The Homotopy Lie Algebra of Symplectomorphism Groups of 3-Fold Blowups of (S2×S2,σstd⊕σstd)

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 $Symp(S^2\times S^2\#3{\overline{\mathbb{C}\mathbb{P}}}^2,\omega )$. So far, the monotone case was studied by Evans [6], who proved that this group is contractible. Moreover, Li, Li, and Wu [13] showed that the group ${Symp}_h(S^2\times S^2\#3{\overline{\mathbb{C}\mathbb{P}}}^2,\omega )$ of symplectomorphisms that act trivially on homology is always connected, and recently, in [14], they 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 $Symp(S^2\times S^2\#3{\overline{\mathbb{C}\mathbb{P}}}^2,\omega )$. Our study depends on Karshon’s classification of Hamiltonian circle actions and the inflation technique introduced by Lalonde and McDuff. As an application, we deduce the rank of the homotopy groups of $Symp(\mathbb{C}{\mathbb{P}}^2\#5{\overline{\mathbb{C}\mathbb{P}}}^2,\tilde{\omega })$ in the case of small blowups.