that it remains constant for all $c$ such that $c_{crit}\le c<w_M$. In this paper, we compute the rational homotopy type, the minimal model and the cohomology with rational coefficients of ${Emb}_{\omega }\left(B^4\left(c\right),M\right)$ in the remaining case of $c$ with $c_{crit}\le c<w_M$. 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.

Détail

DOI : 10.2140/gt.2009.13.1177

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

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