##
**The homotopy Lie algebra of symplectomorphism groups of 3-fold blowups of \((S^2\times S^2,\sigma_{\text{std}}\oplus\sigma_{\text{std}})\).**
*(English)*
Zbl 1437.53066

Authors’ abstract: “We consider the 3-point blowup of the manifold \((S^2\times S^2, \sigma\oplus\sigma)\), where \(\sigma\) is the standard symplectic form that gives area 1 to the sphere \(S^2\), and study its group of symplectomorphisms \(\operatorname{Symp}( S^2\times S^2\#3\overline{\mathbb{C}\mathbb{P}}^2,\omega)\). So far, the monotone case was studied by J. D. Evans [J. Symplectic Geom. 9, No. 1, 45–82 (2011; Zbl 1242.58004)], who proved that this group is contractible. Moreover, J. Li et al. [Mich. Math. J. 64, No. 2, 319–333 (2015; Zbl 1323.53093)] showed that the group \(\operatorname{Symp}_h( S^2\times S^2\#3\overline{\mathbb{C}\mathbb{P}}^2,\omega)\) of symplectomorphisms that act trivially on homology is always connected, and recently, in [“Symplectic \((-2)\) spheres and the symplectomorphism group of small rational 4-manifolds. I”, arXiv:1611.07436], J. Li and T.-J. Li also computed its fundamental group. We describe, in full detail, the rational homotopy Lie algebra of this group.

We show that some particular circle actions contained in the symplectomorphism group generate its full topology. More precisely, they give the generators of the homotopy graded Lie algebra of \(\operatorname{Symp}( S^2\times S^2\#3\overline{\mathbb{C}\mathbb{P}}^2,\omega)\). Our study depends on Y. Karshon’s classification of Hamiltonian circle actions [Periodic Hamiltonian flows on four-dimensional manifolds. Providence, RI: American Mathematical Society (AMS) (1999; Zbl 0982.70011)], and the inflation technique introduced by F. Lalonde and D. McDuff [Math. Res. Lett. 3, No. 6, 769–778 (1996; Zbl 0874.57019)]. As an application, we deduce the rank of the homotopy groups of \(\operatorname{Symp}( S^2\times S^2\#3\overline{\mathbb{C}\mathbb{P}}^2,\widetilde{\omega})\) in the case of small bloups.”

If \((M,\omega)\) is a closed simply connected symplectic manifold, then the symplectomorphism group \(\operatorname{Symp},(\omega)\) endowed with the standard \(C^\infty\)-topology is an infinite-dimensional Fréchet Lie group. Except for dimension 4, there are not many tools available to compute the homotopy type of the symplectomorphism group. However, in this case, a detailed study of the space \(\mathcal{I}_\omega\) of almost complex structures compatible with \(\omega\) produce several results. Among the authors of such a result are: M. Gromov, M. Abreu and D. McDuff [J. Am. Math. Soc. 13, No. 4, 971–1009 (2000; Zbl 0965.57031)], Li et al. [loc. cit]. To get these results, the authors study the space of tamed almost complex structurures using a fine decomposition via smooth rational curves and a relative version of the infinite-dimensional Alexander duality. As pointed out in the abstract, the goal of this very ample and interesting paper is to give a complete description of the homotopy Lie algebra of the symplectomorphism group \(\operatorname{Symp}_h( S^2\times S^2\#3\overline{\mathbb{C}\mathbb{P}}^2,\omega)\) for some \(\omega\). This work adds to a series of projects in the area, focusing on the symplectic blowups of the symplectic manifold \(M_\mu=(S^2\times S^2,\mu\sigma\oplus \sigma)\), \(\mu\geq1\). More precisely, let \(\widetilde{M}_{\mu,c_1,c_2,c_3}\) be obtained from \(M_\mu\) by performing three succesive blowups of capacities \(c_1,c_2, c_3\), with \(\mu\geq 1>c_1+c_2>c_2+c_3>c_1>c_2>c_3\). Let \(G_{c_1,c_2,c_3}=G_{\mu=1,c_1,c_2,g_3}\) denote the group of symplectomorphisms of \(\widetilde{M}_{c_1,c_2,c_3}=\widetilde{M}_{\mu=1,c_1,c_2,c_3}\) acting of \(\omega\) on homology and let \(\mathcal{I}_{c_1,c_2,c_3}\) be the space of \(\omega\)-compatible almost complex structures on \(M_{c_1,c_2,c_3}\). The main theorem of the article gives a complete description of the homotopy Lie algebra of \(G_{c_1,c_2,c_3}\).

Theorem A 1.1. Let \(\widetilde{M}_{c_1,c_2,c_3}\) denote the symplectic manifold obtained from \((S^2 \times S^2,\sigma\oplus\sigma)\) by performing three successive blowups of capacities of \(c_1,c_2\) and \(c_3\) in above conditions. Let \(G_{c_1,c_2,c_3}\) denote the group of symplectomorphisms of \(\widetilde{M}_{c_1,c_2,c_3}\) acting trivially on homologies. Then \(\pi_\ast (G_{c_1,c_2,c_3})\otimes \mathbb{Q}\) is isomorphic to an algebra generated by nine elements of degree 1 satisfying a set of relations between their Lie products (specified in Section 3.5). In particular, \(\pi_1(G_{c_1,c_2,c_3})\otimes \mathbb{Q}\simeq \mathbf{Q}^9\).

