The homotopy type of the space of symplectic balls in $S^2 \times S^2$ above the critical value

Abstract

We compute in this note the full homotopy type of the space of symplectic embeddings of the standard ball $B^4(c) \subset \R^4$ (where $c= \pi r^2$ is the capacity of the standard ball of radius $r$) into the 4-dimensional rational symplectic manifold $M_{\mu}= (S^2 \times S^2, \mu \om_0 \oplus \om_0)$ where $\om_0$ is the area form on the sphere with total area 1 and $\mu$ belongs to the interval $(1,2]$. We know, by the work of Lalonde-Pinsonnault, that this space retracts to the space of symplectic frames of $M_{\mu}$ for any value of $c$ smaller than the critical value $\mu -1$, and that its homotopy type does change when $c$ crosses that value. In this paper, we compute the homotopy type for the case $c \ge \mu-1$ and prove that it is not the type of a finite CW-complex.