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

Anjos, Sílvia; Eden, Sinan

Mich. Math. J. 68, No. 1, 71-126 (2019).

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 \(\mu=1\).

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

Citations:

Zbl 1242.58004; Zbl 1323.53093; Zbl 0982.70011; Zbl 0874.57019; Zbl 0965.57031

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