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 B⁴(c)⊂R⁴ into 4-dimensional rational symplectic manifolds, where c=πr² 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_μ=S²×S², μω₀⊕ω₀, where ω₀ 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 λ=μ−1, which is the difference of areas between the two S²-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 μ 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 $\tilde{M}$, and the space of symplectic embedded balls in M.