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 $B^4(c) \subset \R^4$ 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.Comment: Typos corrected, 2 minor corrections in the text. Numbering consistant with the published versio

Polarizations and symplectic isotopies

“…For instance Theorem 4 can be easily adapted to five balls and Theorem 3 to other ellipsoids (e.g., (a + , a − ) ≈ (2π, π 2 )), other spaces such as S 2 × S 2 with split symplectic forms (see [21] Theorem 10) or to results in the directions of Lalonde-Pinsonnault [1,12]. Our aim is rather to present a method, fundamentally based on the fact that since open domains can be made out of symplectic hypersurfaces, these two classes of objects should share some rigidity or flexibility properties.…”

Quantitative symplectic geometry

“…Since then, lots of new capacities have been defined [16,30,32,44,49,59,60,90,99] and they were further studied in [1,2,8,9,17,26,21,28,31,35,37,38,41,42,43,46,48,50,52,56,57,58,61,62,63,64,65,66,68,74,75,76,88,89,91,92,94,97,98]. Surveys on symplectic capacities are [45,50,55,69,…”

The homotopy type of the space of symplectic balls in rational ruled 4–manifolds

“….c; /I Z p / is at least one dimensional. Then using again the Serre spectral sequence of the fibration (13) and an argument similar to the one used in Corollary 7.1 we obtain the desired result.…”

“…Note that under the restriction > 1, the group of Hamiltonian diffeomorphisms is equal to the full group of symplectic diffeomorphisms. On the other hand, using J -holomorphic techniques, it was proved in Lalonde and Pinsonnault [13] that the stabilizer of this action, ie the subgroup of symplectic diffeomorphisms of M i that preserve (not necessarily pointwise) a symplectically Denote by B 0 and F 0 in H 2 .M 0 ; Z/ the classes of the first and second factor respectively. Denote by F 1 the fiber of the fibration M 1 .D CP 2 # CP 2 / !…”