×

The topology of the space of symplectic balls in rational 4-manifolds. (English) Zbl 1063.57023

The authors study the space of symplectic embeddings of 4-dimensional symplectic balls into symplectic manifolds. More precisely, let \(B(c)\) be a 4-ball in \({\mathbb R}^4\) of radius \(r\) and capacity \(c=\pi r^2\) equipped with restriction of the standard symplectic form from \({\mathbb R}^4\). Let \(\omega\) be the standard symplectic form on \(S^2\) with area equal to 1, and let \(M_{\mu}\) be the symplectic manifold \((S^2\times S^2, \mu p_1^*\omega+p_2^*\omega)\), where \(p_1, p_2: S^2\times S^2\to S^2\) are the obvious projections. Let \(\text{Emb}_{\omega}(c, \mu)\) be the space of all symplectic embeddings of \(B(c)\) in \(M_{\mu}\), equipped with \(C^{\infty}\)-topology. Denote by \(i_c, c<1\) the standard symplectic embedding of \(B(c)\) into \(M_{\mu}\). Denote by \(\widetilde M_{\mu, c}\) the blow-up of \(M_{\mu}\) along the ball \(i_c\) and by \(\text{Symp}(\widetilde M_{\mu,c})\) the group of its symplectomorphisms.
Typical results are the following: Theorem 1.1. The group \(\text{Symp}(\widetilde M_{\mu=1,c})\) has the homotopy type of the semidirect product \(T^2\times {\mathbb Z}/2{\mathbb Z}\). It is interesting, however, that, for \(\mu>1\), the homotopy type of \(\text{Symp}(\widetilde M_{\mu,c})\) changes when \(c\) passes the critical value \(\mu-1\).
Theorem 1.4. For \(\mu\in(1,2)\), the rational cohomology ring of \(\text{Symp}(\widetilde M_{\mu,c})\) is isomorphic to the ring \(\Lambda(\alpha_1, \alpha_2, \alpha_3)\otimes {\mathbb Q}[\varepsilon]\) where \(\deg \alpha_i=1\) and \(\deg \varepsilon=4\) for \(c\in(0, \mu-1)\), \(\deg \varepsilon=2\) for \(c\in (\mu-1,1)\). This result extends the result of M. Abreu [Invent. Math. 131, No. 1, 1–23 (1998; Zbl 0902.53025)] to blown-up spaces.
Corollary 1.5. For any given value of \(\mu\in(1,2)\), the restriction map \(\text{Emb}_{\omega}(c', \mu) \to \text{Emb}_{\omega}(c, \mu)\) is not a homotopy equivalence whenever \(c<\mu-1\) and \(c'>\mu-1\).
Theorem 1.6. Given \(\mu, \mu'\in(1,2]\), \(c,c'\in (0,1]\), assume that \(c\geq\mu-1\) and \(c'\geq\mu'-1\). Then there are natural homotopy equivalences \(\text{ Emb}_{\omega}(c', \mu') \to \text{Emb}_{\omega}(c, \mu)\) and \(\text{Symp}(\widetilde M_{\mu', c'})\to \text{Symp}(\widetilde M_{\mu, c})\).

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
53D35 Global theory of symplectic and contact manifolds
55P62 Rational homotopy theory
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms

Citations:

Zbl 0902.53025
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] M. Abreu, Topology of symplectomorphism groups of \(S^2\times S^2\) , Invent. Math. 131 (1998), 1–23. · Zbl 0902.53025 · doi:10.1007/s002220050196
[2] M. Abreu and D. McDuff, Topology of symplectomorphism groups of rational ruled surfaces , J. Amer. Math. Soc. 13 (2000), 971–1009. JSTOR: · Zbl 0965.57031 · doi:10.1090/S0894-0347-00-00344-1
[3] S. Anjos, Homotopy type of symplectomorphism groups of \(S^2\times S^2\) , Geom. Topol. 6 (2002), 195–218. · Zbl 1023.57021 · doi:10.2140/gt.2002.6.195
[4] M. Audin, The Topology of Torus Actions on Symplectic Manifolds , Progr. Math. 93 , Birkhäuser, Basel, 1991. · Zbl 0726.57029
[5] H. Cartan and J.-P. Serre, Espaces fibrés et groupes d’homotopie, II: Applications , C. R. Acad. Sci. Paris 234 (1952), 393–395. · Zbl 0049.40101
[6] M. Gromov, Pseudoholomorphic curves in symplectic manifolds , Invent. Math. 82 (1985), 307–347. · Zbl 0592.53025 · doi:10.1007/BF01388806
[7] H. Hofer, V. Lizan, and J.-C. Sikorav, On genericity for holomorphic curves in four-dimensional almost-complex manifolds , J. Geom. Anal. 7 (1997), 149–159. · Zbl 0911.53014 · doi:10.1007/BF02921708
[8] F. Lalonde, Isotopy of symplectic balls, Gromov’s radius and the structure of ruled symplectic \(4\)-manifolds , Math. Ann. 300 (1994), 273–296. · Zbl 0812.53032 · doi:10.1007/BF01450487
[9] F. Lalonde and D. McDuff, “\(J\)-curves and the classification of rational and ruled symplectic \(4\)-manifolds” in Contact and Symplectic Geometry (Cambridge, 1994) , Publ. Newton Inst. 8 , Cambridge Univ. Press, Cambridge, 1996, 3–42. · Zbl 0867.53028
[10] T. J. Li and A. Liu, General wall crossing formula , Math. Res. Lett. 2 (1995), 797–810. · Zbl 0871.57017
[11] D. McDuff, Blow ups and symplectic embeddings in dimension \(4\) , Topology 30 (1991), 409–421. · Zbl 0731.53035 · doi:10.1016/0040-9383(91)90021-U
[12] –. –. –. –., “From Symplectic Deformation to Isotopy” in Topics in Symplectic \(4\)-Manifolds (Irvine, Calif., 1996) , First Int. Press Lect. Ser. 1 , Internat. Press, Cambridge, Mass., 1998, 85–99. · Zbl 0928.57018
[13] –. –. –. –., Almost complex structures on \(S^2\times S^2\) , Duke Math. J. 101 (2000), 135–177. · Zbl 0974.53020 · doi:10.1215/S0012-7094-00-10116-0
[14] –. –. –. –., “Symplectomorphism groups and almost complex structures” in Essays on Geometry and Related Topics, Vols. 1, 2 , Monogr. Enseign. Math. 38 , Enseign. Math., Geneva, 2001, 527–556. · Zbl 1010.53064
[15] D. McDuff and L. Polterovich, Symplectic packings and algebraic geometry , appendix by Y. Karshon, Invent. Math. 115 (1994), 405–434. · Zbl 0833.53028 · doi:10.1007/BF01231766
[16] G. Meigniez, Submersions, fibrations and bundles , Trans. Amer. Math. Soc. 354 (2002), 3771–3787. JSTOR: · Zbl 1001.55016 · doi:10.1090/S0002-9947-02-02972-0
[17] M. Pinsonnault, Embeddings of symplectic balls in rational ruled \(4\)-manifolds , in preparation. · Zbl 1151.57031
[18] C. H. Taubes, Counting pseudo-holomorphic submanifolds in dimension \(4\) , J. Differential Geom. 44 (1996), 818–893. · Zbl 0883.57020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.