The topology of the space of symplectic balls in rational 4-manifolds

We study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball $B^4(c) \subset \mathbf{R}^4$ into 4-dimensional rational symplectic manifolds. We compute the rational homotopy groups of that space when the 4-manifold has the form $M_{\mu}=(S^2\times S^2,\mu\omega_0\oplus\omega_0)$ where $\omega_0$ is the area form on the sphere with total area 1 and $\mu$ belongs to the interval $[1,2]$. We show that, when $\mu$ is 1, this space retracts to the space of symplectic frames, for any value of $c$. However, for any given $1<\mu<2$, the rational homotopy type of that space changes as $c$ crosses the critical parameter $c_{\mathrm{crit}}=\mu-1$, which is the difference of areas between the two $S^2$ factors. We prove moreover that the full homotopy type of that space changes only at that value, i.e the restriction map between these spaces is a homotopy equivalence as long as these values of $c$ remain either below or above that critical value.

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