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

Let $M:=(M^{4},\omega)$ be a 4-dimensional rational ruled symplectic manifold and denote by $w_{M}$ its Gromov width. Let $\mathrm{Emb}_{\omega}(B^{4}(c),M)$ be the space of symplectic embeddings of the standard ball $B^4(c)\subset \mathbf{R}^4$ of radius $r$ and of capacity $c:= \pi r^2$ into $(M,\omega)$. By the work of Lalonde and Pinsonnault, we know that there exists a critical capacity $c_{\mathrm{crit}} \in (0,w_{M}]$ such that, for all $c\in(0,c_{\mathrm{crit}})$, the embedding space $\mathrm{Emb}_{\omega}(B^{4}(c),M)$ is homotopy equivalent to the space of symplectic frames $\mathrm{SFr}(M)$. We also know that the homotopy type of $\mathrm{Emb}_{\omega}(B^{4}(c),M)$ changes when $c$ reaches $c_{\mathrm{crit}}$ and that it remains constant for all $c \in [c_{\mathrm{crit}},w_{M})$. In this paper, we compute the rational homotopy type, the minimal model, and the cohomology with rational coefficients of $\mathrm{Emb}_{\omega}(B^{4}(c),M)$ in the remaining case $c \in [c_{\mathrm{crit}},w_{M})$. In particular, we show that it does not have the homotopy type of a finite CW-complex.

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