Remark: The authors explain why it is sufficient to consider the values of \(c_1,c_2,c_3\) in this range and \(\mu=1\).

We show that some particular circle actions contained in the symplectomorphism group generate its full topology. More precisely, they give the generators of the homotopy graded Lie algebra of \(\operatorname{Symp}( S^2\times S^2\#3\overline{\mathbb{C}\mathbb{P}}^2,\omega)\). Our study depends on Y. Karshon’s classification of Hamiltonian circle actions [Periodic Hamiltonian flows on four-dimensional manifolds. Providence, RI: American Mathematical Society (AMS) (1999; Zbl 0982.70011)], and the inflation technique introduced by F. Lalonde and D. McDuff [Math. Res. Lett. 3, No. 6, 769–778 (1996; Zbl 0874.57019)]. As an application, we deduce the rank of the homotopy groups of \(\operatorname{Symp}( S^2\times S^2\#3\overline{\mathbb{C}\mathbb{P}}^2,\widetilde{\omega})\) in the case of small bloups.”

If \((M,\omega)\) is a closed simply connected symplectic manifold, then the symplectomorphism group \(\operatorname{Symp},(\omega)\) endowed with the standard \(C^\infty\)-topology is an infinite-dimensional Fréchet Lie group. Except for dimension 4, there are not many tools available to compute the homotopy type of the symplectomorphism group. However, in this case, a detailed study of the space \(\mathcal{I}_\omega\) of almost complex structures compatible with \(\omega\) produce several results. Among the authors of such a result are: M. Gromov, M. Abreu and D. McDuff [J. Am. Math. Soc. 13, No. 4, 971–1009 (2000; Zbl 0965.57031)], Li et al. [loc. cit]. To get these results, the authors study the space of tamed almost complex structurures using a fine decomposition via smooth rational curves and a relative version of the infinite-dimensional Alexander duality. As pointed out in the abstract, the goal of this very ample and interesting paper is to give a complete description of the homotopy Lie algebra of the symplectomorphism group \(\operatorname{Symp}_h( S^2\times S^2\#3\overline{\mathbb{C}\mathbb{P}}^2,\omega)\) for some \(\omega\). This work adds to a series of projects in the area, focusing on the symplectic blowups of the symplectic manifold \(M_\mu=(S^2\times S^2,\mu\sigma\oplus \sigma)\), \(\mu\geq1\). More precisely, let \(\widetilde{M}_{\mu,c_1,c_2,c_3}\) be obtained from \(M_\mu\) by performing three succesive blowups of capacities \(c_1,c_2, c_3\), with \(\mu\geq 1>c_1+c_2>c_2+c_3>c_1>c_2>c_3\). Let \(G_{c_1,c_2,c_3}=G_{\mu=1,c_1,c_2,g_3}\) denote the group of symplectomorphisms of \(\widetilde{M}_{c_1,c_2,c_3}=\widetilde{M}_{\mu=1,c_1,c_2,c_3}\) acting of \(\omega\) on homology and let \(\mathcal{I}_{c_1,c_2,c_3}\) be the space of \(\omega\)-compatible almost complex structures on \(M_{c_1,c_2,c_3}\). The main theorem of the article gives a complete description of the homotopy Lie algebra of \(G_{c_1,c_2,c_3}\).

Theorem A 1.1. Let \(\widetilde{M}_{c_1,c_2,c_3}\) denote the symplectic manifold obtained from \((S^2 \times S^2,\sigma\oplus\sigma)\) by performing three successive blowups of capacities of \(c_1,c_2\) and \(c_3\) in above conditions. Let \(G_{c_1,c_2,c_3}\) denote the group of symplectomorphisms of \(\widetilde{M}_{c_1,c_2,c_3}\) acting trivially on homologies. Then \(\pi_\ast (G_{c_1,c_2,c_3})\otimes \mathbb{Q}\) is isomorphic to an algebra generated by nine elements of degree 1 satisfying a set of relations between their Lie products (specified in Section 3.5). In particular, \(\pi_1(G_{c_1,c_2,c_3})\otimes \mathbb{Q}\simeq \mathbf{Q}^9\).

Remark: The authors explain why it is sufficient to consider the values of \(c_1,c_2,c_3\) in this range and \(\mu=1\).

