Paper detail

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 $B^4(c) \subset \R^4$ into 4-dimensional rational symplectic manifolds. We compute the rational homotopy groups of that space when the 4-manifold has the form $M_λ= (S^2 \times S^2, μω_0 \oplus ω_0)$ where $ω_0$ is the area form on the sphere with total area 1 and $μ$ belongs to the interval $[1,2]$. We show that, when $μ$ is 1, this space retracts to the space of symplectic frames, for any value of $c$. However, for any given $1 < μ< 2$, the rational homotopy type of that space changes as $c$ crosses the critical parameter $c_{crit} = μ- 1$, 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.