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

Abstract

We study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball $B^4(c)\subset {\mathbb{R}}^4$ into 4-dimensional rational symplectic manifolds. We compute the rational homotopy groups of that space when the 4-manifold has the form $M_{\lambda }=(S^2\times S^2,\mu {\omega }_0\oplus {\omega }_0)$ 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 $c_{crit}=\mu -1$, 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.

Typos corrected, 2 minor corrections in the text. Numbering consistant with the published version

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