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 of R-4 into 4-dimensional rational symplectic manifolds, where c = pi r(2) is the capacity of the standard ball of radius r. We compute the rational homotopy groups of that space when the 4-manifold has the form M-mu = (S-2 x S-2, muomega(0) circle plus omega(0)), where omega(0) is the area form on the sphere with total area I 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 < p less than or equal to 2, the rational homotopy type of that space changes as c crosses the critical parameter lambda = 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, that is that 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. The same methods apply to all other values of p and other rational 4-manifolds as well. The methods rely on two different tools: the study of the action of symplectic groups on the stratified space of almost complex structures developed by Gromov, Abreu, and McDuff and the analysis of the relations between the group corresponding to a manifold M, the group corresponding to its blow-up (M) over tilde, and the space of symplectic embedded balls in M.