Reviewer: Ioan Pop (Iaşi)

### MSC:

53D35 | Global theory of symplectic and contact manifolds |

57R17 | Symplectic and contact topology in high or arbitrary dimension |

58D05 | Groups of diffeomorphisms and homeomorphisms as manifolds |

### Keywords:

symplectomorphism; homotopy algbra of symplectocomorphism; almost complex structures; structure of \(J\)-holomorphic curves; configurations of \(J\)-holomorphic curves; homotopy type of symplectomorphism groups
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\textit{S. Anjos} and \textit{S. Eden}, Mich. Math. J. 68, No. 1, 71--126 (2019; Zbl 1437.53066)

### References:

[1] | M. Abreu, G. Granja, and N. Kitchloo, Compatible complex structures on symplectic rational ruled surfaces, Duke Math. J. 148 (2009), 539-600. · Zbl 1171.53053 |

[2] | M. Abreu and D. McDuff, Topology of symplectomorphism groups of rational ruled surfaces, J. Amer. Math. Soc. 13 (2000), 971-1009. · Zbl 0965.57031 |

[3] | S. Anjos and M. Pinsonnault, The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of the projective plane, Math. Z. 275 (2013), 245-292. · Zbl 1282.57032 |

[4] | O. Buse, Negative inflation and stability in symplectomorphism groups of ruled surfaces, J. Symplectic Geom. 9 (2011), no. 2, 147-160. · Zbl 1233.53010 |

[5] | A. C. da Silva, Lectures on symplectic geometry, Lecture Notes in Math., 1764, Springer-Verlag, Berlin Heidelberg, 2001, 2008 (corrected printing). · Zbl 1016.53001 |

[6] | J. D. Evans, Symplectic mapping class groups of some Stein and rational surfaces, J. Symplectic Geom. 9 (2011), no. 1, 45-82. · Zbl 1242.58004 |

[7] | Y. Félix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, Graduate Texts in Math., 205, Springer-Verlag, New York, 2001. |

[8] | M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347. · Zbl 0592.53025 |

[9] | T. S. Holm and L. Kessler, Circle actions on symplectic four-manifolds, arXiv:1507.05972v1. · Zbl 1489.53113 |

[10] | Y. Karshon, Periodic Hamiltonian flows on four dimensional manifolds, Mem. Amer. Math. Soc. 141 (1999), no. 672. · Zbl 0982.70011 |

[11] | F. Lalonde and D. McDuff, The classification of ruled symplectic 4-manifolds, Math. Res. Lett. 3 (1996), 769-778. · Zbl 0874.57019 |

[12] | F. Lalonde and M. Pinsonnault, The topology of the space of symplectic balls in rational 4-manifolds, Duke Math. J. 122 (2004), no. 2, 347-397. · Zbl 1063.57023 |

[13] | J. Li, T. J. Li, and W. Wu, The symplectic mapping class group of \(\mathbb{CP}^2\#n\overline{\mathbb{CP}^2}\) with \(n\leq4\), Michigan Math. J. 64 (2015), no. 2, 319-333. · Zbl 1323.53093 |

[14] | J. Li, T. J. Li, and W. Wu, Symplectic \(-2\) spheres and the symplectomorphism group of small rational 4-manifolds, arXiv:1611.07436. |

[15] | T. J. Li and A. Liu, General wall crossing formula, Math. Res. Lett. 2 (1995), 797-810. · Zbl 0871.57017 |

[16] | T. J. Li and A. Liu, Symplectic structure on ruled surfaces and a generalized adjunction formula, Math. Res. Lett. 2 (1995), 453-471. · Zbl 0855.53019 |

[17] | T. J. Li and A. Liu, Uniqueness of symplectic canonical class, surface cone and symplectic cone of 4-manifolds with \(b^+=1\), J. Differential Geom. 58 (2001), no. 2, 331-370. · Zbl 1051.57035 |

[18] | D. McDuff, From symplectic deformation to isotopy, Topics in symplectic 4-manifolds (Irvine, CA, 1996), pp. 85-99, Internat. Press, Cambridge, MA, 1996. · Zbl 0928.57018 |

[19] | D. McDuff, Symplectomorphism groups and almost complex structures, Essays on geometry and related topics, vol. 1, 2, Monogr. Enseign. Math., 38, pp. 527-556, Enseignement Math., Geneva, 2001. · Zbl 1010.53064 |

[20] | D. McDuff, The symplectomorphism group of a blow-up, Geom. Dedicata 132 (2008), 1-29. · Zbl 1155.53055 |

[21] | D. McDuff and D. A. Salamon, J-holomorphic curves and quantum cohomology, Univ. Lecture Ser., 6, American Mathematical Society, Providence, RI, 1994. · Zbl 0809.53002 |

[22] | D. McDuff and S. Tolman, Polytopes with mass linear functions, part I, Int. Math. Res. Not. 8 (2010), 1506-1574. · Zbl 1202.52010 |

[23] | J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965), no. 2, 211-264. · Zbl 0163.28202 |

[24] | S. Novikov, The Cartan-Serre theorem and intrinsic homology, Uspekhi Mat. Nauk 21 (1966), no. 5, 217-232. |

[25] | M. Pinsonnault, Symplectomorphism groups and embeddings of balls into rational ruled surfaces, Compos. Math. 144 (2008), no. 3, 787-810. · Zbl 1151.57031 |

[26] | P. Seidel, Lectures on four-dimensional Dehn twists, Symplectic 4-manifolds and algebraic surfaces, Lecture Notes in Math., 1938, pp. 231-267, Springer, Berlin, 2008. · Zbl 1152.53069 |

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