The topology of the space of symplectic balls in rational 4-manifolds
By Francois M. Lalonde, Martin Pinsonnault. 2002
61 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/d55f53b2f9baed4747b37e6db312359431d388d1
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- A compact symplectic four-manifold admits only finitely many inequivalent toric actions (2006, background): “Homotopy theoretic properties of symplectomorphism groups have been studied in [1, 2, 7, 13, 22, 35]; see the survey [30].”
- Symplectic mapping class groups of some Stein and rational surfaces (2009, methodology): “Kählerform, (due to Gromov, [Gro85]), • Symp(D2,ω) is weakly homotopy equivalent to T2 ⋊Z/2, where D2 is the 2-point blow-up of CP2 and ω is the anticanonical Kähler form (due to Lalonde-Pinsonnault, [LP04]), • Sympc(C2) is contractible, where ωis the product symplectic form (due to Gromov, [Gro85])”
- Quantitative symplectic geometry (2005, unknown): “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].”