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

References in corpus (4)

• From symplectic deformation to isotopy
• Symplectomorphism groups and almost complex structures
• Homotopy type of symplectomorphism groups of S^2 X S^2
• Almost Complex Structures on $S^2\times S^2$

Cited by in corpus (24)

• Some 6-dimensional Hamiltonian S^1 manifolds
• Lectures on Groups of Symplectomorphisms
• The homotopy type of the space of symplectic balls in rational ruled 4-manifolds
• The Hofer question on intermediate symplectic capacities
• A survey of the topological properties of symplectomorphism groups
• Lagrangian blow-ups, blow-downs, and applications to real packing
• Symplectic $(-2)$-spheres and the symplectomorphism group of small rational 4-manifolds, II
• Stability of the symplectomorphism group of rational surfaces
• Maximal compact tori in the Hamiltonian groups of 4-dimensional symplectic manifolds
• Topology of symplectomorphism groups and ball-swappings
• Classifying semi-free Hamiltonian $S^1$-manifolds
• The symplectic mapping class group of $\CC P^2 n{\bar{\CC P^2}}$ with $n\le 4$
• Fundamental group of Symp(M,w) with no circle action
• Toric symplectic ball packing
• Topology of spaces of equivariant symplectic embeddings
• Exact Lagrangians in $A_n$-surface singularities
• The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of the projective plane
• Moment maps, symplectomorphism groups and compatible complex structures
• On the rank of ${\pi }_1(\text{Ham})$
• Symplectic isotopy on non-minimal ruled surfaces
• A compact symplectic four-manifold admits only finitely many inequivalent toric actions
• Symplectomorphisms of some Weinstein 4-manifolds
• The homotopy type of the space of symplectic balls in $S^2\times S^2$ above the critical value
• Evaluation fibrations and topology of symplectomorphisms