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

Abstract

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.