$(S^2 \times S^2, σ_{std} \oplus σ_{std}) $的3重爆破的辛同构群的同伦李代数
The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of $(S^2 \times S^2, σ_{std} \oplus σ_{std}) $
摘要：
考虑流形$ (S^2 \times S^2, σ\oplus σ)$的3点爆破，其中$σ$是球面$S^2$面积为1的标准辛形式，并研究了其辛同胚群$\rm{Symp} ( S^2 \times S^2 \#\, 3\overline{\mathbb C\mathbb P}\,\!^2, ω)$ .到目前为止，J. Evans研究了单调的情况，并证明了这个群是可收缩的。此外，J. Li，T. J. Li和W.吴表明，组Symp $_{h}(S^2 \times S^2 \#\, 3\overline{ \mathbb C\mathbb P}\,\!^2,ω) $的symplectomorphisms的作用平凡的同源性总是连接，最近他们还计算了其基本群。我们描述，在充分的细节，合理的同伦李代数这个群体。我们表明，一些特定的圆行动中所包含的syplectomorphism群生成其完整的拓扑结构。更精确地说，他们给出了$\rm{Symp} (S^2 \times S^2 \#\, 3\overline{ \mathbb C\mathbb P}\,\!^2, ω)$的同伦分次李代数的生成元。我们的研究依赖于Karshon对Hamilton圈作用量的分类和Lalonde-McDuff引入的暴胀技术。作为应用，我们推导了小爆破情形下$\rm{Symp}({\mathbb C\mathbb P}^2 \#\, 5\overline{\mathbb C\mathbb P}\,\!^2, \tilde ω)$同伦群的秩。
英文摘要：
We consider the 3-point blow-up of the manifold $ (S^2 \times S^2, σ\oplus σ)$ where $σ$ is the standard symplectic form which gives area 1 to the sphere $S^2$, and study its group of symplectomorphisms $\rm{Symp} ( S^2 \times S^2 \#\, 3\overline{\mathbb C\mathbb P}\,\!^2, ω)$. So far, the monotone case was studied by J. Evans and he proved that this group is contractible. Moreover, J. Li, T. J. Li and W. Wu showed that the group Symp$_{h}(S^2 \times S^2 \#\, 3\overline{ \mathbb C\mathbb P}\,\!^2,ω) $ of symplectomorphisms that act trivially on homology is always connected and recently 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 $\rm{Symp} (S^2 \times S^2 \#\, 3\overline{ \mathbb C\mathbb P}\,\!^2, ω)$. Our study depends on Karshon's classification of Hamiltonian circle actions and the inflation technique introduced by Lalonde-McDuff. As an application, we deduce the rank of the homotopy groups of $\rm{Symp}({\mathbb C\mathbb P}^2 \#\, 5\overline{\mathbb C\mathbb P}\,\!^2, \tilde ω)$, in the case of small blow-ups.
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